Measure-Valued Branching Markov Processes [2 ed.] 9783662669099, 9783662669105

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Probability Theory and Stochastic Modelling  103

Zenghu Li

Measure-Valued Branching Markov Processes Second Edition

Probability Theory and Stochastic Modelling Volume 103

Editors-in-Chief Peter W. Glynn, Stanford University, Stanford, CA, USA Andreas E. Kyprianou, University of Bath, Bath, UK Yves Le Jan, Université Paris-Saclay, Orsay, France Kavita Ramanan, Brown University, Providence, RI, USA Advisory Editors Søren Asmussen, Aarhus University, Aarhus, Denmark Martin Hairer, Imperial College, London, UK Peter Jagers, Chalmers University of Technology, Gothenburg, Sweden Ioannis Karatzas, Columbia University, New York, NY, USA Frank P. Kelly, University of Cambridge, Cambridge, UK Bernt Øksendal, University of Oslo, Oslo, Norway George Papanicolaou, Stanford University, Stanford, CA, USA Etienne Pardoux, Aix Marseille Université, Marseille, France Edwin Perkins, University of British Columbia, Vancouver, Canada Halil Mete Soner, Princeton University, Princeton, NJ, USA

Probability Theory and Stochastic Modelling publishes cutting-edge research monographs in probability and its applications, as well as postgraduate-level textbooks that either introduce the reader to new developments in the field, or present a fresh perspective on fundamental topics. Books in this series are expected to follow rigorous mathematical standards, and all titles will be thoroughly peer-reviewed before being considered for publication. Probability Theory and Stochastic Modelling covers all aspects of modern probability theory including:       

Gaussian processes Markov processes Random fields, point processes, and random sets Random matrices Statistical mechanics, and random media Stochastic analysis High-dimensional probability

as well as applications that include (but are not restricted to):  Branching processes, and other models of population growth  Communications, and processing networks  Computational methods in probability theory and stochastic processes, including simulation  Genetics and other stochastic models in biology and the life sciences  Information theory, signal processing, and image synthesis  Mathematical economics and finance  Statistical methods (e.g. empirical processes, MCMC)  Statistics for stochastic processes  Stochastic control, and stochastic differential games  Stochastic models in operations research and stochastic optimization  Stochastic models in the physical sciences Probability Theory and Stochastic Modelling is a merger and continuation of Springer’s Stochastic Modelling and Applied Probability and Probability and Its Applications series.

Zenghu Li

Measure-Valued Branching Markov Processes Second Edition

Zenghu Li School of Mathematical Sciences Beijing Normal University Beijing, China

ISSN 2199-3149 (electronic) ISSN 2199-3130 Probability Theory and Stochastic Modelling ISBN 978-3-662-66910-5 (eBook) ISBN 978-3-662-66909-9 https://doi.org/10.1007/978-3-662-66910-5 Mathematics Subject Classification (2020): 60-02, 60J80, 60G57, 60J70, 60J85, 60J35, 60J40 © Springer-Verlag GmbH Germany, part of Springer Nature 2011, 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer-Verlag GmbH, DE, part of Springer Nature. The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

Preface to the Second Edition

A considerable number of extensions and improvements have been made in this second edition. The most obvious change is that the section on one-dimensional stochastic equations has been extended to a chapter. There is another new chapter dealing with stochastic flows by means of stochastic equations and non-local branching superprocesses. Stochastic equations have provided powerful tools in the study of continuous-state branching processes without or with immigration. The two new chapters reflect some developments of the subject in the past decade. The chapter on state-dependent immigration has been rewritten completely to treat general canonical entrance rules that are not necessarily entrance laws. Within the chapters already present in the first edition, new sections have been inserted to deal with estimates of variations of the transition probabilities, upper and lower bounds for cumulant semigroups, stationary distributions and ergodicities of immigration superprocesses. New results and examples have also been added in many other places. A number of typos and inaccuracies have been corrected. The new material has been selected according to the same principles as the first edition of the book, that is, to give a compact and rigorous treatment of the basic theory of measure-valued branching processes and immigration processes. The important developments in continuous-state branching processes with competition have not been included as one can find nice treatments of them in the monograph by Pardoux (2016). There are some other interesting new results that are not discussed in the main text because the research still needs time to mature. The reader can find details on such results in the notes and comments sections. For many years, I have benefited from discussions with colleagues and students at Beijing Normal University. I am particularly grateful to Professor Mufa Chen for his advice on the coupling and distance methods, which have been incorporated into this edition. The material in this book has also been used in courses I gave in other places, including Peking University, the University of Verona and the University of Extremadura. I take this opportunity to acknowledge the friends in those institutions for their comments and suggestions. I thank Professors Zhenqing Chen, Renming Song and Jiangang Ying for helpful discussions on the general theory of Markov processes. I am grateful to Professor Matyas Barczy for his suggestions on the treatv

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Preface to the Second Edition

ments of distributional properties of continuous-state branching processes. I would like to thank Professors Vladimir Vatutin and Xiaowen Zhou for their comments on the literature. I am indebted to the Laboratory of Mathematics and Complex Systems (Ministry of Education) for providing me the research facilities. My special thanks go to the series editors of Springer, Richard Kruel and Marina Reizakis, for their patience and suggestions. This edition has been prepared with the support of the National Key R&D Program and the Natural Science Foundation of China. Finally, I thank my family for their continuing support. Beijing, China

Zenghu Li October 6, 2022

Preface to the First Edition

The books by Athreya and Ney (1972), Harris (1963) and Jagers (1975) contain a lot about finite-dimensional branching processes and their applications. Measure-valued branching processes with abstract underlying spaces were constructed in Watanabe (1968), who showed those processes arose as high-density limits of branching particle systems. The connection of measure-valued branching processes with stochastic evolution equations was investigated in Dawson (1975). A special class of measurevalued branching processes are known as Dawson–Watanabe superprocesses, which have been undergoing rapid development thanks to the contributions of a great number of researchers. The developments have been stimulated from different subjects including classical branching processes, interacting particle systems, stochastic partial differential equations and nonlinear partial differential equations. The study of superprocesses leads to a better understanding of results in those subjects as well. We refer the reader to Dawson (1992, 1993), Dynkin (1994, 2002), Etheridge (2000), Le Gall (1999) and Perkins (1995, 2002) for detailed treatments of different aspects of the developments in the past decades. Branching processes give the mathematical modeling for populations evolving randomly in isolated environments. A useful and realistic modification of the branching model is the addition of immigration from outside sources. From the viewpoint of applications, branching models allowing immigration are clearly of great importance and physical appeal; see, e.g., Athreya and Ney (1972). This modification is also familiar in the setting of measure-valued processes; see, e.g., Dawson (1993), Dawson and Ivanoff (1978) and Dynkin (1991a). The main purpose of this book is to give a compact and rigorous treatment of the basic theory of measure-valued branching processes and immigration processes. In the first part of the book, we give an analytic construction of Dawson–Watanabe superprocesses with general branching mechanisms. The spatial motions of those processes can be general Borel right processes in Lusin topological spaces. We show that the superprocesses arise as high-density limits of branching particle systems, giving the intuitive interpretations of the former. Under natural assumptions, it is shown that the superprocesses have Borel right realizations. From the general model, we use transformations to derive the existence and regularity of several different forms of the superprocesses including those in spaces of tempered meavii

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Preface to the First Edition

sures, multitype models, age-structured models and time-inhomogeneous models. This unified treatment of the different models simplifies their constructions and gives useful perspectives for their properties. When the underlying space shrinks to a single point, the superprocess reduces to a one-dimensional continuous-state branching process. We briefly discuss extinction probabilities and limit theorems related to the latter. The theory of the one-dimensional processes requires much less prerequisite knowledge and is helpful for the reader in developing their intuitions for superprocesses. Under Feller type assumptions, several martingale problems for superprocesses are formulated and their equivalence are established. The martingale measures induced by those martingale problems are not necessarily orthogonal, but they are still worthy. To make the book essentially self-contained, overlaps of the first part with Dawson (1993) and Dynkin (1994) cannot be avoided completely, but we have made them as little as possible. In the second part of the book we investigate the immigration structures associated with measure-valued branching processes. For that purpose, we first give some characterizations of entrance laws for those processes. We define immigration processes in an axiomatic way using skew convolution semigroups as in Li (1995/6). It is then proved that the skew convolution semigroups associated with a given measure-valued branching process are in one-to-one correspondence with its infinitely divisible probability entrance laws. The immigration superprocess has regularities similar to those of the Dawson–Watanabe superprocess if the corresponding probability entrance law is closable. Instead of establishing the results by repeating the techniques in the first part, we concentrate on the genuinely new or different aspects of the immigration processes and develop the theory on the bases of the processes without immigration. In this way, we hope to give the book a more compact and unified form. The concept of skew convolution semigroups can actually be introduced in an abstract setting. Roughly speaking, such a semigroup gives the law of evolution of a system with branching structure under the perturbation of random extra forces. The immigration process is only a special case of this formulation. There is another special case investigated by Bogachev and Röckner (1995) and Bogachev et al. (1996), who formulated Ornstein–Uhlenbeck type processes on Hilbert spaces using generalized Mehler semigroups. Skew convolution semigroups were also used in Dawson and Li (2006) to study the affine Markov processes introduced in mathematical finance. In the last part of the book, we briefly discuss characterizations of generalized Mehler semigroups and properties of the corresponding Ornstein–Uhlenbeck type processes. We also show that a typical class of those processes arise as fluctuation limits of immigration superprocesses. The main theory of Dawson–Watanabe superprocesses and immigration superprocesses is developed for general branching mechanisms that are not necessarily decomposable into local and non-local parts. Most of the results were obtained before only for specific classes of branching mechanisms. The emphasis here is the basic structures and regularities, rather than intensive properties of specific models. The setting of Borel right processes we have chosen is very convenient for the development of the theory. The title of the book stresses the applications of techniques from the theory of general Markov processes. Our main references for those are

Preface to the First Edition

ix

Ethier and Kurtz (1986) and Sharpe (1988). In the appendix we give a summary of the basic concepts and results that are frequently used. We hope the summary will help the reader in a quick start of the main parts of the book. In the last section of each chapter, comments on the history and recent development are given. This book can be used as a reference for the basics of Dawson–Watanabe superprocesses and immigration superprocesses. It can also be used in a course for graduate students specializing in probability and stochastic processes. I would like to express my sincere thanks to Professor Zikun Wang for his advice and encouragement given to me for many years. I am deeply grateful to Professor Mufa Chen for his enormous help in my work. My special thanks are given to Professors Donald A. Dawson, Eugene B. Dynkin and Tokuzo Shiga, from whom I learned the theory of measure-valued processes. I have also benefited from stimulating discussions on this subject with many other experts, including Professors Patrick J. Fitzsimmons, Klaus Fleischmann, Luis G. Gorostiza, Zhiming Ma, Hao Wang, Shinzo Watanabe, Jie Xiong and Xiaowen Zhou. I thank Professors Marco Fuhrman, Michael Röckner, Byron Schmuland, Wei Sun and Fengyu Wang for their advice on generalized Mehler semigroups. I am very grateful to Professors Peter Jagers, Thomas G. Kurtz, Jean-François Le Gall and Renming Song for valuable comments on earlier versions of this book. I want to thank Professors Wenming Hong, Yanxia Ren, Yongjin Wang, Kainan Xiang and Mei Zhang for helpful discussions. The material in this book has been used for graduate courses in Beijing Normal University. I am indebted to my colleagues and students here, who provide a very pleasant research environment. In particular, I thank Congzao Dong, Hui He, Chunhua Ma, Rugang Ma, Li Wang and Xu Yang for reading the manuscript carefully and pointing out numerous typos and errors. I would like to express sincere gratitude to Dr. Marina Reizakis, the PIA series editor at Springer, for her advice and help. I want to thank the Natural Science Foundation and the Ministry of Education of China, who have supported my research in the past years. Finally I thank my wife and my son for their continuing moral support. Beijing, China

Zenghu Li May 18, 2010

Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Conventions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1

Random Measures on Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Borel Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Laplace Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Poisson Random Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Infinitely Divisible Random Measures . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Lévy–Khintchine Type Representations . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 9 13 16 21 28

2

Measure-Valued Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Integral Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Dawson–Watanabe Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Examples of Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Some Moment Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Variations of Transition Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 36 41 48 50 58 62

3

One-Dimensional Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Continuous-State Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Long-Time Evolution Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Immigration and Conditioned Processes . . . . . . . . . . . . . . . . . . . . . . . . 3.4 More Conditional Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Scaling Limits of Discrete Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 72 75 80 86 94 xi

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Contents

4

Branching Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.1 Particle Systems with Local Branching . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 Scaling Limits of Local Branching Systems . . . . . . . . . . . . . . . . . . . . . 104 4.3 General Branching Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.4 Scaling Limits of General Branching Systems . . . . . . . . . . . . . . . . . . . 112 4.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5

Basic Regularities of Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.1 Right Continuous Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.2 The Strong Markov Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3 Borel Right Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4 Weighted Occupation Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.5 A Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.6 Bounds for the Cumulant Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6

Constructions by Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.1 Spaces of Tempered Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2 Multitype Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.3 Two-Type Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.4 A Change of the Probability Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.5 Time-Inhomogeneous Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7

Martingale Problems of Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.1 The Differential Evolution Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.2 Generators and Martingale Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.3 Worthy Martingale Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.4 A Stochastic Convolution Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 7.5 Transforms by Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

8

Entrance Laws and Kuznetsov Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.1 Some Simple Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.2 Minimal Probability Entrance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.3 Infinitely Divisible Probability Entrance Laws . . . . . . . . . . . . . . . . . . 215 8.4 Kuznetsov Measures and Excursion Laws . . . . . . . . . . . . . . . . . . . . . . 221 8.5 Cluster Representations of the Process . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.6 Super-Absorbing-Barrier Brownian Motions . . . . . . . . . . . . . . . . . . . . 234 8.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

9

Structures of Independent Immigration . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.1 Skew Convolution Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.2 Properties of Transition Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 247 9.3 Regular Immigration Superprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.4 Characterizations by Martingale Problems . . . . . . . . . . . . . . . . . . . . . . 256

Contents

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9.5 Constructions of the Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.6 Stationary Distributions and Ergodicities . . . . . . . . . . . . . . . . . . . . . . . 269 9.7 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 10 One-Dimensional Stochastic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 10.1 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 279 10.2 The Lamperti Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 10.3 Distributional Properties of Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 10.4 Local and Global Maximal Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 10.5 A Generalized CBI-process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 10.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 11

Path-Valued Processes and Stochastic Flows . . . . . . . . . . . . . . . . . . . . . . . 309 11.1 Path-Valued Growing Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 11.2 The Total Population Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 11.3 Construction by Stochastic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 321 11.4 A Stochastic Flow of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 11.5 The Excursion Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 11.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

12

State-Dependent Immigration Structures . . . . . . . . . . . . . . . . . . . . . . . . . 337 12.1 Inhomogeneous Immigration Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 12.2 Predictable Immigration Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 12.3 State-Dependent Immigration Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 12.4 Changes of the Branching Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 359 12.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

13

Generalized Ornstein–Uhlenbeck Processes . . . . . . . . . . . . . . . . . . . . . . . 365 13.1 Generalized Mehler Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 13.2 Gaussian Type Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 13.3 Non-Gaussian Type Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 13.4 Extensions of Centered Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 13.5 Construction of the Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 13.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

14

Small-Branching Fluctuation Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 14.1 The Brownian Immigration Superprocess . . . . . . . . . . . . . . . . . . . . . . 391 14.2 Stochastic Processes in Nuclear Spaces . . . . . . . . . . . . . . . . . . . . . . . . 393 14.3 Fluctuation Limits in the Schwartz Space . . . . . . . . . . . . . . . . . . . . . . . 400 14.4 Fluctuation Limits in Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 407 14.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

xiv

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Contents

Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 A.1 Measurable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 A.2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 A.3 Right Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 A.4 Ray–Knight Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 A.5 Entrance Space and Entrance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 A.6 Concatenations and Weak Generators . . . . . . . . . . . . . . . . . . . . . . . . . . 433 A.7 Time–Space Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

Conventions and Notations

We say “positive” for “≥ 0” and “strictly positive” for “> 0”. The following notations are frequently used: N: R: C: R+ : R𝑑 : |·|: 𝑎∧𝑏: 𝑎∨𝑏: ⌊𝑥⌋ : 1𝐴 : 𝛿𝑥 : supp(𝜇) : a.s. : 𝐴 := 𝐵 :

set of positive integers; real line, i.e., 1-dimensional Euclidean space; complex plane; positive half real line; 𝑑-dimensional Euclidean space; Euclidean norm of R𝑑 ; minimum of 𝑎 and 𝑏; maximum of 𝑎 and 𝑏; largest integer not exceeding 𝑥; indicator function of set 𝐴; Dirac measure at point 𝑥; closed support of measure 𝜇; “almost sure” or “almost surely”; 𝐴 is defined by 𝐵.

For any 𝑎 ≤ 𝑏 ∈ R we understand that ∫

∫

𝑏

=− 𝑎

∫

∫

𝑎

and

= 𝑏

(𝑎,𝑏]

∞

∫ =

𝑎

. (𝑎,∞)

Other notations are explained as they first appear.

xv

Chapter 1

Random Measures on Metric Spaces

In this chapter, we discuss the basic properties of Laplace functionals of random measures, which provide an important tool in the study of measure-valued processes. In particular, we give some characterizations of the convergence of random measures in terms of their Laplace functionals. Based on these results, a general representation for the distributions of infinitely divisible random measures is established. We also give some characterizations of continuous functions on the positive half line with Lévy–Khintchine type representations.

1.1 Borel Measures Given a class 𝒢 of functions on or subsets of some space 𝐸, let 𝜎(𝒢) denote the 𝜎-algebra on 𝐸 generated by 𝒢. If ℱ is a class of functions, we define the classes bℱ = { 𝑓 ∈ ℱ : 𝑓 is bounded} and pℱ = { 𝑓 ∈ ℱ : 𝑓 is positive}. Let R denote the real line and let R+ = [0, ∞) denote the positive half line. For a topological space 𝐸, let ℬ(𝐸) denote the 𝜎-algebra on 𝐸 generated by the class of open sets, which is referred to as the Borel 𝜎-algebra. A real function defined on 𝐸 is called a Borel function if it is measurable with respect to ℬ(𝐸). We also use ℬ(𝐸) to denote the set of Borel functions on 𝐸. Let 𝐵(𝐸) = bℬ(𝐸) denote the Banach space of bounded Borel functions on 𝐸 endowed with the supremum/uniform norm ∥ · ∥. For any 𝑎 ≥ 0, let 𝐵 𝑎 (𝐸) be the set of functions 𝑓 ∈ 𝐵(𝐸) satisfying ∥ 𝑓 ∥ ≤ 𝑎. Let 𝐶 (𝐸) denote the space of bounded continuous real functions on 𝐸. We use the superscript “+” to denote the subsets of positive elements of the function spaces, and the superscript “++” is used to denote those of positive elements bounded away from zero, e.g., 𝐵(𝐸) + , 𝐶 (𝐸) ++ . If a metric 𝑑 is specified on 𝐸, we denote by 𝐶𝑢 (𝐸) := 𝐶𝑢 (𝐸, 𝑑) the subset of 𝐶 (𝐸) of 𝑑-uniformly continuous real functions. If 𝐸 is a locally compact space, then 𝐶0 (𝐸) denotes the space of functions in 𝐶 (𝐸) vanishing at infinity. Therefore 𝐶0 (𝐸) = 𝐶 (𝐸) when 𝐸 is compact. © Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_1

1

2

1 Random Measures on Metric Spaces

A Borel measure, or simply a measure, on some topological space 𝐸 means a measure on the space (𝐸, ℬ(𝐸)). We write 𝜇( 𝑓 ) or ⟨𝜇, 𝑓 ⟩ for the integral of a function 𝑓 with respect to a measure 𝜇 if the integral exists. The unit measure concentrated at a point 𝑥 ∈ 𝐸 is denoted by 𝛿 𝑥 . A measure 𝜇 on 𝐸 is said to be purely Í atomic if it has the decomposition 𝜇 = 𝑖 𝑎 𝑖 𝛿 𝑥𝑖 for countable families {𝑎 𝑖 } ⊂ [0, ∞) and {𝑥𝑖 } ⊂ 𝐸. We say 𝜇 is a diffuse measure if it does not charge any singleton. Theorem 1.1 Suppose that (𝐸, 𝑑) is a metric space and 𝐴 is a non-empty subset of 𝐸. For 𝑥 ∈ 𝐸 let 𝑑 (𝑥, 𝐴) = inf{𝑑 (𝑥, 𝑦) : 𝑦 ∈ 𝐴}. Then we have |𝑑 (𝑥, 𝐴) − 𝑑 (𝑦, 𝐴)| ≤ 𝑑 (𝑥, 𝑦),

𝑥, 𝑦 ∈ 𝐸 .

(1.1)

In particular, 𝑥 ↦→ 𝑑 (𝑥, 𝐴) is a uniformly continuous function on 𝐸. Proof For any 𝑥, 𝑦 ∈ 𝐸 and 𝑧 ∈ 𝐴 we have 𝑑 (𝑥, 𝐴) − 𝑑 (𝑦, 𝑧) ≤ 𝑑 (𝑥, 𝑧) − 𝑑 (𝑦, 𝑧) ≤ 𝑑 (𝑥, 𝑦). Then we take the supremum over 𝑧 ∈ 𝐴 in both sides to get 𝑑 (𝑥, 𝐴) − 𝑑 (𝑦, 𝐴) ≤ 𝑑 (𝑥, 𝑦). By the symmetry of 𝑑 (·, ·) we have 𝑑 (𝑦, 𝐴) − 𝑑 (𝑥, 𝐴) ≤ 𝑑 (𝑥, 𝑦). Combining the two preceding inequalities gives (1.1).

□

Corollary 1.2 For any metric space (𝐸, 𝑑), we have 𝜎(𝐶𝑢 (𝐸) + ) = 𝜎(𝐶𝑢 (𝐸)) = ℬ(𝐸). Proof Since a continuous function is Borel, we have 𝜎(𝐶𝑢 (𝐸) + ) ⊂ 𝜎(𝐶𝑢 (𝐸)) ⊂ ℬ(𝐸). Given a proper open subset 𝐺 ⊂ 𝐸, let 𝑓𝑛 (𝑥) = (1 ∧ 𝑑 (𝑥, 𝐺 𝑐 )) 1/𝑛 for 𝑥 ∈ 𝐸 and 𝑛 ≥ 1. By Theorem 1.1 we have { 𝑓𝑛 } ⊂ 𝐶𝑢 (𝐸) + . It is easy to see 𝑓𝑛 → 1𝐺 as 𝑛 → ∞, implying 𝐺 ∈ 𝜎(𝐶𝑢 (𝐸) + ). But we have 𝐸 ∈ 𝜎(𝐶𝑢 (𝐸) + ) clearly, so 𝜎(𝐶𝑢 (𝐸) + ) contains all open subsets of 𝐸. This implies ℬ(𝐸) ⊂ 𝜎(𝐶𝑢 (𝐸) + ). Then □ 𝜎(𝐶𝑢 (𝐸) + ) = 𝜎(𝐶𝑢 (𝐸)) = ℬ(𝐸). For a topological space 𝐸, let 𝑀 (𝐸) denote the space of finite Borel measures on 𝐸 and let 𝑃(𝐸) be the subset of 𝑀 (𝐸) consisting of probability measures. We say a sequence {𝜇 𝑛 } ⊂ 𝑀 (𝐸) converges weakly to 𝜇 ∈ 𝑀 (𝐸) and write lim𝑛→∞ 𝜇 𝑛 = 𝜇 or 𝜇 𝑛 → 𝜇 if lim𝑛→∞ 𝜇 𝑛 ( 𝑓 ) = 𝜇( 𝑓 ) for every 𝑓 ∈ 𝐶 (𝐸). The weak convergence is a topological concept. For functions 𝑓1 , . . . , 𝑓 𝑘 ∈ 𝐶 (𝐸) and open sets 𝐺 1 , . . . , 𝐺 𝑘 ⊂ R let 𝑈 ( 𝑓1 , . . . , 𝑓 𝑘 ; 𝐺 1 , . . . , 𝐺 𝑘 ) = {𝜈 ∈ 𝑀 (𝐸) : 𝜈( 𝑓𝑖 ) ∈ 𝐺 𝑖 , 1 ≤ 𝑖 ≤ 𝑘 }.

(1.2)

1.1 Borel Measures

3

It is easy to show that the family 𝒰0 of sets 𝑈 ( 𝑓1 , . . . , 𝑓 𝑘 ; 𝐺 1 , . . . , 𝐺 𝑘 ) obtained by varying 𝑘 ≥ 1 and {( 𝑓1 , 𝐺 1 ), . . . , ( 𝑓 𝑘 , 𝐺 𝑘 )} satisfies the axioms of a base for a topology of 𝑀 (𝐸), which is called the topology of weak convergence. It is clear that lim𝑛→∞ 𝜇 𝑛 = 𝜇 if and only if 𝜇 𝑛 converges to 𝜇 in this topology. In the sequel, we assume 𝑀 (𝐸) is furnished with this topology of weak convergence. Proposition 1.3 Let (𝐸, 𝑑) be a metric space and let 𝒢 be a set of functions on 𝐸 which is closed under bounded pointwise convergence. Then (1) 𝐶𝑢 (𝐸) ⊂ 𝒢 implies 𝐵(𝐸) ⊂ 𝒢; (2) 𝐶𝑢 (𝐸) ++ ⊂ 𝒢 implies 𝐵(𝐸) + ⊂ 𝒢. Proof Since 𝐶𝑢 (𝐸) is a vector space which contains 1𝐸 and is closed under multiplication, the first assertion follows from Proposition A.2 and Corollary 1.2. Under the condition of the second assertion, we have 𝐶𝑢 (𝐸) ⊂ { 𝑓 : e 𝑓 ∈ 𝐶𝑢 (𝐸) ++ } ⊂ { 𝑓 : e 𝑓 ∈ 𝒢}. Then the first assertion implies 𝐵(𝐸) ⊂ { 𝑓 : e 𝑓 ∈ 𝒢}. In particular, for any ℎ ∈ 𝐵(𝐸) ++ we have log ℎ ∈ 𝐵(𝐸) ⊂ { 𝑓 : e 𝑓 ∈ 𝒢} and hence ℎ = elog ℎ ∈ 𝒢. This proves 𝐵(𝐸) ++ ⊂ 𝒢, implying the second assertion since 𝒢 is closed under bounded pointwise convergence. □ Corollary 1.4 Let (𝐸, 𝑑) be a metric space and let 𝜇, 𝜈 ∈ 𝑀 (𝐸). If 𝜇( 𝑓 ) = 𝜈( 𝑓 ) for every 𝑓 ∈ 𝐶𝑢 (𝐸), we have 𝜇 = 𝜈. Proof Let 𝒢 be the family of functions 𝑓 ∈ 𝐵(𝐸) such that 𝜇( 𝑓 ) = 𝜈( 𝑓 ). Then 𝐶𝑢 (𝐸) ⊂ 𝒢. By the dominated convergence theorem it is easy to show that 𝒢 is closed under bounded pointwise convergence. Consequently, we have 𝐵(𝐸) ⊂ 𝒢 by Proposition 1.3. □ Corollary 1.5 Let (𝐸, 𝑑) be a metric space. Then for every 𝑓 ∈ 𝐵(𝐸) the mapping 𝜇 ↦→ 𝜇( 𝑓 ) from 𝑀 (𝐸) to R is Borel measurable. Proof Let 𝒢 be the family of functions 𝑓 ∈ 𝐵(𝐸) such that 𝜇 ↦→ 𝜇( 𝑓 ) is Borel measurable. Then 𝒢 is closed under bounded pointwise convergence. For any 𝑓 ∈ 𝐶𝑢 (𝐸) the mapping 𝜇 ↦→ 𝜇( 𝑓 ) is continuous and hence Borel measurable. In other □ words, we have 𝐶𝑢 (𝐸) ⊂ 𝒢. Then Proposition 1.3 implies 𝐵(𝐸) ⊂ 𝒢. Theorem 1.6 Suppose that (𝐸, 𝑑) is a metric space. For any 𝜇 ∈ 𝑀 (𝐸) and any sequence {𝜇 𝑛 } ⊂ 𝑀 (𝐸) the following statements are equivalent: (1) lim𝑛→∞ 𝜇 𝑛 = 𝜇; (2) lim𝑛→∞ 𝜇 𝑛 ( 𝑓 ) = 𝜇( 𝑓 ) for every 𝑓 ∈ 𝐶𝑢 (𝐸); (3) lim𝑛→∞ 𝜇 𝑛 (1) = 𝜇(1) and lim sup𝑛→∞ 𝜇 𝑛 (𝐶) ≤ 𝜇(𝐶) for every closed set 𝐶 ⊂ 𝐸; (4) lim𝑛→∞ 𝜇 𝑛 (1) = 𝜇(1) and lim inf 𝑛→∞ 𝜇 𝑛 (𝐺) ≥ 𝜇(𝐺) for every open set 𝐺 ⊂ 𝐸; (5) lim𝑛→∞ 𝜇 𝑛 (𝐵) = 𝜇(𝐵) for every 𝐵 ∈ ℬ(𝐸) with 𝜇(𝜕𝐵) = 0, where 𝜕𝐵 is the boundary of 𝐵.

4

1 Random Measures on Metric Spaces

Proof The results are obvious if 𝜇(1) = 0. If 𝜇(1) > 0, then there is an index 𝑛0 ≥ 1 such that 𝜇 𝑛 (1) > 0 for all 𝑛 ≥ 𝑛0 . Let 𝜇ˆ = 𝜇(1) −1 𝜇 ∈ 𝑃(𝐸) and let 𝜇ˆ 𝑛 = 𝜇 𝑛 (1) −1 𝜇 𝑛 ∈ 𝑃(𝐸) for 𝑛 ≥ 𝑛0 . It is easy to see that lim𝑛→∞ 𝜇 𝑛 = 𝜇 if and only if lim𝑛→∞ 𝜇 𝑛 (1) = 𝜇(1) and lim𝑛→∞ 𝜇ˆ 𝑛 = 𝜇. ˆ Then the theorem follows from the results in the special case of probability measures; see, e.g., Ethier and Kurtz (1986, □ p. 108) and Parthasarathy (1967, pp. 41–42). Theorem 1.7 Suppose that 𝐸 is a Borel subspace of a metrizable topological space 𝐹. Let 𝜇 ∈ 𝑀 (𝐸) and let {𝜇 𝑛 } ⊂ 𝑀 (𝐸) be a sequence. Let 𝜈 and 𝜈𝑛 denote respectively the extensions of 𝜇 and 𝜇 𝑛 to 𝐹 such that 𝜈(𝐹 \ 𝐸) = 𝜈𝑛 (𝐹 \ 𝐸) = 0. Then lim𝑛→∞ 𝜈𝑛 = 𝜈 in 𝑀 (𝐹) if and only if lim𝑛→∞ 𝜇 𝑛 = 𝜇 in 𝑀 (𝐸). Proof Since the restriction of a bounded continuous function is also a bounded continuous function, lim𝑛→∞ 𝜇 𝑛 = 𝜇 in 𝑀 (𝐸) implies lim𝑛→∞ 𝜈𝑛 = 𝜈 in 𝑀 (𝐹). For the converse, suppose that lim𝑛→∞ 𝜈𝑛 = 𝜈 in 𝑀 (𝐹). Then we have lim 𝜇 𝑛 (𝐸) = lim 𝜈𝑛 (𝐹) = 𝜈(𝐹) = 𝜇(𝐸). 𝑛→∞

𝑛→∞

For any closed subset 𝐶 of 𝐸, there is a closed subset 𝐷 of 𝐹 such that 𝐶 = 𝐷 ∩ 𝐸. It follows that lim sup 𝜇 𝑛 (𝐶) = lim sup 𝜈𝑛 (𝐷) ≤ 𝜈(𝐷) = 𝜇(𝐶). 𝑛→∞

𝑛→∞

Then lim𝑛→∞ 𝜇 𝑛 = 𝜇 in 𝑀 (𝐸) by Theorem 1.6.

□

If 𝐸 is a separable metric space, its topology can be defined by a totally bounded metric (𝑥, 𝑦) ↦→ 𝑑 (𝑥, 𝑦). Indeed, 𝐸 is homeomorphic to a subset of the countable product space [0, 1] ∞ furnished with the product metric; see, e.g., Kelley (1955, p. 125). Then the set of uniformly continuous functions 𝐶𝑢 (𝐸) endowed with the supremum norm ∥ · ∥ is a separable Banach space; see, e.g., Parthasarathy (1967, p. 43). Theorem 1.8 If 𝐸 is a separable metric space, then 𝑀 (𝐸) is separable. Proof Let 𝑄 be a countable dense subset of [0, ∞) and 𝐹 a countable dense subset of 𝐸. We claim that the countable set 𝑀1 :=

𝑛 n ∑︁

𝛼𝑖 𝛿 𝑥𝑖 : 𝑥1 , . . . , 𝑥 𝑛 ∈ 𝐹; 𝛼1 , . . . , 𝛼𝑛 ∈ 𝑄; 𝑛 ≥ 1

o

𝑖=1

is dense in 𝑀 (𝐸). To see this we first fix a totally bounded metric 𝑑 on 𝐸 compatible with its topology. Then for each integer 𝑛 ≥ 1 the space 𝐸 has a finite covering {𝐵𝑛,𝑖 : 𝑖 = 1, . . . , 𝑝 𝑛 } consisting of open balls of radius 1/𝑛. Take 𝑥 𝑛,𝑖 ∈ 𝐵𝑛,𝑖 ∩ 𝐹 for 𝑖 = 1, . . . , 𝑝 𝑛 . Let 𝐴𝑛,1 = 𝐵𝑛,1 and let 𝐴𝑛,𝑖 = 𝐵𝑛,𝑖 \ (𝐵𝑛,1 ∪ · · · ∪ 𝐵𝑛,𝑖−1 ) for 𝑖 = 2, . . . , 𝑝 𝑛 . Given Í 𝑝𝑛 𝜇 ∈ 𝑀 (𝐸) we take 𝛼𝑛,𝑖 ∈ 𝑄 so that |𝛼𝑛,𝑖 − 𝜇( 𝐴𝑛,𝑖 )| ≤ 1/𝑛𝑝 𝑛 𝛼𝑛,𝑖 𝛿 𝑥𝑛,𝑖 ∈ 𝑀1 . Then for any 𝑓 ∈ 𝐶𝑢 (𝐸) we have and define 𝜇 𝑛 = 𝑖=1

1.1 Borel Measures

5

|𝜇 𝑛 ( 𝑓 ) − 𝜇( 𝑓 )| ≤ ≤

𝑝𝑛 ∑︁ 𝑖=1 𝑝𝑛 ∑︁

|𝛼𝑛,𝑖 𝑓 (𝑥 𝑛,𝑖 ) − 𝜇( 𝑓 1 𝐴𝑛,𝑖 )| |𝛼𝑛,𝑖 − 𝜇( 𝐴𝑛,𝑖 )|| 𝑓 (𝑥 𝑛,𝑖 )|

𝑖=1 𝑝𝑛 ∑︁

+

|𝜇( 𝐴𝑛,𝑖 ) 𝑓 (𝑥 𝑛,𝑖 ) − 𝜇( 𝑓 1 𝐴𝑛,𝑖 )|

𝑖=1

≤ ∥𝑓∥ ≤

𝑝𝑛 𝑝𝑛 ∑︁ ∑︁ 1 + sup | 𝑓 (𝑥 𝑛,𝑖 ) − 𝑓 (𝑦)|𝜇( 𝐴𝑛,𝑖 ) 𝑛𝑝 𝑛 𝑖=1 𝑦 ∈ 𝐴𝑛,𝑖 𝑖=1

∥𝑓∥ + sup | 𝑓 (𝑥) − 𝑓 (𝑦)|𝜇(𝐸). 𝑛 𝑑 ( 𝑥,𝑦) ≤2/𝑛

By the uniform continuity of 𝑓 ∈ 𝐶𝑢 (𝐸), the right-hand side of the above inequality tends to zero as 𝑛 → ∞. Then 𝑀1 is dense in 𝑀 (𝐸). □ Theorem 1.9 Let (𝐸, 𝑑) be a separable and totally bounded metric space and let 𝑆(𝐸, 𝑑) = { 𝑓0 , 𝑓1 , 𝑓2 , . . .} be a dense sequence in 𝐶𝑢 (𝐸) with 𝑓0 ≡ 1. Then 𝜇 𝑛 → 𝜇 in 𝑀 (𝐸) if and only if 𝜇 𝑛 ( 𝑓𝑖 ) → 𝜇( 𝑓𝑖 ) for every 𝑓𝑖 ∈ 𝑆(𝐸, 𝑑). Proof It is clear that 𝜇 𝑛 → 𝜇 in 𝑀 (𝐸) implies 𝜇 𝑛 ( 𝑓𝑖 ) → 𝜇( 𝑓𝑖 ) for every 𝑓𝑖 ∈ 𝑆(𝐸, 𝑑). Conversely, suppose that 𝜇 𝑛 ( 𝑓𝑖 ) → 𝜇( 𝑓𝑖 ) for every 𝑓𝑖 ∈ 𝑆(𝐸, 𝑑). For 𝑓 ∈ 𝐶𝑢 (𝐸) and 𝑓𝑖 ∈ 𝑆(𝐸, 𝑑) we have |𝜇 𝑛 ( 𝑓 ) − 𝜇( 𝑓 )| ≤ 𝜇 𝑛 (| 𝑓 − 𝑓𝑖 |) + 𝜇(| 𝑓 − 𝑓𝑖 |) + |𝜇 𝑛 ( 𝑓𝑖 ) − 𝜇( 𝑓𝑖 )| ≤ ∥ 𝑓 − 𝑓𝑖 ∥ [𝜇 𝑛 (1) + 𝜇(1)] + |𝜇 𝑛 ( 𝑓𝑖 ) − 𝜇( 𝑓𝑖 )|. Since there is a sequence { 𝑓 𝑘𝑖 } ⊂ 𝑆(𝐸, 𝑑) satisfying ∥ 𝑓 𝑘𝑖 − 𝑓 ∥ → 0, it is easy to conclude |𝜇 𝑛 ( 𝑓 ) − 𝜇( 𝑓 )| → 0. Then 𝜇 𝑛 → 𝜇 by Theorem 1.6. □ Corollary 1.10 In the setup of Theorem 1.9, let 𝑆1 (𝐸, 𝑑) = {ℎ0 , ℎ1 , ℎ2 , . . .} be a dense sequence in { 𝑓 ∈ 𝐶𝑢 (𝐸) + : ∥ 𝑓 ∥ ≤ 1} with ℎ0 ≡ 1. Then 𝜇 𝑛 → 𝜇 in 𝑀 (𝐸) if and only if 𝜇 𝑛 (ℎ𝑖 ) → 𝜇(ℎ𝑖 ) for every ℎ𝑖 ∈ 𝑆1 (𝐸, 𝑑). Proof If 𝜇 𝑛 → 𝜇 in 𝑀 (𝐸), we clearly have 𝜇 𝑛 (ℎ𝑖 ) → 𝜇(ℎ𝑖 ) for every ℎ𝑖 ∈ 𝑆1 (𝐸, 𝑑). Conversely, suppose that 𝜇 𝑛 (ℎ𝑖 ) → 𝜇(ℎ𝑖 ) for every ℎ𝑖 ∈ 𝑆1 (𝐸, 𝑑). Then 𝜇 𝑛 ( 𝑓 ) → 𝜇( 𝑓 ) for every 𝑓 ∈ 𝒬, where 𝒬 = {𝑎ℎ𝑖 + 𝑏ℎ 𝑗 : ℎ𝑖 , ℎ 𝑗 ∈ 𝑆1 (𝐸, 𝑑) and 𝑎, 𝑏 are rationals} is a countable dense subset of 𝐶𝑢 (𝐸). Then we have 𝜇 𝑛 → 𝜇 in 𝑀 (𝐸) by Theorem 1.9. □ Given a separable metric space 𝐸, we fix a totally bounded metric 𝑑 compatible with its topology and let 𝑆1 (𝐸, 𝑑) be as in Corollary 1.10. Then a metric 𝜌 on 𝑀 (𝐸) is defined by ∞ ∑︁ 1 (1 ∧ |𝜇(ℎ𝑖 ) − 𝜈(ℎ𝑖 )|), 𝜌(𝜇, 𝜈) = 𝑖 2 𝑖=0

𝜇, 𝜈 ∈ 𝑀 (𝐸).

(1.3)

6

1 Random Measures on Metric Spaces

This metric is compatible with the weak convergence topology of 𝑀 (𝐸). In other words, we have 𝜇 𝑛 → 𝜇 in 𝑀 (𝐸) if and only if 𝜌(𝜇 𝑛 , 𝜇) → 0. The countable family 𝒰1 of sets 𝑈 (ℎ0 , ℎ1 , . . . , ℎ 𝑘 ; (𝑎 0 , 𝑏 0 ), (𝑎 1 , 𝑏 1 ), . . . , (𝑎 𝑘 , 𝑏 𝑘 )) obtained by varying the integer 𝑘 ≥ 1, the functions ℎ𝑖 ∈ 𝑆1 (𝐸, 𝑑) and the pairs of rationals 𝑎 𝑖 < 𝑏 𝑖 is a countable base of the topology of 𝑀 (𝐸). Theorem 1.11 For a separable metric space 𝐸 we have ℬ(𝑀 (𝐸)) = 𝜎({𝜇 ↦→ 𝜇( 𝑓 ) : 𝑓 ∈ 𝐶 (𝐸) + }) = 𝜎({𝜇 ↦→ 𝜇( 𝑓 ) : 𝑓 ∈ 𝐶 (𝐸)}). Proof It is easy to see that ℬ0 := 𝜎({𝜇 ↦→ 𝜇( 𝑓 ) : 𝑓 ∈ 𝐶 (𝐸) + }) contains the countable family 𝒰1 . Since every open subset of 𝑀 (𝐸) is the union of some elements of this family, all those open subsets belong to ℬ0 and hence ℬ(𝑀 (𝐸)) ⊂ ℬ0 ⊂ 𝜎({𝜇 ↦→ 𝜇( 𝑓 ) : 𝑓 ∈ 𝐶 (𝐸)}). On the other hand, for any 𝑓 ∈ 𝐶 (𝐸) the mapping 𝜇 ↦→ 𝜇( 𝑓 ) is continuous on 𝑀 (𝐸). Then we have 𝜎({𝜇 ↦→ 𝜇( 𝑓 ) : 𝑓 ∈ 𝐶 (𝐸)}) ⊂ □ ℬ(𝑀 (𝐸)). Corollary 1.12 If 𝐸 is a separable metric space, then ℬ(𝑀 (𝐸)) = 𝜎({𝜇 ↦→ 𝜇( 𝑓 ) : 𝑓 ∈ 𝐵(𝐸)}) = 𝜎({𝜇 ↦→ 𝜇( 𝐴) : 𝐴 ∈ ℬ(𝐸)}). Proof Let ℬ1 = 𝜎({𝜇 → ↦ 𝜇( 𝑓 ) : 𝑓 ∈ 𝐵(𝐸)}) and ↦ 𝜇( 𝐴) : 𝐴 ∈ ℬ(𝐸)}). ℬ2 = 𝜎({𝜇 → Then ℬ1 ⊃ ℬ2 obviously. By Theorem 1.11 it is easy to see ℬ(𝑀 (𝐸)) ⊂ ℬ1 . Then Corollary 1.5 implies ℬ(𝑀 (𝐸)) = ℬ1 . For a simple function 𝑓 ∈ 𝐵(𝐸), the mapping 𝜇 ↦→ 𝜇( 𝑓 ) is clearly measurable with respect to ℬ2 . By an approximation argument one sees 𝜇 ↦→ 𝜇( 𝑓 ) is measurable with respect to ℬ2 for an arbitrary □ 𝑓 ∈ 𝐵(𝐸). Then ℬ2 ⊃ ℬ1 . Theorem 1.13 Suppose that (𝐹, ℱ) is a general measurable space and 𝐸 is a separable metric space. For 𝐴 ∈ ℬ(𝐸) and 𝜇 ∈ 𝑀 (𝐸) write 𝑙 𝐴 (𝜇) = 𝜇( 𝐴). Then 𝜓 is a measurable map from (𝐹, ℱ) to (𝑀 (𝐸), ℬ(𝑀 (𝐸))) if and only if for every 𝐴 ∈ ℬ(𝐸) the composition 𝑙 𝐴 ◦ 𝜓 is a measurable real function on (𝐹, ℱ). Proof Suppose that 𝜓 is a measurable map from (𝐹, ℱ) to (𝑀 (𝐸), ℬ(𝑀 (𝐸))). By Corollary 1.12 the real function 𝑙 𝐴 on 𝑀 (𝐸) is Borel for every 𝐴 ∈ ℬ(𝐸). Then the composition 𝑙 𝐴 ◦ 𝜓 is a measurable function on (𝐹, ℱ). Conversely, suppose for every 𝐴 ∈ ℬ(𝐸) the composition 𝑙 𝐴 ◦ 𝜓 is a measurable function on (𝐹, ℱ). Then for 𝐵 ∈ ℬ(R) we have 𝜓 −1 (𝑙 −1 𝐴 (𝐵)) ∈ ℱ, so 𝜓 −1 ({𝑙 −1 𝐴 (𝐵) : 𝐴 ∈ ℬ(𝐸), 𝐵 ∈ ℬ(R)}) ⊂ ℱ.

1.1 Borel Measures

7

It follows that ℱ ⊃ 𝜎(𝜓 −1 ({𝑙 −1 𝐴 (𝐵) : 𝐴 ∈ ℬ(𝐸), 𝐵 ∈ ℬ(R)})) = 𝜓 −1 (𝜎({𝑙 −1 𝐴 (𝐵) : 𝐴 ∈ ℬ(𝐸), 𝐵 ∈ ℬ(R)})) −1 = 𝜓 (𝜎({𝑙 𝐴 : 𝐴 ∈ ℬ(𝐸)})) = 𝜓 −1 (ℬ(𝑀 (𝐸))). Then 𝜓 is a measurable map from (𝐹, ℱ) to (𝑀 (𝐸), ℬ(𝑀 (𝐸))).

□

Theorem 1.14 If 𝐸 is a compact metric space, then 𝑀 (𝐸) is a locally compact separable and metrizable space. Moreover, for any 𝑏 ≥ 0 the set 𝑀𝑏 := {𝜇 ∈ 𝑀 (𝐸) : 𝜇(𝐸) ≤ 𝑏} is compact. Proof Since 𝐸 is a compact metric space, it is separable. Then 𝑀 (𝐸) is separable by Theorem 1.8. Let 𝑑 be a metric on 𝐸 for its topology and let 𝑆1 (𝐸, 𝑑) be as in Corollary 1.10. The topology of 𝑀 (𝐸) can be defined by the metric 𝜌 given by (1.3). For any 𝜇 ∈ 𝑀 (𝐸) let 𝑇 (𝜇) = (𝜇(ℎ0 ), 𝜇(ℎ1 ), 𝜇(ℎ2 ), . . .). It is easy to see that 𝑇 is a homeomorphism between 𝑀 (𝐸) and a subset of the countable product space R∞ +. Observe that 𝑇 (𝑀𝑏 ) ⊂ [0, 𝑏] ∞ ⊂ R+∞ . We claim that 𝑇 (𝑀𝑏 ) is closed in [0, 𝑏] ∞ . To see this, suppose that {𝜇 𝑛 } ⊂ 𝑀𝑏 and 𝑇 (𝜇 𝑛 ) → (𝛼0 , 𝛼1 , 𝛼2 , . . .) in [0, 𝑏] ∞ . We need to show (𝛼0 , 𝛼1 , 𝛼2 , . . .) ∈ 𝑇 (𝑀𝑏 ). For 𝑓 ∈ 𝐶𝑢 (𝐸) + satisfying ∥ 𝑓 ∥ ≤ 1 let {ℎ𝑖𝑘 } ⊂ 𝑆1 (𝐸, 𝑑) be a sequence such that ∥ℎ𝑖𝑘 − 𝑓 ∥ → 0 as 𝑘 → ∞. For 𝑛 ≥ 𝑚 ≥ 1 we have |𝜇 𝑛 ( 𝑓 ) − 𝜇 𝑚 ( 𝑓 )| ≤ ∥ 𝑓 − ℎ𝑖𝑘 ∥ [𝜇 𝑛 (1) + 𝜇 𝑚 (1)] + |𝜇 𝑛 (ℎ𝑖𝑘 ) − 𝜇 𝑚 (ℎ𝑖𝑘 )| and hence lim sup |𝜇 𝑛 ( 𝑓 ) − 𝜇 𝑚 ( 𝑓 )| ≤ 2𝛼0 ∥ 𝑓 − ℎ𝑖𝑘 ∥. 𝑚,𝑛→∞

Then letting 𝑘 → ∞ gives lim sup |𝜇 𝑛 ( 𝑓 ) − 𝜇 𝑚 ( 𝑓 )| = 0. 𝑚,𝑛→∞

By linearity, the above relation holds for all 𝑓 ∈ 𝐶𝑢 (𝐸), so the limit 𝜆( 𝑓 ) = lim𝑛→∞ 𝜇 𝑛 ( 𝑓 ) exists for each 𝑓 ∈ 𝐶𝑢 (𝐸). Clearly, 𝑓 ↦→ 𝜆( 𝑓 ) is a positive linear functional on 𝐶𝑢 (𝐸). By the Riesz representation theorem, there exists a 𝜇 ∈ 𝑀 (𝐸) such that 𝜇( 𝑓 ) = 𝜆( 𝑓 ) for every 𝑓 ∈ 𝐶𝑢 (𝐸). In particular, 𝜇(ℎ𝑖 ) = 𝜆(ℎ𝑖 ) = 𝛼𝑖 for all 𝑖 ≥ 0. It follows that 𝜇(1) = 𝛼0 = lim𝑛→∞ 𝜇 𝑛 (1) ≤ 𝑏 and hence 𝜇 ∈ 𝑀𝑏 . This shows (𝛼0 , 𝛼1 , 𝛼2 , . . .) = 𝑇 (𝜇) ∈ 𝑇 (𝑀𝑏 ). Then 𝑇 (𝑀𝑏 ) is a closed subset of [0, 𝑏] ∞ . Since [0, 𝑏] ∞ is compact, so is 𝑇 (𝑀𝑏 ). It follows that 𝑀𝑏 is compact and 𝑀 (𝐸) □ locally compact. Corollary 1.15 Let 𝐸 be a compact metric space and let 𝑀¯ (𝐸) := 𝑀 (𝐸) ∪ {Δ} be the one-point compactification of 𝑀 (𝐸). Then 𝜇 𝑛 → Δ if and only if 𝜇 𝑛 (𝐸) → ∞. Proof It is easy to see that { 𝑀¯ (𝐸) \ 𝑀𝑏 : 𝑏 ≥ 0} is a local base at Δ. Then the □ assertion is evident.

8

1 Random Measures on Metric Spaces

A metrizable space 𝐸 is called a Lusin topological space if it is homeomorphic to a Borel subset of a compact metric space. Such a space is clearly separable. A measurable space (𝐹, ℱ) is called a Lusin measurable space if it is measurably isomorphic to (𝐸, ℬ(𝐸)) with 𝐸 being a Lusin topological space. Theorem 1.16 If 𝐸 is a Lusin topological space, then 𝑀 (𝐸) is a Lusin topological space. Proof Since 𝐸 is a Lusin topological space, we may embed it into some compact metric space 𝐹 as a Borel subset. Theorem 1.7 implies that 𝑀 (𝐸) is homeomorphic to 𝑀0 := {𝜇 ∈ 𝑀 (𝐹) : 𝜇(𝐹 \ 𝐸) = 0}. By Corollary 1.12 the mapping 𝜇 ↦→ 𝜇(𝐹 \ 𝐸) is ℬ(𝑀 (𝐹))-measurable. Then 𝑀0 is a Borel subset of the locally compact separable and metrizable space 𝑀 (𝐹), which is an open subset of its one-point compactification 𝑀¯ (𝐹) := 𝑀 (𝐹) ∪ {Δ}. Therefore 𝑀 (𝐸) is homeomorphic to a Borel subset of the compact metrizable space 𝑀¯ (𝐹). □ A subset of a topological space is called a 𝐺 𝛿 set if it can be expressed as the intersection of a countable number of open sets. It is well known that a space is homeomorphic to a complete metric space if and only if it is a 𝐺 𝛿 set in some complete metric space, and in this case it is a 𝐺 𝛿 set in every complete metric space into which it is topologically embedded; see, e.g., Kelley (1955, pp. 207–208). Theorem 1.17 Suppose that 𝐸 is a separable and complete metric space. Then 𝑀 (𝐸) is homeomorphic to a separable and complete metric space. Proof By Theorem 1.8, the metric space 𝑀 (𝐸) is a separable. Let 𝑑 be a totally bounded metric on 𝐸 for its topology. Then its completion 𝐹 is compact and 𝐸 is a 𝐺 𝛿 subset of the compact metric space 𝐹. Choose a decreasing sequence of open ¯ sets 𝐺 1 ⊃ 𝐺 2 ⊃ · · · in 𝐹 such that ∩∞ 𝑘=1 𝐺 𝑘 = 𝐸. Let 𝑀0 , 𝑀 (𝐹) and 𝑀 (𝐹) be defined as in the proof of Theorem 1.16. It is clear that 𝑀0 = =

∞ Ù

{𝜇 ∈ 𝑀 (𝐹) : 𝜇(𝐹 \ 𝐺 𝑘 ) = 0}

𝑘=1 ∞ Ù ∞ Ù

{𝜇 ∈ 𝑀 (𝐹) : 𝜇(𝐹 \ 𝐺 𝑘 ) < 1/𝑖}.

𝑘=1 𝑖=1

If 𝜇 𝑛 → 𝜇 in 𝑀 (𝐹) and 𝜇 𝑛 (𝐹 \ 𝐺 𝑘 ) ≥ 1/𝑖, by Theorem 1.6 we have 𝜇(𝐹 \ 𝐺 𝑘 ) ≥ lim sup 𝜇 𝑛 (𝐹 \ 𝐺 𝑘 ) ≥ 1/𝑖. 𝑛→∞

This shows {𝜇 ∈ 𝑀 (𝐹) : 𝜇(𝐹 \ 𝐺 𝑘 ) ≥ 1/𝑖} is closed in 𝑀¯ (𝐹), so 𝑀0 is a 𝐺 𝛿 set in 𝑀¯ (𝐹). Thus 𝑀 (𝐸) is homeomorphic to a separable and complete subset of a compact metric space. □ Example 1.1 Let [0, 1] ∞ be the countable product of the unit interval furnished with the product metric 𝑞. Suppose that (𝐸, 𝑑) is a separable metric space with the dense sequence 𝐹 := {𝑥1 , 𝑥2 , . . .}. For any 𝑥 ∈ 𝐸 write

1.2 Laplace Functionals

9

𝑔(𝑥) = (1 ∧ 𝑑 (𝑥, 𝑥 1 ), 1 ∧ 𝑑 (𝑥, 𝑥 2 ), . . .). Then 𝑔 is a homeomorphism between 𝐸 and 𝑔(𝐸) ⊂ [0, 1] ∞ . This homeomorphism induces a totally bounded metric on 𝐸 compatible with its original topology. For 𝜀 > 0 let 𝑔(𝐹) 𝜀 = {𝑦 ∈ [0, 1] ∞ : 𝑞(𝑦, 𝑔(𝑥𝑖 )) < 𝜀 for some 𝑖 ≥ 1}. Clearly, each 𝑔(𝐹) 𝜀 is an open set in the compact metric space [0, 1] ∞ . If (𝐸, 𝑑) is 1/𝑛 . Consequently, a complete separable complete in addition, then 𝑔(𝐸) = ∩∞ 𝑛=1 𝑔(𝐹) metric space is a Lusin topological space.

1.2 Laplace Functionals In this section, we assume 𝐸 is a Lusin topological space. Recall that 𝑀 (𝐸) is the space of finite measures on 𝐸 equipped with the topology of weak convergence. Given a finite measure 𝑄 on 𝑀 (𝐸), we define the Laplace functional 𝐿 𝑄 of 𝑄 by ∫ (1.4) 𝐿𝑄 ( 𝑓 ) = 𝑓 ∈ 𝐵(𝐸) + . e−𝜈 ( 𝑓 ) 𝑄(d𝜈), 𝑀 (𝐸)

Theorem 1.18 A finite measure on 𝑀 (𝐸) is uniquely determined by the restriction of its Laplace functional to 𝐶 (𝐸) ++ . Proof Suppose that 𝑄 1 and 𝑄 2 are finite measures on 𝑀 (𝐸) and 𝐿 𝑄1 ( 𝑓 ) = 𝐿 𝑄2 ( 𝑓 ) for all 𝑓 ∈ 𝐶 (𝐸) ++ . Then for any 𝑓 ∈ 𝐶 (𝐸) + we have 𝐿 𝑄1 ( 𝑓 ) = lim 𝐿 𝑄1 ( 𝑓 + 1/𝑛) = lim 𝐿 𝑄2 ( 𝑓 + 1/𝑛) = 𝐿 𝑄2 ( 𝑓 ). 𝑛→∞

𝑛→∞

Let 𝒦 = {𝜈 ↦→ e−𝜈 ( 𝑓 ) : 𝑓 ∈ 𝐶 (𝐸) + } and let ℒ = {𝐹 ∈ 𝐵(𝑀 (𝐸)) : 𝑄 1 (𝐹) = 𝑄 2 (𝐹)}. Then 𝒦 is closed under multiplication and ℒ is a monotone vector space containing 𝒦. By Theorem 1.11 it is easy to show 𝜎(𝒦) = ℬ(𝑀 (𝐸)). Then Proposition A.1 implies ℒ ⊃ b𝜎(𝒦) = 𝐵(𝑀 (𝐸)). This proves the desired result.□ Theorem 1.19 Let 𝑄 1 , 𝑄 2 , . . . and 𝑄 be finite measures on 𝑀 (𝐸). If 𝑄 𝑛 → 𝑄 weakly, then 𝐿 𝑄𝑛 ( 𝑓 ) → 𝐿 𝑄 ( 𝑓 ) for 𝑓 ∈ 𝐶 (𝐸) + . Conversely, if 𝐿 𝑄𝑛 ( 𝑓 ) → 𝐿 𝑄 ( 𝑓 ) for all 𝑓 ∈ 𝐶 (𝐸) ++ ∪ {0}, then 𝑄 𝑛 → 𝑄 weakly. Proof If 𝑄 𝑛 → 𝑄 weakly, we have lim𝑛→∞ 𝐿 𝑄𝑛 ( 𝑓 ) = 𝐿 𝑄 ( 𝑓 ) for 𝑓 ∈ 𝐶 (𝐸) + clearly. Now assume lim𝑛→∞ 𝐿 𝑄𝑛 ( 𝑓 ) = 𝐿 𝑄 ( 𝑓 ) for all 𝑓 ∈ 𝐶 (𝐸) ++ ∪ {0}. Let 𝐹 be a compact metric space such that 𝐸 is embedded into 𝐹 as a Borel subset. Then 𝑀 (𝐹) is a locally compact separable and metrizable space. Let 𝑀¯ (𝐹) = 𝑀 (𝐹) ∪ {Δ} be its one-point compactification, which is a compact metrizable space by Theorem 1.14. We identify 𝑀 (𝐸) with the Borel subset of 𝑀 (𝐹) consisting of measures supported by 𝐸 and regard 𝑄 1 , 𝑄 2 , . . . and 𝑄 as finite measures on 𝑀¯ (𝐹). Then

10

1 Random Measures on Metric Spaces

lim 𝑄 𝑛 ( 𝑀¯ (𝐹)) = lim 𝑄 𝑛 (𝑀 (𝐸)) = lim 𝐿 𝑄𝑛 (0) = 𝐿 𝑄 (0), 𝑛→∞

𝑛→∞

(1.5)

𝑛→∞

and hence {𝑄 𝑛 } ⊂ 𝑀 ( 𝑀¯ (𝐹)) is a bounded sequence. By another application of Theorem 1.14 we conclude that {𝑄 𝑛 } is relatively compact. Let {𝑄 𝑛𝑘 } ⊂ {𝑄 𝑛 } be a subsequence that converges to some 𝑄¯ ∈ 𝑀 ( 𝑀¯ (𝐹)). By (1.5) we have ¯ 𝑀¯ (𝐹)) = lim 𝑄 𝑛𝑘 ( 𝑀¯ (𝐹)) = 𝐿 𝑄 (0). 𝑄(

(1.6)

𝑘→∞

Moreover, for any 𝑓¯ ∈ 𝐶 (𝐹) ++ , ∫ ∫ ¯ ¯ e−𝜈 ( 𝑓 ) 𝑄(d𝜈) = lim ¯ (𝐹) 𝑀

𝑘→∞

¯ (𝐹) 𝑀

¯

e−𝜈 ( 𝑓 ) 𝑄 𝑛𝑘 (d𝜈) = 𝐿 𝑄 ( 𝑓 ),

(1.7)

¯ where 𝑓 = 𝑓¯| 𝐸 denotes the restriction of 𝑓¯ to 𝐸 and e−Δ( 𝑓 ) = 0 by convention. By ¯ letting 𝑓 → 0 uniformly in (1.7) we find 𝑄(𝑀 (𝐹)) = 𝐿 𝑄 (0), so 𝑄¯ is supported by 𝑀 (𝐹). From (1.7) we have ∫ ∫ ¯ ¯ e−𝜈 ( 𝑓 ) 𝑄(d𝜈). = 𝐿𝑄 ( 𝑓 ) = e−𝜈 ( 𝑓 ) 𝑄(d𝜈) 𝑀 (𝐹)

𝑀 (𝐸)

Then the uniqueness of Laplace functionals implies 𝑄¯ is supported by 𝑀 (𝐸) and its restriction to 𝑀 (𝐸) coincides with 𝑄. By Theorem 1.7 we have lim𝑛→∞ 𝑄 𝑛𝑘 = 𝑄 weakly on 𝑀 (𝐸). In the same way, one shows that every convergent subsequence of □ {𝑄 𝑛 } has the same limit 𝑄. Thus lim𝑛→∞ 𝑄 𝑛 = 𝑄 weakly on 𝑀 (𝐸). A Lusin topological space 𝐸 with the Borel 𝜎-algebra is isomorphic to a compact metric space with the Borel 𝜎-algebra. Indeed, a complete separable metric space is at most of the cardinality of the continuum and two Borel subsets of complete separable metric spaces are isomorphic if and only if they have the same cardinality; see, e.g., Parthasarathy (1967, pp. 8–14). Consequently, (𝐸, ℬ(𝐸)) is in fact isomorphic to a compact subset of the real line with its Borel 𝜎-algebra. Then we can and do introduce a metric 𝑟 into 𝐸 so that (𝐸, 𝑟) becomes a compact metric space while the Borel 𝜎-algebra induced by 𝑟 coincides with ℬ(𝐸). Let 𝑆2 (𝐸, 𝑟) be a dense sequence in 𝐶 (𝐸, 𝑟) ++ including all strictly positive rationals and let 𝑆¯2 (𝐸, 𝑟) = 𝑆2 (𝐸, 𝑟) ∪{0}. Proposition 1.20 Suppose that 𝐿 is a functional on 𝑆¯2 (𝐸, 𝑟) and there is a sequence { 𝑓𝑛 } ⊂ 𝑆2 (𝐸, 𝑟) such that lim𝑛→∞ 𝑓𝑛 = 0 in bounded pointwise convergence and lim𝑛→∞ 𝐿 ( 𝑓𝑛 ) = 𝐿(0). If there is a sequence of finite measures {𝑄 𝑛 } on 𝑀 (𝐸) such that lim 𝐿 𝑄𝑛 ( 𝑓 ) = 𝐿( 𝑓 ), 𝑛→∞

𝑓 ∈ 𝑆¯2 (𝐸, 𝑟),

(1.8)

then there is a finite measure 𝑄 on 𝑀 (𝐸) such that 𝐿 𝑄 ( 𝑓 ) = 𝐿 ( 𝑓 ) for every 𝑓 ∈ 𝑆¯2 (𝐸, 𝑟) and lim𝑛→∞ 𝑄 𝑛 = 𝑄 weakly on 𝑀 (𝐸, 𝑟).

1.2 Laplace Functionals

11

Proof This is a modification of the proof of Theorem 1.19. By Theorem 1.14, the space 𝑀 (𝐸, 𝑟) is locally compact, separable and metrizable. Let 𝑀¯ (𝐸, 𝑟) = 𝑀 (𝐸, 𝑟) ∪ {Δ} be its one-point compactification. Then (1.8) implies that {𝑄 𝑛 } is a bounded sequence of measures on 𝑀¯ (𝐸, 𝑟). By Theorem 1.14 the sequence is relatively compact in 𝑀 ( 𝑀¯ (𝐸, 𝑟)). Choose any subsequence {𝑄 𝑛𝑘 } ⊂ {𝑄 𝑛 } that converges to a finite measure 𝑄 ∈ 𝑀 ( 𝑀¯ (𝐸, 𝑟)). Then 𝑄( 𝑀¯ (𝐸, 𝑟)) = lim 𝑄 𝑛𝑘 ( 𝑀¯ (𝐸, 𝑟)) = lim 𝐿 𝑄𝑛𝑘 (0) = 𝐿 (0). By (1.8) for any 𝑓 ∈ 𝑆2 (𝐸, 𝑟) we have ∫ ∫ e−𝜈 ( 𝑓 ) 𝑄(d𝜈) = lim ¯ (𝐸,𝑟) 𝑀

𝑘→∞

¯ (𝐸,𝑟) 𝑀

e−𝜈 ( 𝑓 ) 𝑄 𝑛𝑘 (d𝜈) = 𝐿 ( 𝑓 ),

where e−Δ( 𝑓 ) = 0 by convention. It follows that ∫ e−𝜈 ( 𝑓𝑛 ) 𝑄(d𝜈) = lim 𝐿 ( 𝑓𝑛 ) = 𝐿 (0). 𝑄(𝑀 (𝐸, 𝑟)) = lim 𝑛→∞

(1.9)

𝑘→∞

𝑘→∞

¯ (𝐸,𝑟) 𝑀

(1.10)

(1.11)

𝑛→∞

In view of (1.9) and (1.11) we have 𝑄({Δ}) = 0, so (1.10) implies 𝐿 𝑄 ( 𝑓 ) = 𝐿( 𝑓 ) for 𝑓 ∈ 𝑆2 (𝐸, 𝑟). By Theorem 1.7 we have lim 𝑘→∞ 𝑄 𝑛𝑘 = 𝑄 weakly on 𝑀 (𝐸, 𝑟). In the same way, if {𝑄 𝑛′ 𝑘 } ⊂ {𝑄 𝑛 } is another subsequence converging to a finite measure 𝑄 ′ on 𝑀¯ (𝐸, 𝑟), then 𝑄 ′ ({Δ}) = 0 and 𝐿 𝑄′ ( 𝑓 ) = 𝐿( 𝑓 ) for 𝑓 ∈ 𝑆2 (𝐸, 𝑟). Consequently, ∫ ∫ e−𝜈 ( 𝑓 ) 𝑄 ′ (d𝜈) e−𝜈 ( 𝑓 ) 𝑄(d𝜈) = 𝑀 (𝐸)

𝑀 (𝐸)

first for 𝑓 ∈ 𝑆2 (𝐸, 𝑟) and then for all 𝑓 ∈ 𝐶 (𝐸, 𝑟) + by dominated convergence, so 𝑄 = 𝑄 ′ by Theorem 1.18. Therefore we must have lim𝑛→∞ 𝑄 𝑛 = 𝑄 weakly on 𝑀 (𝐸, 𝑟). □ Theorem 1.21 Let {𝑄 𝑛 } be a sequence of finite measures on 𝑀 (𝐸) and let 𝐿 be a functional on 𝐵(𝐸) + continuous with respect to bounded pointwise convergence. If lim𝑛→∞ 𝐿 𝑄𝑛 ( 𝑓 ) = 𝐿 ( 𝑓 ) for all 𝑓 ∈ 𝐵(𝐸) + , then there is a finite measure 𝑄 on 𝑀 (𝐸) such that 𝐿 = 𝐿 𝑄 and lim𝑛→∞ 𝑄 𝑛 = 𝑄 by weak convergence. Proof By Proposition 1.20, there is a finite measure 𝑄 on 𝑀 (𝐸) such that 𝐿 𝑄 ( 𝑓 ) = 𝐿 ( 𝑓 ) for all 𝑓 ∈ 𝑆¯2 (𝐸, 𝑟) and lim𝑛→∞ 𝑄 𝑛 = 𝑄 weakly on 𝑀 (𝐸, 𝑟). Let 𝒢 = { 𝑓 ∈ 𝐵(𝐸) + : 𝐿 𝑄 ( 𝑓 ) = 𝐿( 𝑓 )}. Then 𝑆¯2 (𝐸, 𝑟) ⊂ 𝒢. Since both 𝑓 ↦→ 𝐿 ( 𝑓 ) and 𝑓 ↦→ 𝐿 𝑄 ( 𝑓 ) are continuous in bounded pointwise convergence and 𝑆¯2 (𝐸, 𝑟) is dense in 𝐶 (𝐸, 𝑟) + , we have 𝐶 (𝐸, 𝑟) + ⊂ 𝒢, so Proposition 1.3 implies 𝐵(𝐸) + ⊂ 𝒢. That is, 𝐿 𝑄 ( 𝑓 ) = 𝐿( 𝑓 ) for all 𝑓 ∈ 𝐵(𝐸) + . It then follows that lim𝑛→∞ 𝐿 𝑄𝑛 ( 𝑓 ) = 𝐿 𝑄 ( 𝑓 ) for all 𝑓 ∈ 𝐵(𝐸) + . By Theorem 1.19, we have lim𝑛→∞ 𝑄 𝑛 = 𝑄 weakly on 𝑀 (𝐸). □ Corollary 1.22 Let {𝑄 𝑛 } be a sequence of finite measures on 𝑀 (𝐸). If 𝐿 𝑄𝑛 ( 𝑓 ) → 𝐿 ( 𝑓 ) uniformly in 𝑓 ∈ 𝐵 𝑎 (𝐸) + for each 𝑎 ≥ 0, then there is a finite measure 𝑄 on 𝑀 (𝐸) such that 𝐿 = 𝐿 𝑄 and lim𝑛→∞ 𝑄 𝑛 = 𝑄 by weak convergence.

12

1 Random Measures on Metric Spaces

Theorem 1.23 Suppose that (𝐹, ℱ) is a measurable space and to each 𝑧 ∈ 𝐹 there corresponds a finite measure 𝑄 𝑧 (d𝜈) on 𝑀 (𝐸). If 𝑧 ↦→ 𝐿 𝑄𝑧 ( 𝑓 ) is ℱ-measurable for every 𝑓 ∈ 𝐶 (𝐸) + , then 𝑄 𝑧 (d𝜈) is a kernel from (𝐹, ℱ) to (𝑀 (𝐸), ℬ(𝑀 (𝐸))). Proof Let ℒ denote the set of functions 𝐺 ∈ 𝐵(𝑀 (𝐸)) such that 𝑧 ↦→ 𝑄 𝑧 (𝐺) is ℱ-measurable. Then ℒ ⊃ 𝒦 := {𝜈 ↦→ e−𝜈 ( 𝑓 ) : 𝑓 ∈ 𝐶 (𝐸) + }. By Proposition A.1 and Theorem 1.11 we have ℒ ⊃ b𝜎(𝒦) = 𝐵(𝑀 (𝐸)). Then 𝑄 𝑧 (d𝜈) is a kernel from (𝐹, ℱ) to (𝑀 (𝐸), ℬ(𝑀 (𝐸))). □ Let 𝑀 (𝐸) ◦ = 𝑀 (𝐸) \ {0}, where 0 is the null measure. We often use a variation of the Laplace functional in dealing with 𝜎-finite measures on 𝑀 (𝐸) ◦ . A typical case is considered in the following: Theorem 1.24 Let 𝑄 1 and 𝑄 2 be two 𝜎-finite measures on 𝑀 (𝐸) ◦ . If for every 𝑓 ∈ 𝐶 (𝐸) + , ∫ ∫
−𝜈 ( 𝑓 )
𝑄 1 (d𝜈) = 1 − e−𝜈 ( 𝑓 ) 𝑄 2 (d𝜈) 1−e (1.12) 𝑀 (𝐸) ◦

𝑀 (𝐸) ◦

and the value is finite, then we have 𝑄 1 = 𝑄 2 . Proof By setting 𝑄 1 ({0}) = 𝑄 2 ({0}) = 0 we extend 𝑄 1 and 𝑄 2 to 𝜎-finite measures on 𝑀 (𝐸). Taking the difference of (1.12) for 𝑓 and 𝑓 + 1 we obtain ∫ ∫

e−𝜈 ( 𝑓 ) 1 − e−𝜈 (1) 𝑄 2 (d𝜈). e−𝜈 ( 𝑓 ) 1 − e−𝜈 (1) 𝑄 1 (d𝜈) = 𝑀 (𝐸)

𝑀 (𝐸)

Then the result of Theorem 1.18 implies that

1 − e−𝜈 (1) 𝑄 1 (d𝜈) = 1 − e−𝜈 (1) 𝑄 2 (d𝜈) as finite measures on 𝑀 (𝐸). Since 1 − e−𝜈 (1) is strictly positive on 𝑀 (𝐸) ◦ , it follows that 𝑄 1 = 𝑄 2 as 𝜎-finite measures on 𝑀 (𝐸) ◦ . □ For any integer 𝑚 ≥ 1, we can also consider the Laplace functionals of finite measures on the product space 𝑀 (𝐸) 𝑚 . The results proved above can be modified obviously to the multi-dimensional setting. In particular, we have the following: Theorem 1.25 Let 𝑄 1 , 𝑄 2 , . . . and 𝑄 be finite measures on 𝑀 (𝐸) 𝑚 . Then 𝑄 𝑛 → 𝑄 weakly if and only if 𝑚 n ∑︁ o 𝜈𝑖 ( 𝑓𝑖 ) 𝑄 𝑛 (d𝜈1 , . . . , d𝜈𝑚 ) exp −

∫ lim 𝑛→∞

𝑀 (𝐸) 𝑚

𝑖=1

∫ = 𝑀 (𝐸) 𝑚

for all { 𝑓1 , . . . , 𝑓𝑚 } ⊂ 𝐶 (𝐸) + .

𝑚 o n ∑︁ 𝜈𝑖 ( 𝑓𝑖 ) 𝑄(d𝜈1 , . . . , d𝜈𝑚 ) exp − 𝑖=1

1.3 Poisson Random Measures

13

Suppose that ℎ ∈ pℬ(𝐸) is a strictly positive function and let 𝑀ℎ (𝐸) be the space of Borel measures 𝜇 on 𝐸 satisfying 𝜇(ℎ) < ∞, which is sometimes referred to as the space of tempered measures. A topology on 𝑀ℎ (𝐸) can be defined by the convention: 𝜇 𝑛 → 𝜇 in 𝑀ℎ (𝐸) if and only if 𝜇 𝑛 (ℎ 𝑓 ) → 𝜇(ℎ 𝑓 ) for all 𝑓 ∈ 𝐶 (𝐸). In particular, if ℎ is a strictly positive continuous function on 𝐸, we have 𝜇 𝑛 → 𝜇 in 𝑀ℎ (𝐸) if and only if 𝜇 𝑛 ( 𝑓 ) → 𝜇( 𝑓 ) for all 𝑓 ∈ 𝐶ℎ (𝐸), where 𝐶ℎ (𝐸) is the set of continuous functions 𝑓 on 𝐸 such that | 𝑓 | ≤ const. · ℎ. A random variable 𝑋 taking values in 𝑀ℎ (𝐸) is also called a random measure on 𝐸. Let 𝐵 ℎ (𝐸) be the set of functions 𝑓 ∈ ℬ(𝐸) satisfying | 𝑓 | ≤ const. · ℎ. Given a finite measure 𝑄 on 𝑀ℎ (𝐸), we define the Laplace functional 𝐿 𝑄 of 𝑄 by ∫ (1.13) e−𝜈 ( 𝑓 ) 𝑄(d𝜈), 𝐿𝑄 ( 𝑓 ) = 𝑓 ∈ 𝐵 ℎ (𝐸) + . 𝑀ℎ (𝐸)

This is a generalization of (1.4). By increasing limits we can easily extend the Laplace functional to all functions 𝑓 ∈ 𝐵(𝐸) + , or even to all 𝑓 ∈ pℬ(𝐸), with the convention e−∞ = 0. We shall make those extensions whenever they are needed. The Laplace functional of a random measure 𝑋 taking values in 𝑀ℎ (𝐸) means the Laplace functional of its distribution on 𝑀ℎ (𝐸). It is easy to see that the mapping 𝜇(d𝑥) ↦→ ℎ(𝑥)𝜇(d𝑥) defines a homeomorphism between 𝑀ℎ (𝐸) and 𝑀 (𝐸). Then the results proved above can also be modified to the space 𝑀ℎ (𝐸). If 𝐸 = {𝑎 1 , . . . , 𝑎 𝑑 } is a finite set containing 𝑑 elements, the mapping 𝜇 ↦→ (𝜇({𝑎 1 }), . . . , 𝜇({𝑎 𝑑 })) gives a homeomorphism between 𝑀 (𝐸) and R+𝑑 . Then the results for 𝑀 (𝐸) can be restated for the space R+𝑑 . In particular, we define the Laplace transform of a finite measure 𝐺 on R+𝑑 by ∫ 𝜆 ∈ R+𝑑 , 𝐿 𝐺 (𝜆) = e− ⟨𝜆,𝑢⟩ 𝐺 (d𝑢), (1.14) R+𝑑

where ⟨·, ·⟩ denotes the Euclidean inner product of R𝑑 . This is essentially a special form of the Laplace functional defined by (1.4).

1.3 Poisson Random Measures Suppose that 𝐸 is a Lusin topological space. Let ℎ ∈ pℬ(𝐸) be a strictly positive function and let 𝜆 ∈ 𝑀ℎ (𝐸). A random measure 𝑋 on 𝐸 taking values in 𝑀ℎ (𝐸) is called a Poisson random measure with intensity 𝜆 provided:

14

1 Random Measures on Metric Spaces

(1) for each 𝐵 ∈ ℬ(𝐸) with 𝜆(𝐵) < ∞, the random variable 𝑋 (𝐵) has the Poisson distribution with parameter 𝜆(𝐵), that is, P{𝑋 (𝐵) = 𝑛} =

𝜆(𝐵) 𝑛 −𝜆(𝐵) e , 𝑛!

𝑛 = 0, 1, 2, . . . ;

(2) if 𝐵1 , . . . , 𝐵𝑛 ∈ ℬ(𝐸) are disjoint and 𝜆(𝐵𝑖 ) < ∞ for each 𝑖 = 1, . . . , 𝑛, then 𝑋 (𝐵1 ), . . . , 𝑋 (𝐵𝑛 ) are mutually independent random variables. In this case, we call 𝑋˜ := 𝑋 − 𝜆 the compensated Poisson random measure. Theorem 1.26 A random measure 𝑋 on 𝐸 is Poissonian with intensity 𝜆 ∈ 𝑀ℎ (𝐸) if and only if its Laplace functional is given by  ∫  − 𝑓 ( 𝑥) E exp{−𝑋 ( 𝑓 )} = exp − (1 − e )𝜆(d𝑥) , 𝑓 ∈ 𝐵 ℎ (𝐸) + . (1.15) 𝐸

Proof Suppose that 𝑋 is a Poisson random measure on 𝐸 with intensity 𝜆. Let 𝐵1 , . . . , 𝐵𝑛 ∈ ℬ(𝐸) be disjoint sets satisfying 𝜆(𝐵𝑖 ) < ∞ for each 𝑖 = 1, . . . , 𝑛. For any constants 𝛼1 , . . . , 𝛼𝑛 ≥ 0 we can use the above two properties to see  ∑︁   ∑︁  𝑛 𝑛 E exp − 𝛼𝑖 𝑋 (𝐵𝑖 ) = exp − (1 − e−𝛼𝑖 )𝜆(𝐵𝑖 ) . 𝑖=1

(1.16)

𝑖=1

Then we get (1.15) by approximating 𝑓 ∈ 𝐵 ℎ (𝐸) + by simple functions and using the dominated convergence theorem. Conversely, if the Laplace functional Í𝑛 of 𝑋 is given by (1.15), we may apply the equality to the simple function 𝑓 = 𝑖=1 𝛼𝑖 1 𝐵𝑖 to get (1.16). Then 𝑋 satisfies the above two properties in the definition of a Poisson random measure on 𝐸 with intensity 𝜆. □ Corollary 1.27 Suppose that 𝑋1 and 𝑋2 are independent Poisson random measures on 𝐸 with intensities 𝜆1 and 𝜆2 ∈ 𝑀ℎ (𝐸), respectively. Then 𝑋1 + 𝑋2 is a Poisson random measure on 𝐸 with intensity 𝜆1 + 𝜆2 . Theorem 1.28 For any 𝜆 ∈ 𝑀ℎ (𝐸), there exists a Poisson random measure with intensity 𝜆. Proof We assume 𝜆 ≠ 0 to avoid triviality. Let {𝐸 1 , 𝐸 2 , . . .} ⊂ ℬ(𝐸) be a sequence of disjoint sets such that 𝐸 = ∪∞ 𝑖=1 𝐸 𝑖 and 0 < 𝜆(𝐸 𝑖 ) < ∞. For each 𝑖 ≥ 1 let 𝜂𝑖 be a Poisson random variable with parameter 𝜆(𝐸 𝑖 ) and let {𝜉𝑖1 , 𝜉𝑖2 , . . .} be a sequence of random variables on 𝐸 with identical distribution 𝜆(𝐸 𝑖 ) −1 𝜆| 𝐸𝑖 , where 𝜆| 𝐸𝑖 denotes the restriction of 𝜆 to 𝐸 𝑖 . Suppose that {𝜂𝑖 , 𝜉𝑖 𝑗 : 𝑖, 𝑗 = 1, 2, . . .} are mutually Then we can define a 𝜎-finite random measure on 𝐸 by Í independent. Í 𝜂𝑖 𝑋 := ∞ 𝛿 . For 𝑓 ∈ 𝐵 ℎ (𝐸) + we have 𝜉 𝑖𝑗 𝑖=1 𝑗=1  ∑︁  𝜂𝑖 ∞ ∑︁ E exp{−𝑋 ( 𝑓 )} = E exp − 𝑓 (𝜉𝑖 𝑗 ) 𝑖=1 𝑗=1

1.3 Poisson Random Measures

15 ∞ ∑︁ ∞ Ö

 ∑︁  𝑛 𝜆(𝐸 𝑖 ) 𝑛 E exp − 𝑓 (𝜉𝑖 𝑗 ) 𝑛! 𝑖=1 𝑛=0 𝑗=1 ∫ 𝑛 ∞ ∞ Ö ∑︁ 1 e− 𝑓 ( 𝑥) 𝜆(d𝑥) = e−𝜆(𝐸𝑖 ) 𝑛! 𝐸𝑖 𝑖=1 𝑛=0   ∫ ∞ Ö − 𝑓 ( 𝑥) = exp − 𝜆(𝐸 𝑖 ) + e 𝜆(d𝑥) 𝐸𝑖 𝑖=1   ∫
= exp − 1 − e− 𝑓 ( 𝑥) 𝜆(d𝑥) .

=

e−𝜆(𝐸𝑖 )

𝐸

Thus 𝑋 is a Poisson random measure on 𝐸 with intensity 𝜆.

□

Proposition 1.29 Suppose that 𝑋 is a Poisson random measure on 𝐸 with intensity 𝜆 ∈ 𝑀ℎ (𝐸). Let 𝑋˜ = 𝑋 − 𝜆. Then for 𝑓 , 𝑔 ∈ 𝐵 ℎ (𝐸) + we have: (1) E[𝑋 (𝑔)e−𝑋 ( 𝑓 ) ] = 𝜆(𝑔e− 𝑓 )E[e−𝑋 ( 𝑓 ) ]; (2) E[𝑋 (𝑔) 2 e−𝑋 ( 𝑓 ) ] = [𝜆(𝑔 2 e− 𝑓 ) + 𝜆(𝑔e− 𝑓 ) 2 ]E[e−𝑋 ( 𝑓 ) ]; (3) E[ 𝑋˜ ( 𝑓 ) 4 ] = 𝜆( 𝑓 4 ) + 3𝜆( 𝑓 2 ) 2 . Proof For any 𝜃 ≥ 0 we may apply (1.15) to the function 𝑥 ↦→ 𝑓 (𝑥) + 𝜃𝑔(𝑥) to get  ∫      E exp{−𝑋 ( 𝑓 + 𝜃𝑔)} = exp − 1 − e− 𝑓 ( 𝑥)−𝜃𝑔 ( 𝑥) 𝜆(d𝑥) . 𝐸

By differentiating both sides with respect to 𝜃 ≥ 0 at zero we get (1). The other two results can be obtained in similar ways. □ Theorem 1.30 Suppose that 𝜆 is a finite measure on 𝐸 and 𝜇 is a probability measure on (0, ∞). Then there is a probability measure 𝑄 on 𝑀 (𝐸) with Laplace functional given by  ∫  ∫ ∞ −𝑢 𝑓 ( 𝑥)
𝐿 𝑄 ( 𝑓 ) = exp − 𝜆(d𝑥) 1−e 𝜇(d𝑢) . (1.17) 0

𝐸

Proof We assume 𝜆 ≠ 0 to avoid triviality. Let 𝜂 be a Poisson random variable with parameter 𝜆(𝐸) and let {𝜉1 , 𝜉2 , . . .} be a sequence of random variables on 𝐸 identically distributed according to 𝜆(𝐸) −1 𝜆. In addition, let {𝜃 1 , 𝜃 2 , . . .} be a sequence of random variables with identical distribution 𝜇. Suppose that {𝜂, 𝜉 𝑗 , 𝜃 𝑗 : 𝑗 = 1, 2, . . .} are Í 𝜂mutually independent. Then a finite random measure on 𝐸 is defined by 𝑋 = 𝑗=1 𝜃 𝑗 𝛿 𝜉 𝑗 . As in the proof of Theorem 1.28 it is easy to see 

∫

E exp{−𝑋 ( 𝑓 )} = exp −

∫

1−e

𝜆(d𝑥) 𝐸

∞ −𝑢 𝑓 ( 𝑥)


 𝜇(d𝑢) .

0

for 𝑓 ∈ 𝐵(𝐸) + . Then 𝑋 has distribution 𝑄 on 𝑀 (𝐸) given by (1.17).

□

16

1 Random Measures on Metric Spaces

A random measure 𝑋 on 𝐸 with distribution 𝑄 given by (1.17) is called a compound Poisson random measure with intensity 𝜆 and height distribution 𝜇. The intuitive meanings of the parameters are clear from the construction of the random measure given in the above proof.

1.4 Infinitely Divisible Random Measures Let 𝐸 be a Lusin topological space. For probability measures 𝑄 1 and 𝑄 2 on 𝑀 (𝐸), the product 𝑄 1 × 𝑄 2 is a probability measure on 𝑀 (𝐸) 2 . The image of 𝑄 1 × 𝑄 2 under the mapping (𝜇1 , 𝜇2 ) ↦→ 𝜇1 + 𝜇2 is called the convolution of 𝑄 1 and 𝑄 2 and is denoted by 𝑄 1 ∗ 𝑄 2 , which is a probability measure on 𝑀 (𝐸). According to the definition, for any 𝐹 ∈ bℬ(𝑀 (𝐸)) we have ∫ 𝐹 (𝜇) (𝑄 1 ∗ 𝑄 2 ) (d𝜇) 𝑀 (𝐸) ∫ = 𝐹 (𝜇1 + 𝜇2 )𝑄 1 (d𝜇1 )𝑄 2 (d𝜇2 ). (1.18) 𝑀 (𝐸) 2

Clearly, if 𝑋1 and 𝑋2 are independent random measures on 𝐸 with distributions 𝑄 1 and 𝑄 2 on 𝑀 (𝐸), respectively, the random measure 𝑋1 + 𝑋2 has distribution 𝑄 1 ∗𝑄 2 . It is easy to show that 𝐿 𝑄1 ∗𝑄2 ( 𝑓 ) = 𝐿 𝑄1 ( 𝑓 )𝐿 𝑄2 ( 𝑓 ),

𝑓 ∈ 𝐵(𝐸) + .

(1.19)

Let 𝑄 ∗0 = 𝛿0 and define 𝑄 ∗𝑛 = 𝑄 ∗(𝑛−1) ∗ 𝑄 inductively for integers 𝑛 ≥ 1. We say a probability distribution 𝑄 on 𝑀 (𝐸) is infinitely divisible if for each integer 𝑛 ≥ 1, there is a probability 𝑄 𝑛 such that 𝑄 = 𝑄 ∗𝑛 𝑛 . In this case, we call 𝑄 𝑛 the 𝑛-th root of 𝑄. A random measure 𝑋 on 𝐸 is said to be infinitely divisible if its distribution on 𝑀 (𝐸) is infinitely divisible. Example 1.2 Poisson random measures and compound Poisson random measures are infinitely divisible. The 𝑛-th roots of their distributions can be obtained by replacing the intensity 𝜆 with 𝜆/𝑛. The main purpose of this section is to give a characterization for the class of infinitely divisible probability measures on 𝑀 (𝐸). Let ℐ(𝐸) denote the convex cone of all functionals 𝑈 on 𝐵(𝐸) + with the representation ∫
1 − e−𝜈 ( 𝑓 ) 𝐿(d𝜈), 𝑓 ∈ 𝐵(𝐸) + , (1.20) 𝑈 ( 𝑓 ) = 𝜆( 𝑓 ) + 𝑀 (𝐸) ◦

where 𝜆 ∈ 𝑀 (𝐸) and (1 ∧ 𝜈(1))𝐿(d𝜈) is a finite measure on 𝑀 (𝐸) ◦ . Let 𝑆2 (𝐸, 𝑟) be defined as in Section 1.2. Proposition 1.31 The measures 𝜆 and 𝐿 in (1.20) are uniquely determined by the functional 𝑈 ∈ ℐ(𝐸).

1.4 Infinitely Divisible Random Measures

17

Proof Suppose that 𝑈 can also be represented by (1.20) with (𝜆, 𝐿) replaced by (𝛾, 𝐺). For any constant 𝜃 ≥ 0, we can evaluate 𝑈 ( 𝑓 + 𝜃) − 𝑈 (𝜃) with the two representations and get ∫
1 − e−𝜈 ( 𝑓 ) e−𝜈 ( 𝜃) 𝐺 (d𝜈) 𝛾( 𝑓 ) + 𝑀 (𝐸) ◦ ∫
1 − e−𝜈 ( 𝑓 ) e−𝜈 ( 𝜃) 𝐿(d𝜈). = 𝜆( 𝑓 ) + 𝑀 (𝐸) ◦

Letting 𝜃 → ∞ gives 𝛾( 𝑓 ) = 𝜆( 𝑓 ), and hence 𝛾 = 𝜆. Then the above equality and □ Theorem 1.24 imply e−𝜈 ( 𝜃) 𝐺 (d𝜈) = e−𝜈 ( 𝜃) 𝐿(d𝜈), and so 𝐺 (d𝜈) = 𝐿(d𝜈). Proposition 1.32 Suppose that 𝐽 is a functional on 𝑆2 (𝐸, 𝑟) and there is a sequence { 𝑓𝑛 } ⊂ 𝑆2 (𝐸, 𝑟) such that lim𝑛→∞ 𝑓𝑛 = 0 in bounded pointwise convergence and lim𝑛→∞ 𝐽 ( 𝑓𝑛 ) = 0. If there is a sequence {𝑈𝑛 } ⊂ ℐ(𝐸) such that lim 𝑈𝑛 ( 𝑓 ) = 𝐽 ( 𝑓 ), 𝑛→∞

𝑓 ∈ 𝑆2 (𝐸, 𝑟),

(1.21)

then 𝐽 is the restriction to 𝑆2 (𝐸, 𝑟) of a functional 𝑈 ∈ ℐ(𝐸). Proof Recall that 𝑃(𝐸) denotes the space of probability measures on 𝐸. For 𝜈 ∈ 𝑀 (𝐸) ◦ let |𝜈| = 𝜈(𝐸) and 𝜈ˆ = |𝜈| −1 𝜈. The mapping 𝐹 : 𝜈 ↦→ (|𝜈|, 𝜈) ˆ is clearly a homeomorphism between 𝑀 (𝐸) ◦ and the product space (0, ∞) × 𝑃(𝐸). For 0 ≤ 𝑢 ≤ ∞, 𝜋 ∈ 𝑃(𝐸) and 𝑓 ∈ 𝐵(𝐸) + let   (1 − e−𝑢 ) −1 (1 − e−𝑢 𝜋 ( 𝑓 ) )   𝜉 (𝑢, 𝜋, 𝑓 ) = 𝜋( 𝑓 )   1

if 0 < 𝑢 < ∞, if 𝑢 = 0, if 𝑢 = ∞.

(1.22)

Suppose that 𝑈𝑛 ∈ ℐ(𝐸) is given by (1.21) with (𝜆, 𝐿) replaced by (𝜆 𝑛 , 𝐿 𝑛 ). Let 𝜋 𝑛 ∈ 𝑃(𝐸) be such that 𝜆 𝑛 = |𝜆 𝑛 |𝜋 𝑛 and let 𝐻𝑛 (d𝑢, d𝜋) be the image of 𝐿 𝑛 (d𝜈) under the mapping 𝐹. Define the finite measure 𝐺 𝑛 (d𝑢, d𝜋) on [0, ∞] × 𝑃(𝐸) by 𝐺 𝑛 ({(0, 𝜋 𝑛 )}) = |𝜆 𝑛 |, 𝐺 𝑛 (({0, ∞} × 𝑃(𝐸)) \ {(0, 𝜋 𝑛 )}) = 0 and 𝐺 𝑛 (d𝑢, d𝜋) = (1 − e−𝑢 )𝐻𝑛 (d𝑢, d𝜋) for 0 < 𝑢 < ∞ and 𝜋 ∈ 𝑃(𝐸). Then we have ∫ ∫ 𝑓 ∈ 𝐵(𝐸) + . 𝜉 (𝑢, 𝜋, 𝑓 )𝐺 𝑛 (d𝑢, d𝜋), 𝑈𝑛 ( 𝑓 ) = [0,∞]

𝑃 (𝐸)

By (1.21) it is evident that {𝐺 𝑛 } is a bounded sequence in 𝑀 ( [0, ∞] × 𝑃(𝐸)). Let (𝐸, 𝑟) be the compact metric space as described in Section 1.2. Then Theorem 1.14 implies that {𝐺 𝑛 } viewed as a sequence of measures on [0, ∞] × 𝑃(𝐸, 𝑟) is relatively compact. Take any subsequence {𝐺 𝑛𝑘 } ⊂ {𝐺 𝑛 } such that lim 𝑘→∞ 𝐺 𝑛𝑘 = 𝐺 weakly for a finite measure 𝐺 on [0, ∞] × 𝑃(𝐸, 𝑟). For any 𝑓 ∈ 𝑆2 (𝐸, 𝑟) the mapping (𝑢, 𝜋) ↦→ 𝜉 (𝑢, 𝜋, 𝑓 ) is clearly continuous on [0, ∞] × 𝑃(𝐸, 𝑟). By (1.21) we have ∫ ∫ 𝐽( 𝑓 ) = 𝑓 ∈ 𝑆2 (𝐸, 𝑟). 𝜉 (𝑢, 𝜋, 𝑓 )𝐺 (d𝑢, d𝜋), [0,∞]

𝑃 (𝐸)

18

1 Random Measures on Metric Spaces

Observe also that lim𝑛→∞ 𝐽 ( 𝑓𝑛 ) = 0 implies 𝐺 ({∞} × 𝑃(𝐸)) = 0. Then the desired □ conclusion follows by a change of the integration variable. Theorem 1.33 Suppose that 𝑈 is a functional on 𝐵(𝐸) + continuous with respect to bounded pointwise convergence. If there is a sequence {𝑈𝑛 } ⊂ ℐ(𝐸) such that 𝑈 ( 𝑓 ) = lim𝑛→∞ 𝑈𝑛 ( 𝑓 ) for all 𝑓 ∈ 𝐵(𝐸) + , then 𝑈 ∈ ℐ(𝐸). Proof This is similar to the proof of Theorem 1.21 with an application of Proposi□ tion 1.32. Corollary 1.34 Suppose that {𝑈𝑛 } ⊂ ℐ(𝐸) and 𝑈 is a functional on 𝐵(𝐸) + . If 𝑈𝑛 ( 𝑓 ) → 𝑈 ( 𝑓 ) uniformly in 𝑓 ∈ 𝐵 𝑎 (𝐸) + for each 𝑎 ≥ 0, then 𝑈 ∈ ℐ(𝐸). Corollary 1.35 Suppose that {𝑈𝑛 } ⊂ ℐ(𝐸) and there is a measure 𝜋 ∈ 𝑀 (𝐸) such that 𝑈𝑛 ( 𝑓 ) ≤ 𝜋( 𝑓 ) for all 𝑛 ≥ 1 and 𝑓 ∈ 𝐵(𝐸) + . If 𝑈 ( 𝑓 ) = lim𝑛→∞ 𝑈𝑛 ( 𝑓 ) for all 𝑓 ∈ 𝐵(𝐸) + , then we have 𝑈 ∈ ℐ(𝐸). Proof Let 𝑈𝑛 be given by (1.20) with (𝜆, 𝐿) replaced by (𝜆 𝑛 , 𝐿 𝑛 ). For 𝑓 ∈ 𝐵(𝐸) + we can use monotone convergence to see ∫ 𝜈( 𝑓 )𝐿 𝑛 (d𝜈) = lim 𝜃 −1𝑈𝑛 (𝜃 𝑓 ) ≤ 𝜋( 𝑓 ). 𝜆𝑛 ( 𝑓 ) + 𝜃 ↓0

𝑀 (𝐸) ◦

Consequently, for any 𝑓 and 𝑔 ∈ 𝐵(𝐸) + we have ∫ |𝑈𝑛 ( 𝑓 ) − 𝑈𝑛 (𝑔)| ≤ 𝜆 𝑛 (| 𝑓 − 𝑔|) +

𝜈(| 𝑓 − 𝑔|)𝐿 𝑛 (d𝜈)

𝑀 (𝐸) ◦

≤ 𝜋(| 𝑓 − 𝑔|), and hence |𝑈 ( 𝑓 ) − 𝑈 (𝑔)| ≤ 𝜋(| 𝑓 − 𝑔|). Then 𝑈 is continuous in bounded pointwise □ convergence and so 𝑈 ∈ ℐ(𝐸) by Theorem 1.33. Suppose that 𝑈 ∈ ℐ(𝐸) has the representation (1.20). Let 𝑁 (d𝜈) be a Poisson random measure on 𝑀 (𝐸) ◦ with intensity measure 𝐿. By Theorem 1.26 it is easy to show that ∫ (1.23) 𝑋 := 𝜆 + 𝜈𝑁 (d𝜈) 𝑀 (𝐸) ◦

defines a random measure on 𝐸 with − log 𝐿 𝑋 = 𝑈. Let 𝑄 be the distribution of 𝑋 on 𝑀 (𝐸). By the same reasoning, for any integer 𝑛 ≥ 1 there is a probability measure 𝑄 𝑛 on 𝑀 (𝐸) satisfying − log 𝐿 𝑄𝑛 = 𝑛−1𝑈, implying 𝑄 ∗𝑛 𝑛 = 𝑄. Therefore 𝑄 is infinitely divisible. We write 𝑄 = 𝐼 (𝜆, 𝐿) if 𝑄 is an infinitely divisible probability measure on 𝑀 (𝐸) with 𝑈 = − log 𝐿 𝑄 ∈ ℐ(𝐸) represented by (1.20). The construction (1.23) for the corresponding random measure is also called a cluster representation. Theorem 1.36 The equation 𝑈 = − log 𝐿 𝑄 establishes a one-to-one correspondence between functionals 𝑈 ∈ ℐ(𝐸) and infinitely divisible probability measures 𝑄 on 𝑀 (𝐸).

1.4 Infinitely Divisible Random Measures

19

Proof By the above comments we only need to show if 𝑄 is an infinitely divisible probability measure on 𝑀 (𝐸), then 𝑈 := − log 𝐿 𝑄 ∈ ℐ(𝐸). For 𝑛 ≥ 1 let 𝑄 𝑛 be the 𝑛-th root of 𝑄. Then ∫
−1 𝑈 ( 𝑓 ) = lim 𝑛[1 − e−𝑛 𝑈 ( 𝑓 ) ] = lim 1 − e−𝜈 ( 𝑓 ) 𝑛𝑄 𝑛 (d𝜈) 𝑛→∞

𝑛→∞

𝑀 (𝐸) ◦

and the convergence is uniform in 𝑓 ∈ 𝐵 𝑎 (𝐸) + for each 𝑎 ≥ 0. By Corollary 1.34 □ we have 𝑈 ∈ ℐ(𝐸). Theorem 1.37 Suppose that 𝑉 : 𝑓 ↦→ 𝑣(·, 𝑓 ) is an operator on 𝐵(𝐸) + such that 𝑣(𝑥, ·) ∈ ℐ(𝐸) for all 𝑥 ∈ 𝐸. Then we have the representation ∫
(1.24) 𝑣(𝑥, 𝑓 ) = 𝜆(𝑥, 𝑓 ) + 1 − e−𝜈 ( 𝑓 ) 𝐿(𝑥, d𝜈), 𝑓 ∈ 𝐵(𝐸) + , 𝑀 (𝐸) ◦

where 𝜆(𝑥, d𝑦) is a bounded kernel on 𝐸 and (1 ∧ 𝜈(1))𝐿 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ . Proof Under the assumption, for any fixed 𝑥 ∈ 𝐸 we have the representation (1.24), where 𝜆(𝑥, ·) ∈ 𝑀 (𝐸) and (1 ∧ 𝜈(1))𝐿(𝑥, d𝜈) is a finite measure on 𝑀 (𝐸) ◦ . For every 𝑓 ∈ 𝐵(𝐸) + , 𝑥 ↦→ 𝑤(𝑥, 𝑓 ) := 2𝑣(𝑥, 𝑓 + 1) − 𝑣(𝑥, 𝑓 + 2) − 𝑣(𝑥, 𝑓 ) is a bounded Borel function on 𝐸. Observe also that ∫
2 𝑤(𝑥, 𝑓 ) = e−𝜈 ( 𝑓 ) 1 − e−𝜈 (1) 𝐿 (𝑥, d𝜈). 𝑀 (𝐸) ◦

By Theorem 1.23 we see that (1 − e−𝜈 (1) ) 2 𝐿(𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ . In view of (1.24), ∫
1 − e−𝜈 (1) 𝐿 (𝑥, d𝜈) 𝑥 ↦→ 𝑀 (𝐸) ◦

is a bounded function on 𝐸. It follows that (1∧𝜈(1))𝐿(𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ . By another application of the relation (1.24) one sees that 𝑥 ↦→ 𝜆(𝑥, 𝑓 ) is a bounded Borel function on 𝐸 for every 𝑓 ∈ 𝐵(𝐸) + . Then 𝜆(𝑥, d𝑦) is a bounded kernel on 𝐸. □ Theorem 1.38 If 𝑈 ∈ ℐ(𝐸) and if 𝑉 : 𝑓 ↦→ 𝑣(·, 𝑓 ) is an operator on 𝐵(𝐸) + such that 𝑣(𝑥, ·) ∈ ℐ(𝐸) for all 𝑥 ∈ 𝐸, then 𝑈 ◦ 𝑉 ∈ ℐ(𝐸). Proof By Theorem 1.37 it is easy to see that for any 𝜇 ∈ 𝑀 (𝐸) the functional 𝑓 ↦→ 𝜇(𝑉 𝑓 ) belongs to ℐ(𝐸). Then there is an infinitely divisible probability measure 𝑄(𝜇, ·) on 𝑀 (𝐸) satisfying − log 𝐿 𝑄 ( 𝜇,·) ( 𝑓 ) = 𝜇(𝑉 𝑓 ). By Theorem 1.23 we see that 𝑄(𝜇, d𝜈) is a probability kernel on 𝑀 (𝐸). Let 𝐺 be the infinitely divisible probability measure on 𝑀 (𝐸) with − log 𝐿 𝐺 = 𝑈 and define

20

1 Random Measures on Metric Spaces

∫ 𝐺 (d𝜇)𝑄(𝜇, d𝜈),

𝑄(d𝜈) =

𝜈 ∈ 𝑀 (𝐸).

𝑀 (𝐸)

It is not hard to show that − log 𝐿 𝑄 = 𝑈 ◦ 𝑉. By the same reasoning, for each integer 𝑛 ≥ 1 there is a probability measure 𝑄 𝑛 such that − log 𝐿 𝑄𝑛 = 𝑛−1𝑈 ◦ 𝑉. Then 𝑄 = 𝑄 ∗𝑛 𝑛 and hence 𝑄 is infinitely divisible. By Theorem 1.36 we conclude that □ 𝑈 ◦ 𝑉 ∈ ℐ(𝐸). The following result gives a useful method for the calculation of the moments of infinitely divisible probability measures on 𝑀 (𝐸). Proposition 1.39 Let 𝑈 ∈ ℐ(𝐸) be given by (1.20). Then for any 𝑓 ∈ 𝐵(𝐸) + we have ∫ d 𝑈 (𝜃 𝑓 ) = 𝜆( 𝑓 ) + 𝜈( 𝑓 )e−𝜃 𝜈 ( 𝑓 ) 𝐿(d𝜈) (1.25) d𝜃 𝑀 (𝐸) ◦ and d𝑛 𝑈 (𝜃 𝑓 ) = (−1) 𝑛−1 d𝜃 𝑛

∫

𝜈( 𝑓 ) 𝑛 e−𝜃 𝜈 ( 𝑓 ) 𝐿(d𝜈)

(1.26)

𝑀 (𝐸) ◦

for 0 < 𝜃 < ∞ and 𝑛 = 2, 3, . . .. Proof For any 𝑛 ≥ 1 the function 𝑧 ↦→ 𝑧 𝑛 e−𝑧 achieves its maximal value on [0, ∞) at 𝑧 = 𝑛. It follows that 𝜈( 𝑓 ) 𝑛 e−𝜃 𝜈 ( 𝑓 ) ≤ 𝑛𝑛 𝜃 0−𝑛 e−𝑛 ,

𝜃 ≥ 𝜃 0 > 0, 𝜈 ∈ 𝑀 (𝐸).

Fix 𝜃 0 > 0 and 𝑓 ∈ 𝐵(𝐸) + and let 𝐹𝑛 (𝜈) = 𝑛𝑛 𝜃 0−𝑛 e−𝑛 ∧ 𝜈( 𝑓 ) 𝑛 . It is easy to see that 𝐿 (𝐹𝑛 ) < ∞ and 𝜈( 𝑓 ) 𝑛 e−𝜃 𝜈 ( 𝑓 ) ≤ 𝐹𝑛 (𝜈),

𝜃 ≥ 𝜃 0 , 𝜈 ∈ 𝑀 (𝐸).

Then we have (1.25) and (1.26) by dominated convergence.

□

To close this section we give a characterization of infinitely divisible probability measures on the positive half line R+ = [0, ∞). Write 𝜓 ∈ ℐ if 𝜆 ↦→ 𝜓(𝜆) is a positive function on [0, ∞) with the representation ∫ ∞ 𝜓(𝜆) = 𝛽𝜆 + (1 − e−𝜆𝑢 )𝑙 (d𝑢), (1.27) 0

where 𝛽 ≥ 0 and (1∧𝑢)𝑙 (d𝑢) is a finite measure on (0, ∞). By applying Theorem 1.36 to the case where 𝐸 is a singleton, we have the following: Theorem 1.40 The relation 𝜓 = − log 𝐿 𝜇 establishes a one-to-one correspondence between the functions 𝜓 ∈ ℐ and infinitely divisible probability measures 𝜇 on [0, ∞).

1.5 Lévy–Khintchine Type Representations

21

Example 1.3 Let 𝑏 > 0 and 𝛼 > 0. The Gamma distribution 𝛾 on [0, ∞) with parameters (𝑏, 𝛼) is defined by ∫ 𝛼𝑏 𝛾(𝐵) = 𝑥 𝑏−1 e−𝛼𝑥 d𝑥, 𝐵 ∈ ℬ[0, ∞). Γ(𝑏) 𝐵 This reduces to the exponential distribution when 𝑏 = 1. The Gamma distribution has Laplace transform 𝐿 𝛾 (𝜆) =

 𝛼 𝑏 , 𝛼+𝜆

𝜆 ≥ 0.

It is easily seen that 𝛾 is infinitely divisible and its 𝑛-th root is the Gamma distribution with parameters (𝑏/𝑛, 𝛼). Example 1.4 For 𝑐 > 0 and 0 < 𝛼 < 1 the function 𝜆 ↦→ 𝑐𝜆 𝛼 admits the representation (1.27). Indeed, it is easy to show ∫ ∞ d𝑢 𝛼 (1 − e−𝜆𝑢 ) 1+𝛼 , 𝜆𝛼 = 𝜆 ≥ 0. (1.28) Γ(1 − 𝛼) 0 𝑢 The infinitely divisible probability measure 𝜈 on [0, ∞) satisfying − log 𝐿 𝜈 (𝜆) = 𝑐𝜆 𝛼 is known as the one-sided stable distribution with index 0 < 𝛼 < 1. This distribution does not charge zero and is absolutely continuous with respect to the Lebesgue measure on (0, ∞) with continuous density. For 𝛼 = 1/2 it has density 2 𝑐 𝑞(𝑥) := √ 𝑥 −3/2 e−𝑐 /4𝑥 , 2 𝜋

𝑥 > 0.

For a general index the density can be given using an infinite series; see, e.g., Sato (1999, p. 88).

1.5 Lévy–Khintchine Type Representations In this section, we present some criteria for continuous functions on R+ = [0, ∞) to have Lévy–Khintchine type representations. The results are useful in the study of high-density limits of discrete branching processes. For an interval 𝑇 ⊂ R let 𝒞(𝑇) denote the set of continuous (not necessarily bounded) functions on 𝑇. For any 𝑐 ≥ 0 we define the difference operator Δ𝑐 by Δ𝑐 𝑓 (𝜆) = 𝑓 (𝜆 + 𝑐) − 𝑓 (𝜆),

𝜆, 𝜆 + 𝑐 ∈ 𝑇, 𝑓 ∈ 𝒞(𝑇).

Let Δ0𝑐 be the identity and define Δ𝑛𝑐 = Δ𝑛−1 𝑐 Δ𝑐 for 𝑛 ≥ 1 inductively. Then we have Δ𝑚 𝑐

𝑓 (𝜆) = (−1)

𝑚

𝑚   ∑︁ 𝑚

𝑖 𝑖=0

(−1) 𝑖 𝑓 (𝜆 + 𝑖𝑐).

(1.29)

22

1 Random Measures on Metric Spaces

We call 𝜃 ∈ 𝒞[0, ∞) a completely monotone function if it satisfies (−1) 𝑖 Δ𝑖𝑐 𝜃 (𝜆) ≥ 0,

𝜆 ≥ 0, 𝑐 ≥ 0, 𝑖 = 0, 1, 2, . . . .

(1.30)

The Bernstein polynomials of a function 𝑓 ∈ 𝒞[0, 1] are given by 𝐵 𝑓 ,𝑚 (𝑠) =

𝑚   ∑︁ 𝑚

𝑖

𝑖 Δ1/𝑚 𝑓 (0)𝑠𝑖 ,

0 ≤ 𝑠 ≤ 1, 𝑚 = 1, 2, . . . .

𝑖=0

It is well known that 𝐵 𝑓 ,𝑚 (𝑠) → 𝑓 (𝑠),

𝑠 ∈ [0, 1],

(1.31)

uniformly as 𝑚 → ∞; see, e.g., Feller (1971, p. 222). Theorem 1.41 A function 𝜃 ∈ 𝒞[0, ∞) is the Laplace transform of a finite measure 𝐺 on [0, ∞) if and only if it is completely monotone. Proof Suppose that 𝜃 ∈ 𝒞[0, ∞) is the Laplace transform of a finite measure 𝐺 on [0, ∞). Clearly, we have (1.30) because ∫ e−𝜆𝑢 (1 − e−𝑐𝑢 ) 𝑖 𝐺 (d𝑢). (−1) 𝑖 Δ𝑖𝑐 𝜃 (𝜆) = [0,∞)

Conversely, suppose that (1.30) holds. For fixed 𝑎 > 0, we let 𝛾 𝑎 (𝑠) = 𝜃 (𝑎 − 𝑎𝑠) for 0 ≤ 𝑠 ≤ 1. The complete monotonicity of 𝜃 implies Δ𝑖1/𝑚 𝛾 𝑎 (0) ≥ 0,

𝑖 = 0, 1, . . . , 𝑚.

Then the Bernstein polynomial 𝐵 𝛾𝑎 ,𝑚 (𝑠) has positive coefficients, so 𝐵 𝛾𝑎 ,𝑚 (e−𝜆/𝑎 ) is the Laplace transform of a finite measure 𝐺 𝑎,𝑚 on [0, ∞). By Theorem 1.21, 𝜃 (𝜆) = lim lim 𝐵 𝛾𝑎 ,𝑚 (e−𝜆/𝑎 ),

𝜆 ≥ 0,

𝑎→∞ 𝑚→∞

is the Laplace transform of a finite measure on [0, ∞).

□

Corollary 1.42 A function 𝜃 ∈ 𝒞[0, ∞) is the Laplace transform of a finite measure 𝐺 on [0, ∞) if and only if it is infinitely differentiable in (0, ∞) and (−1) 𝑖 𝐷 𝑖 𝜃 (𝜆) ≥ 0,

𝜆 > 0, 𝑖 = 0, 1, 2, . . . ,

(1.32)

where 𝐷 𝑖 denotes the 𝑖-th derivative. Proof Suppose that 𝜃 ∈ 𝒞[0, ∞) is the Laplace transform of a finite measure 𝐺 on [0, ∞). Then (1.32) holds because ∫ e−𝜆𝑢 𝑢 𝑖 𝐺 (d𝑢). (−1) 𝑖 𝐷 𝑖 𝜃 (𝜆) = [0,∞)

1.5 Lévy–Khintchine Type Representations

23

Conversely, suppose that 𝜃 ∈ 𝒞[0, ∞) satisfies (1.32). Since the operators 𝐷 𝑖 and Δ𝑐 are interchangeable, for any positive sequence {𝑐 𝑖 } it is easy to see that (−1) 𝑖 Δ𝑐1 · · · Δ𝑐𝑖 𝜃 (𝜆) ≥ 0,

𝜆 ≥ 0, 𝑖 = 0, 1, 2, . . . .

Then 𝜃 is completely monotone. By Theorem 1.41, it is the Laplace transform of a finite measure on [0, ∞). □ Now let us consider a general Lévy–Khintchine type representation for continuous functions on [0, ∞). For 𝑢 ≥ 0 and 𝜆 ≥ 0 let 𝜉 𝑛 (𝑢, 𝜆) = e−𝜆𝑢 − 1 − (1 + 𝑢 𝑛 ) −1

𝑛−1 ∑︁ (−𝜆𝑢) 𝑖 , 𝑖! 𝑖=1

𝑛 = 1, 2, . . . .

We are interested in functions 𝜙 ∈ 𝒞[0, ∞) with the representation 𝜙(𝜆) =

𝑛−1 ∑︁

∫

𝜉 𝑛 (𝑢, 𝜆) (1 − e−𝑢 ) −𝑛 𝐺 (d𝑢),

𝑎 𝑖 𝜆𝑖 +

𝜆 ≥ 0,

(1.33)

[0,∞)

𝑖=0

where 𝑛 ≥ 1 is an integer, {𝑎 0 , . . . , 𝑎 𝑛−1 } is a set of constants and 𝐺 (d𝑢) is a finite measure on [0, ∞). The value at 𝑢 = 0 of the integrand in (1.33) is defined by continuity as (−𝜆) 𝑛 /𝑛!. Lemma 1.43 A function 𝜂 ∈ 𝒞[0, ∞) is a polynomial of degree less than 𝑛 ≥ 1 if and only if Δ𝑛𝑐 𝜂(0) = 0 for all 𝑐 ≥ 0. Proof If 𝜂 ∈ 𝒞[0, ∞) is a polynomial of degree less than 𝑛 ≥ 1, one sees easily Δ𝑛𝑐 𝜂(0) = 0 for all 𝑐 ≥ 0. For the converse, suppose that 𝜂 ∈ 𝒞[0, ∞) and Δ𝑛𝑐 𝜂(0) = 0 for all 𝑐 ≥ 0. Fix 𝑎 > 0 and let 𝜂 𝑎 (𝑠) = 𝜂(𝑎𝑠) for 0 ≤ 𝑠 ≤ 1. Since 0 ≤ 𝑐 ≤ 𝑛−1 ,

Δ𝑛𝑐 𝜂 𝑎 (0) = 0,

the polynomials of 𝜂 𝑎 have degree less than 𝑛, that is, 𝐵 𝜂𝑎 ,𝑚 (𝑠) =

𝑛−1 ∑︁

𝑏 𝑖(𝑚) 𝑠𝑖 ,

𝑚 = 𝑛, 𝑛 + 1, . . . .

𝑖=0

The coefficients 𝑏 𝑖(𝑚) here can be represented as linear combinations of 𝐵 𝜂𝑎 ,𝑚 (1/𝑛), 𝐵 𝜂𝑎 ,𝑚 (2/𝑛), . . . , 𝐵 𝜂𝑎 ,𝑚 (𝑛/𝑛). By (1.31) the limits lim 𝑏 (𝑚) 𝑚→∞ 𝑖 exist and hence 𝜂 𝑎 (𝑠) =

Í𝑛−1 𝑖=0

= 𝑏𝑖 ,

𝑖 = 0, 1, . . . , 𝑛 − 1

𝑏 𝑖 𝑠𝑖 for 0 ≤ 𝑠 ≤ 1. Setting 𝑎 𝑖 = 𝑎 −1 𝑏 𝑖 we get

24

1 Random Measures on Metric Spaces

𝜂(𝑠) =

𝑛−1 ∑︁

𝑎 𝑖 𝑠𝑖 ,

0 ≤ 𝑠 ≤ 𝑎.

𝑖=0

Clearly, this formula holds in fact for all 𝑠 ≥ 0.

□

Theorem 1.44 A function 𝜙 ∈ 𝒞[0, ∞) has the representation (1.33) if and only if for every 𝑐 ≥ 0 the function 𝜃 𝑐 (𝜆) := (−1) 𝑛 Δ𝑐𝑛 𝜙(𝜆),

𝜆≥0

(1.34)

is the Laplace transform of a finite measure on [0, ∞). Proof Suppose that 𝜙 is given by (1.33). Using (1.29) it is easy to see ∫ e−𝜆𝑢 (1 − e−𝑐𝑢 ) 𝑛 (1 − e−𝑢 ) −𝑛 𝐺 (d𝑢), 𝜃 𝑐 (𝜆) = [0,∞)

where the integrand is defined as 𝑐 𝑛 at 𝑢 = 0 by continuity. Thus 𝜃 𝑐 is the Laplace transform of a finite measure on [0, ∞). Conversely, assume 𝜃 𝑐 is the Laplace transform of a finite measure 𝐺 𝑐 on [0, ∞), that is, ∫ (1.35) 𝜆 ≥ 0. e−𝜆𝑢 𝐺 𝑐 (d𝑢), 𝜃 𝑐 (𝜆) = [0,∞)

From (1.29) and the relation (−1) 𝑛 Δ1𝑛 𝜃 𝑐 (𝜆) = Δ1𝑛 Δ𝑛𝑐 𝜙(𝜆) = Δ𝑐𝑛 Δ1𝑛 𝜙(𝜆) = (−1) 𝑛 Δ𝑛𝑐 𝜃 1 (𝜆) it follows that ∫

e−𝜆𝑢 (1 − e−𝑢 ) 𝑛 𝐺 𝑐 (d𝑢) =

∫

e−𝜆𝑢 (1 − e−𝑐𝑢 ) 𝑛 𝐺 (d𝑢), [0,∞)

[0,∞)

where 𝐺 = 𝐺 1 . Therefore 𝐺 𝑐 (d𝑢) = (1 − e−𝑐𝑢 ) 𝑛 (1 − e−𝑢 ) −𝑛 𝐺 (d𝑢), by the uniqueness of the Laplace transform. Let ∫ 𝜙0 (𝜆) = 𝜉 𝑛 (𝑢, 𝜆) (1 − e−𝑢 ) −𝑛 𝐺 (d𝑢),

0 0. Example 1.5 Suppose that 𝑔 is a probability generating function such that 𝛽 := 𝑔 ′ (1−) < ∞. Let 𝛼 𝑘 = 𝑘 and 𝑔 𝑘 (𝑧) = 𝑔(𝑧). Then the sequence 𝜓 𝑘 (𝜆) defined by (1.39) converges to 𝛽𝜆 as 𝑘 → ∞. Example 1.6 For any 0 < 𝛼 ≤ 1 the function 𝜓(𝜆) = 𝜆 𝛼 has the representation (1.27). For 𝛼 = 1 this is trivial, and for 0 < 𝛼 < 1 it follows from (1.28). Let 𝜓 𝑘 (𝜆) be defined by (1.39) with 𝛼 𝑘 = 𝑘 𝛼 and 𝑔 𝑘 (𝑧) = 1 − (1 − 𝑧) 𝛼 . Then 𝜓 𝑘 (𝜆) = 𝜆 𝛼 for 0 ≤ 𝜆 ≤ 𝑘. In the study of limit theorems of branching models, we shall also need to consider the limit of another function sequence defined as follows. Let {𝛼 𝑘 } and {𝑔 𝑘 } be given as above and let 𝜙 𝑘 (𝜆) = 𝛼 𝑘 [𝑔 𝑘 (1 − 𝜆/𝑘) − (1 − 𝜆/𝑘)],

0 ≤ 𝜆 ≤ 𝑘.

(1.40)

Theorem 1.47 If the sequence {𝜙 𝑘 } defined by (1.40) converges to some 𝜙 ∈ 𝒞[0, ∞), then the limit function has the representation ∫ ∞ 𝜆𝑢  e−𝜆𝑢 − 1 + 𝜙(𝜆) = 𝑎𝜆 + 𝑐𝜆2 + 𝑚(d𝑢), (1.41) 1 + 𝑢2 0 where 𝑐 ≥ 0 and 𝑎 are constants, and 𝑚(d𝑢) is a 𝜎-finite measure on (0, ∞) satisfying ∫ ∞ (1 ∧ 𝑢 2 )𝑚(d𝑢) < ∞. (1.42) 0

Proof Since 𝜙(0) = lim 𝑘→∞ 𝜙 𝑘 (0) = 0, arguing as in the proof of Theorem 1.46 we see that 𝜙 has the representation (1.33) with 𝑛 = 2 and 𝑎 0 = 0, which can be rewritten in the equivalent form (1.41). □ There are some frequently used variations of the representation (1.41). One of those is the following equivalent representation: ∫ ∞
2 𝜙(𝜆) = 𝑏 1 𝜆 + 𝑐𝜆 + e−𝜆𝑢 − 1 + 𝜆𝑢1 {𝑢≤1} 𝑚(d𝑢), (1.43) 0

where ∫ 𝑏1 = 𝑎 + 0

∞



 𝑢 − 𝑢1 𝑚(d𝑢). {𝑢 ≤1} 1 + 𝑢2

1.5 Lévy–Khintchine Type Representations

27

In particular, if the measure 𝑚(d𝑢) satisfies the integrability condition ∫ ∞ (𝑢 ∧ 𝑢 2 ) 𝑚(d𝑢) < ∞,

(1.44)

0

we have 𝜙(𝜆) = 𝑏𝜆 + 𝑐𝜆2 +

∞

∫


e−𝜆𝑢 − 1 + 𝜆𝑢 𝑚(d𝑢),

(1.45)

0

where ∫ 𝑏=𝑎− 0

∞

𝑢3 𝑚(d𝑢). 1 + 𝑢2

Proposition 1.48 The function 𝜙 ∈ 𝒞[0, ∞) with the representation (1.41) is locally Lipschitz if and only if (1.44) holds. Proof By the dominated convergence theorem, we can differentiate both sides of (1.43) to obtain, for each 𝜆 > 0, ∫ 1 ∫ ∞
𝜙 ′ (𝜆) = 𝑏 1 + 2𝑐𝜆 + 𝑢 1 − e−𝜆𝑢 𝑚(d𝑢) − 𝑢e−𝜆𝑢 𝑚(d𝑢). 0

1

Then we use monotone convergence to the two integrals to get ∫ ∞ ′ 𝜙 (0+) = 𝑏 1 − 𝑢𝑚(d𝑢). 1

𝜙 ′ (0+)

If 𝜙 is locally Lipschitz, we have > −∞ and the integral on the right-hand side is finite. This together with (1.42) implies (1.44). Conversely, if (1.44) holds, then 𝜙 ′ is bounded on each bounded interval and so 𝜙 is locally Lipschitz. □ Corollary 1.49 If the sequence {𝜙 𝑘 } defined by (1.40) is uniformly Lipschitz on each bounded interval and 𝜙 𝑘 (𝜆) → 𝜙(𝜆) for all 𝜆 ≥ 0 as 𝑘 → ∞, then the limit function has the representation (1.45). Example 1.7 Suppose that 𝑔 is a probability generating function such that 𝑔 ′ (1−) = 1 and 𝑐 := 𝑔 ′′ (1−)/2 < ∞. Let 𝛼 𝑘 = 𝑘 2 and 𝑔 𝑘 (𝑧) = 𝑔(𝑧). By Taylor’s expansion it is easy to show that the sequence 𝜙 𝑘 (𝜆) defined by (1.40) converges to 𝑐𝜆2 as 𝑘 → ∞. Example 1.8 For 0 < 𝛼 < 1 the function 𝜙(𝜆) = −𝜆 𝛼 has the representation (1.41). This follows from (1.28) as we notice ∫ ∞ ∫ ∞ 1  d𝑢 𝑢  d𝑢 = < ∞. 1 + 𝑢 2 𝑢 1+𝛼 1 + 𝑢2 𝑢 𝛼 0 0 The function is the limit of the sequence 𝜙 𝑘 (𝜆) defined by (1.40) with 𝛼 𝑘 = 𝑘 𝛼 and 𝑔 𝑘 (𝑧) = 1 − (1 − 𝑧) 𝛼 .

28

1 Random Measures on Metric Spaces

Example 1.9 For any 1 ≤ 𝛼 ≤ 2 the function 𝜙(𝜆) = 𝜆 𝛼 can be represented in the form of (1.45). In particular, for 1 < 𝛼 < 2 we have ∫ ∞ 𝛼(𝛼 − 1) d𝑢 𝜆𝛼 = (e−𝜆𝑢 − 1 + 𝜆𝑢) 1+𝛼 , 𝜆 ≥ 0. Γ(2 − 𝛼) 0 𝑢 Let 𝜙 𝑘 (𝜆) be defined by (1.40) with 𝛼 𝑘 = 𝛼𝑘 𝛼 and 𝑔 𝑘 (𝑧) = 𝑧 + 𝛼−1 (1 − 𝑧) 𝛼 . Then 𝜙 𝑘 (𝜆) = 𝜆 𝛼 for 0 ≤ 𝜆 ≤ 𝑘. Example 1.10 The function 𝜙(𝜆) = 𝜆 log 𝜆 has the representation (1.41). In fact, we have ∫ ∞ ∫ ∞ d𝑣 d𝑢 −𝜆𝑢 (e−𝑣 − 1 + 𝑣1 {𝑣 ≤𝜆} ) 2 (e − 1 + 𝜆𝑢1 {𝑢≤1} ) 2 = 𝜆 𝑢 𝑣 0 0 ∫ 𝜆 d𝑣 = ℎ𝜆 + 𝜆 = ℎ𝜆 + 𝜆 log 𝜆, 𝑣 1 where ∫

∞

ℎ=

(e−𝑣 − 1 + 𝑣1 {𝑣 ≤1} )

0

d𝑣 . 𝑣2

Then 𝜙 has representation (1.43), which is equivalent to (1.41). Clearly, this function cannot be represented by (1.45). For sufficiently large 𝑘 ≥ 1, let 𝜙 𝑘 (𝜆) be defined by (1.40) with 𝛼 𝑘 = 𝑘 (log 𝑘 − 1) and 𝑔 𝑘 (𝑧) = 𝑧 + 𝑘𝛼−1 𝑘 (1 − 𝑧) log[𝑘 (1 − 𝑧)]. It is easy to see that 𝜙 𝑘 (𝜆) = 𝜆 log 𝜆 for 0 ≤ 𝜆 ≤ 𝑘.

1.6 Notes and Comments For the theory of convergence of probability measures on metric spaces we refer to Billingsley (1999), Ethier and Kurtz (1986) and Parthasarathy (1967). A standard reference for random measures is Kallenberg (1975). A slightly different form of Proposition 1.20 was given in Dynkin (1989a). Theorem 1.21 can be found in Dynkin (1989b, 1991a). See Dynkin (1989b) for another proof of Theorem 1.24. Some earlier forms of Theorem 1.33 were given in Silverstein (1969) and Watanabe (1968). Functions in the class ℐ defined by (1.27) are known as Bernstein functions. We refer the reader to Schilling et al. (2012) for a systematic treatment of those functions. Theorems 1.41 and 1.44 and their proofs are from Li (1991). Theorem 1.41 can also be derived from the results of Widder (1931). Corollary 1.42 is a famous result of Bernstein; see, e.g., Feller (1971, p. 439) or Berg et al. (1984, p. 135).

1.6 Notes and Comments

29

The Laplace transform provides an important tool for the study of probability measures on the half line R+ . To characterize a probability measure 𝜇 on the real line R one usually uses its characteristic function defined by ∫ ˆ = e𝑖𝑡 𝑥 𝜇(d𝑥), 𝜇(𝑡) 𝑡 ∈ R. R

It is well known that the probability measure is infinitely divisible if and only if its characteristic function is given by the Lévy–Khintchine formula:   ∫
2 𝑖𝑡 𝑦 (1.46) e − 1 − 𝑖𝑡𝑦1 { |𝑦 | ≤1} 𝐿(d𝑦) , ˆ = exp 𝑖𝑎𝑡 − 𝑐𝑡 + 𝜇(𝑡) R\{0}

where 𝑐 ≥ 0 and 𝑎 are constants and 𝐿(d𝑦) is a 𝜎-finite (Lévy) measure on R \ {0} such that ∫ (1 ∧ 𝑦 2 )𝐿 (d𝑦) < ∞. R\{0}

For the infinitely divisible probability measure 𝜇 given by (1.46) we have ∫ |𝑥|𝜇(d𝑥) < ∞ R

if and only ∫

(|𝑦| ∧ 𝑦 2 )𝐿 (d𝑦) < ∞;

R\{0}

see, e.g., Sato (1999, p. 163). In this case, we can rewrite the Lévy–Khintchine representation as   ∫
2 𝑖𝑡 𝑦 𝜇(𝑡) (1.47) ˆ = exp 𝑖𝑏𝑡 − 𝑐𝑡 + e − 1 − 𝑖𝑡𝑦 𝐿 (d𝑦) , R\{0}

where ∫ 𝑏=

𝑥𝜇(d𝑥). R

In particular, a one-sided stable distribution with index 1 < 𝛼 < 2 is obtained by taking 𝑏 = 𝑐 = 0 and 𝐿(d𝑦) = ℎ𝑦 −1−𝛼 1 {𝑦>0} d𝑦 for some constant ℎ > 0 in (1.47).

Chapter 2

Measure-Valued Branching Processes

A measure-valued process describes the evolution of a population that evolves according to the law of chance. In this chapter we provide some basic characterizations and constructions for measure-valued branching processes. In particular, we establish a one-to-one correspondence between those processes and cumulant semigroups. Some results for nonlinear integral evolution equations are proved, which lead to an analytic construction of a class of measure-valued branching processes, the socalled Dawson–Watanabe superprocesses. We shall construct the superprocesses for admissible killing densities and general branching mechanisms that are not necessarily decomposable into local and non-local parts. A number of moment formulas are proved. We also give some estimates for the variations of the transition probabilities with different initial states in Wasserstein and total variation distances.

2.1 Definitions and Basic Properties Suppose that 𝐸 is a Lusin topological space and (𝑄 𝑡 )𝑡 ≥0 is a conservative transition semigroup on 𝑀 (𝐸). We say (𝑄 𝑡 )𝑡 ≥0 satisfies the branching property provided 𝑄 𝑡 (𝜇1 + 𝜇2 , ·) = 𝑄 𝑡 (𝜇1 , ·) ∗ 𝑄 𝑡 (𝜇2 , ·),

𝑡 ≥ 0, 𝜇1 , 𝜇2 ∈ 𝑀 (𝐸).

(2.1)

In this case, it is easy to see that 𝑄 𝑡 (𝜇, ·) is an infinitely divisible probability measure on 𝑀 (𝐸) for all 𝑡 ≥ 0 and 𝜇 ∈ 𝑀 (𝐸). Given the transition semigroup (𝑄 𝑡 )𝑡 ≥0 , for 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + let ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝛿 𝑥 , d𝜈), 𝑉𝑡 𝑓 (𝑥) = − log 𝑥 ∈ 𝐸. (2.2) 𝑀 (𝐸)

We say (𝑄 𝑡 )𝑡 ≥0 satisfies the regular branching property if for every 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + the function 𝑉𝑡 𝑓 belongs to 𝐵(𝐸) + and © Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_2

31

32

2 Measure-Valued Branching Processes

∫

e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈) = exp{−𝜇(𝑉𝑡 𝑓 )},

𝜇 ∈ 𝑀 (𝐸).

(2.3)

𝑀 (𝐸)

Clearly, (𝑄 𝑡 )𝑡 ≥0 has the branching property (2.1) if it satisfies the regular branching property (2.3). Theorem 2.1 If (𝑄 𝑡 )𝑡 ≥0 satisfies the branching property (2.1), then for any probability measures 𝑁1 and 𝑁2 on 𝑀 (𝐸) we have (𝑁1 ∗ 𝑁2 )𝑄 𝑡 = (𝑁1 𝑄 𝑡 ) ∗ (𝑁2 𝑄 𝑡 ),

𝑡 ≥ 0.

(2.4)

Proof For any 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + , ∫ e−𝜈 ( 𝑓 ) (𝑁1 ∗ 𝑁2 )𝑄 𝑡 (d𝜈) 𝑀 (𝐸) ∫ ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈) = (𝑁1 ∗ 𝑁2 ) (d𝜇) (𝐸) 𝑀 (𝐸) 𝑀 ∫ ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇1 + 𝜇2 , d𝜈) = 𝑁1 (d𝜇1 )𝑁2 (d𝜇2 ) 2 ∫𝑀 (𝐸) ∫𝑀 (𝐸) = 𝑁1 (d𝜇1 )𝑁2 (d𝜇2 ) e−𝜈1 ( 𝑓 )−𝜈2 ( 𝑓 ) 𝑄 𝑡 (𝜇1 , d𝜈1 )𝑄 𝑡 (𝜇2 , d𝜈2 ) 𝑀 ∫𝑀 (𝐸) 2 ∫(𝐸) 2 e−𝜈2 ( 𝑓 ) (𝑁2 𝑄 𝑡 ) (d𝜈2 ). = e−𝜈1 ( 𝑓 ) (𝑁1 𝑄 𝑡 ) (d𝜈1 ) 𝑀 (𝐸)

𝑀 (𝐸)

Then (2.4) follows by the uniqueness of the Laplace functional.

□

Proposition 2.2 If (𝑄 𝑡 )𝑡 ≥0 satisfies the branching property (2.1) and 𝐾 is an infinitely divisible probability measure on 𝑀 (𝐸), then 𝐾𝑄 𝑡 is an infinitely divisible probability measure on 𝑀 (𝐸) for any 𝑡 ≥ 0. Proof For 𝑛 ≥ 1 let 𝐾𝑛 be the 𝑛-th root of 𝐾. By applying (2.4) inductively we have □ (𝐾𝑛 𝑄 𝑡 ) ∗𝑛 = (𝐾𝑛∗𝑛 )𝑄 𝑡 = 𝐾𝑄 𝑡 . Then 𝐾𝑄 𝑡 is infinitely divisible. Suppose that 𝑇 is an interval on the real line and (ℱ𝑡 )𝑡 ∈𝑇 is a filtration. A Markov process {(𝑋𝑡 , ℱ𝑡 ) : 𝑡 ∈ 𝑇 } in 𝑀 (𝐸) with transition semigroup (𝑄 𝑡 )𝑡 ≥0 satisfying the branching property (2.1) is called a measure-valued branching process (MBprocess). In particular, we call {(𝑋𝑡 , ℱ𝑡 ) : 𝑡 ∈ 𝑇 } a regular MB-process if (𝑄 𝑡 )𝑡 ≥0 satisfies the regular branching property defined by (2.2) and (2.3). Theorem 2.3 Suppose that {(𝑋𝑡 , ℱ𝑡 ) : 𝑡 ∈ 𝑇 } and {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ∈ 𝑇 } are two independent MB-processes with transition semigroup (𝑄 𝑡 )𝑡 ≥0 . Let 𝑍𝑡 = 𝑋𝑡 + 𝑌𝑡 and ℋ𝑡 = 𝜎(ℱ𝑡 ∪ 𝒢𝑡 ). Then {(𝑍𝑡 , ℋ𝑡 ) : 𝑡 ∈ 𝑇 } is also an MB-process with transition semigroup (𝑄 𝑡 )𝑡 ≥0 . Proof Let 𝑟 ≤ 𝑡 ∈ 𝑇 and suppose 𝐹 ∈ bℱ𝑟 and 𝐺 ∈ b𝒢𝑟 . For any 𝑓 ∈ 𝐵(𝐸) + we use the independence of {(𝑋𝑡 , ℱ𝑡 ) : 𝑡 ∈ 𝑇 } and {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ∈ 𝑇 } and the branching property (2.1) to see that

2.1 Definitions and Basic Properties

33

i

h

E 𝐹𝐺 exp{−𝑍𝑡 ( 𝑓 )} i i h h = E 𝐹 exp{−𝑋𝑡 ( 𝑓 )} E 𝐺 exp{−𝑌𝑡 ( 𝑓 )} i i h ∫ h ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡−𝑟 (𝑋𝑟 , d𝜈) E 𝐺 e−𝜈 ( 𝑓 ) 𝑄 𝑡−𝑟 (𝑌𝑟 , d𝜈) = E 𝐹 𝑀 (𝐸) 𝑀 ∫(𝐸) h i −𝜈 ( 𝑓 ) e 𝑄 𝑡−𝑟 (𝑍𝑟 , d𝜈) . = E 𝐹𝐺 𝑀 (𝐸)

Then Proposition A.1 implies h ∫ E 𝐻 exp{−𝑍𝑡 ( 𝑓 )} = E 𝐻 h

i

e−𝜈 ( 𝑓 ) 𝑄 𝑡−𝑟 (𝑍𝑟 , d𝜈)

i

𝑀 (𝐸)

for any 𝐻 ∈ bℋ𝑟 . This gives the desired result.

□

Recall that ℐ(𝐸) denotes the convex cone of functionals on 𝐵(𝐸) + with the representation (1.20). Let (𝑉𝑡 )𝑡 ≥0 be a family of operators on 𝐵(𝐸) + and let 𝑣 𝑡 (𝑥, 𝑓 ) = 𝑉𝑡 𝑓 (𝑥). We call (𝑉𝑡 )𝑡 ≥0 a cumulant semigroup provided: (1) 𝑣 𝑡 (𝑥, ·) ∈ ℐ(𝐸) for all 𝑡 ≥ 0 and 𝑥 ∈ 𝐸; (2) 𝑉𝑟 𝑉𝑡 = 𝑉𝑟+𝑡 for every 𝑟, 𝑡 ≥ 0. By Theorem 1.37, if (𝑉𝑡 )𝑡 ≥0 is a cumulant semigroup, each operator 𝑉𝑡 has the canonical representation ∫
(2.5) 𝑉𝑡 𝑓 (𝑥) = 𝜆 𝑡 (𝑥, 𝑓 ) + 1 − e−𝜈 ( 𝑓 ) 𝐿 𝑡 (𝑥, d𝜈), 𝑓 ∈ 𝐵(𝐸) + , 𝑀 (𝐸) ◦

where 𝜆 𝑡 (𝑥, d𝑦) is a bounded kernel on 𝐸 and [1 ∧ 𝜈(1)] 𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ . Theorem 2.4 The relation (2.3) establishes a one-to-one correspondence between cumulant semigroups (𝑉𝑡 )𝑡 ≥0 on 𝐵(𝐸) + and transition semigroups (𝑄 𝑡 )𝑡 ≥0 on 𝑀 (𝐸) satisfying the regular branching property. Proof Suppose that (𝑉𝑡 )𝑡 ≥0 is a cumulant semigroup. By Theorem 1.36 we see that (2.3) defines an infinitely divisible probability measure 𝑄 𝑡 (𝜇, ·) on 𝑀 (𝐸). From 𝑉𝑟 𝑉𝑡 = 𝑉𝑟+𝑡 we have 𝑄 𝑟 𝑄 𝑡 = 𝑄 𝑟+𝑡 . That is, (𝑄 𝑡 )𝑡 ≥0 is a transition semigroup on 𝑀 (𝐸). Conversely, suppose that (𝑄 𝑡 )𝑡 ≥0 is a transition semigroup on 𝑀 (𝐸) satisfying the regular branching property. Then 𝑄 𝑡 (𝜇, ·) is an infinitely divisible probability measure on 𝑀 (𝐸). This is true in particular for 𝜇 = 𝛿 𝑥 , and so 𝑉𝑡 𝑓 (𝑥) has the representation (2.5) by Theorems 1.36 and 1.37. The semigroup property of (𝑉𝑡 )𝑡 ≥0 follows from that of (𝑄 𝑡 )𝑡 ≥0 . □ Example 2.1 Let 𝑀𝑎 (𝐸) and 𝑀𝑑 (𝐸) denote respectively the subset of 𝑀 (𝐸) of purely atomic measures and that of diffuse measures. Then each 𝜇 ∈ 𝑀 (𝐸) has the unique decomposition 𝜇 = 𝜇 𝑎 + 𝜇 𝑑 for 𝜇 𝑎 ∈ 𝑀𝑎 (𝐸) and 𝜇 𝑑 ∈ 𝑀𝑑 (𝐸). The mappings 𝜇 ↦→ 𝜇 𝑎 and 𝜇 ↦→ 𝜇 𝑑 are measurable; see Kallenberg (1975, pp. 10–11). Take two

34

2 Measure-Valued Branching Processes

distinct real constants 𝑐 𝑎 and 𝑐 𝑑 and let 𝑄 𝑡 (𝜇, ·) be the unit mass concentrated at e𝑐𝑎 𝑡 𝜇 𝑎 + e𝑐𝑑 𝑡 𝜇 𝑑 . Then (𝑄 𝑡 )𝑡 ≥0 satisfies the branching property (2.1), but it is not regular in the sense of (2.3). Theorem 2.5 Suppose that 𝐸 is a compact metric space. If (𝑉𝑡 )𝑡 ≥0 is a cumulant semigroup on 𝐸 preserving 𝐶 (𝐸) ++ and 𝑉𝑡 𝑓 (𝑥) → 𝑓 (𝑥) pointwise as 𝑡 → 0 for every 𝑓 ∈ 𝐶 (𝐸) ++ , then (2.3) defines a Feller semigroup (𝑄 𝑡 )𝑡 ≥0 on 𝑀 (𝐸). Conversely, if (𝑄 𝑡 )𝑡 ≥0 is a Feller semigroup having the branching property (2.1), then it satisfies the regular branching property (2.3) with cumulant semigroup (𝑉𝑡 )𝑡 ≥0 preserving 𝐶 (𝐸) ++ and 𝑉𝑡 𝑓 (𝑥) → 𝑓 (𝑥) pointwise as 𝑡 → 0 for every 𝑓 ∈ 𝐶 (𝐸) ++ . Proof If (𝑉𝑡 )𝑡 ≥0 is a cumulant semigroup on 𝐸 that preserves 𝐶 (𝐸) ++ and 𝑉𝑡 𝑓 (𝑥) → 𝑓 (𝑥) pointwise as 𝑡 → 0 for every 𝑓 ∈ 𝐶 (𝐸) ++ , it is easy to see that (2.3) defines a Feller semigroup on 𝑀 (𝐸). For the converse, suppose that (𝑄 𝑡 )𝑡 ≥0 is a Feller semigroup on 𝑀 (𝐸) having the branching property (2.1). Given 𝑓 ∈ 𝐵(𝐸) + we ++ + define Í𝑛𝑉𝑡 𝑓 (𝑥) by (2.2). For any 𝑓 ∈ 𝐶 (𝐸) we clearly have 𝑉𝑡 𝑓 ∈ 𝐶 (𝐸) . If 𝜇 = 𝑖=1 ( 𝑝 𝑖 /𝑞 𝑖 )𝛿 𝑥𝑖 for 𝑥𝑖 ∈ 𝐸 and integers 𝑝 𝑖 and 𝑞 𝑖 ≥ 1, we have (2.3) by easy calculations based on (2.1). By an approximating argument, the equality holds for all 𝜇 ∈ 𝑀 (𝐸) and 𝑓 ∈ 𝐶 (𝐸) ++ . The extension from 𝑓 ∈ 𝐶 (𝐸) ++ to 𝑓 ∈ 𝐵(𝐸) + is immediate by Proposition 1.3. Then (𝑄 𝑡 )𝑡 ≥0 satisfies the regular branching property. If there exists an 𝑓 ∈ 𝐶 (𝐸) ++ such that 𝑉𝑡 𝑓 ∉ 𝐶 (𝐸) ++ , the compactness of 𝐸 assures the existence of a point 𝑥0 ∈ 𝐸 satisfying 𝑉𝑡 𝑓 (𝑥0 ) = 0, so the function 𝜇 ↦→ exp{−𝜇(𝑉𝑡 𝑓 )} does not belong to 𝐶0 (𝑀 (𝐸)), yielding a contradiction. Then (𝑉𝑡 )𝑡 ≥0 preserves 𝐶 (𝐸) ++ . Since 𝑄 𝑡 𝐹 (𝜇) → 𝐹 (𝜇) pointwise as 𝑡 → 0 for every 𝐹 ∈ 𝐶0 (𝑀 (𝐸)), we have 𝑉𝑡 𝑓 (𝑥) → 𝑓 (𝑥) pointwise as 𝑡 → 0 for every 𝑓 ∈ 𝐶 (𝐸) ++ .□ In the rest of the book, we will only consider regular MB-processes and will omit the adjective “regular”. By (2.3) it is easy to see that 𝑄 𝑡 (0, {0}) = 1 for every 𝑡 ≥ 0. More generally, we have 𝑄 𝑡 (𝜇, {0}) = e−𝜇 ( 𝑣¯𝑡 ) ,

𝑡 ≥ 0, 𝜇 ∈ 𝑀 (𝐸),

(2.6)

where 𝑣¯ 𝑡 (𝑥) = lim 𝑉𝑡 𝜆(𝑥) ∈ [0, ∞],

𝑥 ∈ 𝐸.

(2.7)

𝜆→∞

From (2.6) we see that 𝑡 ↦→ 𝑣¯ 𝑡 (𝑥) is decreasing for every 𝑥 ∈ 𝐸. Observe also that 𝑄 𝑡 (𝜇, {0}) > 0 for every 𝜇 ∈ 𝑀 (𝐸) if and only if 𝑣¯ 𝑡 is a bounded function on 𝐸. Let (𝑄 ◦𝑡 )𝑡 ≥0 denote the restriction of (𝑄 𝑡 )𝑡 ≥0 to 𝑀 (𝐸) ◦ . Proposition 2.6 Suppose that (𝑄 𝑡 )𝑡 ≥0 is defined by (2.3) with (𝑉𝑡 )𝑡 ≥0 given by (2.5). If 𝑁 = 𝐼 (𝜂, 𝐻) is an infinitely divisible probability measure on 𝑀 (𝐸), then 𝑁𝑄 𝑡 = 𝐼 (𝜂𝑡 , 𝐻𝑡 ) is infinitely divisible for every 𝑡 ≥ 0, where ∫ ∫ (2.8) 𝜂(d𝑦)𝐿 𝑡 (𝑦, ·) + 𝐻𝑄 ◦𝑡 . 𝜂𝑡 = 𝜂(d𝑦)𝜆 𝑡 (𝑦, ·) and 𝐻𝑡 = 𝐸

𝐸

2.1 Definitions and Basic Properties

35

Proof We first note that 𝑁𝑄 𝑡 is infinitely divisible by Proposition 2.2. For 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + we have ∫ e−𝜈 ( 𝑓 ) 𝑁𝑄 𝑡 (d𝜈) − log 𝑀 (𝐸) ∫
1 − e−𝜈 (𝑉𝑡 𝑓 ) 𝐻 (d𝜈) = 𝜂(𝑉𝑡 𝑓 ) + ◦ 𝑀 (𝐸) ∫ ∫ ∫
𝜂(d𝑦)𝜆 𝑡 (𝑦, 𝑓 ) + 1 − e−𝜈 ( 𝑓 ) 𝐿 𝑡 (𝑦, d𝜈) 𝜂(d𝑦) = ◦ 𝑀 (𝐸) 𝐸 𝐸 ∫ ◦ −𝜈 ( 𝑓 )
+ 𝐻𝑄 𝑡 (d𝜈). 1−e 𝑀 (𝐸) ◦

Then 𝑁𝑄 𝑡 = 𝐼 (𝜂𝑡 , 𝐻𝑡 ) with (𝜂𝑡 , 𝐻𝑡 ) given by (2.8).

□

Corollary 2.7 Suppose that (𝑄 𝑡 )𝑡 ≥0 is defined by (2.3) with (𝑉𝑡 )𝑡 ≥0 given by (2.5). Then for any 𝑡 ≥ 𝑟 ≥ 0 and 𝑥 ∈ 𝐸 we have ∫ (2.9) 𝜆𝑟+𝑡 (𝑥, ·) = 𝜆𝑟 (𝑥, d𝑦)𝜆 𝑡 (𝑦, ·) 𝐸

and ∫

∫ 𝜆𝑟 (𝑥, d𝑦)𝐿 𝑡 (𝑦, ·) +

𝐿 𝑟+𝑡 (𝑥, ·) = 𝐸

𝐿 𝑟 (𝑥, d𝜇)𝑄 ◦𝑡 (𝜇, ·).

(2.10)

𝑀 (𝐸) ◦

Proof This follows by applying Proposition 2.6 to the infinitely divisible probability measure 𝑄 𝑟 (𝛿 𝑥 , ·) on 𝑀 (𝐸). □ Corollary 2.8 For any 𝑥 ∈ 𝐸 the family of 𝜎-finite measures 𝐿(𝑥) = {𝐿 𝑡 (𝑥, ·) : 𝑡 > 0} defined by (2.5) constitute an entrance rule for the restricted semigroup (𝑄 ◦𝑡 )𝑡 ≥0 . We call 𝐿 (𝑥) = {𝐿 𝑡 (𝑥, ·) : 𝑡 > 0} the canonical entrance rule at 𝑥 ∈ 𝐸 defined by (𝑉𝑡 )𝑡 ≥0 . Let 𝐸𝐶 be the set of points 𝑥 ∈ 𝐸 such that 𝜆 𝑡 (𝑥, 𝐸) = 0 for every 𝑡 > 0. Then 𝑥 ∈ 𝐸𝐶 if and only if ∫
1 − e−𝜈 ( 𝑓 ) 𝐿 𝑡 (𝑥, d𝜈), 𝑡 > 0, 𝑓 ∈ 𝐵(𝐸) + . (2.11) 𝑉𝑡 𝑓 (𝑥) = 𝑀 (𝐸) ◦

By Corollary 2.7, for 𝑥 ∈ 𝐸𝐶 the canonical entrance rule 𝐿(𝑥) = {𝐿 𝑡 (𝑥, ·) : 𝑡 > 0} is an entrance law for (𝑄 ◦𝑡 )𝑡 ≥0 . If the function 𝑣¯ 𝑡 defined by (2.7) is finite on 𝐸 for every 𝑡 > 0, we clearly have 𝐸𝐶 = 𝐸 and 𝑣¯ 𝑡 (𝑥) = 𝐿 𝑡 (𝑥, 𝑀 (𝐸) ◦ ) for every 𝑥 ∈ 𝐸. Example 2.2 Let 𝐸 = {0} be a singleton. In this case, we understand 𝑀 (𝐸) = [0, ∞). For 𝜆 ≥ 0 let 𝑣 0 (𝜆) = 𝜆 and ∫ ∞ 𝜆 , 𝑡 > 0. (2.12) 𝑣 𝑡 (𝜆) = (1 − e−𝜆𝑢 )𝑡 −2 e−𝑢/𝑡 u. = 1 + 𝑡𝜆 0 Then (𝑣 𝑡 )𝑡 ≥0 is a one-dimensional cumulant semigroup.

36

2 Measure-Valued Branching Processes

Example 2.3 Let 𝐸 = [0, ∞) and let (𝑣 𝑡 )𝑡 ≥0 be defined as in Example 2.2. For 𝑥 ∈ 𝐸 and 𝑓 ∈ 𝐵(𝐸) + let 𝑉𝑡 𝑓 (𝑥) = 1 {𝑥+𝑡 0} defined by (2.5) is regular. Example 2.4 Let 𝐸 = [0, ∞) and let (𝑣 𝑡 )𝑡 ≥0 be defined as in Example 2.2. For 𝑥 ∈ 𝐸 and 𝑓 ∈ 𝐵(𝐸) + let 𝑉𝑡 𝑓 (𝑥) = 1 {𝑥+𝑡 0} be the canonical entrance rule defined from this cumulant semigroup. In view of (2.12), we have ∫ ∞ 𝑉𝑡 𝑓 (0) = 1 {𝑡 0} is not regular.

2.2 Integral Evolution Equations Let 𝐸 be a Lusin topological space. Suppose that 𝜉 = (Ω, ℱ, ℱ𝑡 , 𝜉𝑡 , P 𝑥 ) is a Borel right process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 . Let {𝐾 (𝑡) : 𝑡 ≥ 0} be a continuous additive functional of 𝜉 which is admissible in the sense that each 𝜔 ↦→ 𝐾𝑡 (𝜔) is measurable with respect to the 𝜎-algebra ℱ 0 := 𝜎({𝜉𝑡 : 𝑡 ≥ 0}) and   𝑡 → 0. (2.13) 𝑘 (𝑡) := sup P 𝑥 𝐾 (𝑡) → 0, 𝑥 ∈𝐸

For any 𝛽 ∈ 𝐵(𝐸) we write ∫ 𝐾𝑡 (𝛽) =

𝑡

𝛽(𝜉 𝑠 )𝐾 (d𝑠),

𝑡 ≥ 0.

0

Let bℰ(𝐾) denote the set of functions 𝛽 ∈ 𝐵(𝐸) such that 𝑡 ↦→ e−𝐾𝑡 (𝛽) is a locally bounded stochastic process. Recall that ∥ · ∥ denotes the supremum norm of functions on 𝐸.

2.2 Integral Evolution Equations

37

Proposition 2.9 Let 𝑓 ∈ 𝐵(𝐸) and 𝑏, 𝛽 ∈ bℰ(𝐾). If the two locally bounded functions ℎ, 𝑢 ∈ ℬ([0, ∞) × 𝐸) satisfy ∫ 𝑡   −𝐾 (𝛽)  −𝐾𝑠 (𝛽) 𝑡 𝑓 (𝜉𝑡 ) + P 𝑥 𝑢(𝑡, 𝑥) = P 𝑥 e e ℎ(𝑡 − 𝑠, 𝜉 𝑠 )𝐾 (d𝑠) , (2.14) 0

they also satisfy  ∫ 𝑡   e−𝐾𝑠 (𝑏) ℎ(𝑡 − 𝑠, 𝜉 𝑠 )𝐾 (d𝑠) 𝑢(𝑡, 𝑥) = P 𝑥 e−𝐾𝑡 (𝑏) 𝑓 (𝜉𝑡 ) + P 𝑥 0  ∫ 𝑡 −𝐾𝑠 (𝑏) e [𝛽(𝜉 𝑠 ) − 𝑏(𝜉 𝑠 )]𝑢(𝑡 − 𝑠, 𝜉 𝑠 )𝐾 (d𝑠) . − P𝑥

(2.15)

0

Proof Let 𝐾𝑡𝑟 (𝛽) = 𝐾𝑡 (𝛽) −𝐾𝑟 (𝛽) for 𝑡 ≥ 𝑟 ≥ 0. Since 𝑠 ↦→ ℱ𝑠 is a right continuous filtration, the process 𝑠 ↦→ P 𝑥 [e−𝐾𝑡 (𝛽) 𝑓 (𝜉𝑡 )|ℱ𝑠 ] = e𝐾𝑠 (𝛽) P 𝑥 [e−𝐾𝑡 (𝛽) 𝑓 (𝜉𝑡 )|ℱ𝑠 ] 𝑠

is a.s. right continuous. Let 𝑔 = 𝛽 − 𝑏 ∈ bℰ(𝐾). By the Markov property of 𝜉, ∫

   𝑔 ( 𝜉𝑠 )e−𝐾𝑠 (𝑏) P 𝜉𝑠 e−𝐾𝑡−𝑠 (𝛽) 𝑓 ( 𝜉𝑡−𝑠 ) 𝐾 (d𝑠) 0 ∫  𝑡   𝑠 𝑔 ( 𝜉𝑠 )e−𝐾𝑠 (𝑏) P 𝑥 e−𝐾𝑡 (𝛽) 𝑓 ( 𝜉𝑡 ) |ℱ𝑠 𝐾 (d𝑠) = P𝑥 0   ∑︁ 𝑛 ∫ 𝑖𝑡/𝑛   𝑖𝑡/𝑛 𝑔 ( 𝜉𝑠 )e−𝐾𝑠 (𝑏) P 𝑥 e−𝐾𝑡 (𝛽) 𝑓 ( 𝜉𝑡 ) |ℱ𝑖𝑡/𝑛 𝐾 (d𝑠) = lim P 𝑥 𝑡

P𝑥

𝑛→∞ 𝑖=1 𝑛 ∑︁

= lim



P𝑥 P𝑥

𝑛→∞ 𝑖=1 𝑛 ∑︁

(𝑖−1) 𝑡/𝑛  ∫ 𝑖𝑡/𝑛

∫

𝑖𝑡/𝑛

𝑔 ( 𝜉𝑠 )e−𝐾𝑠 (𝑏) e−𝐾𝑡

(𝑖−1) 𝑡/𝑛 𝑖𝑡/𝑛 𝑖𝑡/𝑛

(𝛽)

 𝑓 ( 𝜉𝑡 ) 𝐾 (d𝑠) ℱ𝑖𝑡/𝑛

= lim P𝑥 𝑔 ( 𝜉𝑠 )e−𝐾𝑠 (𝑏) e−𝐾𝑡 (𝛽) 𝑓 ( 𝜉𝑡 ) 𝐾 (d𝑠) 𝑛→∞ (𝑖−1) 𝑡/𝑛 𝑖=1 ∫ 𝑡  𝑠 𝑔 ( 𝜉𝑠 )e−𝐾𝑠 (𝑏) e−𝐾𝑡 (𝛽) 𝑓 ( 𝜉𝑡 ) 𝐾 (d𝑠) = P𝑥   ∫0 𝑡 𝑠 𝑔 ( 𝜉𝑠 )e−𝐾𝑡 (𝑏) e−𝐾𝑡 (𝑔) 𝑓 ( 𝜉𝑡 ) 𝐾 (d𝑠) = P𝑥 n 0
o = P 𝑥 𝑓 ( 𝜉𝑡 )e−𝐾𝑡 (𝑏) 1 − e−𝐾𝑡 (𝑔) .



By similar calculations we have   ∫ 𝑡−𝑠  𝑔 ( 𝜉𝑠 )e−𝐾𝑠 (𝑏) P 𝜉𝑠 e−𝐾𝑟 (𝛽) ℎ (𝑡 − 𝑠 − 𝑟 , 𝜉𝑟 ) 𝐾 (d𝑟) 𝐾 (d𝑠) 0 0 ∫  ∫ 𝑡−𝑠 𝑡 𝑠 −𝐾𝑠 (𝑏) 𝑔 ( 𝜉𝑠 )e 𝐾 (d𝑠) e−𝐾𝑠+𝑟 (𝛽) ℎ (𝑡 − 𝑠 − 𝑟 , 𝜉𝑠+𝑟 ) 𝐾 (𝑠 + d𝑟) P𝑥   ∫0 𝑡 ∫0 𝑡 𝑠 −𝐾𝑠 (𝑏) P𝑥 e−𝐾𝑟 (𝛽) ℎ (𝑡 − 𝑟 , 𝜉𝑟 ) 𝐾 (d𝑟) 𝑔 ( 𝜉𝑠 )e 𝐾 (d𝑠) 𝑠   ∫0 𝑡 ∫ 𝑟 𝑠 𝑔 ( 𝜉𝑠 )e−𝐾𝑟 (𝑔) 𝐾 (d𝑠) P𝑥 ℎ(𝑡 − 𝑟 , 𝜉𝑟 )e−𝐾𝑟 (𝑏) 𝐾 (d𝑟) 0   ∫0 𝑡 −𝐾𝑟 (𝑔)
−𝐾𝑟 (𝑏) P𝑥 𝐾 (d𝑟) . 1−e ℎ(𝑡 − 𝑟 , 𝜉𝑟 )e

∫ P𝑥 = = = =

𝑡

0

38

2 Measure-Valued Branching Processes

Then we add up both sides of the two equations and use (2.14) to get (2.15).

□

In the sequel, we assume 𝛽 ∈ bℰ(𝐾) and 𝑓 ↦→ 𝜙(·, 𝑓 ) is an operator from 𝐵(𝐸) + into 𝐵(𝐸) which is bounded on 𝐵 𝑎 (𝐸) + for every 𝑎 ≥ 0. For 𝑓 ∈ 𝐵(𝐸) + we consider the integral evolution equation  ∫ 𝑡   𝑣 𝑡 (𝑥) = P 𝑥 e−𝐾𝑡 (𝛽) 𝑓 (𝜉𝑡 ) − P 𝑥 e−𝐾𝑠 (𝛽) 𝜙(𝜉 𝑠 , 𝑣 𝑡−𝑠 )𝐾 (d𝑠) . (2.16) 0

For convenience of statement of the results, we formulate the following conditions: Condition 2.10 There is a constant 𝐿 ≥ 0 such that −𝜙(𝑥, 𝑓 ) ≤ 𝐿 ∥ 𝑓 ∥ for 𝑥 ∈ 𝐸 and 𝑓 ∈ 𝐵(𝐸) + . Condition 2.11 For every 𝑎 ≥ 0 there is a constant 𝐿 𝑎 ≥ 0 such that sup |𝜙(𝑥, 𝑓 ) − 𝜙(𝑥, 𝑔)| ≤ 𝐿 𝑎 ∥ 𝑓 − 𝑔∥,

𝑓 , 𝑔 ∈ 𝐵 𝑎 (𝐸) + .

𝑥 ∈𝐸

Proposition 2.12 Let 𝑟 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + . Then (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥) satisfies (2.16) for 𝑡 ≥ 0 if and only if it satisfies the equation for 0 ≤ 𝑡 ≤ 𝑟 and (𝑡, 𝑥) ↦→ 𝑣 𝑟+𝑡 (𝑥) satisfies   𝑣𝑟+𝑡 ( 𝑥) = P 𝑥 e−𝐾𝑡 (𝛽) 𝑣𝑟 ( 𝜉𝑡 ) − P 𝑥

∫

𝑡

 e−𝐾𝑠 (𝛽) 𝜙 ( 𝜉𝑠 , 𝑣𝑟+𝑡−𝑠 ) 𝐾 (d𝑠) .

(2.17)

0

Proof Suppose that (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥) satisfies (2.16) for 0 ≤ 𝑡 ≤ 𝑟 and (𝑡, 𝑥) ↦→ 𝑣 𝑟+𝑡 (𝑥) satisfies (2.17) for 𝑡 ≥ 0. Then we have ∫ 𝑡    e−𝐾𝑠 (𝛽) 𝜙 ( 𝜉𝑠 , 𝑣𝑟+𝑡−𝑠 ) 𝐾 (d𝑠) 𝑣𝑟+𝑡 ( 𝑥) = P 𝑥 e−𝐾𝑡 (𝛽) 𝑣𝑟 ( 𝜉𝑡 ) − P 𝑥 0 o n   = P 𝑥 e−𝐾𝑡 (𝛽) P 𝜉𝑡 e−𝐾𝑟 (𝛽) 𝑓 ( 𝜉𝑟 ) ∫ 𝑟   e−𝐾𝑠 (𝛽) 𝜙 ( 𝜉𝑠 , 𝑣𝑟−𝑠 ) 𝐾 (d𝑠) − P 𝑥 e−𝐾𝑡 (𝛽) P 𝜉𝑡 0 ∫ 𝑡  − P𝑥 e−𝐾𝑠 (𝛽) 𝜙 ( 𝜉𝑠 , 𝑣𝑟+𝑡−𝑠 ) 𝐾 (d𝑠) 0 ∫ 𝑟   −𝐾 (𝛽) 𝑟+𝑡 𝑓 ( 𝜉𝑟+𝑡 ) ] − P 𝑥 = P𝑥 e e−𝐾𝑡+𝑠 (𝛽) 𝜙 ( 𝜉𝑡+𝑠 , 𝑣𝑟−𝑠 ) 𝐾 (𝑡 + d𝑠) 0 ∫ 𝑡  e−𝐾𝑠 (𝛽) 𝜙 ( 𝜉𝑠 , 𝑣𝑟+𝑡−𝑠 ) 𝐾 (d𝑠) − P𝑥 0  ∫ 𝑟+𝑡   −𝐾 (𝛽) 𝑟+𝑡 = P𝑥 e 𝑓 ( 𝜉𝑟+𝑡 ) ] − P 𝑥 e−𝐾𝑠 (𝛽) 𝜙 ( 𝜉𝑠 , 𝑣𝑟+𝑡−𝑠 ) 𝐾 (d𝑠) 𝑡  ∫ 𝑡 −𝐾𝑠 (𝛽) e 𝜙 ( 𝜉𝑠 , 𝑣𝑟+𝑡−𝑠 ) 𝐾 (d𝑠) − P𝑥 0   ∫ 𝑟+𝑡  = P 𝑥 e−𝐾𝑟+𝑡 (𝛽) 𝑓 ( 𝜉𝑟+𝑡 ) ] − P 𝑥 e−𝐾𝑠 (𝛽) 𝜙 ( 𝜉𝑠 , 𝑣𝑟+𝑡−𝑠 ) 𝐾 (d𝑠) . 0

Therefore (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥) satisfies (2.16) for 𝑡 ≥ 0. For the converse, suppose that (2.16) holds for 𝑡 ≥ 0. The equation certainly holds for 0 ≤ 𝑡 ≤ 𝑟. By calculations □ similar to the above we see (𝑡, 𝑥) ↦→ 𝑣 𝑟+𝑡 (𝑥) satisfies (2.17).

2.2 Integral Evolution Equations

39

Corollary 2.13 If for every 𝑓 ∈ 𝐵(𝐸) + there is a unique locally bounded positive solution (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥, 𝑓 ) to (2.16), then the operators 𝑉𝑡 : 𝑓 ↦→ 𝑣 𝑡 (·, 𝑓 ) on 𝐵(𝐸) + constitute a semigroup. Proof Fix 𝑟 ≥ 0 and define 𝑢 𝑡 = 𝑣 𝑡 for 0 ≤ 𝑡 ≤ 𝑟 and 𝑢𝑟+𝑡 = 𝑣 𝑡 (·, 𝑣 𝑟 ) for 𝑡 ≥ 0. By Proposition 2.12 we see (𝑡, 𝑥) ↦→ 𝑢 𝑡 (𝑥) solves (2.16) for 𝑡 ≥ 0. Then the uniqueness of the solution implies 𝑢𝑟+𝑡 = 𝑣 𝑟+𝑡 for all 𝑡 ≥ 0. This gives the semigroup property □ of (𝑉𝑡 )𝑡 ≥0 . Proposition 2.14 Suppose that Condition 2.10 holds. Then there is an increasing function 𝑡 ↦→ 𝐶 (𝑡) on [0, ∞) such that for any locally bounded positive solution (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥, 𝑓 ) to (2.16) we have sup ∥𝑣 𝑠 (·, 𝑓 ) ∥ ≤ 𝐶 (𝑡) ∥ 𝑓 ∥,

𝑡 ≥ 0.

(2.18)

0≤𝑠 ≤𝑡

Proof Let 𝑡 ↦→ 𝑙 (𝑡) be an increasing function such that e−𝐾𝑡 (𝛽) ≤ 𝑙 (𝑡) for all 𝑡 ≥ 0. By (2.16) and Condition 2.10 we have ∫ 𝑡  ∥𝑣 𝑡 (·, 𝑓 ) ∥ ≤ 𝑙 (𝑡) ∥ 𝑓 ∥ + 𝐿𝑙 (𝑡) sup P 𝑥 ∥𝑣 𝑡−𝑠 (·, 𝑓 ) ∥𝐾 (d𝑠) . 𝑥 ∈𝐸

0

It follows that sup ∥𝑣 𝑠 (·, 𝑓 ) ∥ ≤ 𝑙 (𝑡) ∥ 𝑓 ∥ + 𝐿𝑘 (𝑡)𝑙 (𝑡) sup ∥𝑣 𝑠 (·, 𝑓 ) ∥. 0≤𝑠 ≤𝑡

0≤𝑠 ≤𝑡

Let 𝛿 > 0 be sufficiently small so that 𝐿𝑘 (𝛿)𝑙 (𝛿) < 1. For 0 ≤ 𝑡 ≤ 𝛿 the above inequality implies  −1  sup ∥𝑣 𝑠 (·, 𝑓 ) ∥ ≤ 𝑙 (𝑡) 1 − 𝐿𝑘 (𝑡)𝑙 (𝑡) ∥ 𝑓 ∥. 0≤𝑠 ≤𝑡

Then the desired result follows by Proposition 2.12 and a successive application of the above estimate. □ Proposition 2.15 If Condition 2.11 holds, there is at most one locally bounded positive solution (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥, 𝑓 ) to (2.16). Proof Suppose that (𝑡, 𝑥) ↦→ 𝑢 𝑡 (𝑥) and (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥) are two locally bounded positive solutions of (2.16). Let ℎ𝑡 (𝑥) = 𝑢 𝑡 (𝑥) − 𝑣 𝑡 (𝑥) and let 𝑙 (𝑡) be as in the proof of Proposition 2.14. For fixed 𝑇 > 0 we can use Proposition 2.14 to find a constant 𝑎 ≥ 0 such that ∥𝑢 𝑡 ∥ ≤ 𝑎 and ∥𝑣 𝑡 ∥ ≤ 𝑎 for all 0 ≤ 𝑡 ≤ 𝑇. By (2.16) and Condition 2.11 we have ∫ 𝑡  ∥ℎ𝑡 ∥ ≤ 𝑙 (𝑡)P 𝑥 |𝜙(𝜉 𝑠 , 𝑢 𝑡−𝑠 ) − 𝜙(𝜉 𝑠 , 𝑣 𝑡−𝑠 )|𝐾 (d𝑠) 0 ∫ 𝑡  ∥ℎ𝑡−𝑠 ∥𝐾 (d𝑠) . ≤ 𝐿 𝑎 𝑙 (𝑡) sup P 𝑥 𝑥 ∈𝐸

0

40

2 Measure-Valued Branching Processes

Then it is easy to get sup ∥ℎ 𝑠 ∥ ≤ 𝐿 𝑎 𝑘 (𝑡)𝑙 (𝑡) sup ∥ℎ 𝑠 ∥, 0≤𝑠 ≤𝑡

0 ≤ 𝑡 ≤ 𝑇.

0≤𝑠 ≤𝑡

Take 0 < 𝛿 ≤ 𝑇 so that 𝐿 𝑎 𝑘 (𝛿)𝑙 (𝛿) < 1. The above inequality implies ∥ℎ𝑡 ∥ = 0 and hence 𝑢 𝑡 = 𝑣 𝑡 for 0 ≤ 𝑡 ≤ 𝛿. Then an application of Proposition 2.12 gives the □ uniqueness of the solution to (2.16).

Proposition 2.16 Let {𝜙 𝑛 } be a sequence of operators from 𝐵(𝐸) + into 𝐵(𝐸) satisfying Conditions 2.10 and 2.11 with the constants 𝐿 and 𝐿 𝑎 independent of 𝑛 ≥ 1. Suppose that lim𝑛→∞ 𝜙 𝑛 (𝑥, 𝑓 ) = 𝜙(𝑥, 𝑓 ) uniformly on 𝐸 × 𝐵 𝑎 (𝐸) + for every 𝑎 ≥ 0 and for each 𝑓𝑛 ∈ 𝐵(𝐸) + there is a unique locally bounded positive solution 𝑡 ↦→ 𝑣 𝑛 (𝑡) = 𝑣 𝑛 (𝑡, 𝑥) to the equation   𝑣𝑛 (𝑡 , 𝑥) = P 𝑥 e−𝐾𝑡 (𝛽) 𝑓 𝑛 ( 𝜉𝑡 ) − P 𝑥

∫

𝑡

 e−𝐾𝑠 (𝛽) 𝜙𝑛 ( 𝜉𝑠 , 𝑣𝑛 (𝑡 − 𝑠)) 𝐾 (d𝑠) .

(2.19)

0

If lim𝑛→∞ 𝑓𝑛 = 𝑓 in the supremum norm, then the limit lim𝑛→∞ 𝑣 𝑛 (𝑡, 𝑥) = 𝑣 𝑡 (𝑥) exists and is uniform on [0, 𝑇] × 𝐸 for every 𝑇 ≥ 0, and (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥) is a solution of (2.16). Proof Choose a sufficiently large constant 𝑎 ≥ 0 so that { 𝑓𝑛 } ⊂ 𝐵 𝑎 (𝐸) + . By Proposition 2.14 there is an increasing function 𝑡 ↦→ 𝐶 (𝑡) on [0, ∞) such that sup ∥𝑣 𝑛 (𝑠) ∥ ≤ 𝐶 (𝑡) ∥ 𝑓𝑛 ∥ ≤ 𝑎𝐶 (𝑡),

𝑡 ≥ 0.

0≤𝑠 ≤𝑡

Fix 𝑇 > 0 and let 𝑐 = 𝑎𝐶 (𝑇). For 𝜀 > 0 let 𝑁 = 𝑁 (𝜀, 𝑐) be an integer such that ∥ 𝑓𝑛 − 𝑓 ∥ ≤ 𝜀 and ∥𝜙 𝑛 (·, ℎ) − 𝜙(·, ℎ) ∥ ≤ 𝜀 for 𝑛 ≥ 𝑁 and ℎ ∈ 𝐵𝑐 (𝐸) + . Let 𝑙 (𝑡) be as in the proof of Proposition 2.14 and let 𝐻𝑡 (𝑛1 , 𝑛2 ) = sup ∥𝑣 𝑛2 (𝑠) − 𝑣 𝑛1 (𝑠) ∥. 0≤𝑠 ≤𝑡

By (2.19) and Condition 2.11 we have 𝐻𝑡 (𝑛1 , 𝑛2 ) ≤ 2𝑙 (𝑡) [1 + 𝑘 (𝑡)]𝜀 + 𝐿 𝑐 𝑘 (𝑡)𝑙 (𝑡)𝐻𝑡 (𝑛1 , 𝑛2 ) for 0 ≤ 𝑡 ≤ 𝑇 and 𝑛1 , 𝑛2 ≥ 𝑁. Take 0 < 𝛿 ≤ 𝑇 so that 𝐿 𝑐 𝑘 (𝛿)𝑙 (𝛿) < 1. The above inequality implies 𝐻𝑡 (𝑛1 , 𝑛2 ) ≤ 2𝑙 (𝑡) [1 + 𝑘 (𝑡)] [1 − 𝐿 𝑐 𝑘 (𝑡)𝑙 (𝑡)] −1 𝜀 for 0 ≤ 𝑡 ≤ 𝛿. Then 𝑣 𝑛 (𝑡, 𝑥) converges uniformly on [0, 𝛿]×𝐸. By repeating the above arguments and applying Proposition 2.12 we see the limit lim𝑛→∞ 𝑣 𝑛 (𝑡, 𝑥) = 𝑣 𝑡 (𝑥) exists and is uniform on [0, 𝑇] × 𝐸. Then letting 𝑛 → ∞ in (2.19) we obtain (2.16).□

2.3 Dawson–Watanabe Superprocesses

41

2.3 Dawson–Watanabe Superprocesses In this section we give the construction of a general class of Dawson–Watanabe superprocesses. For this purpose we need to discuss the existence of solutions of some nonlinear integral evolution equations which define cumulant semigroups. Let 𝐸 be a Lusin topological space. Suppose that 𝜉 is a Borel right process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 and {𝐾 (𝑡) : 𝑡 ≥ 0} is a continuous admissible additive functional of 𝜉. Lemma 2.17 Suppose that 𝑏 ∈ 𝐵(𝐸) and 𝛾(𝑥, d𝑦) is a bounded kernel on 𝐸. Then for each 𝑓 ∈ 𝐵(𝐸) there is a unique locally bounded solution (𝑡, 𝑥) ↦→ 𝜋𝑡 𝑓 (𝑥) to the linear evolution equation ∫ 𝑡  𝛾(𝜉 𝑠 , 𝜋𝑡−𝑠 𝑓 )𝐾 (d𝑠) 𝜋𝑡 𝑓 (𝑥) = P 𝑥 𝑓 (𝜉𝑡 ) + P 𝑥 0 ∫ 𝑡  − P𝑥 (2.20) 𝑏(𝜉 𝑠 )𝜋𝑡−𝑠 𝑓 (𝜉 𝑠 )𝐾 (d𝑠) , 0

which defines a locally bounded semigroup (𝜋𝑡 )𝑡 ≥0 of kernels on 𝐸. Proof Let 𝑏 + = 0 ∨ 𝑏 and 𝑏 − = 0 ∨ (−𝑏). By Proposition A.42 there is a unique locally bounded solution (𝑡, 𝑥) ↦→ 𝜋𝑡 𝑓 (𝑥) to the equation  ∫ 𝑡  −𝐾 (𝑏+ )  −𝐾𝑠 (𝑏+ ) 𝑡 𝜋𝑡 𝑓 (𝑥) = P 𝑥 e e 𝛾(𝜉 𝑠 , 𝜋𝑡−𝑠 𝑓 )𝐾 (d𝑠) 𝑓 (𝜉𝑡 ) + P 𝑥 0  ∫ 𝑡 + + P𝑥 e−𝐾𝑠 (𝑏 ) 𝑏 − (𝜉 𝑠 )𝜋𝑡−𝑠 𝑓 (𝜉 𝑠 )𝐾 (d𝑠) , 0

which defines a locally bounded semigroup (𝜋𝑡 )𝑡 ≥0 of kernels on 𝐸. By Proposition 2.9 the above equation is equivalent to (2.20). □ Suppose that 𝜌(𝑥, d𝑦) is a bounded kernel on 𝐸 and 𝜈(1)𝑅(𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ . We consider a function 𝑏 ∈ 𝐵(𝐸) and an operator 𝑓 ↦→ 𝜓(·, 𝑓 ) on 𝐵(𝐸) + with the representation ∫
𝜓(𝑥, 𝑓 ) = 𝜌(𝑥, 𝑓 ) + 1 − e−𝜈 ( 𝑓 ) 𝑅(𝑥, d𝜈). (2.21) 𝑀 (𝐸) ◦

From 𝜌(𝑥, d𝑦) and 𝑅(𝑥, d𝜈) we can define the bounded kernel 𝛾0 (𝑥, d𝑦) on 𝐸 by ∫ 𝜈(d𝑦)𝑅(𝑥, d𝜈). 𝛾0 (𝑥, d𝑦) = 𝜌(𝑥, d𝑦) + (2.22) 𝑀 (𝐸) ◦

Let 𝛽 ≥ 0 be a constant such that 𝑏(𝑥) ≤ 𝛽 for all 𝑥 ∈ 𝐸. For fixed 𝑓 ∈ 𝐵(𝐸) + set 𝑢 0 (𝑡, 𝑥) = 0 and define 𝑢 𝑛 (𝑡, 𝑥) = 𝑢 𝑛 (𝑡, 𝑥, 𝑓 ) inductively by

42

2 Measure-Valued Branching Processes −𝐾𝑡 (𝛽)

𝑢 𝑛+1 (𝑡, 𝑥) = P 𝑥 [e ∫ + P𝑥

∫



𝑡 −𝐾𝑠 (𝛽)

𝜓(𝜉 𝑠 , 𝑢 𝑛 (𝑡 − 𝑠))𝐾 (d𝑠)  𝑡 −𝐾𝑠 (𝛽) [𝛽 − 𝑏(𝜉 𝑠 )]𝑢 𝑛 (𝑡 − 𝑠, 𝜉 𝑠 )𝐾 (d𝑠) . (2.23) e 𝑓 (𝜉𝑡 )] + P 𝑥

e

0

0

Proposition 2.18 For every 𝑓 ∈ 𝐵(𝐸) + there is a unique locally bounded positive solution (𝑡, 𝑥) ↦→ 𝑢 𝑡 (𝑥, 𝑓 ) to the evolution equation ∫ 𝑡  [𝜓(𝜉 𝑠 , 𝑢 𝑡−𝑠 ) − 𝑏(𝜉 𝑠 )𝑢 𝑡−𝑠 (𝜉 𝑠 )]𝐾 (d𝑠) . (2.24) 𝑢 𝑡 (𝑥) = P 𝑥 [ 𝑓 (𝜉𝑡 )] + P 𝑥 0

Moreover, we have 𝜋𝑡 𝑓 (𝑥) ≥ 𝑢 𝑡 (𝑥, 𝑓 ) = ↑lim𝑛→∞ 𝑢 𝑛 (𝑡, 𝑥, 𝑓 ) for all 𝑡 ≥ 0 and 𝑥 ∈ 𝐸, where (𝜋𝑡 )𝑡 ≥0 is the semigroup defined by (2.20) with 𝛾 = 𝛾0 given by (2.22).

Proof The operator 𝑓 ↦→ 𝜓(·, 𝑓 ) − 𝑏 𝑓 clearly satisfies Condition 2.11 with 𝐿 𝑎 = ∥𝑏∥ + ∥𝛾0 (·, 1) ∥ for all 𝑎 ≥ 0. By Proposition 2.15 there is at most one locally bounded positive solution to (2.24). We next claim that 0 ≤ 𝑢 𝑛−1 (𝑡, 𝑥, 𝑓 ) ≤ 𝑢 𝑛 (𝑡, 𝑥, 𝑓 ) ≤ 𝜋𝑡 𝑓 (𝑥),

𝑡 ≥ 0, 𝑥 ∈ 𝐸

(2.25)

for every 𝑛 ≥ 1. By Proposition 2.9 we can also define (𝑡, 𝑥) ↦→ 𝜋𝑡 𝑓 (𝑥) by the evolution equation ∫ 𝑡  −𝐾𝑠 (𝛽) −𝐾𝑡 (𝛽) 𝛾0 (𝜉 𝑠 , 𝜋𝑡−𝑠 𝑓 )𝐾 (d𝑠) 𝜋𝑡 𝑓 (𝑥) = P 𝑥 [e e 𝑓 (𝜉𝑡 )] + P 𝑥 0  ∫ 𝑡 + P𝑥 e−𝐾𝑠 (𝛽) [𝛽 − 𝑏(𝜉 𝑠 )]𝜋𝑡−𝑠 𝑓 (𝜉 𝑠 )𝐾 (d𝑠) . (2.26) 0

Then for 𝑛 = 1 the inequalities in (2.25) are trivial. Suppose they are true for some 𝑛 ≥ 1. By the monotonicity of the operator 𝑓 ↦→ 𝜓(·, 𝑓 ) + (𝛽 − 𝑏) 𝑓 we have 0 ≤ 𝑢 𝑛 (𝑡, 𝑥, 𝑓 ) ≤ 𝑢 𝑛+1 (𝑡, 𝑥, 𝑓 ) ≤ 𝑣(𝑡, 𝑥, 𝑓 ), where  ∫ 𝑡 e−𝐾𝑠 (𝛽) 𝜓(𝜉 𝑠 , 𝜋𝑡−𝑠 𝑓 )𝐾 (d𝑠) 𝑣(𝑡, 𝑥, 𝑓 ) = P 𝑥 [e−𝐾𝑡 (𝛽) 𝑓 (𝜉𝑡 )] + P 𝑥 0  ∫ 𝑡 −𝐾𝑠 (𝛽) (2.27) [𝛽 − 𝑏(𝜉 𝑠 )]𝜋𝑡−𝑠 𝑓 (𝜉 𝑠 )𝐾 (d𝑠) . + P𝑥 e 0

In view of (2.26) and (2.27) we have 𝑣(𝑡, 𝑥, 𝑓 ) ≤ 𝜋𝑡 𝑓 (𝑥). Then (2.25) holds for all 𝑛 ≥ 1. Let 𝑢 𝑡 (𝑥, 𝑓 ) = ↑lim𝑛→∞ 𝑢 𝑛 (𝑡, 𝑥, 𝑓 ). From (2.23) we see that (𝑡, 𝑥) ↦→ 𝑢 𝑡 (𝑥, 𝑓 ) is a locally bounded positive solution of

2.3 Dawson–Watanabe Superprocesses

43

∫

𝑡

e−𝐾𝑠 (𝛽) 𝜓(𝜉 𝑠 , 𝑢 𝑡−𝑠 )𝐾 (d𝑠) 𝑢 𝑡 (𝑥) = P 𝑥 [e−𝐾𝑡 (𝛽) 𝑓 (𝜉𝑡 )] + P 𝑥 0 ∫ 𝑡  −𝐾𝑠 (𝛽) + P𝑥 e [𝛽 − 𝑏(𝜉 𝑠 )]𝑢 𝑡−𝑠 (𝜉 𝑠 )𝐾 (d𝑠) ,



0

which is equivalent to (2.24) by Proposition 2.9.

□

Proposition 2.19 In the case where 𝐾 (d𝑠) = d𝑠 is the Lebesgue measure, we have 𝑢 𝑡 (𝑥, 𝑓 ) = ↑lim𝑛→∞ 𝑢 𝑛 (𝑡, 𝑥, 𝑓 ) uniformly on [0, 𝑇] × 𝐸 × 𝐵 𝑎 (𝐸) + for every 𝑇 ≥ 0 and 𝑎 ≥ 0. Proof Let 𝐷 𝑛 (𝑡) = sup0≤𝑠 ≤𝑡 ∥𝑢 𝑛 (𝑠) − 𝑢 𝑛−1 (𝑠) ∥. In the present case, we can rewrite (2.23) as ∫ 𝑡 ∫ 𝜓(𝑦, 𝑢 𝑛 (𝑠))𝑃𝑡−𝑠 (𝑥, d𝑦) 𝑢 𝑛+1 (𝑡, 𝑥) = e−𝛽𝑡 𝑃𝑡 𝑓 (𝑥) + e−𝛽 (𝑡−𝑠) d𝑠 𝐸 0∫ ∫ 𝑡 e−𝛽 (𝑡−𝑠) d𝑠 [𝛽 − 𝑏(𝑦)]𝑢 𝑛 (𝑠, 𝑦)𝑃𝑡−𝑠 (𝑥, d𝑦). + 0

𝐸

From this it is easy to get ∫

𝑡

𝐷 𝑛 (𝑡) ≤ (𝛽 + ∥𝑏∥ + ∥𝛾0 (·, 1) ∥) ≤ (𝛽 + ∥𝑏∥ + ∥𝛾0 (·, 1) ∥) 2

𝐷 𝑛−1 (𝑠1 )d𝑠1 ∫ 𝑠1 d𝑠1 𝐷 𝑛−2 (𝑠2 )d𝑠2

∫0

𝑡

0

≤ ···

0

∫ ≤ (𝛽 + ∥𝑏∥ + ∥𝛾0 (·, 1) ∥)

𝑛−1

1 (𝛽 + ∥𝑏∥ + ∥𝛾0 (·, 1) ∥) (𝑛 − 1)!

∫

𝑠1

0 𝑛−1 𝑛−1

𝑡

𝑠𝑛−2

∥ 𝑓 ∥d𝑠 𝑛−1

···

d𝑠1 0

≤

∫

𝑡

0

∥ 𝑓 ∥,

and hence 𝐷 (𝑡) :=

∞ ∑︁

 𝐷 𝑛 (𝑡) ≤ ∥ 𝑓 ∥exp (𝛽 + ∥𝑏∥ + ∥𝛾0 (·, 1) ∥)𝑡 < ∞.

𝑛=1

Then lim𝑛→∞ 𝑢 𝑛 (𝑡, 𝑥, 𝑓 ) = 𝑢 𝑡 (𝑥, 𝑓 ) uniformly on [0, 𝑇] × 𝐸 × 𝐵 𝑎 (𝐸) + .

□

Now we consider a more general operator 𝑓 ↦→ 𝜙(·, 𝑓 ) as follows. Let 𝑏 ∈ 𝐵(𝐸) and 𝑐 ∈ 𝐵(𝐸) + . Let 𝜂(𝑥, d𝑦) be a bounded kernel on 𝐸 and 𝐻 (𝑥, d𝜈) a 𝜎-finite kernel from 𝐸 to 𝑀 (𝐸) ◦ . Suppose that ∫   sup 𝜈(1) ∧ 𝜈(1) 2 + 𝜈 𝑥 (1) 𝐻 (𝑥, d𝜈) < ∞, (2.28) 𝑥 ∈𝐸

𝑀 (𝐸) ◦

where 𝜈 𝑥 (d𝑦) denotes the restriction of 𝜈(d𝑦) to 𝐸 \ {𝑥}. For 𝑥 ∈ 𝐸 and 𝑓 ∈ 𝐵(𝐸) + write

44

2 Measure-Valued Branching Processes

𝜙(𝑥, 𝑓 ) = 𝑏(𝑥) 𝑓 (𝑥) + 𝑐(𝑥) 𝑓 (𝑥) 2 − 𝜂(𝑥, 𝑓 ) +

∫ 𝐾 (𝑥, 𝜈, 𝑓 )𝐻 (𝑥, d𝜈), (2.29) 𝑀 (𝐸) ◦

where 𝐾 (𝑥, 𝜈, 𝑓 ) = e−𝜈 ( 𝑓 ) − 1 + 𝜈({𝑥}) 𝑓 (𝑥). By Taylor’s expansion we have 1 𝐾 (𝑥, 𝜈, 𝑓 ) = −𝜈 𝑥 ( 𝑓 ) + e−𝜃 𝜈( 𝑓 ) 2 , 2 where 0 ≤ 𝜃 ≤ 𝜈( 𝑓 ). Observe also that |𝐾 (𝑥, 𝜈, 𝑓 )| ≤ 𝜈( 𝑓 ) + 𝜈({𝑥}) 𝑓 (𝑥). Then the last integral on the right-hand side of (2.29) is bounded on 𝐸 × 𝐵 𝑎 (𝐸) + for every 𝑎 ≥ 0. Moreover, we can rewrite (2.29) as ∫ 𝜙(𝑥, 𝑓 ) = 𝑏(𝑥) 𝑓 (𝑥) + 𝑐(𝑥) 𝑓 (𝑥) 2 − 𝛾(𝑥, 𝑓 ) + 𝐾 (𝜈, 𝑓 )𝐻 (𝑥, d𝜈), (2.30) 𝑀 (𝐸) ◦

where 𝐾 (𝜈, 𝑓 ) = e−𝜈 ( 𝑓 ) − 1 + 𝜈( 𝑓 ) and ∫ 𝛾(𝑥, d𝑦) = 𝜂(𝑥, d𝑦) +

𝜈 𝑥 (d𝑦)𝐻 (𝑥, d𝜈).

(2.31)

𝑀 (𝐸) ◦

For each integer 𝑛 ≥ 1 define ∫ 𝜙 𝑛 (𝑥, 𝑓 ) = 𝑏(𝑥) 𝑓 (𝑥) + 2𝑛𝑐(𝑥) 𝑓 (𝑥) + 𝜈( 𝑓 )ℎ 𝑛 (𝜈)𝐻 (𝑥, d𝜈) 𝑀 (𝐸) ◦ ∫ − 𝑓 (𝑦)𝛾(𝑥, d𝑦) − 2𝑛2 𝑐(𝑥) (1 − e− 𝑓 ( 𝑥)/𝑛 ) ∫𝐸
− 1 − e−𝜈 ( 𝑓 ) ℎ 𝑛 (𝜈)𝐻 (𝑥, d𝜈), (2.32) 𝑀 (𝐸) ◦

where ℎ 𝑛 (𝜈) = 1 ∧ (𝑛𝜈(1)). It is easy to see 𝜙 𝑛 (𝑥, 𝑓 ) → 𝜙(𝑥, 𝑓 ) increasingly as 𝑛 → ∞. For 𝑛 ≥ 1 and 𝑓 ∈ 𝐵(𝐸) + we consider the equation ∫ 𝑡  𝑣(𝑡, 𝑥) = P 𝑥 [ 𝑓 (𝜉𝑡 )] − P 𝑥 𝜙 𝑛 (𝜉 𝑠 , 𝑣(𝑡 − 𝑠))𝐾 (d𝑠) . (2.33) 0

This is clearly a special case of (2.24). By Proposition 2.18 there is a unique locally bounded positive solution (𝑡, 𝑥) ↦→ 𝑣 𝑛 (𝑡, 𝑥, 𝑓 ) to (2.33).

2.3 Dawson–Watanabe Superprocesses

45

Proposition 2.20 Suppose that 𝜙 and 𝛾 are defined respectively by (2.30) and (2.31). Let (𝜋𝑡 )𝑡 ≥0 be defined by (2.20). Then for every 𝑓 ∈ 𝐵(𝐸) + there is a unique locally bounded positive solution (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥, 𝑓 ) to  ∫ 𝑡 𝜙(𝜉 𝑠 , 𝑣 𝑡−𝑠 )𝐾 (d𝑠) , 𝑡 ≥ 0, 𝑥 ∈ 𝐸 . (2.34) 𝑣 𝑡 (𝑥) = P 𝑥 𝑓 (𝜉𝑡 ) − P 𝑥 0

Moreover, we have 𝜋𝑡 𝑓 (𝑥) ≥ 𝑣 𝑡 (𝑥, 𝑓 ) = ↓lim𝑛→∞ 𝑣 𝑛 (𝑡, 𝑥, 𝑓 ) for 𝑡 ≥ 0 and 𝑥 ∈ 𝐸. Proof Since 𝜙 𝑛 (𝑥, 𝑓 ) is increasing in 𝑛 ≥ 1 and 𝑓 ∈ 𝐵(𝐸) + , by Proposition 2.18 we see 𝑣 𝑛 (𝑡, 𝑥, 𝑓 ) is decreasing in 𝑛 ≥ 1. Let 𝑣 𝑡 (𝑥, 𝑓 ) = lim𝑛→∞ 𝑣 𝑛 (𝑡, 𝑥, 𝑓 ) ≤ 𝜋𝑡 (𝑥, 𝑓 ). In view of (2.32) and (2.33), we conclude by dominated convergence that (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥, 𝑓 ) is a locally bounded positive solution of (2.34). For 𝑎 ≥ 0 and 𝑓 , 𝑔 ∈ 𝐵 𝑎 (𝐸) + we can use (2.30) to see |𝜙(𝑥, 𝑓 ) − 𝜙(𝑥, 𝑔)| ≤ (∥𝑏∥ + 2𝑎∥𝑐∥) ∥ 𝑓 − 𝑔∥ + 𝛾(𝑥, 1) ∥ 𝑓 − 𝑔∥ ∫ + |𝜈( 𝑓 − 𝑔) + e−𝜈 ( 𝑓 ) − e−𝜈 (𝑔) |𝐻 (𝑥, d𝜈). 𝑀 (𝐸) ◦

By the mean-value theorem we have 𝜈( 𝑓 − 𝑔) + e−𝜈 ( 𝑓 ) − e−𝜈 (𝑔) = 𝜈( 𝑓 − 𝑔) (1 − e−𝜃 ), where 𝜈( 𝑓 ∧ 𝑔) ≤ 𝜃 ≤ 𝜈( 𝑓 ∨ 𝑔) ≤ 𝑎𝜈(1). It follows that |𝜈( 𝑓 − 𝑔) + e−𝜈 ( 𝑓 ) − e−𝜈 (𝑔) | ≤ ∥ 𝑓 − 𝑔∥ (𝜈(1) ∧ 𝑎𝜈(1) 2 ). Then 𝑓 ↦→ 𝜙(·, 𝑓 ) satisfies Condition 2.11 for some constant 𝐿 𝑎 ≥ 0 and the □ uniqueness of the solution of (2.34) follows by Proposition 2.15. Theorem 2.21 Let 𝜙 be given by (2.29) or (2.30). For every 𝑓 ∈ 𝐵(𝐸) + let (𝑡, 𝑥) ↦→ 𝑉𝑡 𝑓 (𝑥) denote the unique locally bounded positive solution of (2.34). Then the operators (𝑉𝑡 )𝑡 ≥0 constitute a cumulant semigroup. Proof By (2.23) and Theorem 1.38 one checks inductively 𝑢 𝑛 (𝑡, 𝑥, ·) ∈ ℐ(𝐸) for each 𝑛 ≥ 1. Now Corollary 1.35 and Propositions 2.18 and 2.20 imply first 𝑢 𝑡 (𝑥, ·) ∈ ℐ(𝐸) for the solution of (2.24), and then 𝑣 𝑡 (𝑥, ·) ∈ ℐ(𝐸) for the solution of (2.34). The semigroup property of (𝑉𝑡 )𝑡 ≥0 follows from Corollary 2.13. □ Let 𝜙 be given by (2.29) or (2.30) and let (𝑉𝑡 )𝑡 ≥0 be the cumulant semigroup defined by (2.34). Then we can define a Markov transition semigroup (𝑄 𝑡 )𝑡 ≥0 on 𝑀 (𝐸) by ∫ 𝑓 ∈ 𝐵(𝐸) + . e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈) = exp{−𝜇(𝑉𝑡 𝑓 )}, (2.35) 𝑀 (𝐸)

If 𝑋 is a Markov process in 𝑀 (𝐸) with transition semigroup (𝑄 𝑡 )𝑡 ≥0 , we call it a Dawson–Watanabe superprocess with parameters (𝜉, 𝐾, 𝜙), or simply a (𝜉, 𝐾, 𝜙)superprocess, where 𝜉 is the spatial motion, 𝐾 is the killing functional or killing

46

2 Measure-Valued Branching Processes

density, and 𝜙 is the branching mechanism. If 𝐾 (d𝑠) = d𝑠 is the Lebesgue measure, we call 𝑋 a (𝜉, 𝜙)-superprocess. In this case, we can rewrite (2.34) as ∫ 𝑡 ∫ d𝑠 𝜙(𝑦, 𝑣 𝑠 )𝑃𝑡−𝑠 (𝑥, d𝑦), 𝑥 ∈ 𝐸, 𝑡 ≥ 0. (2.36) 𝑣 𝑡 (𝑥) = 𝑃𝑡 𝑓 (𝑥) − 0

𝐸

We say the branching mechanism is spatially constant if 𝑓 ↦→ 𝜙(·, 𝑓 ) maps constant functions to constant functions. In Chapter 4 we shall give some intuitive interpretations of the superprocesses in terms of limit theorems of branching particle systems. Theorem 2.22 A realization {𝑋𝑡 : 𝑡 ≥ 0} of the (𝜉, 𝐾, 𝜙)-superprocess is right continuous in probability. Proof Let 𝑓 ∈ 𝐶 (𝐸) + . Since 𝜉 is right continuous, the map 𝑡 ↦→ 𝑃𝑡 𝑓 (𝑥) is right continuous for every 𝑥 ∈ 𝐸, so (2.34) implies lim𝑡→0 𝑉𝑡 𝑓 (𝑥) = 𝑓 (𝑥). From (2.35) we get ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈) = exp{−𝜇( 𝑓 )}. lim 𝑡→0

𝑀 (𝐸)

Then we have lim𝑡→0 𝑄 𝑡 (𝜇, ·) = 𝛿 𝜇 weakly. For any 𝜀 > 0 let 𝐵(𝜇, 𝜀) 𝑐 = {𝜈 ∈ 𝑀 (𝐸) : 𝜌(𝜈, 𝜇) > 𝜀}, where 𝜌 is the metric on 𝑀 (𝐸) defined by (1.3). Then we infer lim𝑡→0 𝑄 𝑡 (𝜇, 𝐵(𝜇, 𝜀) 𝑐 ) = 0. Using the Markov property of 𝑋 and the dominated convergence theorem we get lim P{𝜌(𝑋𝑡 , 𝑋𝑟 ) > 𝜀} = lim P{𝑄 𝑡−𝑟 (𝑋𝑟 , 𝐵(𝑋𝑟 , 𝜀) 𝑐 )} = 0 𝑡 ↓𝑟

𝑡 ↓𝑟

for every 𝑟 ≥ 0. Therefore 𝑡 ↦→ 𝑋𝑡 is right continuous in probability.

□

Let us consider the special case of a (𝜉, 𝜙)-superprocess. Given a function 𝑏 ∈ 𝐵(𝐸), we define a locally bounded semigroup of Borel kernels (𝑃𝑡𝑏 )𝑡 ≥0 on 𝐸 by the following Feynman–Kac formula: i h ∫𝑡 𝑥 ∈ 𝐸, 𝑓 ∈ 𝐵(𝐸). (2.37) 𝑃𝑡𝑏 𝑓 (𝑥) = P 𝑥 e− 0 𝑏 ( 𝜉𝑠 )d𝑠 𝑓 (𝜉𝑡 ) , For the (𝜉, 𝜙)-superprocess we can rewrite (2.20) as ∫ 𝑡 𝜋𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) + 𝑃𝑡−𝑠 (𝛾 − 𝑏)𝜋 𝑠 𝑓 (𝑥)d𝑠,

𝑡 ≥ 0, 𝑥 ∈ 𝐸,

(2.38)

0

where 𝛾 is defined by (2.31). By Proposition A.50, the above equation is equivalent to ∫ 𝑡 𝑏 𝑏 𝛾𝜋 𝑠 𝑓 (𝑥)d𝑠, 𝜋𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) + 𝑃𝑡−𝑠 (2.39) 𝑡 ≥ 0, 𝑥 ∈ 𝐸 . 0

2.3 Dawson–Watanabe Superprocesses

47

By Proposition A.42 we have 𝜋𝑡 𝑓 (𝑥) = 𝑃𝑡𝑏 𝑓 (𝑥) +

∞ ∫ ∑︁

∫

𝑡

0

𝑛=1

∫

𝑠1

𝑠𝑛−1

𝑏 𝛾𝑃𝑠𝑏1 −𝑠2 𝑃𝑡−𝑠 1 0 · · · 𝛾𝑃𝑠𝑏𝑛−1 −𝑠𝑛 𝛾𝑃𝑠𝑏𝑛 𝑓 (𝑥)d𝑠 𝑛 .

d𝑠2 · · ·

d𝑠1 0

(2.40) +

Let 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)] and let 𝑐+0 = 0 ∨ 𝑐 0 . Then ∥𝜋𝑡 ∥ ≤ e𝑐0 𝑡 ≤ e𝑐0 𝑡 for 𝑡 ≥ 0 by Theorem A.53.

Theorem 2.23 Suppose that 𝜙 and 𝛾 are defined respectively by (2.30) and (2.31). Let (𝜋𝑡 )𝑡 ≥0 be defined by (2.38). Then (2.36) is equivalent to the evolution equation ∫ 𝑡 ∫ 𝜙0 (𝑦, 𝑣 𝑠 )𝜋𝑡−𝑠 (𝑥, d𝑦), 𝑥 ∈ 𝐸, 𝑡 ≥ 0, (2.41) 𝑣 𝑡 (𝑥) = 𝜋𝑡 𝑓 (𝑥) − d𝑠 0

𝐸

where 𝜙0 (𝑦, 𝑓 ) = 𝑐(𝑦) 𝑓 (𝑦) 2 +

∫ 𝐾 (𝜈, 𝑓 )𝐻 (𝑦, d𝜈).

(2.42)

𝑀 (𝐸) ◦

Proof We first show (2.36) implies (2.41). By applying Proposition 2.9 to (2.36) we have ∫ 𝑡   𝑏 𝑣 𝑡 (𝑥) = 𝑃𝑡𝑏 𝑓 (𝑥) − 𝜙(𝑣 𝑠 ) − 𝑏𝑣 𝑠 (𝑥)d𝑠. 𝑃𝑡−𝑠 0

This combined with (2.39) implies ∫ ∫ 𝑡 𝑏 𝑣 𝑡 (𝑥) = 𝜋𝑡 𝑓 (𝑥) − 𝑃𝑡−𝑠 𝜙0 (𝑣 𝑠 ) (𝑥)d𝑠 + 0

𝑡 𝑏 𝛾(𝑣 𝑠 − 𝜋 𝑠 𝑓 ) (𝑥)d𝑠. 𝑃𝑡−𝑠

0

Then we use the above relation inductively to see ∫ 𝑡 𝑏 𝜙 (𝑣 𝑠1 ) (𝑥)d𝑠1 + 𝑤 𝑛 (𝑡, 𝑥) 𝑃𝑡−𝑠 𝑣 𝑡 (𝑥) = 𝜋𝑡 𝑓 (𝑥) − 1 0 0 ∫ 𝑠𝑖−1 ∫ ∫ 𝑛 𝑡 𝑠1 ∑︁ 𝑏 d𝑠1 𝛾𝑃𝑠𝑏1 −𝑠2 − d𝑠2 · · · 𝑃𝑡−𝑠 1 𝑖=2

0

0

0

· · · 𝛾𝑃𝑠𝑏𝑖−1 −𝑠𝑖 𝑔𝑠𝑖 (𝑥)d𝑠𝑖 , where 𝑔𝑠𝑖 (𝑥) = 𝜙0 (𝑥, 𝑣 𝑠𝑖 ) and ∫

∫

𝑡

d𝑠1 · · ·

𝑤𝑛 (𝑡 , 𝑥) = 0

0

𝑠𝑛−1 𝑏 𝑃𝑡−𝑠 𝛾 · · · 𝑃𝑠𝑏𝑛−1 −𝑠𝑛 𝛾 (𝑣𝑠𝑛 − 𝜋𝑠𝑛 𝑓 ) ( 𝑥)d𝑠𝑛 . 1

Since 0 ≤ 𝑃𝑠𝑏 𝑓 (𝑥) ≤ 𝜋 𝑠 𝑓 (𝑥) and 0 ≤ 𝑣 𝑠 (𝑥) ≤ 𝜋 𝑠 𝑓 (𝑥), we have

(2.43)

48

2 Measure-Valued Branching Processes 𝑛 𝑐0+ 𝑡

∫

∥𝑤 𝑛 (𝑡, ·) ∥ ≤ ∥ 𝑓 ∥ ∥𝛾(·, 1) ∥ e

+

≤ ∥ 𝑓 ∥ ∥𝛾(·, 1) ∥ 𝑛 e𝑐0 𝑡

∫

𝑡

𝑠𝑛−1

d𝑠2 · · ·

d𝑠1 0

0 𝑡𝑛

∫

𝑠1

d𝑠 𝑛 0

. 𝑛!

Then letting 𝑛 → ∞ in (2.43) and using (2.40) we obtain (2.41). The uniqueness of the solution to (2.41) follows from Gronwall’s inequality by standard arguments. Then the two equations are equivalent. □

2.4 Examples of Superprocesses The (𝜉, 𝐾, 𝜙)- and (𝜉, 𝜙)-superprocesses we have constructed are quite wide. From these one can derive the existence of various special classes of superprocesses. Some special cases of the parameters are discussed in the following examples. Example 2.5 Let | · | and ⟨·, ·⟩ denote respectively the Euclidean norm and inner product of R𝑑 . For each 1 ≤ 𝑖 ≤ 𝑑 suppose that 𝜆 ↦→ 𝜙𝑖 (𝜆) is a function on R+𝑑 with the representation ∫
2 𝜙𝑖 (𝜆) = 𝑏 𝑖 𝜆𝑖 + 𝑐 𝑖 𝜆𝑖 − ⟨𝜂𝑖 , 𝜆⟩ + e− ⟨𝑧,𝜆⟩ − 1 + 𝑧𝑖 𝜆𝑖 𝐻𝑖 (d𝑧), (2.44) R+𝑑 \{0}

where 𝑐 𝑖 ≥ 0 and 𝑏 𝑖 are constants, 𝜂𝑖 ∈ R+𝑑 is a vector, and 𝐻𝑖 (d𝑧) is a 𝜎-finite measure on R+𝑑 \ {0} such that ∫  ∑︁  |𝑧| ∧ |𝑧| 2 + 𝑧 𝑗 𝐻𝑖 (d𝑧) < ∞. R+𝑑 \{0}

𝑗≠𝑖

By Proposition 2.20 and Theorem 2.21 for any 𝜆 ∈ R+𝑑 there is a unique locally bounded vector-valued solution 𝑡 ↦→ 𝑣(𝑡, 𝜆) ∈ R+𝑑 to the evolution equation system ∫ 𝑡 𝑣 𝑖 (𝑡, 𝜆) = 𝜆𝑖 − 𝜙𝑖 (𝑣(𝑠, 𝜆))d𝑠, 𝑡 ≥ 0, 𝑖 = 1, . . . , 𝑑, (2.45) 0

and there is a Feller transition semigroup (𝑄 𝑡 )𝑡 ≥0 on R+𝑑 defined by ∫ e− ⟨𝑦,𝜆⟩ 𝑄 𝑡 (𝑥, d𝑦) = e− ⟨𝑥,𝑣 (𝑡 ,𝜆) ⟩ , 𝜆, 𝑥 ∈ R+𝑑 .

(2.46)

R+𝑑

From (2.45) we see that 𝑡 ↦→ 𝑣 𝑖 (𝑡, 𝜆) is continuously differentiable. Then we can rewrite the equation into the equivalent differential form d𝑣 𝑖 (𝑡, 𝜆) = −𝜙𝑖 (𝑣(𝑡, 𝜆)), d𝑡

𝑣 𝑖 (0, 𝜆) = 𝜆𝑖 ,

𝑖 = 1, . . . , 𝑑.

2.4 Examples of Superprocesses

49

A Markov process in R+𝑑 with transition semigroup (𝑄 𝑡 )𝑡 ≥0 given by (2.46) is called a continuous-state branching process (CB-process). Example 2.6 By a super-Brownian motion we mean a superprocess with Brownian motion as underlying spatial motion. A particular super-Brownian motion is described as follows. Let 𝜉 be a standard Brownian motion in R. It is well known that 𝜉 has a continuous local time {2𝑙 (𝑡, 𝑦) : 𝑡 ≥ 0, 𝑦 ∈ R}, that is, ∫ 𝑡 ∫ (2.47) 𝑡 ≥ 0, 𝐵 ∈ ℬ(R); 1 𝐵 (𝜉 𝑠 )d𝑠 = 2𝑙 (𝑡, 𝑦)d𝑦, 0

𝐵

see, e.g., Ikeda and Watanabe (1989, p. 113). Let 𝜌 ∈ 𝑀 (R) and define the continuous additive functional 𝑡 ↦→ 𝐾 (𝑡) by ∫ 𝑡 ≥ 0. 2𝑙 (𝑡, 𝑦) 𝜌(d𝑦), 𝐾 (𝑡) = R

Then we have ∫

∫ P 𝑥 [𝐾 (𝑡)] =

𝑡

𝜌(d𝑦) 0

R

√ 2𝑡 𝑔𝑠 (𝑦 − 𝑥)d𝑠 ≤ √ 𝜌(R), 𝜋

where 𝑔𝑡 (𝑧) = √

1 2𝜋𝑡

exp{−𝑧2 /2𝑡},

𝑡 > 0, 𝑧 ∈ R.

(2.48)

Thus 𝑡 ↦→ 𝐾 (𝑡) is admissible. In this case, we can rewrite (2.34) as ∫ 𝑡 ∫ 𝑣 𝑡 (𝑥) = 𝑃𝑡 𝑓 (𝑥) − d𝑠 𝜙(𝑦, 𝑣 𝑡−𝑠 )𝑔𝑠 (𝑦 − 𝑥) 𝜌(d𝑦). 0

R

The corresponding (𝜉, 𝐾, 𝜙)-superprocess is called a catalytic super-Brownian motion with catalyst measure 𝜌(d𝑦). Example 2.7 Let 𝑏 ∈ 𝐵(𝐸) and 𝑐 ∈ 𝐵(𝐸) + . Let (𝑧 ∧ 𝑧2 )𝑚(𝑥, d𝑧) be a bounded kernel from 𝐸 to (0, ∞). We define a Borel function (𝑥, 𝜆) ↦→ 𝜙(𝑥, 𝜆) on 𝐸 × [0, ∞) by ∫ ∞ 𝜙(𝑥, 𝜆) = 𝑏(𝑥)𝜆 + 𝑐(𝑥)𝜆2 + (e−𝑧𝜆 − 1 + 𝑧𝜆)𝑚(𝑥, d𝑧). (2.49) 0

Then (𝑥, 𝑓 ) ↦→ 𝜙(𝑥, 𝑓 (𝑥)) can be represented in the form (2.29) or (2.30). In this case, we say the corresponding superprocess has a local branching mechanism. If there is a 𝑐 ∈ 𝐵(𝐸) + such that 𝜙(𝑥, 𝜆) = 𝑐(𝑥)𝜆2 for all 𝑥 ∈ 𝐸 and 𝜆 ≥ 0, we say the superprocess has a binary local branching mechanism. Example 2.8 Let (𝑥, 𝑓 ) ↦→ 𝜓(𝑥, 𝑓 ) be given by (2.21) and let (𝑥, 𝜆) ↦→ 𝜙(𝑥, 𝜆) be given by (2.49). Then the operator 𝑓 ↦→ 𝜙(·, 𝑓 (·)) − 𝜓(·, 𝑓 ) can be represented in the form (2.29) or (2.30), so it defines a branching mechanism. A branching mechanism

50

2 Measure-Valued Branching Processes

of this type is said to be decomposable with local part 𝜙 and non-local part 𝜓. A superprocess with such a branching mechanism is referred to as a (𝜉, 𝐾, 𝜙, 𝜓)superprocess. In the special case of the Lebesgue killing density 𝐾 (d𝑠) = d𝑠, we call it a (𝜉, 𝜙, 𝜓)-superprocess. Of course, the expression 𝜙(·, 𝑓 (·)) − 𝜓(·, 𝑓 ) of a decomposable branching mechanism is not unique. Example 2.9 Let 𝜋(𝑥, d𝑦) be a probability kernel on 𝐸. Suppose that 𝛽 ∈ 𝐵(𝐸) + and 𝑧𝑛(𝑥, d𝑧) is a bounded kernel from 𝐸 to (0, ∞). Given the function ∫ ∞ (2.50) (1 − e−𝑧𝜆 )𝑛(𝑥, d𝑧), 𝑥 ∈ 𝐸, 𝜆 ≥ 0, 𝜁 (𝑥, 𝜆) = 𝛽(𝑥)𝜆 + 0

we can define a non-local branching mechanism by 𝜓(𝑥, 𝑓 ) = 𝜁 (𝑥, 𝜋(𝑥, 𝑓 )),

𝑥 ∈ 𝐸, 𝑓 ∈ 𝐵(𝐸) + .

(2.51)

If 𝜁 (𝑥, 𝑦, 𝜆) is given by (2.50) with 𝑥 ∈ 𝐸 replaced by (𝑥, 𝑦) ∈ 𝐸 2 , we can define another special non-local branching mechanism by ∫ 𝑥 ∈ 𝐸, 𝑓 ∈ 𝐵(𝐸) + . (2.52) 𝜁 (𝑥, 𝑦, 𝑓 (𝑦))𝜋(𝑥, d𝑦), 𝜓(𝑥, 𝑓 ) = 𝐸

Example 2.10 Let 1 < 𝛼 < 2 be a constant and let 𝜋0 be a diffuse probability measure on 𝐸. We can define a branching mechanism on 𝐸 by ∫ 𝜙(𝑥, 𝑓 ) =

1



exp{−𝑢 𝑓 (𝑥) − 𝑢 2 𝜋0 ( 𝑓 )} − 1 + 𝑢 𝑓 (𝑥)

0

In fact, it is easy to see that ∫ 𝜙(𝑥, 𝑓 ) =



 d𝑢 . 𝑢 1+𝛼

 e−𝜈 ( 𝑓 ) − 1 + 𝜈({𝑥}) 𝑓 (𝑥) 𝐻 (𝑥, d𝜈),

𝑀 (𝐸) ◦

where 𝐻 (𝑥, d𝜈) is the image of 𝑢 −1−𝛼 d𝑢 under the mapping 𝑢 ↦→ 𝑢𝛿 𝑥 + 𝑢 2 𝜋0 of (0, 1] into 𝑀 (𝐸) ◦ . This branching mechanism cannot be decomposed into local and non-local parts.

2.5 Some Moment Formulas In this section, we prove some moment formulas and give some applications. We start with a general MB-process. Suppose that 𝐸 is a Lusin topological space and (𝑄 𝑡 )𝑡 ≥0 is the transition semigroup of the process defined by (2.3) with (𝑉𝑡 )𝑡 ≥0 given by (2.5).

2.5 Some Moment Formulas

51

Proposition 2.24 The probability measure 𝑄 𝑡 (𝜇, ·) has finite first-moments for every 𝑡 ≥ 0 and 𝜇 ∈ 𝑀 (𝐸) if and only if 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ for every 𝑡 ≥ 0. In this case, we have ∫ 𝜈( 𝑓 )𝑄 𝑡 (𝜇, d𝜈) = 𝜇(𝜋𝑡 𝑓 ), 𝜇 ∈ 𝑀 (𝐸), 𝑓 ∈ 𝐵(𝐸), (2.53) 𝑀 (𝐸)

where (𝜋𝑡 )𝑡 ≥0 is a semigroup of bounded kernels on 𝐸 defined by ∫ 𝜋𝑡 𝑓 (𝑥) = 𝜆 𝑡 (𝑥, 𝑓 ) + 𝜈( 𝑓 )𝐿 𝑡 (𝑥, d𝜈), 𝑥 ∈ 𝐸, 𝑓 ∈ 𝐵(𝐸).

(2.54)

𝑀 (𝐸) ◦

Proof We first define the kernel 𝜋𝑡 (𝑥, d𝑦) on 𝐸 using (2.54) for 𝑓 ∈ 𝐵(𝐸) + and allowing infinite values for both sides. Writing 𝑣 𝑡 (𝑥, 𝑓 ) = 𝑉𝑡 𝑓 (𝑥), we have 𝜋𝑡 𝑓 (𝑥) = lim𝑛→∞ 𝑛𝑣 𝑡 (𝑥, 𝑓 /𝑛) increasingly by (2.5). From (2.3) we get ∫ 𝑛(1 − e−𝜈 ( 𝑓 /𝑛) )𝑄 𝑡 (𝜇, d𝜈) = 𝑛(1 − exp{−𝜇(𝑣 𝑡 (·, 𝑓 /𝑛))}). 𝑀 (𝐸)

Then (2.53) holds for 𝑓 ∈ 𝐵(𝐸) + by monotone convergence if infinite values are allowed. Suppose that 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ . Then 𝜋𝑡 (𝑥, d𝑦) is a bounded kernel on 𝐸, so 𝜇(𝜋𝑡 𝑓 ) < ∞ for 𝑓 ∈ 𝐵(𝐸) + and 𝜇 ∈ 𝑀 (𝐸). This implies 𝑄 𝑡 (𝜇, ·) has finite first-moments given by (2.53). Conversely, suppose that the probability measures 𝑄 𝑡 (𝜇, ·) have finite first-moments. Then 𝜇(𝜋𝑡 𝑓 ) < ∞ for 𝑓 ∈ 𝐵(𝐸) + and 𝜇 ∈ 𝑀 (𝐸), and so 𝜋𝑡 𝑓 ∈ 𝐵(𝐸) + , implying 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ . The extensions of (2.53) and (2.54) to 𝑓 ∈ 𝐵(𝐸) are immediate. The semigroup property of (𝜋𝑡 )𝑡 ≥0 follows from that of (𝑄 𝑡 )𝑡 ≥0 and the relation (2.53). □ Corollary 2.25 Suppose that 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ for every 𝑡 ≥ 0. Then for any 𝑓 , 𝑔 ∈ 𝐵(𝐸) + we have |𝑉𝑡 𝑓 (𝑥) − 𝑉𝑡 𝑔(𝑥)| ≤ 𝜋𝑡 (𝑥, | 𝑓 − 𝑔|),

𝑡 ≥ 0, 𝑥 ∈ 𝐸,

(2.55)

where (𝜋𝑡 )𝑡 ≥0 is defined by (2.54). Proof By the canonical representation (2.5) we have ∫ |𝑉𝑡 𝑓 (𝑥) − 𝑉𝑡 𝑔(𝑥)| ≤ 𝜆 𝑡 (𝑥, | 𝑓 − 𝑔|) + |e−𝜈 ( 𝑓 ) − e−𝜈 ( 𝑓 ) |𝐿 𝑡 (𝑥, d𝜈) ◦ ∫𝑀 (𝐸) ≤ 𝜆 𝑡 (𝑥, | 𝑓 − 𝑔|) + 𝜈(| 𝑓 − 𝑔|)𝐿 𝑡 (𝑥, d𝜈). 𝑀 (𝐸) ◦

Then (2.55) follows from (2.54).

□

52

2 Measure-Valued Branching Processes

Let 𝑋 be an MB-process with transition semigroup (𝑄 𝑡 )𝑡 ≥0 and let (𝜋𝑡 )𝑡 ≥0 be defined by (2.54). If 𝜋𝑡 1(𝑥) ≤ 1 for all 𝑡 ≥ 0 and 𝑥 ∈ 𝐸, we say 𝑋 is subcritical. If 𝜋𝑡 1(𝑥) = 1 for all 𝑡 ≥ 0 and 𝑥 ∈ 𝐸, we say 𝑋 is critical. If 𝜋𝑡 1(𝑥) ≥ 1 for all 𝑡 ≥ 0 and 𝑥 ∈ 𝐸, we say 𝑋 is supercritical. The meanings of the concepts are made clear by (2.53). Proposition 2.26 Suppose 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ for every 𝑡 ≥ 0. Then for 𝑡 ≥ 0, 𝜇 ∈ 𝑀 (𝐸) and ( 𝑓 , 𝑔) ∈ 𝐵(𝐸) + × 𝐵(𝐸) we have ∫ 𝑔 𝜈(𝑔)e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈) = exp{−𝜇(𝑉𝑡 𝑓 )}𝜇(𝑉𝑡 𝑓 ), (2.56) 𝑀 (𝐸)

where 𝑔

∫

𝑉𝑡 𝑓 (𝑥) = 𝜆 𝑡 (𝑥, 𝑔) +

𝜈(𝑔)e−𝜈 ( 𝑓 ) 𝐿 𝑡 (𝑥, d𝜈),

𝑥 ∈ 𝐸.

(2.57)

𝑀 (𝐸) ◦

Proof By Proposition 2.24 the left-hand side of (2.56) is finite. For any ( 𝑓 , 𝑔) ∈ 𝑔 𝐵(𝐸) + × 𝐵(𝐸) + let 𝑉𝑡 𝑓 (𝑥) = (d/d𝜃)𝑉𝑡 ( 𝑓 + 𝜃𝑔) (𝑥)| 𝜃=0+ . We get (2.56) and (2.57) by differentiating both sides of (2.3) and (2.5), respectively. The relations for 𝑔 ∈ 𝐵(𝐸) □ follow by linearity. Now let us consider the case of a Dawson–Watanabe superprocess. Let 𝜉 be a Borel right process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 and with resolvent (𝑈 𝛼 ) 𝛼>0 defined by (A.6). Let 𝑡 ↦→ 𝐾 (𝑡) be a continuous admissible additive functional of 𝜉 and let 𝜙 be a branching mechanism given by (2.29) or (2.30). Proposition 2.27 For the (𝜉, 𝐾, 𝜙)-superprocess, we have (2.53) and (2.54) with (𝜋𝑡 )𝑡 ≥0 defined by (2.20) and (2.31). Proof By the proof of Proposition 2.24 we have 𝜋𝑡 𝑓 (𝑥) = lim𝑛→∞ 𝑛𝑣 𝑡 (𝑥, 𝑓 /𝑛) increasingly for 𝑓 ∈ 𝐵(𝐸) + . Then one can see from (2.30) and (2.34) that (𝑡, 𝑥) ↦→ 𝜋𝑡 𝑓 (𝑥) is the unique locally bounded solution of (2.20) and (2.31). The extension to □ 𝑓 ∈ 𝐵(𝐸) is immediate. Corollary 2.28 For the (𝜉, 𝜙)-superprocess, we have (2.53) and (2.54) with (𝜋𝑡 )𝑡 ≥0 defined by (2.31) and (2.38). In particular, if 𝜙 is the local branching mechanism given by (2.49), the two equalities hold with 𝜋𝑡 = 𝑃𝑡𝑏 for all 𝑡 ≥ 0. If 𝑏(𝑥) ≥ 𝛾(𝑥, 1) for all 𝑥 ∈ 𝐸, then (𝜋𝑡 )𝑡 ≥0 is a Borel right semigroup by Theorem A.44. In this case, the (𝜉, 𝐾, 𝜙)-superprocess is subcritical. If (𝑃𝑡 )𝑡 ≥0 is conservative and 𝑏(𝑥) = 𝛾(𝑥, 1) for all 𝑥 ∈ 𝐸, then (𝜋𝑡 )𝑡 ≥0 is a conservative semigroup and the superprocess is critical. If (𝑃𝑡 )𝑡 ≥0 is conservative and 𝑏(𝑥) ≤ 𝛾(𝑥, 1) for all 𝑥 ∈ 𝐸, then 𝜋𝑡 1(𝑥) ≥ 1 for all 𝑡 ≥ 0 and 𝑥 ∈ 𝐸 and the (𝜉, 𝐾, 𝜙)superprocess is supercritical.

2.5 Some Moment Formulas

53

𝑔

Proposition 2.29 Let 𝑉𝑡 𝑓 (𝑥) be defined by (2.57) from the canonical representation 𝑔 of the cumulant semigroup of the (𝜉, 𝐾, 𝜙)-superprocess. Then (𝑡, 𝑥) ↦→ 𝑉𝑡 𝑓 (𝑥) is the unique locally bounded solution of ∫ 𝑡  𝑔 𝑔 𝑉𝑡 𝑓 (𝑥) = P 𝑥 𝑔(𝜉𝑡 ) − P 𝑥 (2.58) 𝜓(𝜉 𝑠 , 𝑉𝑡−𝑠 𝑓 , 𝑉𝑡−𝑠 𝑓 )𝐾 (d𝑠) 0

and ( 𝑓 , 𝑔) ↦→ 𝜓(·, 𝑓 , 𝑔) is the operator from 𝐵(𝐸) + × 𝐵(𝐸) to 𝐵(𝐸) defined by ∫ 𝜓(𝑥, 𝑓 , 𝑔) = 𝑏(𝑥)𝑔(𝑥) + 2𝑐(𝑥) 𝑓 (𝑥)𝑔(𝑥) − 𝑔(𝑦)𝛾(𝑥, d𝑦) 𝐸 ∫
+ (2.59) 𝜈(𝑔) 1 − e−𝜈 ( 𝑓 ) 𝐻 (𝑥, d𝜈). 𝑀 (𝐸) ◦

Proof Using the notation introduced in the proof of Proposition 2.26, for ( 𝑓 , 𝑔) ∈ 𝐵(𝐸) + × 𝐵(𝐸) + we get (2.58) by differentiating both sides of (2.34). For 𝑔 ∈ 𝐵(𝐸) the relation follows by linearity. For any 𝑟 ≥ 0 it is not hard to show that (2.58) holds for all 𝑡 ≥ 0 if and only if it holds for 0 ≤ 𝑡 ≤ 𝑟 and  ∫ 𝑡 𝑔 𝑔 𝑔 𝑉𝑟+𝑡 𝑓 (𝑥) = P 𝑥 𝑉𝑟 𝑓 (𝜉𝑡 ) − P 𝑥 𝜓(𝜉 𝑠 , 𝑉𝑟+𝑡−𝑠 𝑓 , 𝑉𝑟+𝑡−𝑠 𝑓 )𝐾 (d𝑠) 0

holds for 𝑡 ≥ 0. Based on this, the uniqueness of the solution to (2.58) follows by □ arguments similar to those in the proofs of Propositions 2.15 and 2.20. Corollary 2.30 Let ( 𝑓 , 𝑔) ∈ 𝐵(𝐸) + × 𝐵(𝐸) and let (𝑡, 𝑥) ↦→ 𝑉𝑡 𝑓 (𝑥) be defined by 𝑔

𝑔 𝑉𝑡

𝑔

(2.58). Then we have 𝑉𝑟+𝑡 𝑓 (𝑥) = 𝑉𝑟

𝑓

𝑉𝑡 𝑓 (𝑥) for all 𝑟, 𝑡 ≥ 0 and 𝑥 ∈ 𝐸.

Proof For any ( 𝑓 , 𝑔) ∈ 𝐵(𝐸) + ×𝐵(𝐸) we can use Proposition 2.29 and the semigroup property of (𝑄 𝑡 )𝑡 ≥0 to see ∫ 𝜈(𝑔)e−𝜈 ( 𝑓 ) 𝑄 𝑟+𝑡 (𝜇, d𝜈) 𝑀 (𝐸) ∫ ∫ 𝜈(𝑔)e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜂, d𝜈). = 𝑄 𝑟 (𝜇, d𝜂) 𝑀 (𝐸)

𝑀 (𝐸)

By applying (2.56) to both sides for 𝜇 = 𝛿 𝑥 we obtain the desired equality.

□

For a (𝜉, 𝜙)-superprocess, the solution of (2.58) can be approximated by an iterating sequence. In this case, we can rewrite the equation as ∫ 𝑡 ∫ 𝑔 𝑔 d𝑠 𝑉𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑔(𝑥) − (2.60) 𝜓(𝑦, 𝑉𝑠 𝑓 , 𝑉𝑠 𝑓 )𝑃𝑡−𝑠 (𝑥, d𝑦). 0

𝐸

54

2 Measure-Valued Branching Processes 𝑔

Proposition 2.31 Let (𝑡, 𝑥) ↦→ 𝑉𝑡 𝑓 (𝑥) be defined by (2.60). Let 𝑣 0 (𝑡, 𝑥) = 0 and define 𝑣 𝑛 (𝑡, 𝑥) = 𝑣 𝑛 (𝑡, 𝑥, 𝑓 , 𝑔) inductively by ∫ 𝑡 ∫ (2.61) d𝑠 𝜓(𝑦, 𝑉𝑠 𝑓 , 𝑣 𝑛 )𝑃𝑡−𝑠 (𝑥, d𝑦). 𝑣 𝑛+1 (𝑥) = 𝑃𝑡 𝑔(𝑥) − 0

𝐸 𝑔

Then for every 𝑇 ≥ 0 we have 𝑣 𝑛 (𝑥) → 𝑉𝑡 𝑓 (𝑥) uniformly on [0, 𝑇] × 𝐸. Proof Let 𝐷 𝑛 (𝑡) = sup0≤𝑠 ≤𝑡 ∥𝑣 𝑛 (𝑠) − 𝑣 𝑛−1 (𝑠) ∥. Since (𝑡, 𝑥) ↦→ 𝑉𝑡 𝑓 (𝑥) is locally bounded, by (2.59) and (2.61), for any 𝑇 ≥ 0 there is a constant 𝐿 ≥ 0 such that ∫ 𝑡 1 𝐷 𝑛−1 (𝑠)d𝑠 ≤ · · · ≤ 𝐷 𝑛 (𝑡) ≤ 𝐿 𝐿 𝑛−1 𝑡 𝑛−1 ∥𝑔∥, 0 ≤ 𝑡 ≤ 𝑇 . (𝑛 − 1)! 0 Then the result follows as in the proof of Proposition 2.19.

□

By Corollary 2.28, for the (𝜉, 𝜙)-superprocess we have (2.53) and (2.54) with (𝜋𝑡 )𝑡 ≥0 defined by (2.31) and (2.38). Recall that 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)] and 𝑐+0 = 0 ∨ 𝑐 0 . Proposition 2.32 Let (𝑋𝑡 , 𝒢𝑡 , P) be a (𝜉, 𝜙)-superprocess satisfying P[𝑋0 (1)] < ∞. Suppose that 𝛼 ∈ R and 𝑓 ∈ 𝐵(𝐸) + satisfy 𝜋𝑡 𝑓 (𝑥) ≤ e 𝛼𝑡 𝑓 (𝑥) for all 𝑡 ≥ 0 and 𝑥 ∈ 𝐸. Then 𝑡 ↦→ e−𝛼𝑡 𝑋𝑡 ( 𝑓 ) is a positive (𝒢𝑡 )-supermartingale. Proof By Corollary 2.28, for any 𝑡 ≥ 𝑟 ≥ 0 we have   P e−𝛼𝑡 𝑋𝑡 ( 𝑓 ) 𝒢𝑟 = e−𝛼𝑡 𝑋𝑟 (𝜋𝑡−𝑟 𝑓 ) ≤ e−𝛼𝑟 𝑋𝑟 ( 𝑓 ). Therefore 𝑡 ↦→ e−𝛼𝑡 𝑋𝑡 ( 𝑓 ) is a (𝒢𝑡 )-supermartingale.

□

Corollary 2.33 Let (𝑋𝑡 , 𝒢𝑡 , P) be a right continuous (𝜉, 𝜙)-superprocess satisfying P[𝑋0 (1)] < ∞. Then 𝑡 ↦→ e−𝑐0 𝑡 𝑋𝑡 (1) is a positive (𝒢𝑡 )-supermartingale and for any 𝜆 ≥ 0 we have n o 𝜆P sup e−𝑐0 𝑡 𝑋𝑡 (1) ≥ 𝜆 ≤ P[𝑋0 (1)]. (2.62) 𝑡 ≥0

Proof By Theorem A.53 we have ∥𝜋𝑡 ∥ ≤ e𝑐0 𝑡 for 𝑡 ≥ 0. By Proposition 2.32 the process 𝑡 ↦→ e−𝑐0 𝑡 𝑋𝑡 (1) is a positive (𝒢𝑡 )-supermartingale. Then we have the inequality (2.62). □ Corollary 2.34 Let (𝑋𝑡 , 𝒢𝑡 , P) be a (𝜉, 𝜙)-superprocess satisfying P[𝑋0 (1)] < ∞. Let 𝛼 ≥ 0 and let 𝑓 ∈ 𝐵(𝐸) + be an 𝛼-super-mean-valued function for (𝑃𝑡 )𝑡 ≥0 satisfying 𝜀 := inf 𝑥 ∈𝐸 𝑓 (𝑥) > 0. Then for any 𝛽 ≥ 𝛼 + 𝑐+0 𝜀 −1 ∥ 𝑓 ∥ the process 𝑡 ↦→ e−2𝛽𝑡 𝑋𝑡 ( 𝑓 ) is a positive (𝒢𝑡 )-supermartingale. Proof Since 𝑓 is 𝛼-super-mean-valued for (𝑃𝑡 )𝑡 ≥0 , from (2.38) it follows that ∫ 𝑡 + 𝜋𝑡 𝑓 (𝑥) ≤ 𝑃𝑡 𝑓 (𝑥) + 𝑐 0 ∥ 𝑓 ∥ e𝑐0 𝑠 𝑃𝑡−𝑠 1(𝑥)d𝑠 0

2.5 Some Moment Formulas

55

∫

𝑡 +

e𝑐0 𝑠 𝑃𝑡−𝑠 𝑓 (𝑥)d𝑠 ≤ e 𝛼𝑡 𝑓 (𝑥) + 𝑐+0 𝜀 −1 ∥ 𝑓 ∥ 0 ∫ 𝑡 e𝛽𝑠 𝑓 (𝑥)d𝑠 ≤ e2𝛽𝑡 𝑓 (𝑥). ≤ e𝛽𝑡 𝑓 (𝑥) + 𝛽e𝛽𝑡 0

Then the result follows by Proposition 2.32.

□

Corollary 2.35 Let 𝜙 be a local branching mechanism given by (2.49) and let (𝑋𝑡 , 𝒢𝑡 , P) be a (𝜉, 𝜙)-superprocess satisfying P[𝑋0 (1)] < ∞. Let 𝛼 ≥ 0 and let 𝑓 ∈ 𝐵(𝐸) + be an 𝛼-super-mean-valued function for (𝑃𝑡 )𝑡 ≥0 . Then for any 𝛼1 ≥ 𝛼 + ∥𝑏 − ∥ the process 𝑡 ↦→ e−𝛼1 𝑡 𝑋𝑡 ( 𝑓 ) is a (𝒢𝑡 )-supermartingale. Proof Since 𝑓 ∈ 𝐵(𝐸) + is 𝛼-super-mean-valued for (𝑃𝑡 )𝑡 ≥0 , we have 𝑃𝑡𝑏 𝑓 (𝑥) ≤ e ∥𝑏

− ∥𝑡

𝑃𝑡 𝑓 (𝑥) ≤ e ( ∥𝑏

− ∥+𝛼)𝑡

Then we have the result by Proposition 2.32.

𝑓 (𝑥) ≤ e 𝛼1 𝑡 𝑓 (𝑥). □

Let F be the set of functions 𝑓 ∈ 𝐵(𝐸) that are finely continuous relative to 𝜉. Fix 𝛽 > 0 and let ( 𝐴, 𝒟( 𝐴)) be the weak generator of (𝑃𝑡 )𝑡 ≥0 defined by 𝒟( 𝐴) = 𝑈 𝛽 F and 𝐴 𝑓 = 𝛽 𝑓 − 𝑔 for 𝑓 = 𝑈 𝛽 𝑔 ∈ 𝒟( 𝐴). Theorem 2.36 Suppose that (𝑋𝑡 , 𝒢𝑡 , P) is a progressive realization of the (𝜉, 𝜙)superprocess such that P[𝑋0 (1)] < ∞. Then for any 𝑓 ∈ 𝒟( 𝐴), the process ∫ 𝑡 𝑀𝑡 ( 𝑓 ) := 𝑋𝑡 ( 𝑓 ) − 𝑋0 ( 𝑓 ) − 𝑋𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠, 𝑡 ≥ 0, 0

is a (𝒢𝑡 )-martingale. Proof Let (𝜋𝑡 )𝑡 ≥0 be defined by (2.38). For any 𝑡 ≥ 𝑟 ≥ 0 we use Corollary 2.28 and the Markov property of {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} to see that  ∫ 𝑡    𝑋𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠 𝒢𝑟 P 𝑀𝑡 ( 𝑓 ) 𝒢𝑟 = P 𝑋𝑡 ( 𝑓 ) − 𝑋0 ( 𝑓 ) − 0  ∫ 𝑡−𝑟  = P 𝑋𝑡 ( 𝑓 ) − 𝑋𝑟+𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠 𝒢𝑟 0 ∫ 𝑟 𝑋𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠 − 𝑋0 ( 𝑓 ) − 0 ∫ 𝑡−𝑟 𝑋𝑟 (𝜋 𝑠 ( 𝐴 + 𝛾 − 𝑏) 𝑓 )d𝑠 = 𝑋𝑟 (𝜋𝑡−𝑟 𝑓 ) − 0∫ 𝑟 − 𝑋0 ( 𝑓 ) − 𝑋𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠 0∫ 𝑟 = 𝑋𝑟 ( 𝑓 ) − 𝑋0 ( 𝑓 ) − 𝑋𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠, 0

where we have also used Theorem A.59 for the last equality. This gives the martingale □ property of {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0}.

56

2 Measure-Valued Branching Processes

We next give some second-moment formulas. For simplicity we only consider the (𝜉, 𝜙)-superprocess. In this case, the semigroup (𝜋𝑡 )𝑡 ≥0 is defined by (2.38). We shall need the integral condition ∫ (2.63) 𝜈(1) 2 𝐻 (𝑥, d𝜈) < ∞. sup 𝑥 ∈𝐸

𝑀 (𝐸) ◦

Proposition 2.37 Suppose that (2.63) holds. Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup of the (𝜉, 𝜙)-superprocess. Then for 𝑡 > 0, 𝑥 ∈ 𝐸 and 𝑓 ∈ 𝐵(𝐸) we have ∫ 𝑡 ∫ ∫ 2 d𝑠 𝑞(𝑦, 𝜋 𝑠 𝑓 )𝜋𝑡−𝑠 (𝑥, d𝑦), 𝜈( 𝑓 ) 𝐿 𝑡 (𝑥, d𝜈) = 𝑀 (𝐸)

0

𝐸

where (𝜋𝑡 )𝑡 ≥0 is defined by (2.38) and 𝑞(𝑦, 𝑓 ) = 2𝑐(𝑦) 𝑓 (𝑦) 2 +

∫

𝜈( 𝑓 ) 2 𝐻 (𝑦, d𝜈).

(2.64)

𝑀 (𝐸) ◦

Proof We first assume 𝑓 ∈ 𝐵(𝐸) + . By applying Proposition 1.39 to (2.5), for any 𝜃 > 0 we can define the function 𝑢 𝑡′ (𝑥, 𝜃) := (d/d𝜃)𝑣 𝑡 (𝑥, 𝜃 𝑓 ), which is given by ∫ ′ 𝜈( 𝑓 )e−𝜃 𝜈 ( 𝑓 ) 𝐿 𝑡 (𝑥, d𝜈). (2.65) 𝑢 𝑡 (𝑥, 𝜃) = 𝜆 𝑡 (𝑥, 𝑓 ) + 𝑀 (𝐸) ◦

Then we differentiate both sides of (2.41) to obtain ∫ 𝑡 ∫ 𝑢 𝑡′ (𝑥, 𝜃) = 𝜋𝑡 𝑓 (𝑥) − 2 𝑑𝑠 𝑐(𝑦)𝑣 𝑠 (𝑦, 𝜃 𝑓 )𝑢 𝑠′ (𝑦, 𝜃)𝜋𝑡−𝑠 (𝑥, d𝑦) 𝐸 ∫ 𝑡 ∫ 0 ℎ 𝑠 (𝑦, 𝜃, 𝑓 )𝜋𝑡−𝑠 (𝑥, d𝑦), (2.66) − 𝑑𝑠 0

𝐸

where ∫


𝜈(𝑢 𝑠′ (·, 𝜃)) 1 − e−𝜈 (𝑣𝑠 (·, 𝜃 𝑓 )) 𝐻 (𝑦, d𝜈).

ℎ 𝑠 (𝑦, 𝜃, 𝑓 ) = 𝑀 (𝐸) ◦

For any 𝜃 > 0 let 𝑢 𝑡′′ (𝑥, 𝜃) = (d2 /d𝜃 2 )𝑣 𝑡 (𝑥, 𝜃 𝑓 ). By Proposition 1.39, ∫ ′′ 𝜈( 𝑓 ) 2 e−𝜃 𝜈 ( 𝑓 ) 𝐿 𝑡 (𝑥, d𝜈). 𝑢 𝑡 (𝑥, 𝜃) = −

(2.67)

𝑀 (𝐸) ◦

On the other hand, from (2.66) we have ∫ 𝑡 ∫   𝑑𝑠 𝑢 𝑡′′ (𝑥, 𝜃) = −2 𝑐(𝑦) 𝑢 𝑠′ (𝑦, 𝜃) 2 + 𝑣 𝑠 (𝑦, 𝜃 𝑓 )𝑢 𝑠′′ (𝑦, 𝜃) 𝜋𝑡−𝑠 (𝑥, d𝑦) ∫ 0𝑡 ∫ 𝐸 ℎ 𝑠′ (𝑦, 𝜃, 𝑓 )𝜋𝑡−𝑠 (𝑥, d𝑦) − 𝑑𝑠 0

𝐸

2.5 Some Moment Formulas

57

where ℎ 𝑠′ (𝑦, 𝜃,

∫

𝜈(𝑢 𝑠′ (·, 𝜃)) 2 e−𝜈 (𝑣𝑠 (·, 𝜃 𝑓 )) 𝐻 (𝑦, d𝜈)

𝑓) = 𝑀 ∫ (𝐸) ◦

+


𝜈(𝑢 𝑠′′ (·, 𝜃)) 1 − e−𝜈 (𝑣𝑠 (·, 𝜃 𝑓 )) 𝐻 (𝑦, d𝜈).

𝑀 (𝐸) ◦

By the dominated convergence theorem we have ∫ 𝑡 ∫ 𝑞(𝑦, 𝜋 𝑠 𝑓 )𝜋𝑡−𝑠 (𝑥, d𝑦). lim 𝑢 𝑡′′ (𝑥, 𝜃) = − 𝑑𝑠 𝜃→0

0

𝐸

From this and (2.67) we get the desired equality for 𝑓 ∈ 𝐵(𝐸) + . The extension to □ 𝑓 ∈ 𝐵(𝐸) is elementary. Proposition 2.38 Suppose that (2.63) holds. Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup of the (𝜉, 𝜙)-superprocess. Then for 𝑡 ≥ 0, 𝜇 ∈ 𝑀 (𝐸) and 𝑓 ∈ 𝐵(𝐸) we have ∫ 𝑡 ∫ ∫ 2 2 𝜈( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈) = 𝜇(𝜋𝑡 𝑓 ) + d𝑠 𝑞(𝑦, 𝜋 𝑠 𝑓 )𝜇𝜋𝑡−𝑠 (d𝑦), (2.68) 𝑀 (𝐸)

0

𝐸

where (𝜋𝑡 )𝑡 ≥0 is defined by (2.38) and 𝑞(𝑦, 𝑓 ) is defined by (2.64). Proof Let 𝑢 𝑡′ (𝑥, 𝜃) and 𝑢 𝑡′′ (𝑥, 𝜃) be defined as in the proof of Proposition 2.37. In view of (2.35), we have ∫ 𝜈( 𝑓 ) 2 e−𝜃 𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈) 𝑀 (𝐸)

  = 𝜇(𝑢 𝑡′ (·, 𝜃)) 2 − 𝜇(𝑢 𝑡′′ (·, 𝜃)) exp{−𝜇(𝑣 𝑡 (·, 𝜃))}. By letting 𝜃 → 0 in the above equation we obtain (2.68), first for 𝑓 ∈ 𝐵(𝐸) + and then for 𝑓 ∈ 𝐵(𝐸). □ Corollary 2.39 Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup of the (𝜉, 𝜙)-superprocess with local branching mechanism given by (2.49) and assume ∫ ∞ ′′ (2.69) 𝑧 2 𝑚(𝑥, d𝑧) 𝑥 ↦→ 𝜙 (𝑥, 0) := 2𝑐(𝑥) + 0

is bounded on 𝐸. Then for 𝑡 ≥ 0, 𝜇 ∈ 𝑀 (𝐸) and 𝑓 ∈ 𝐵(𝐸) we have ∫ ∫ 𝑡 ∫ 𝑏 𝜈( 𝑓 ) 2 𝑄 𝑡 (𝜇, d𝜈) = 𝜇(𝑃𝑡𝑏 𝑓 ) 2 + d𝑠 (d𝑥). 𝜙 ′′ (𝑥, 0)𝑃𝑠𝑏 𝑓 (𝑥) 2 𝜇𝑃𝑡−𝑠 𝑀 (𝐸)

0

𝐸

Example 2.11 Suppose that 𝑋 = (𝑊, 𝒢, 𝒢𝑡 , 𝑋𝑡 , Q 𝜇 ) is a (𝜉, 𝜙)-superprocess with binary local branching mechanism 𝜙(𝑥, 𝜆) = 𝑐(𝑥)𝜆2 /2. Let (𝑉𝑡 )𝑡 ≥0 denote the cumulant semigroup of 𝑋. Fix 𝑓 ∈ 𝐵(𝐸) + and define 𝑣 𝑡(𝑛) (𝑥) = (−1) 𝑛−1

𝜕𝑛 𝑉 (𝜃 𝑓 ) (𝑥) . 𝑡 𝜃=0+ 𝜕𝜃 𝑛

58

2 Measure-Valued Branching Processes

Then we have 𝑣 𝑡(1) (𝑥) = 𝑃𝑡 𝑓 (𝑥) and 𝑣 𝑡(𝑛) (𝑥)

=

∫ 𝑛−1  ∑︁ 𝑛−1 𝑘

𝑡

𝑃𝑡−𝑠 (𝑐𝑣 𝑠(𝑘) 𝑣 𝑠(𝑛−𝑘) ) (𝑥)d𝑠

0

𝑘=1

for 𝑛 = 2, 3, . . .. The moments of 𝑋 are determined by Q 𝜇 [𝑋𝑡 ( 𝑓 )] = 𝜇(𝑃𝑡 𝑓 ) and Q 𝜇 [𝑋𝑡 ( 𝑓 ) 𝑛 ] =

 𝑛−1  ∑︁ 𝑛−1 𝑘

𝜇(𝑣 𝑡(𝑛−𝑘) )Q 𝜇 [𝑋𝑡 ( 𝑓 ) 𝑘 ].

𝑘=0

2.6 Variations of Transition Probabilities In this section we give some estimates for the variations of transition probabilities of the MB-process with different initial states. For 𝜇, 𝜈 ∈ 𝑀 (𝐸) let |𝜇 − 𝜈| denote the total variation of the signed measure 𝜇 − 𝜈. Then ∥𝜇 − 𝜈∥ := |𝜇 − 𝜈|(𝐸) is the total variation norm of 𝜇 − 𝜈. For a function 𝐹 on 𝑀 (𝐸) the Lipschitz constant 𝐿 var (𝐹) relative to the total variation norm is defined by  (2.70) 𝐿 var (𝐹) = sup ∥𝜇 − 𝜈∥ −1 |𝐹 (𝜇) − 𝐹 (𝜈)| : 𝜇 ≠ 𝜈 ∈ 𝑀 (𝐸) . A coupling of two probability measures 𝑄 1 and 𝑄 2 on 𝑀 (𝐸) means a probability measure 𝑃 on 𝑀 (𝐸) 2 with marginals 𝑃(· × 𝐸) = 𝑄 1 (·) and 𝑃(𝐸 × ·) = 𝑄 2 (·). The Wasserstein distance 𝑊1 (𝑄 1 , 𝑄 2 ) between 𝑄 1 and 𝑄 2 is defined by ∫ 𝑊1 (𝑄 1 , 𝑄 2 ) = inf ∥𝜇 − 𝜈∥𝑃(d𝜇, d𝜈), (2.71) 𝑃

𝑀 (𝐸) 2

where 𝑃 runs over all couplings of 𝑄 1 and 𝑄 2 . The integrand (𝜇, 𝜈) ↦→ ∥𝜇 − 𝜈∥ in (2.71) is measurable with respect to the Borel 𝜎-algebra ℬ(𝑀 (𝐸) 2 ) = ℬ(𝑀 (𝐸)) 2 . In fact, by the regularity of 𝜇 and 𝜈 we have  ∥𝜇 − 𝜈∥ = sup 𝜇( 𝑓 ) − 𝜈( 𝑓 ) : 𝑓 ∈ 𝐶 (𝐸), ∥ 𝑓 ∥ ≤ 1 , so the function (𝜇, 𝜈) ↦→ ∥𝜇 − 𝜈∥ on 𝑀 (𝐸) 2 is lower semi-continuous. The next theorem gives useful estimates for the variations in Wasserstein distance 𝑊1 of the transition probabilities of the MB-process. Theorem 2.40 Suppose that 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ for every 𝑡 ≥ 0. Then for 𝜇, 𝜈 ∈ 𝑀 (𝐸) we have |(𝜇 − 𝜈) (𝜋𝑡 1)| ≤ 𝑊1 (𝑄 𝑡 (𝜇, ·), 𝑄 𝑡 (𝜈, ·)) ≤ |𝜇 − 𝜈|(𝜋𝑡 1), where (𝜋𝑡 )𝑡 ≥0 is the semigroup of bounded kernels on 𝐸 defined by (2.54).

(2.72)

2.6 Variations of Transition Probabilities

59

Proof Clearly, for any coupling 𝑄 𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) of 𝑄 𝑡 (𝜇, d𝛾1 ) and 𝑄 𝑡 (𝜈, d𝛾2 ) we have ∫ ∥𝛾1 − 𝛾2 ∥𝑄 𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) 𝑀 (𝐸) 2 ∫ ≥ [𝛾1 (1) − 𝛾2 (1)]𝑄 𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) ∫ 𝑀 (𝐸) 2 = 𝛾(1) [𝑄 𝑡 (𝜇, d𝛾) − 𝑄 𝑡 (𝜈, d𝛾)] = (𝜇 − 𝜈) (𝜋𝑡 1). 𝑀 (𝐸)

It follows that 𝑊1 (𝑄 𝑡 (𝜇, ·), 𝑄 𝑡 (𝜈, ·)) ≥ (𝜇 − 𝜈) (𝜋𝑡 1). By symmetry, we get 𝑊1 (𝑄 𝑡 (𝜇, ·), 𝑄 𝑡 (𝜈, ·)) ≥ (𝜈 − 𝜇) (𝜋𝑡 1). Then the lower bound in (2.72) holds. Let (𝜇−𝜈)+ and (𝜇−𝜈)− denote the upper and lower variations of the signed measure 𝜇−𝜈 in its Jordan–Hahn decomposition, respectively. Let 𝜇∧𝜈 = 𝜇−(𝜇−𝜈)+ = 𝜈−(𝜇−𝜈)− . Let 𝑃𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) be the image of the product measure 𝑄 𝑡 (𝜇 ∧ 𝜈, d𝜂0 )𝑄 𝑡 ((𝜇 − 𝜈)+ , d𝜂1 )𝑄 𝑡 ((𝜇 − 𝜈)− , d𝜂2 ) under the mapping (𝜂0 , 𝜂1 , 𝜂2 ) ↦→ (𝛾1 , 𝛾2 ) := (𝜂0 + 𝜂1 , 𝜂0 + 𝜂2 ). By the branching property (2.1) one can see that 𝑃𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) is a coupling of 𝑄 𝑡 (𝜇, d𝛾1 ) and 𝑄 𝑡 (𝜈, d𝛾2 ). Then we have ∫ ∥𝛾1 − 𝛾2 ∥𝑃𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) 𝑊1 (𝑄 𝑡 (𝜇, ·), 𝑄 𝑡 (𝜈, ·)) ≤ ∫𝑀 (𝐸) 2 ∫ = 𝑄 𝑡 (𝜇 ∧ 𝜈, d𝜂0 ) 𝑄 𝑡 ((𝜇 − 𝜈)+ , d𝜂1 ) 𝑀 (𝐸) 𝑀 (𝐸) ∫ ∥𝜂1 − 𝜂2 ∥𝑄 𝑡 ((𝜇 − 𝜈)− , d𝜂2 ) 𝑀 (𝐸) ∫ ∫ 𝑄 𝑡 (𝜇 ∧ 𝜈, d𝜂0 ) ≤ 𝑄 𝑡 ((𝜇 − 𝜈)+ , d𝜂1 ) 𝑀 (𝐸) 𝑀 (𝐸) ∫ [𝜂1 (1) + 𝜂2 (1)]𝑄 𝑡 ((𝜇 − 𝜈)− , d𝜂2 ) 𝑀 (𝐸) ∫ = 𝜂(1)𝑄 𝑡 (|𝜇 − 𝜈|, d𝜂) = |𝜇 − 𝜈|(𝜋𝑡 1). 𝑀 (𝐸)

This gives the upper bound in (2.72).

□

Corollary 2.41 In the setup of Theorem 2.40, for any 𝐹 ∈ 𝐵(𝑀 (𝐸)) we have 𝐿 var (𝑄 𝑡 𝐹) ≤ ∥𝜋𝑡 1∥𝐿 var (𝐹). Proof Let 𝑄 𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) be any coupling of 𝑄 𝑡 (𝜇, d𝛾1 ) and 𝑄 𝑡 (𝜈, d𝛾2 ) for 𝜇, 𝜈 ∈ 𝑀 (𝐸). Clearly, we have ∫ [𝐹 (𝛾1 ) − 𝐹 (𝛾2 )]𝑄 𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) |𝑄 𝑡 𝐹 (𝜇) − 𝑄 𝑡 𝐹 (𝜈)| ≤ 𝑀 (𝐸) 2 ∫ ∥𝛾1 − 𝛾2 ∥𝑄 𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ). ≤ 𝐿 var (𝐹) 𝑀 (𝐸) 2

60

2 Measure-Valued Branching Processes

Then we see by Theorem 2.40 that |𝑄 𝑡 𝐹 (𝜇) − 𝑄 𝑡 𝐹 (𝜈)| ≤ 𝐿 var (𝐹)𝑊1 (𝑄 𝑡 (𝜇, ·), 𝑄 𝑡 (𝜈, ·)) ≤ 𝐿 var (𝐹)|𝜇 − 𝜈|(𝜋𝑡 1) ≤ ∥𝜋𝑡 1∥𝐿 var (𝐹) ∥𝜇 − 𝜈∥. This implies the desired estimate.

□

Corollary 2.42 In the setup of Theorem 2.40, we have 𝑊1 (𝑄 𝑡 (𝜇, ·), 𝑄 𝑡 (𝜈, ·)) = (𝜇 − 𝜈) (𝜋𝑡 1) for 𝜇 ≥ 𝜈 ∈ 𝑀 (𝐸). Corollary 2.43 In the setup of Theorem 2.40, we have 𝑊1 (𝑄 𝑡 (𝜇, ·), 𝛿0 ) = 𝜇(𝜋𝑡 1) for 𝜇 ∈ 𝑀 (𝐸). Corollary 2.44 In the setup of Theorem 2.40, we have 𝑊1 (𝑄 𝑡 (𝜇, ·), 𝛿0 ) → 0 as 𝑡 → ∞ for every 𝜇 ∈ 𝑀 (𝐸) if and only if lim𝑡→∞ 𝜇(𝜋𝑡 1) = 0 for every 𝜇 ∈ 𝑀 (𝐸). Clearly, if the condition of Corollary 2.44 is satisfied, the MB-process is ergodic in the Wasserstein distance 𝑊1 with 𝛿0 as its unique stationary distribution. The following example shows that the condition of the corollary may not be satisfied even if 𝜋𝑡 𝑓 (𝑥) → 0 pointwise as 𝑡 → ∞. Example 2.12 Let (𝑃𝑡 )𝑡 ≥0 be the Borel right semigroup on 𝐸 0 := (0, ∞) defined by 𝑃𝑡 𝑓 (𝑥) = 𝑓 (𝑥 − 𝑡)1 {𝑡 0 and let 𝐹1 = ∥𝐹 ∥ −1 𝐹. For any 𝜇, 𝜈 ∈ 𝑀 (𝐸) we can use Theorem 2.45 to see |𝑄 𝑡 𝐹 (𝜇) − 𝑄 𝑡 𝐹 (𝜈)| = ∥𝐹 ∥|𝑄 𝑡 𝐹1 (𝜇) − 𝑄 𝑡 𝐹1 (𝜈)| ≤ ∥𝐹 ∥ ∥𝑄 𝑡 (𝜇, ·) − 𝑄 𝑡 (𝜈, ·) ∥ ≤ 2∥ 𝑣¯ 𝑡 ∥ ∥𝐹 ∥ ∥𝜇 − 𝜈∥. Then the desired estimate holds.

□

Corollary 2.47 In the setup of Theorem 2.45, for any 𝜇 ∈ 𝑀 (𝐸) we have ∥𝑄 𝑡 (𝜇, ·) − 𝛿0 ∥ = 2(1 − e−𝜇 ( 𝑣¯𝑡 ) ) ≤ 2𝜇( 𝑣¯ 𝑡 ). Corollary 2.48 In the setup of Theorem 2.45, we have ∥𝑄 𝑡 (𝜇, ·) −𝛿0 ∥ → 0 as 𝑡 → ∞ for every 𝜇 ∈ 𝑀 (𝐸) if and only if lim𝑡→∞ 𝑣¯ 𝑡 (𝑥) = 0 decreasingly for every 𝑥 ∈ 𝐸. By Corollary 2.46, if 𝑣¯ 𝑡 is bounded on 𝐸 for every 𝑡 > 0, then (𝑄 𝑡 )𝑡 ≥0 possesses the strong Feller property, that is, the operators (𝑄 𝑡 )𝑡 >0 map bounded Borel functions on 𝑀 (𝐸) into functions continuous in the total variation distance.

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2 Measure-Valued Branching Processes

2.7 Notes and Comments The one-to-one correspondence stated in Theorem 2.4 was established in Watanabe (1968) under some stronger assumptions. Theorem 2.5 was also proved in Watanabe (1968). Jiřina (1964) studied the extinction problem of discrete-time branching processes taking values of finite measures on the positive half line. A class of superprocesses over compact metric spaces were constructed in Watanabe (1968), where it was shown those processes arise as high-density limits of branching particle systems. Silverstein (1969) constructed more general superprocesses with decomposable branching mechanisms; see also Dawson et al. (2002c) and Dynkin (1993a). Some inhomogeneous superprocesses with general branching mechanisms were constructed in Dynkin (1994), who assumed the existence of a càdlàg realization of the underlying spatial motion and a technical condition on the tail behavior of the kernel in the expression of the branching mechanism. The superprocesses constructed in Dynkin (1994) are not necessarily conservative. Dawson et al. (1998) proved that a general class of local branching (𝜉, 𝜙, 𝐾)-superprocesses with a fixed underlying spatial motion 𝜉 depend on the parameters (𝜙, 𝐾) continuously and the superprocesses with Lebesgue killing density constitute a dense subset of the class. Leduc (2000) constructed some Hunt superprocesses under a second-moment condition. For spatially constant local branching mechanisms, the superprocess was constructed in Le Gall (1999) and Le Gall and Le Jan (1998b) by using path-valued processes known as Lévy snakes. Our construction of the (𝜉, 𝜙, 𝐾)-superprocess mainly follows Silverstein (1969) and Watanabe (1968). Theorems 2.40 and 2.45 are from Li (2021). Our assumptions on the branching mechanism guarantee that the corresponding superprocesses have finite first-moments in the sense of (2.53). Let 𝑋 = (𝑊, 𝒢, 𝒢𝑡 , 𝑋𝑡 , Q 𝜇 ) be a realization of the (𝜉, 𝐾, 𝜙)-superprocess. For 𝑡 ≥ 0 and 𝜇 ∈ 𝑀 (𝐸) we can define the mean measure 𝐼 𝜇,𝑡 on 𝐸 by 𝐵 ∈ ℬ(𝐸).

𝐼 𝜇,𝑡 (𝐵) = Q 𝜇 [𝑋𝑡 (𝐵)],

The Campbell measure of the random measure 𝑋𝑡 is the unique finite measure 𝑅 𝜇,𝑡 on 𝐸 × 𝑀 (𝐸) such that 𝑅 𝜇,𝑡 (𝐵 × 𝐴) = Q 𝜇 [𝑋𝑡 (𝐵)1 𝐴 (𝑋𝑡 )],

𝐵 ∈ ℬ(𝐸), 𝐴 ∈ ℬ(𝑀 (𝐸)).

In view of (2.56) we have ∫ ∫ 𝑔 𝑔(𝑥)e−𝜈 ( 𝑓 ) 𝑅 𝜇,𝑡 (d𝑥, d𝜈) = exp{−𝜇(𝑉𝑡 𝑓 )}𝜇(𝑉𝑡 𝑓 ), 𝐸

𝑀 (𝐸)

where 𝑓 ∈ 𝐵(𝐸) + and 𝑔 ∈ 𝐵(𝐸). By the existence of regular conditional probabilities, there is a probability kernel 𝐽 𝜇,𝑡 (𝑥, d𝜈) from 𝐸 to 𝑀 (𝐸) such that 𝑅 𝜇,𝑡 (d𝑥, d𝜈) = 𝐼 𝜇,𝑡 (d𝑥)𝐽 𝜇,𝑡 (𝑥, d𝜈),

𝑥 ∈ 𝐸, 𝜈 ∈ 𝑀 (𝐸).

2.7 Notes and Comments

63

The probability measures {𝐽 𝜇,𝑡 (𝑥, ·) : 𝑥 ∈ 𝐸 } are called Palm distributions of 𝑋𝑡 . If (𝜂, 𝑌 ) is a random variable on 𝐸 × 𝑀 (𝐸) distributed according to the Campbell measure 𝑅 𝜇,𝑡 , then 𝜂 is chosen according to the random measure 𝑌 and 𝐽 𝜇,𝑡 (𝑥, ·) is the conditional distribution of 𝑌 given 𝜂 = 𝑥. See Dawson (1993) and Dawson and Perkins (1991) for some applications of the Campbell measure and the Palm distributions in the study of the superprocess. Example 2.1 was given by Dynkin et al. (1994). Rhyzhov and Skorokhod (1970) and Watanabe (1969) constructed multi-dimensional/type CB-processes under conditions on the branching mechanism weaker than those of Example 2.5. CB-processes with countably many types were introduced as super Markov chains in Kyprianou and Palau (2018). Moment formulas for superprocesses as in Example 2.11 were established in Dynkin (1989a) and Konno and Shiga (1988). The precise asymptotics of the moments of spatial branching processes was studied in the recent work of Gonzalez et al. (2022+). A construction for super-Brownian motions was given in Ren (2001) under a weaker admissibility assumption on the killing additive functional. A super-stable process with infinite mean was constructed in Fleischmann and Sturm (2004) by a passage to the limit. The catalyst measure 𝜌(d𝑦) in Example 2.6 can be time dependent. In fact, it can be replaced by a measure-valued process {𝜌𝑡 : 𝑡 ≥ 0}. The study of superprocesses with measure-valued catalysts was initiated by Dawson and Fleischmann (1991, 1992). A binary local branching super-Brownian motion with super-Brownian catalyst was constructed in Dawson and Fleischmann (1997a). The property of persistence (no loss of expected mass in the long-time behavior) of the process with underlying dimensions 𝑑 ≤ 3 was proved in Dawson and Fleischmann (1997a, 1997b) and Etheridge and Fleischmann (1998). This phenomenon is in contrast to the superBrownian motion with Lebesgue catalyst, where persistence only holds in high dimensions. A construction of catalytic super-Brownian motion via collision local times was given in Mörters and Vogt (2005). The long-time behavior of a branching random walk in a random catalytic medium was investigated in Greven et al. (1999). Engländer (2007) gave a survey of some recent topics in spatial branching processes in deterministic and random media. There is another important class of measure-valued Markov processes, the socalled Fleming–Viot superprocesses. A Fleming–Viot superprocess takes values of probability measures and describes the evolution of a genetic system involving mutation, selection and recombination. The Saint-Flour lecture notes of Dawson (1993) provide a complete survey of the literature before 1992 on both Dawson– Watanabe and Fleming–Viot superprocesses. For a survey of the latter see also Ethier and Kurtz (1993). It was conjectured in Ethier and Kurtz (1993) that a Fleming–Viot superprocess is reversible if and only if its mutation operator is of the uniform jump type. This was proved in Li et al. (1999); see also Handa (2002) and Schmuland and Sun (2002). A nice introduction of the theory of superprocesses was given by Etheridge (2000), where Brownian spatial motion was mainly considered. The connections between Dawson–Watanabe and Fleming–Viot superprocesses were investigated in Etheridge and March (1991), Perkins (1992) and Shiga (1990). The two classes of superprocesses model large population systems in which branching

64

2 Measure-Valued Branching Processes

or splitting occurs. The dual phenomenon is coalescence or coagulation. Bertoin (2006) gave a comprehensive account of stochastic models involving fragmentation and coagulation. A kind of generalized Fleming–Viot superprocess arising from coalescent processes was studied in Bertoin and Le Gall (2003, 2005, 2006). Feng (2010) provided an up-to-date account of Fleming–Viot superprocesses and Poisson– Dirichlet type distributions. Durrett (2008) and Ewens (2004) gave comprehensive coverage of mathematical population genetics.

Chapter 3

One-Dimensional Branching Processes

A one-dimensional CB-process is a Markov process with branching property taking values in the positive half line. A more general model is the CB-process with immigration which deals with the situation where immigrants may come from outer sources. In this chapter, we first give some characterizations of the extinction probabilities and evolution rates of CB-processes. Then we prove some conditional limit theorems for the processes that extinguish with strictly positive probability. In particular, we shall see that a class of CB-processes with immigration can be obtained from those without immigration by conditioning on non-extinction. The proofs of those theorems are based on the asymptotic analysis of the cumulant semigroup and are easier than their discrete-state counterparts given as in Athreya and Ney (1972). The greater tractability of the CB-processes arises because both their time and state spaces are smooth, and the distributions which appear are infinitely divisible. In this sense, the continuous-state models provide a more economical way to establish the nicest conditional limit theorems for branching processes. We also show that the CB-process with immigration arises naturally as the scaling limit of a sequence of discrete Galton–Watson branching processes with immigration. The contents of this chapter will be helpful to readers who wish to develop their intuition concerning Dawson–Watanabe superprocesses.

3.1 Continuous-State Branching Processes In this section we prove some basic properties of the one-dimensional CB-process, which is a special case of the process considered in Example 2.5. Suppose that 𝜙 is a branching mechanism defined by ∫ ∞
2 𝜙(𝜆) = 𝑏𝜆 + 𝑐𝜆 + e−𝑧𝜆 − 1 + 𝑧𝜆 𝑚(d𝑧), 𝜆 ≥ 0, (3.1) 0

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_3

65

66

3 One-Dimensional Branching Processes

where 𝑐 ≥ 0 and 𝑏 are constants and (𝑧 ∧ 𝑧2 )𝑚(d𝑧) is a finite measure on (0, ∞). A one-dimensional CB-process has the transition semigroup (𝑄 𝑡 )𝑡 ≥0 defined by ∫ 𝜆 ≥ 0, (3.2) e−𝜆𝑦 𝑄 𝑡 (𝑥, d𝑦) = e−𝑥𝑣𝑡 (𝜆) , [0,∞)

where 𝑡 ↦→ 𝑣 𝑡 (𝜆) is the unique positive solution of ∫ 𝑡 𝑣 𝑡 (𝜆) = 𝜆 − 𝜙(𝑣 𝑠 (𝜆))d𝑠,

𝑡 ≥ 0.

(3.3)

0

As in the case of Dawson–Watanabe superprocesses, the infinite divisibility of the transition semigroup (𝑄 𝑡 )𝑡 ≥0 implies that the cumulant semigroup (𝑣 𝑡 )𝑡 ≥0 can be expressed canonically as ∫ ∞ (1 − e−𝜆𝑢 )𝑙 𝑡 (d𝑢), 𝑡 ≥ 0, 𝜆 ≥ 0, 𝑣 𝑡 (𝜆) = ℎ𝑡 𝜆 + (3.4) 0

where ℎ𝑡 ≥ 0 and 𝑢𝑙 𝑡 (d𝑢) is a finite measure on (0, ∞). From (3.3) we see that 𝑡 ↦→ 𝑣 𝑡 (𝜆) is first continuous and then continuously differentiable. Moreover, we have the backward differential equation: 𝜕 𝑣 𝑡 (𝜆) = −𝜙(𝑣 𝑡 (𝜆)), 𝜕𝑡

𝑣 0 (𝜆) = 𝜆.

(3.5)

By (3.5) and the semigroup property 𝑣 𝑟 ◦ 𝑣 𝑡 = 𝑣 𝑟+𝑡 for 𝑟, 𝑡 ≥ 0 we also have the forward differential equation 𝜕 𝜕 𝑣 𝑡 (𝜆) = −𝜙(𝜆) 𝑣 𝑡 (𝜆), 𝜕𝑡 𝜕𝜆

𝑣 0 (𝜆) = 𝜆.

From a moment formula for general superprocesses we have ∫ ∞ 𝑦𝑄 𝑡 (𝑥, d𝑦) = 𝑥e−𝑏𝑡 , 𝑡 ≥ 0, 𝑥 ≥ 0.

(3.6)

(3.7)

0

We say the CB-process is critical, subcritical or supercritical according as 𝑏 = 0, ≥ 0 or ≤ 0. Proposition 3.1 For every 𝑡 ≥ 0 the function 𝜆 ↦→ 𝑣 𝑡 (𝜆) is strictly increasing on [0, ∞). Proof By the continuity of 𝑡 ↦→ 𝑣 𝑡 (𝜆), for any 𝜆0 > 0 there is a 𝑡 0 > 0 such that 𝑣 𝑡 (𝜆0 ) > 0 for 0 ≤ 𝑡 ≤ 𝑡0 . Then (3.2) implies 𝑄 𝑡 (𝑥, {0}) < 1 for 𝑥 > 0 and 0 ≤ 𝑡 ≤ 𝑡 0 , and so 𝜆 ↦→ 𝑣 𝑡 (𝜆) is strictly increasing for 0 ≤ 𝑡 ≤ 𝑡 0 . By the semigroup property of (𝑣 𝑡 )𝑡 ≥0 we infer 𝜆 ↦→ 𝑣 𝑡 (𝜆) is strictly increasing for all 𝑡 ≥ 0. □ Corollary 3.2 The transition semigroup (𝑄 𝑡 )𝑡 ≥0 defined by (3.2) is a Feller semigroup.

3.1 Continuous-State Branching Processes

67

Proof By Proposition 3.1 for 𝑡 ≥ 0 and 𝜆 > 0 we have 𝑣 𝑡 (𝜆) > 0. From (3.2) we see the operator 𝑄 𝑡 maps {𝑥 ↦→ e−𝜆𝑥 : 𝜆 > 0} to itself. By the Stone–Weierstrass theorem, the linear span of {𝑥 ↦→ e−𝜆𝑥 : 𝜆 > 0} is dense in 𝐶0 (R+ ) in the supremum norm. Then 𝑄 𝑡 maps 𝐶0 (R+ ) to itself. The Feller property of (𝑄 𝑡 )𝑡 ≥0 follows by the continuity of 𝑡 ↦→ 𝑣 𝑡 (𝜆). □ Proposition 3.3 Suppose that 𝜆 > 0 and 𝜙(𝜆) ≠ 0. Then the equation 𝜙(𝑧) = 0 has no root between 𝜆 and 𝑣 𝑡 (𝜆). Moreover, we have ∫ 𝜆 𝑡 ≥ 0. (3.8) 𝜙(𝑧) −1 d𝑧 = 𝑡, 𝑣𝑡 (𝜆)

Proof By (3.1) we see 𝜙(0) = 0 and 𝑧 ↦→ 𝜙(𝑧) is a convex function. Since 𝜙(𝜆) ≠ 0 for some 𝜆 > 0 according to the assumption, the equation 𝜙(𝑧) = 0 has at most one root in (0, ∞). Suppose that 𝜆0 ≥ 0 is a root of 𝜙(𝑧) = 0. Then (3.6) implies 𝑣 𝑡 (𝜆0 ) = 𝜆0 for all 𝑡 ≥ 0. By Proposition 3.1 we have 𝑣 𝑡 (𝜆) > 𝜆 0 for 𝜆 > 𝜆0 and 0 < 𝑣 𝑡 (𝜆) < 𝜆0 for 0 < 𝜆 < 𝜆 0 . Then 𝜆 > 0 and 𝜙(𝜆) ≠ 0 imply there is no root of 𝜙(𝑧) = 0 between 𝜆 and 𝑣 𝑡 (𝜆). From (3.5) we get (3.8). □ Corollary 3.4 Suppose that 𝜙(𝑧0 ) ≠ 0 for some 𝑧 0 > 0. Let 𝜙−1 (0) = inf{𝑧 ≥ 0 : 𝜙(𝑧) > 0} with the convention inf ∅ = ∞. Then lim𝑡→∞ 𝑣 𝑡 (𝜆) = 𝜙−1 (0) increasingly for 0 < 𝜆 < 𝜙−1 (0) and decreasingly for 𝜆 > 𝜙−1 (0). Proof In the case 𝜙−1 (0) = ∞, we have 𝜙(𝑧) < 0 for all 𝑧 > 0. From (3.5) we see that 𝑡 ↦→ 𝑣 𝑡 (𝜆) is increasing. Then (3.8) implies lim𝑡→∞ 𝑣 𝑡 (𝜆) = ∞ for every 𝜆 > 0. In the case 𝜙−1 (0) < ∞, we have 𝜙(𝜙−1 (0)) = 0. Then (3.5) implies 𝑣 𝑡 (𝜙−1 (0)) = 𝜙−1 (0) for all 𝑡 ≥ 0. Moreover, it is easy to see that 𝜙(𝑧) < 0 for 0 < 𝑧 < 𝜙−1 (0) and 𝜙(𝑧) > 0 for 𝑧 > 𝜙−1 (0). From (3.8) we see that lim𝑡→∞ 𝑣 𝑡 (𝜆) = 𝜙−1 (0) increasingly □ for 0 < 𝜆 < 𝜙−1 (0) and decreasingly for 𝜆 > 𝜙−1 (0). Corollary 3.5 Suppose that 𝜙(𝑧0 ) ≠ 0 for some 𝑧0 > 0. Then for any 𝑥 > 0 we have lim 𝑄 𝑡 (𝑥, ·) = e−𝑥 𝜙 𝑡→∞

−1 (0)

𝛿0 + (1 − e−𝑥 𝜙

−1 (0)

)𝛿∞

by weak convergence of probability measures on [0, ∞]. Proof By Theorem 1.14, the space of probability measures on [0, ∞] endowed with the topology of weak convergence is compact and metrizable; see also Parthasarathy (1967, p. 45). Let {𝑡 𝑛 } be any positive sequence such that 𝑡 𝑛 → ∞ and 𝑄 𝑡𝑛 (𝑥, ·) converges to some 𝑄 ∞ (𝑥, ·) weakly as 𝑛 → ∞. By (3.2) and Corollary 3.4, for every 𝜆 > 0 we have ∫ ∫ −𝜆𝑦 e 𝑄 ∞ (𝑥, d𝑦) = lim e−𝜆𝑦 𝑄 𝑡𝑛 (𝑥, d𝑦) [0,∞]

𝑛→∞

= lim e

[0,∞] −𝑥𝑣𝑡𝑛 (𝜆)

= e−𝑥 𝜙

𝑛→∞

where e−𝜆𝑦 = 0 for 𝑦 = ∞ by convention. It follows that

−1 (0)

,

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3 One-Dimensional Branching Processes

∫

e−𝜆𝑦 𝑄 ∞ (𝑥, d𝑦) = e−𝑥 𝜙

𝑄 ∞ (𝑥, {0}) = lim 𝜆→∞

−1 (0)

[0,∞]

and ∫

(1 − e−𝜆𝑦 )𝑄 ∞ (𝑥, d𝑦) = 1 − e−𝑥 𝜙

𝑄 ∞ (𝑥, {∞}) = lim 𝜆→0

−1 (0)

.

[0,∞]

This shows 𝑄 ∞ (𝑥, ·) = e−𝑥 𝜙

−1 (0)

𝛿0 + (1 − e−𝑥 𝜙

−1 (0)

)𝛿∞ ,

which is independent of the choice of the sequence {𝑡 𝑛 }. Then 𝑄 𝑡 (𝑥, ·) converges to 𝑄 ∞ (𝑥, ·) weakly as 𝑡 → ∞. □ Clearly, under the condition of Corollary 3.4, we have: (i) 𝜙−1 (0) = 0 if and only if 𝑏 ≥ 0; (ii) 𝜙−1 (0) > 0 if and only if 𝑏 < 0; (iii) 𝜙−1 (0) = ∞ if and only if 𝜙(𝜆) < 0 for all 𝜆 > 0. Proposition 3.6 For any 𝑡 ≥ 0 and 𝜆 ≥ 0 let 𝑣 𝑡′ (𝜆) = (𝜕/𝜕𝜆)𝑣 𝑡 (𝜆). Then we have  ∫ 𝑡  (3.9) 𝜙 ′ (𝑣 𝑠 (𝜆))d𝑠 , 𝑣 𝑡′ (𝜆) = exp − 0

where 𝜙 ′ (𝜆) = 𝑏 + 2𝑐𝜆 +

∫

∞


𝑧 1 − e−𝑧𝜆 𝑚(d𝑧).

(3.10)

0

Proof Based on (3.3) and (3.5) it is elementary to see that 𝜕 ′ 𝜕 𝜕 𝑣 𝑡 (𝜆) = −𝜙 ′ (𝑣 𝑡 (𝜆))𝑣 𝑡′ (𝜆) = 𝑣 𝑡 (𝜆). 𝜕𝑡 𝜕𝜆 𝜕𝑡 It follows that  𝜕  𝜕 log 𝑣 𝑡′ (𝜆) = 𝑣 𝑡′ (𝜆) −1 𝑣 𝑡′ (𝜆) = −𝜙 ′ (𝑣 𝑡 (𝜆)). 𝜕𝑡 𝜕𝑡 Then we have (3.9) since 𝑣 0′ (𝜆) = 1.

□

Since (𝑄 𝑡 )𝑡 ≥0 is a Feller semigroup by Corollary 3.2, the CB-process has a realization 𝑋 = (Ω, ℱ, ℱ𝑡 , 𝑥(𝑡), Q 𝑥 ) as a Hunt process; see Corollary A.26. Let 𝜏0 = inf{𝑠 ≥ 0 : 𝑥(𝑠) = 0} denote the extinction time of the CB-process. By the strong Markov property, we have Q 𝑥 {𝑥(𝜏0 + 𝑡) = 0 for all 𝑡 ≥ 0} = 1. Proposition 3.7 Let 𝜏 = inf{𝑡 ≥ 0 : 𝑥(𝑡) = 0 or 𝑥(𝑡−) = 0}. Then for any 𝑥 ≥ 0 we have Q 𝑥 {𝜏 = 𝜏0 } = 1. Proof It is clear that 𝜏 ≤ 𝜏0 . For any 𝑘 ≥ 1 let 𝜏𝑘 = inf{𝑡 ≥ 0 : 𝑥(𝑡) < 1/𝑘 }. Then 𝑥(𝜏𝑘 ) ≤ 1/𝑘 and lim 𝑘→∞ 𝜏𝑘 = 𝜏 increasingly. By the quasi-left continuity, we have a.s. 𝑥(𝜏) = lim 𝑘→∞ 𝑥(𝜏𝑘 ) = 0. This implies Q 𝑥 {𝜏0 = 𝜏} = 1. □

3.1 Continuous-State Branching Processes

69

Theorem 3.8 For every 𝑡 ≥ 0 the limit 𝑣¯ 𝑡 := 𝑣 𝑡 (∞) = ↑ lim𝜆→∞ 𝑣 𝑡 (𝜆) exists in (0, ∞]. Moreover, the mapping 𝑡 ↦→ 𝑣¯ 𝑡 is decreasing and for any 𝑡 ≥ 0 and 𝑥 > 0 we have Q 𝑥 {𝜏0 ≤ 𝑡} = Q 𝑥 {𝑥(𝑡) = 0} = exp{−𝑥 𝑣¯ 𝑡 }.

(3.11)

Proof By Proposition 3.1 the limit 𝑣¯ 𝑡 = ↑lim𝜆→∞ 𝑣 𝑡 (𝜆) exists in (0, ∞] for every 𝑡 ≥ 0. For 𝑡 ≥ 𝑟 ≥ 0 we have 𝑣¯ 𝑡 = ↑ lim 𝑣 𝑟 (𝑣 𝑡−𝑟 (𝜆)) = 𝑣 𝑟 ( 𝑣¯ 𝑡−𝑟 ) ≤ 𝑣¯ 𝑟 = ↑ lim 𝑣 𝑟 (𝜆).

(3.12)

𝜆→∞

𝜆→∞

Since zero is a trap for the CB-process, we get (3.11) by letting 𝜆 → ∞ in (3.2). □ The following condition on the branching mechanism 𝜙 is known as Grey’s condition: Condition 3.9 There is some constant 𝜃 > 0 such that ∫ ∞ 𝜙(𝑧) > 0 for 𝑧 ≥ 𝜃 and 𝜙(𝑧) −1 d𝑧 < ∞. 𝜃

Theorem 3.10 We have 𝑣¯ 𝑡 < ∞ for some and hence all 𝑡 > 0 if and only if Condition 3.9 holds. Proof By (3.12) it is easy to see that 𝑣¯ 𝑡 = ↑lim𝜆→∞ 𝑣 𝑡 (𝜆) < ∞ for all 𝑡 > 0 if and only if this holds for some 𝑡 > 0. If Condition 3.9 holds, we can let 𝜆 → ∞ in (3.8) to obtain ∫ ∞ (3.13) 𝜙(𝑧) −1 d𝑧 = 𝑡 𝑣¯𝑡

and hence 𝑣¯ 𝑡 < ∞ for 𝑡 > 0. For the converse, suppose that 𝑣¯ 𝑡 < ∞ for some 𝑡 > 0. By (3.5) there exists some 𝜃 > 0 such that 𝜙(𝜃) > 0, for otherwise we would have 𝑣 𝑡 (𝜆) ≥ 𝜆, yielding a contradiction. Then 𝜙(𝑧) > 0 for all 𝑧 ≥ 𝜃 by the convexity of the branching mechanism. As in the above we see that (3.13) still holds, □ so Condition 3.9 is satisfied. Theorem 3.11 Let 𝑣¯ = ↓lim𝑡→∞ 𝑣¯ 𝑡 ∈ [0, ∞]. Then for any 𝑥 > 0 we have Q 𝑥 {𝜏0 < ∞} = exp{−𝑥 𝑣¯ }.

(3.14)

Moreover, we have 𝑣¯ < ∞ if and only if Condition 3.9 holds, and in this case 𝑣¯ is the largest root of 𝜙(𝑧) = 0. Proof The first assertion follows immediately from Theorem 3.8. By Theorem 3.10 we have 𝑣¯ 𝑡 < ∞ for some and hence all 𝑡 > 0 if and only if Condition 3.9 holds. This is clearly equivalent to 𝑣¯ < ∞. From (3.13) it is easy to see that 𝑣¯ is the largest root □ of 𝜙(𝑧) = 0.

70

3 One-Dimensional Branching Processes

Corollary 3.12 Suppose that 𝜙(𝑧0 ) ≠ 0 for some 𝑧 0 > 0. Then: (i) 𝜙−1 (0) = 𝑣¯ = ∞ if 𝜙(𝜆) < 0 for all 𝜆 > 0; (ii) 0 ≤ 𝜙−1 (0) = 𝑣¯ < ∞ if Condition 3.9 holds; (iii) 0 ≤ 𝜙−1 (0) < 𝑣¯ = ∞ if there is a 𝜃 > 0 such that ∫ ∞ 𝜙(𝑧) −1 d𝑧 = ∞. 𝜙(𝑧) > 0 for 𝑧 ≥ 𝜃 and 𝜃

Corollary 3.13 Suppose that Condition 3.9 holds. Then for any 𝑥 > 0 we have Q 𝑥 {𝜏0 < ∞} = 1 if and only if 𝑏 ≥ 0. Let (𝑄 ◦𝑡 )𝑡 ≥0 be the restriction to (0, ∞) of the semigroup (𝑄 𝑡 )𝑡 ≥0 . The special case of the canonical representation (3.4) with ℎ𝑡 = 0 for all 𝑡 > 0 is particularly interesting. In this case, we have ∫ ∞ 𝑡 > 0, 𝜆 ≥ 0. (3.15) 𝑣 𝑡 (𝜆) = (1 − e−𝜆𝑢 )𝑙 𝑡 (d𝑢), 0

Theorem 3.14 The cumulant semigroup admits the representation (3.15) if and only if ∫ ∞ ′ (3.16) 𝜙 (∞) := 𝑏 + 2𝑐 · ∞ + 𝑧 𝑚(d𝑧) = ∞ 0

with 0 · ∞ = 0 by convention. If condition (3.16) is satisfied, then (𝑙 𝑡 )𝑡 >0 is an entrance law for the restricted semigroup (𝑄 ◦𝑡 )𝑡 ≥0 . Proof From (3.10) it is clear that the limit 𝜙 ′ (∞) = lim𝜆→∞ 𝜙 ′ (𝜆) always exists in (−∞, ∞]. By (3.4) we have ∫ ∞ 𝑣 𝑡′ (𝜆) = ℎ𝑡 + 𝑢e−𝜆𝑢 𝑙 𝑡 (d𝑢), (3.17) 𝑡 ≥ 0, 𝜆 ≥ 0. 0

From (3.9) and (3.17) it follows that ℎ𝑡 =

𝑣 𝑡′ (∞)



∫

𝑡 ′



𝜙 ( 𝑣¯ 𝑠 )d𝑠 .

= exp −

(3.18)

0

Then ℎ𝑡 = 0 for any 𝑡 > 0 implies 𝜙 ′ (∞) = ∞. For the converse, assume that 𝜙 ′ (∞) = ∞. If Condition 3.9 holds, by Theorem 3.10 for every 𝑡 > 0 we have 𝑣¯ 𝑡 < ∞, so ℎ𝑡 = 0 by (3.4). If Condition 3.9 does not hold, then 𝑣¯ 𝑡 = ∞ for 𝑡 > 0 by Theorem 3.10. Then (3.18) implies ℎ𝑡 = 0 for 𝑡 > 0. This proves the first assertion of the theorem. If (𝑣 𝑡 )𝑡 >0 admits the representation (3.15), we can use the semigroup property of (𝑣 𝑡 )𝑡 ≥0 to see ∫ ∞ ∫ ∞ −𝜆𝑢 (1 − e )𝑙𝑟+𝑡 (d𝑢) = (1 − e−𝑢𝑣𝑡 (𝜆) )𝑙𝑟 (d𝑢) 0 ∫ ∞ ∫0 ∞ (1 − e−𝜆𝑢 )𝑄 ◦𝑡 (𝑥, d𝑢) = 𝑙𝑟 (d𝑥) 0

0

3.1 Continuous-State Branching Processes

71

for 𝑟, 𝑡 > 0 and 𝜆 ≥ 0. Then (𝑙 𝑡 )𝑡 >0 is an entrance law for (𝑄 ◦𝑡 )𝑡 ≥0 .

□

Corollary 3.15 If Condition 3.9 holds, the cumulant semigroup admits the representation (3.15) and 𝑡 ↦→ 𝑣¯ 𝑡 = 𝑙 𝑡 (0, ∞) is the unique solution of the differential equation d 𝑣¯ 𝑡 = −𝜙( 𝑣¯ 𝑡 ), d𝑡

𝑡>0

(3.19)

with singular initial condition 𝑣¯ 0+ = ∞. Proof Under Condition 3.9, for every 𝑡 > 0 we have 𝑣¯ 𝑡 < ∞ by Theorem 3.10. Moreover, the condition and the convexity of 𝜆 ↦→ 𝜙(𝜆) imply 𝜙 ′ (∞) = ∞. Then we have the representation (3.15) by Theorem 3.14. The semigroup property of (𝑣 𝑡 )𝑡 ≥0 implies 𝑣¯ 𝑡+𝑠 = 𝑣 𝑡 ( 𝑣¯ 𝑠 ) for 𝑡 > 0 and 𝑠 > 0. Then 𝑡 ↦→ 𝑣¯ 𝑡 satisfies (3.19). From (3.13) it is easy to see 𝑣¯ 0+ = ∞. Suppose that 𝑡 ↦→ 𝑢 𝑡 and 𝑡 ↦→ 𝑣 𝑡 are two solutions to (3.19) with 𝑢 0+ = 𝑣 0+ = ∞. For any 𝜀 > 0 there exists a 𝛿 > 0 such that 𝑢 𝑠 ≥ 𝑣 𝜀 for every 0 < 𝑠 ≤ 𝛿. Since both 𝑡 ↦→ 𝑢 𝑠+𝑡 and 𝑡 ↦→ 𝑣 𝜀+𝑡 are solutions to (3.19), we have 𝑢 𝑠+𝑡 ≥ 𝑣 𝜀+𝑡 for 𝑡 ≥ 0 and 0 < 𝑠 ≤ 𝛿 by Proposition 3.1. Then we can let 𝑠 → 0 and 𝜀 → 0 to see 𝑢 𝑡 ≥ 𝑣 𝑡 for 𝑡 > 0. By symmetry we get the uniqueness of the solution to (3.19). □ Corollary 3.16 Suppose that Condition 3.9 holds. Then for any 𝑡 > 0 the function 𝜆 ↦→ 𝑣 𝑡 (𝜆) is strictly increasing and concave on [0, ∞), and 𝑣¯ is the largest solution of the equation 𝑣 𝑡 (𝜆) = 𝜆. Moreover, we have 𝑣¯ = ↑lim𝑡→∞ 𝑣 𝑡 (𝜆) for 0 < 𝜆 < 𝑣¯ and 𝑣¯ = ↓lim𝑡→∞ 𝑣 𝑡 (𝜆) for 𝜆 > 𝑣¯ . Proof By Corollary 3.15 we have the canonical representation (3.15) for every 𝑡 > 0. Since 𝜆 ↦→ 𝑣 𝑡 (𝜆) is strictly increasing by Proposition 3.1, the measure 𝑙 𝑡 (d𝑢) is non-trivial, so 𝜆 ↦→ 𝑣 𝑡 (𝜆) is strictly concave. The equality 𝑣¯ = 𝑣 𝑡 ( 𝑣¯ ) follows by letting 𝑠 → ∞ in 𝑣¯ 𝑡+𝑠 = 𝑣 𝑡 ( 𝑣¯ 𝑠 ), where 𝑣¯ 𝑡+𝑠 ≤ 𝑣¯ 𝑠 . Then 𝑣¯ is clearly the largest solution to 𝑣 𝑡 (𝜆) = 𝜆. When 𝑏 ≥ 0, we have 𝑣¯ = 0 by Theorem 3.11 and Corollary 3.13. Furthermore, since 𝜙(𝑧) ≥ 0, from (3.5) we see 𝑡 ↦→ 𝑣 𝑡 (𝜆) is decreasing, and hence ↓lim𝑡→∞ 𝑣 𝑡 (𝜆) = ↓lim𝑡→∞ 𝑣¯ 𝑡 = 0. If 𝑏 < 0 and 0 < 𝜆 < 𝑣¯ , we have 𝜆 ≤ 𝑣 𝑡 (𝜆) < 𝑣 𝑡 ( 𝑣¯ ) = 𝑣¯ for all 𝑡 ≥ 0. Then the limit 𝑣 ∞ (𝜆) = ↑lim𝑡 ↑∞ 𝑣 𝑡 (𝜆) exists. From the relation 𝑣 𝑡 (𝑣 𝑠 (𝜆)) = 𝑣 𝑡+𝑠 (𝜆) we have 𝑣 𝑡 (𝑣 ∞ (𝜆)) = 𝑣 ∞ (𝜆), and hence 𝑣 ∞ (𝜆) = 𝑣¯ since 𝑣¯ is the unique solution to 𝑣 𝑡 (𝜆) = 𝜆 in (0, ∞). The assertion for 𝑏 < 0 and 𝜆 > 𝑣¯ can be proved similarly. □ We remark that in Theorem 3.14 one usually cannot extend (𝑙 𝑡 )𝑡 >0 to a 𝜎finite entrance law for the semigroup (𝑄 𝑡 )𝑡 ≥0 on R+ . For example, let us assume Condition 3.9 holds and ( 𝑙¯𝑡 )𝑡 >0 is such an extension. For any 0 < 𝑟 < 𝜀 < 𝑡 we have ∫ ∞ ∫ ∞ ¯𝑙 𝑡 ({0}) ≥ 𝑄 𝑡−𝑟 (𝑥, {0})𝑙𝑟 (d𝑥) ≥ e−𝑥 𝑣¯𝑡−𝜀 𝑙𝑟 (d𝑥) 0 ∫ 0 ∞ −𝑢 𝑣¯𝑡−𝜀 = 𝑣¯ 𝑟 − (1 − e )𝑙𝑟 (d𝑢) = 𝑣¯ 𝑟 − 𝑣 𝑟 ( 𝑣¯ 𝑡−𝜀 ). 0

72

3 One-Dimensional Branching Processes

The right-hand side tends to infinity as 𝑟 → 0. Then 𝑙¯𝑡 (d𝑥) cannot be a 𝜎-finite measure on R+ . Example 3.1 Suppose that there are constants 𝑐 > 0, 0 < 𝛼 ≤ 1 and 𝑏 such that 𝜙(𝜆) = 𝑐𝜆1+𝛼 + 𝑏𝜆. Then Condition 3.9 is satisfied. Let 𝑞 𝛼 (0, 𝑡) = 𝛼𝑡 and, for 𝑏 ≠ 0, 𝑞 𝛼 (𝑏, 𝑡) = 𝑏 −1 (1 − e−𝛼𝑏𝑡 ). By solving the equation 𝜕 𝑣 𝑡 (𝜆) = −𝑐𝑣 𝑡 (𝜆) 1+𝛼 − 𝑏𝑣 𝑡 (𝜆), 𝜕𝑡

𝑣 0 (𝜆) = 𝜆

we get 𝑣 𝑡 (𝜆) = 

e−𝑏𝑡 𝜆 1 + 𝑐𝑞 𝛼 (𝑏, 𝑡)𝜆 𝛼

 1/𝛼 ,

𝑡 ≥ 0, 𝜆 ≥ 0.

(3.20)

Thus 𝑣¯ 𝑡 = 𝑐−1/𝛼 e−𝑏𝑡 𝑞 𝛼 (𝑏, 𝑡) −1/𝛼 for 𝑡 > 0. In particular, if 𝛼 = 1, then (3.15) holds with 𝑙 𝑡 (d𝑢) =

n o e−𝑏𝑡 𝑢 exp − d𝑢, 𝑐𝑞 1 (𝑏, 𝑡) 𝑐2 𝑞 1 (𝑏, 𝑡) 2

𝑡 > 0, 𝑢 > 0.

3.2 Long-Time Evolution Rates In this section we study the long-time asymptotic behavior of the CB-process. This makes sense only in the event of non-extinction, of course. Let (𝑄 𝑡 )𝑡 ≥0 denote the transition semigroup defined by (3.2) and (3.3). Let 𝑋 = (Ω, ℱ, ℱ𝑡 , 𝑥(𝑡), Q 𝑥 ) be a Hunt realization of the CB-process. To avoid triviality, we assume 𝜆 ↦→ 𝜙(𝜆) is strictly convex throughout this section. Recall that 𝑣¯ 𝑡 := 𝑣 𝑡 (∞) = ↑lim𝜆→∞ 𝑣 𝑡 (𝜆) ∈ (0, ∞] and 𝑣¯ = ↓lim𝑡→∞ 𝑣¯ 𝑡 ∈ [0, ∞]. By Proposition 3.1 the function 𝜆 ↦→ 𝑣 𝑡 (𝜆) is strictly increasing on [0, ∞) for each 𝑡 ≥ 0. Let 𝑣 ↦→ 𝜂𝑡 (𝑣) denote its inverse, which is a strictly increasing function on [0, 𝑣¯ 𝑡 ). It is easy to show that 𝜂 𝑠 (𝜂𝑡 (𝑧)) = 𝜂 𝑠+𝑡 (𝑧) for any 𝑠, 𝑡 ≥ 0 and 0 ≤ 𝑧 < 𝑣¯ 𝑠+𝑡 . By Proposition 3.3, if 0 < 𝜆 < 𝑣¯ 𝑡 , then the equation 𝜙(𝑧) = 0 has no root between 𝜆 and 𝜂𝑡 (𝜆). From (3.8) we get ∫ 𝜆

𝜂𝑡 (𝜆)

d𝑧 =− 𝜙(𝑧)

∫

𝜆 𝜂𝑡 (𝜆)

d𝑧 = 𝑡. 𝜙(𝑧)

(3.21)

In view of (3.21), we have lim𝑡→∞ 𝜂𝑡 (𝜆) = 0 decreasingly for 0 < 𝜆 < 𝜙−1 (0).

3.2 Long-Time Evolution Rates

73

Theorem 3.17 Let 0 < 𝜆 < 𝑣¯ and define 𝑊𝑡 = 𝜂𝑡 (𝜆)𝑥(𝑡) for 𝑡 ≥ 0. Then 𝑡 ↦→ e−𝑊𝑡 is an (ℱ𝑡 )-martingale and Q 𝑥 [e−𝜃𝑊𝑡 ] = e−𝑥 𝑓𝑡 ( 𝜃) ,

𝑡 ≥ 0, 𝑥 ≥ 0, 𝜃 ≥ 0,

(3.22)

where 𝑓𝑡 (𝜃) = 𝑣 𝑡 (𝜃𝜂𝑡 (𝜆)). If 𝜂𝑡 (𝜆) and 𝜃𝜂𝑡 (𝜆) belong to (0, 𝜙−1 (0)), we have ∫

𝑓𝑡 ( 𝜃)

𝜆

d𝑧 = 𝜙(𝑧)

∫

𝜃 𝜂𝑡 (𝜆) 𝜂𝑡 (𝜆)

d𝑧 . 𝜙(𝑧)

(3.23)

Proof For any 𝑠, 𝑡 ≥ 0 we can use the Markov property of {𝑥(𝑡) : 𝑡 ≥ 0} to get Q 𝑥 [e−𝑊𝑠+𝑡 |ℱ𝑠 ] = Q 𝑥 [e−𝜂𝑠+𝑡 (𝜆) 𝑥 (𝑠+𝑡) |ℱ𝑠 ] = e−𝑣𝑡 ( 𝜂𝑠+𝑡 (𝜆)) 𝑥 (𝑠) = e−𝜂𝑠 (𝜆) 𝑥 (𝑠) . Then 𝑡 ↦→ e−𝑊𝑡 is an (ℱ𝑡 )-martingale. By similar calculations we get (3.22). If 𝜂𝑡 (𝜆) and 𝜃𝜂𝑡 (𝜆) belong to (0, 𝜙−1 (0)), then 𝜙(𝑧) = 0 has no root between 𝜂𝑡 (𝜆) and 𝜃𝜂𝑡 (𝜆). Observe that ∫

𝑓𝑡 ( 𝜃)

𝜆

d𝑧 = 𝜙(𝑧)

𝜂𝑡 (𝜆)

∫

d𝑧 + 𝜙(𝑧)

𝜆

∫

𝜃 𝜂𝑡 (𝜆) 𝜂𝑡 (𝜆)

d𝑧 + 𝜙(𝑧)

𝑣𝑡 ( 𝜃 𝜂𝑡 (𝜆))

∫

𝜃 𝜂𝑡 (𝜆)

d𝑧 . 𝜙(𝑧)

Then (3.23) follows by (3.8) and (3.21).

□

𝑏𝑡 Proposition 3.18 Suppose that 𝑏 < 0 and 0 < 𝜆 < 𝜙−1 (0). Then ∫ ∞𝜂𝑡 (𝜆) = 𝐾e + 𝑏𝑡 𝑜(e ) as 𝑡 → ∞ for some constant 𝐾 = 𝐾 (𝜆) > 0 if and only if 1 𝑧 log 𝑧𝑚(d𝑧) < ∞.

Proof We first note that 𝜙(𝑧) = 𝑏𝑧 + 𝑜(𝑧) as 𝑧 → 0. From (3.21) it follows that  ∫ 𝜆  1 𝜂𝑡 (𝜆) 𝑏 𝜂𝑡 (𝜆) − d𝑧 = −𝑏𝑡 + log = log 𝑏𝑡 , 𝜙(𝑧) 𝑧 𝜆 𝜆e 𝜂𝑡 (𝜆) where the integrand is positive because of the convexity of 𝑧 ↦→ 𝜙(𝑧). Then 𝑡 ↦→ e−𝑏𝑡 𝜂𝑡 (𝜆) increases, and it remains bounded if and only if  ∫ 𝜆 1 𝑏 − d𝑧 < ∞, 𝜙(𝑧) 𝑧 0 which is equivalent to ∫ 0

𝜆

𝜙(𝑧) − 𝑏𝑧 d𝑧 < ∞. 𝑧2

By (3.1) the value on the left-hand side is equal to ∫ 𝑐𝜆 +

∫

𝜆

d𝑧 0

∞ −𝑧𝑢

e 0


1 − 1 + 𝑧𝑢 2 𝑚(d𝑢) = 𝑐𝜆 + 𝑧

∫

∞

𝑢ℎ𝜆 (𝑢)𝑚(d𝑢), 0

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3 One-Dimensional Branching Processes

where ℎ𝜆 (𝑢) =

1 𝑢

∫

𝜆

(e−𝑧𝑢 − 1 + 𝑧𝑢)

0

∫

d𝑧 = 𝑧2

𝜆𝑢

(e−𝑦 − 1 + 𝑦)

0

d𝑦 𝑦2

is equivalent to 𝜆𝑢/2 as 𝑢 → 0 and equivalent to log 𝑢 as 𝑢 → ∞. Then we have the desired result. □ Theorem 3.19 Suppose that 𝑏 < 0 and 0 < 𝜆 < 𝜙−1 (0). Let 𝑊𝑡 = 𝜂𝑡 (𝜆)𝑥(𝑡) for −1 𝑡 ≥ 0. Then the limit 𝑊 := lim𝑡→∞ 𝑊𝑡 exists a.s. and Q 𝑥 {𝑊 = 0} = e−𝑥 𝜙 (0) for any 𝑥 > 0. Proof By Theorem 3.17 and martingale theory, the limit 𝑌 := lim𝑡→∞ e−𝑊𝑡 a.s. exists; see, e.g., Dellacherie and Meyer (1982, p. 72). Recall that 𝜂𝑡 (𝜆) → 0 decreasingly as 𝑡 → ∞. Then for any 𝜃 > 0 we get from (3.23) that ∫

𝑓𝑡 ( 𝜃)

lim 𝑡→∞

𝜆

∫

d𝑧 = lim 𝜙(𝑧) 𝑡→∞

𝜃 𝜂𝑡 (𝜆) 𝜂𝑡 (𝜆)

d𝑧 = lim 𝜙(𝑧) 𝑡→∞

𝜃 𝜂𝑡 (𝜆)

∫

𝜂𝑡 (𝜆)

d𝑧 1 = log 𝜃, 𝑏𝑧 𝑏

so the limit 𝑓 (𝜃) := lim𝑡→∞ 𝑓𝑡 (𝜃) exists and ∫ 𝜆

𝑓 ( 𝜃)

1 d𝑧 = log 𝜃. 𝜙(𝑧) 𝑏

This equality implies lim 𝜃→0 𝑓 (𝜃) = 0 and lim 𝜃→∞ 𝑓 (𝜃) = 𝜙−1 (0). By (3.22) and the dominated convergence theorem, for any 𝑥 > 0 we have Q 𝑥 [𝑌 𝜃 ] = lim Q 𝑥 [e−𝜃𝑊𝑡 ] = lim e−𝑥 𝑓𝑡 ( 𝜃) = e−𝑥 𝑓 ( 𝜃) . 𝑡→∞

𝑡→∞

It follows that Q 𝑥 {𝑌 = 0} = lim Q 𝑥 [1 − 𝑌 𝜃 ] = lim [1 − e−𝑥 𝑓 ( 𝜃) ] = 0 𝜃→0

𝜃→0

and Q 𝑥 {𝑌 = 1} = lim Q 𝑥 [𝑌 𝜃 ] = lim e−𝑥 𝑓 ( 𝜃) = e−𝑥 𝜙 𝜃→∞

−1 (0)

.

𝜃→∞

Then the desired result follows with 𝑊 = − log 𝑌 . Corollary 3.20 Suppose that 𝑏 < 0 and

∫∞ 1

□

𝑧 log 𝑧𝑚(d𝑧) < ∞. Then the limit

𝑍 := lim𝑡→∞ e𝑏𝑡 𝑥(𝑡) exists a.s. and Q 𝑥 {𝑍 = 0} = e−𝑥 𝜙

−1 (0)

for any 𝑥 > 0.

Theorem 3.19 and Corollary 3.20 characterize the long-time evolution rate of the supercritical branching CB-process. In particular, Corollary 3.20 gives a necessary and sufficient condition for the exponential evolution rate. To get similar results in the critical and subcritical case, we consider a special form of the branching mechanism.

3.3 Immigration and Conditioned Processes

75

If 𝑐 = 0 and if 𝑧𝑚(d𝑧) is a finite measure on (0, ∞), we can rewrite (3.1) as ∫ ∞
𝜙(𝜆) = 𝑏 1 𝜆 − (3.24) 1 − e−𝑧𝜆 𝑚(d𝑧), 0

where ∫

∞

𝑏1 = 𝑏 +

(3.25)

𝑧𝑚(d𝑧). 0

In this case, we have 𝜙(𝜆) = 𝑏 1 𝜆 + 𝑜(𝜆) as 𝜆 → ∞, so Theorem 3.11 implies 𝑣¯ = ∞. The following results can be proved by arguments similar to those in the supercritical case. Proposition 3.21 Suppose that 𝜙 is given by (3.24) and (3.25) with 𝑏 ≥ 0. For any fixed 𝜆 > 0 we have 𝜂𝑡 (𝜆) = 𝐾e𝑏1 𝑡 + 𝑜(e𝑏1 𝑡 ) as 𝑡 → ∞ for some constant 𝐾 > 0 if ∫1 and only if 0 𝑧 log(1/𝑧)𝑚(d𝑧) < ∞. Theorem 3.22 Suppose that 𝜙 is given by (3.24) and (3.25) with 𝑏 ≥ 0. Fix 𝜆 > 0 and let 𝑊𝑡 = 𝜂𝑡 (𝜆)𝑥(𝑡) for 𝑡 ≥ 0. Then the limit 𝑊 := lim𝑡→∞ 𝑊𝑡 exists a.s. and Q 𝑥 {𝑊 = 0} = 0 for any 𝑥 > 0. Corollary 3.23 Suppose that 𝜙 is given by (3.24) and (3.25) with 𝑏 ≥ 0 and ∫1 𝑧 log(1/𝑧)𝑚(d𝑧) < ∞. Then the limit 𝑍 := lim𝑡→∞ e𝑏1 𝑡 𝑥(𝑡) a.s. exists and 0 Q 𝑥 {𝑍 = 0} = 0 for any 𝑥 > 0.

3.3 Immigration and Conditioned Processes We first consider a generalization of the CB-process. Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup defined by (3.2) and (3.3). Suppose that 𝜓 ∈ ℐ is a function with the representation ∫ ∞
𝑧 ≥ 0, 𝜓(𝜆) = 𝛽𝜆 + 1 − e−𝑧𝜆 𝑛(d𝑧), (3.26) 0

where 𝛽 ≥ 0 is a constant and (1 ∧ 𝑧)𝑛(d𝑧) is a finite measure on (0, ∞). By Theorems 1.36 and 1.38 one may see that   ∫ 𝑡 ∫ −𝜆𝑦 𝜆≥0 𝜓(𝑣 𝑠 (𝜆))d𝑠 , e 𝛾𝑡 (d𝑦) = exp − (3.27) [0,∞)

0

defines a family of infinitely divisible probability measures (𝛾𝑡 )𝑡 ≥0 on [0, ∞). Then we can define the probability measures 𝛾

𝑄 𝑡 (𝑥, ·) := 𝑄 𝑡 (𝑥, ·) ∗ 𝛾𝑡 (·),

𝑡, 𝑥 ≥ 0.

(3.28)

76

3 One-Dimensional Branching Processes

It is easily seen that  ∫ ∫ −𝜆𝑦 𝛾 e 𝑄 𝑡 (𝑥, d𝑦) = exp − 𝑥𝑣 𝑡 (𝜆) − [0,∞)



𝑡

𝜓(𝑣 𝑠 (𝜆))d𝑠 .

(3.29)

0 𝛾

Moreover, the kernels (𝑄 𝑡 )𝑡 ≥0 form a Feller transition semigroup on R+ . A Markov process is called a continuous-state branching process with immigration (CBIprocess) with branching mechanism 𝜙 and immigration mechanism 𝜓 if it has 𝛾 transition semigroup (𝑄 𝑡 )𝑡 ≥0 . The intuitive meaning of the CBI-process is clear from (3.28), that is, the immigration at the time interval (0, 𝑡] results in the dis𝛾 tribution 𝛾𝑡 = 𝑄 𝑡 (0, ·). In particular, if 𝑧𝑛(d𝑧) is a finite measure on (0, ∞), we have ∫ ∞ ∫ 𝑡 𝛾 𝑦𝑄 𝑡 (𝑥, d𝑦) = 𝑥e−𝑏𝑡 + 𝜓 ′ (0+) e−𝑏𝑡 d𝑠, (3.30) 0

0

where 𝜓 ′ (0+) = 𝛽 +

∞

∫

𝑧𝑛(d𝑧).

(3.31)

0

Theorem 3.24 Suppose that 𝑏 ≥ 0 and 𝜙(𝜆) ≠ 0 for all 𝜆 > 0. Then the CBI-process 𝛾 with transition semigroup (𝑄 𝑡 )𝑡 ≥0 is ergodic if and only if ∫

𝜆

0

𝜓(𝑧) d𝑧 < ∞ for some 𝜆 > 0. 𝜙(𝑧)

(3.32)

In this case, the unique stationary distribution 𝜂 of the process is given by, for 𝜆 ≥ 0,   ∫ ∞   ∫ 𝜆 𝜓(𝑧) d𝑧 . (3.33) 𝐿 𝜂 (𝜆) = exp − 𝜓(𝑣 𝑠 (𝜆))d𝑠 = exp − 0 0 𝜙(𝑧) Proof Since 𝜙(𝜆) ≥ 0 for all 𝜆 ≥ 0, from (3.5) we see 𝑡 ↦→ 𝑣 𝑡 (𝜆) is decreasing. Then (3.8) implies lim𝑡→∞ 𝑣 𝑡 (𝜆) = 0. By (3.29) we have  ∫ ∞  ∫ −𝜆𝑦 𝛾 lim e 𝑄 𝑡 (𝑥, d𝑦) = exp − 𝜓(𝑣 𝑠 (𝜆))d𝑠 (3.34) 𝑡→∞

[0,∞)

0

for every 𝜆 ≥ 0. A further application of (3.5) gives ∫

∫

𝑡

𝜆

𝜓(𝑣 𝑠 (𝜆))d𝑠 = 𝑣𝑡 (𝜆)

0

𝜓(𝑧) d𝑧. 𝜙(𝑧)

It follows that ∫

∞

∫ 𝜓(𝑣 𝑠 (𝜆))d𝑠 =

0

0

𝜆

𝜓(𝑧) d𝑧, 𝜙(𝑧)

3.3 Immigration and Conditioned Processes

77

which is a continuous function of 𝜆 ≥ 0 if and only if (3.32) holds. In this case, we see by (3.34) and Theorem 1.21 that (3.33) defines a probability measure 𝜂 and 𝛾 𝑄 𝑡 (𝑥, ·) → 𝜂 weakly for every 𝑥 ≥ 0 as 𝑡 → ∞. This proves the desired result. □ Corollary 3.25 Suppose that 𝑏 >∫0. Then the CBI-process with transition semigroup ∞ 𝛾 (𝑄 𝑡 )𝑡 ≥0 is ergodic if and only if 1 log 𝑧𝑛(d𝑧) < ∞. Proof We have 𝜙(𝑧) = 𝑏𝑧 + 𝑜(𝑧) as 𝑧 → 0. Thus (3.32) holds if and only if ∫ 0

𝜆

𝜓(𝑧) d𝑧 < ∞ for some 𝜆 > 0, 𝑧

which is equivalent to ∫ 0

𝜆

d𝑧 𝑧

∫

∞


1 − e−𝑧𝑢 𝑛(d𝑢) =

0

∫

∞

∫ 𝑛(d𝑢)

0

for some 𝜆 > 0. The latter holds if and only if the result by Theorem 3.24.

0

∫∞ 1

𝜆𝑢

1 − e−𝑦 d𝑦 < ∞ 𝑦

log 𝑧𝑛(d𝑧) < ∞. Then we have □

The fact that the CBI-process may have a non-trivial stationary distribution makes it a more interesting model in many respects than the CB-process without immigration. Example 3.2 Suppose that 𝑐 > 0, 0 < 𝛼 ≤ 1 and 𝑏 are constants and let 𝜙(𝜆) = 𝑐𝜆1+𝛼 + 𝑏𝜆 for 𝜆 ≥ 0. In this case the cumulant semigroup (𝑣 𝑡 )𝑡 ≥0 is given by (3.20). Let 𝛽 ≥ 0 and let 𝜓(𝜆) = 𝛽𝜆 𝛼 for 𝜆 ≥ 0. We can use (3.29) to define the transition 𝛾 semigroup (𝑄 𝑡 )𝑡 ≥0 . It is easy to show that ∫ 1 𝛾 −𝑥𝑣𝑡 (𝜆) e−𝜆𝑦 𝑄 𝑡 (𝑥, d𝑦) =  , 𝜆 ≥ 0. (3.35)  𝛽/𝑐 𝛼 e [0,∞) 1 + 𝑐𝑞 𝛼 (𝑏, 𝑡)𝜆 𝛼 Example 3.3 In the special case of 𝛼 = 1, the CBI-process {𝑦(𝑡) : 𝑡 ≥ 0} with 𝛾 transition semigroup (𝑄 𝑡 )𝑡 ≥0 given by (3.35) is a diffusion process. This special CBI-process solves the stochastic differential equation √︁ d𝑦(𝑡) = 2𝑐𝑦(𝑡)d𝐵(𝑡) + (𝛽 − 𝑏𝑦(𝑡))d𝑡, 𝑡 ≥ 0, (3.36) where {𝐵(𝑡) : 𝑡 ≥ 0} is a standard Brownian motion; see, e.g., Ikeda and Watanabe (1989, p. 235) and Shiga and Watanabe (1973). Let 𝐶 2 (R+ ) denote the set of bounded continuous real functions on R+ with bounded continuous derivatives up to the second order. Then this diffusion process has generator 𝐴 determined by 𝐴 𝑓 (𝑥) = 𝑐

d2 d 𝑓 (𝑥) + (𝛽 − 𝑏𝑥) 𝑓 (𝑥), 2 d𝑥 d𝑥

𝑓 ∈ 𝐶 2 (R+ ).

(3.37)

78

3 One-Dimensional Branching Processes

It is known as the Cox–Ingersoll–Ross model (CIR-model) in mathematical finance. In particular, for 𝛽 = 0 the solution of (3.36) is called Feller’s branching diffusion. Recall that (𝑄 ◦𝑡 )𝑡 ≥0 is the restriction to (0, ∞) of the semigroup (𝑄 𝑡 )𝑡 ≥0 . It is easy to check that 𝑄 𝑡𝑏 (𝑥, d𝑦) := e𝑏𝑡 𝑥 −1 𝑦𝑄 ◦𝑡 (𝑥, d𝑦) defines a Markov semigroup on (0, ∞). Let 𝑞 𝑡 (𝜆) = e𝑏𝑡 𝑣 𝑡 (𝜆) and let 𝑞 𝑡′ (𝜆) = (𝜕/𝜕𝜆)𝑞 𝑡 (𝜆). Recall that 𝜆 ↦→ 𝜙 ′ (𝜆) is defined by (3.10). From (3.9) we have  ∫ 𝑡  ′ ′ (3.38) 𝜙0 (𝑣 𝑠 (𝜆))d𝑠 , 𝑞 𝑡 (𝜆) = exp − 0

where 𝜙0′ (𝑧) = 𝜙 ′ (𝑧) − 𝑏. By differentiating both sides of (3.2) we see ∫

∞

e−𝜆𝑦 𝑄 𝑡𝑏 (𝑥, d𝑦) = exp{−𝑥𝑣 𝑡 (𝜆)}𝑞 𝑡′ (𝜆),

𝜆 ≥ 0.

(3.39)

0

By (3.39) it is easy to extend (𝑄 𝑡𝑏 )𝑡 ≥0 to a Feller semigroup on [0, ∞). From (3.38) we have   ∫ 𝑡 ∫ e−𝜆𝑦 𝑄 𝑡𝑏 (𝑥, d𝑦) = exp − 𝑥𝑣 𝑡 (𝜆) − 𝜙0′ (𝑣 𝑠 (𝜆))d𝑠 . (3.40) [0,∞)

0

This is clearly a special case of the semigroup defined by (3.29). Theorem 3.26 Let (ℎ𝑡 , 𝑙 𝑡 )𝑡 ≥0 be defined by the canonical representation (3.4). Then for any 𝑡 ≥ 0 we have 𝑄 𝑡𝑏 (0, d𝑦) = e𝑏𝑡 ℎ𝑡 𝛿0 (d𝑦) + e𝑏𝑡 𝑦𝑙 𝑡 (d𝑦),

𝑦 ≥ 0.

(3.41)

Proof Clearly, the probability measure 𝑄 𝑡𝑏 (0, d𝑦) has Laplace transform 𝑞 𝑡′ (𝜆). By (3.4) and the definition of 𝑞 𝑡 (𝜆) we have ∫ ∞
𝑞 𝑡 (𝜆) = e𝑏𝑡 ℎ𝑡 𝜆 + 1 − e−𝜆𝑢 e𝑏𝑡 𝑙 𝑡 (d𝑢), 0

and hence 𝑞 𝑡′ (𝜆) = e𝑏𝑡 ℎ𝑡 +

∫

∞

𝑢e−𝜆𝑢 e𝑏𝑡 𝑙 𝑡 (d𝑢).

(3.42)

0

Then (3.41) follows.

□

Corollary 3.27 For any 𝑡 > 0 the probability measure 𝑄 𝑡𝑏 (0, ·) is supported by (0, ∞) if and only if 𝜙 ′ (∞) = ∞. In this case, we have 𝑄 𝑡𝑏 (0, d𝑦) = 𝑦e𝑏𝑡 𝑙 𝑡 (d𝑦) for 𝑦 > 0, where (𝑙 𝑡 )𝑡 >0 is defined by (3.15).

3.3 Immigration and Conditioned Processes

79

Theorem 3.28 Suppose that 𝑏 > 0. Then for every 𝜆 ≥ 0 the limit 𝑞 ′ (𝜆) := ↓lim𝑡→∞ 𝑞 𝑡′ (𝜆) exists and is given by  ∫ ∞  ′ ′ 𝜙0 (𝑣 𝑠 (𝜆))d𝑠 , 𝑞 (𝜆) = exp − 𝜆 ≥ 0. (3.43) 0

Moreover, the CBI-process with transition semigroup (𝑄 𝑡𝑏 )𝑡 ≥0 is ergodic if and only ∫∞ if 1 𝑧 log 𝑧𝑚(d𝑧) < ∞. In this case, the unique stationary distribution 𝜂 of the process has Laplace transform 𝐿 𝜂 = 𝑞 ′ given by (3.43). Proof The first assertion is immediate by (3.38). In view of (3.40), the other two □ assertions follow from Theorem 3.24 and Corollary 3.25. Corollary 3.29 Let 𝑞 ′ (𝜆) be defined by (3.43). Then the following properties are (i) 𝑞 ′ (0+) = 𝑞 ′ (0) = 1; (ii) 𝑞 ′ (𝜆) > 0 for some and hence all 𝜆 > 0; equivalent: ∫∞ (iii) 1 𝑧 log 𝑧𝑚(d𝑧) < ∞. ∫∞ Corollary 3.30 Suppose that 𝑏 > 0, 𝜙 ′ (∞) = ∞ and 1 𝑧 log 𝑧𝑚(d𝑧) < ∞. Then the unique stationary distribution 𝜂 of the CBI-process with transition semigroup (𝑄 𝑡𝑏 )𝑡 ≥0 is supported by (0, ∞). Proof Under the conditions, by Corollary 3.27 the probability measure 𝑄 𝑡𝑏 (0, ·) is supported by (0, ∞) for 𝑡 > 0. Then 𝑞 𝑡′ (∞) = 0 by (3.39). In view of (3.38) and (3.43), we have 𝐿 𝜂 (∞) = 𝑞 ′ (∞) = 0, which implies the desired result. □ Now let 𝑋 = (Ω, ℱ, ℱ𝑡 , 𝑥(𝑡), Q 𝑥 ) be a Hunt realization of the CB-process with transition semigroup (𝑄 𝑡 )𝑡 ≥0 . Let 𝜏0 := inf{𝑠 ≥ 0 : 𝑥(𝑠) = 0} denote the extinction time of 𝑋. Theorem 3.31 Suppose that 𝑏 ≥ 0 and Condition 3.9 holds. Then for any 𝑇 ≥ 𝑡 ≥ 0 and 𝑥 > 0, the distribution of 𝑥(𝑡) under Q 𝑥 {·|𝑇 + 𝑟 < 𝜏0 } converges as 𝑟 → ∞ to 𝑄 𝑡𝑏 (𝑥, ·). Proof By Theorem 3.8 and the Markov property of {𝑥(𝑡) : 𝑡 ≥ 0}, for any 𝑟 > 0 we have    Q 𝑥 e−𝜆𝑥 (𝑡) 1 {𝑇+𝑟 0. Then P𝑏,𝑇 𝑥 (d𝜔) = 𝑥 e 𝑥(𝜔, 𝑇) Q 𝑥 (d𝜔) defines a probability measure on (Ω, ℱ𝑇 ), under which {(𝑥(𝑡), ℱ𝑡 ) : 0 ≤ 𝑡 ≤ 𝑇 } is a Markov process with transition semigroup (𝑄 𝑡𝑏 )𝑡 ≥0 given by (3.40).

is carried by {𝑥(𝑇) > 0} ∈ ℱ𝑇 . Then we have Proof Clearly, the measure P𝑏,𝑇 𝑥 > 1 0 ≤ 𝑡 ≤ 𝑇. Let 0 ≤ 𝑟 ≤ 𝑡 ≤ 𝑇 and let 𝐹 be a {𝑥(𝑡) for P𝑏,𝑇 0} every = 𝑥 bounded ℱ𝑟 -measurable random variable. For any 𝑓 ∈ 𝐵(R+ ), by (3.7) and the Markov property under Q 𝑥 we have   −1 𝑏𝑇 P𝑏,𝑇 𝑥 [𝐹 𝑓 (𝑥(𝑡))] = 𝑥 e Q 𝑥 𝐹 𝑓 (𝑥(𝑡))𝑥(𝑇)  = 𝑥 −1 e𝑏𝑡 Q 𝑥 𝐹 𝑓 (𝑥(𝑡))𝑥(𝑡)  𝑏  = 𝑥 −1 e𝑏𝑟 Q 𝑥 𝐹𝑄 𝑡−𝑟 𝑓 (𝑥(𝑟))𝑥(𝑟)  𝑏  = 𝑥 −1 e𝑏𝑇 Q 𝑥 𝐹𝑄 𝑡−𝑟 𝑓 (𝑥(𝑟))𝑥(𝑇)   𝑏 𝑓 (𝑥(𝑟)) . 𝐹𝑄 𝑡−𝑟 = P𝑏,𝑇 𝑥 Then {(𝑥(𝑡), ℱ𝑡 ) : 0 ≤ 𝑡 ≤ 𝑇 } under P𝑏,𝑇 is a Markov process with transition 𝑥 semigroup (𝑄 𝑡𝑏 )𝑡 ≥0 . □ By a modification of the proof of Theorem 3.31 we have the following: Theorem 3.33 Suppose that 𝑏 ≥ 0 and Condition 3.9 holds. Let 𝑥 > 0 and 𝑇 ≥ 0. Then for any bounded ℱ𝑇 -measurable random variable 𝐹 we have lim Q 𝑥 [𝐹 |𝑇 + 𝑟 < 𝜏0 ] = P𝑏,𝑇 𝑥 (𝐹).

𝑟→∞

The above theorem shows that in the critical and subcritical cases the probability measure P𝑏,𝑇 𝑥 is intuitively the law conditioned on large extinction times. Some more conditional limit theorems of the CB-process will be given in the next section.

3.4 More Conditional Limit Theorems Throughout this section, we assume Condition 3.9 is satisfied. Let us consider a Hunt realization 𝑋 = (Ω, ℱ, ℱ𝑡 , 𝑥(𝑡), Q 𝑥 ) of the transition semigroup (𝑄 𝑡 )𝑡 ≥0 defined by (3.2) and (3.3). Then 𝑣¯ 𝑡 := 𝑣 𝑡 (∞) = ↑lim𝜆→∞ 𝑣 𝑡 (𝜆) ∈ (0, ∞) for 𝑡 > 0 by Theorem 3.10 and 𝑣¯ := ↓ lim𝑡→∞ 𝑣¯ 𝑡 ∈ [0, ∞) by Theorem 3.11. Let 𝜏0 := inf{𝑠 ≥ 0 : 𝑥(𝑠) = 0} denote the extinction time. Recall that (𝑄 ◦𝑡 )𝑡 ≥0 denotes the restriction to (0, ∞) of the semigroup (𝑄 𝑡 )𝑡 ≥0 .

3.4 More Conditional Limit Theorems

81

Theorem 3.34 Suppose that 𝑏 > 0. Then the limit 𝑔(𝜆) := ↑lim𝑡→∞ 𝑣¯ −1 𝑡 𝑣 𝑡 (𝜆) exists for every 𝜆 ≥ 0 and 0 = 𝑔(0) = 𝑔(0+) ≤ 𝑔(𝜆) ≤ 𝑔(∞) = 1. Consequently, 𝑣¯ −1 𝑡 𝑙𝑡 converges as 𝑡 → ∞ to a probability measure 𝜋0 on (0, ∞) with Laplace transform 𝐿 𝜋0 (𝜆) = 1 − 𝑔(𝜆). −1 Proof Let 𝑔𝑡 (𝜆) = 𝑣¯ −1 𝑡 𝑣 𝑡 (𝜆) and ℎ 𝑡 (𝜆) = 𝜆 𝑣 𝑡 (𝜆) for 𝜆 ≥ 0. Then 0 ≤ 𝑔𝑡 (𝜆) ≤ 1 and

𝑔𝑡+𝑠 (𝜆) = 𝑣 𝑠 ( 𝑣¯ 𝑡 ) −1 𝑣 𝑠 (𝑣 𝑡 (𝜆)) = ℎ 𝑠 ( 𝑣¯ 𝑡 ) −1 ℎ 𝑠 (𝑣 𝑡 (𝜆))𝑔𝑡 (𝜆)

(3.44)

for 𝑠, 𝑡 > 0. Since 𝑣 𝑡 (0) = 0 and 𝜆 ↦→ 𝑣 𝑡 (𝜆) is a concave function, we have ℎ𝑡′ (𝜆) = 𝜆−2 [𝑣 𝑡′ (𝜆)𝜆 − 𝑣 𝑡 (𝜆)] ≤ 0, so 𝜆 ↦→ ℎ𝑡 (𝜆) is decreasing. Thus 𝑡 ↦→ 𝑔𝑡 (𝜆) is increasing by (3.44). Consequently, the limit 𝑔(𝜆) = ↑lim𝑡→∞ 𝑔𝑡 (𝜆) exists and 0 = 𝑔(0) ≤ 𝑔(𝜆) ≤ 𝑔(∞) = 1. Observe also that −1 −1 𝑔𝑡 (𝑣 𝑠 (𝜆)) = 𝑣¯ −1 𝑡 𝑣 𝑡+𝑠 (𝜆) = 𝑣¯ 𝑡+𝑠 𝑣 𝑡+𝑠 (𝜆) 𝑣¯ 𝑡 𝑣 𝑠 ( 𝑣¯ 𝑡 ) = 𝑔𝑡+𝑠 (𝜆)ℎ 𝑠 ( 𝑣¯ 𝑡 ).

(3.45)

By Theorem 3.11 we have lim𝑡→∞ 𝑣¯ 𝑡 = 0, so lim𝑡→∞ ℎ 𝑠 ( 𝑣¯ 𝑡 ) = 𝑣 𝑠′ (0) = e−𝑏𝑠 . Taking 𝑡 → ∞ in (3.45) gives 𝑔(𝑣 𝑠 (𝜆)) = e−𝑏𝑠 𝑔(𝜆),

𝜆 ≥ 0, 𝑠 ≥ 0.

(3.46)

Then we must have 𝑔(0+) = 𝑔(0) = 0. From the relation ∫ ∞ −1 e−𝜆𝑢 𝑣¯ −1 lim 𝑡 𝑙 𝑡 (d𝑢) = 1 − lim 𝑣¯ 𝑡 𝑣 𝑡 (𝜆) = 1 − 𝑔(𝜆) 𝑡→∞

𝑡→∞

0

we see that 𝑣¯ −1 𝑡 𝑙 𝑡 converges as 𝑡 → ∞ to a probability measure 𝜋0 on [0, ∞) with □ Laplace transform 1 − 𝑔(𝜆). Since 𝑔(∞) = 1, we have 𝜋0 ({0}) = 0. Theorem 3.35 Suppose that 𝑏 > 0. Then for any 0 ≤ 𝜆 ≤ ∞, the limit 𝑞(𝜆) := ↓lim𝑡→∞ 𝑞 𝑡 (𝜆) exists (with 𝑞 𝑡 (∞) = e𝑏𝑡 𝑣¯ 𝑡 by convention). Moreover, we have 𝑞(𝜆) > ∫∞ 0 for some and hence all 0 < 𝜆 ≤ ∞ if and only if 1 𝑧 log 𝑧𝑚(d𝑧) < ∞. Proof By Theorem 3.28 and the dominated convergence theorem it is easy to see 𝑞(𝜆) = ↓lim𝑡→∞ 𝑞 𝑡 (𝜆) for all 0 ≤ 𝜆 < ∞, where ∫ 𝑞(𝜆) =

𝜆

𝑞 ′ (𝑢)d𝑢.

0

This convergence can be extended to 𝜆 = ∞ by Theorem 3.10. Since 𝑞(∞) ≥ 𝑞(𝜆) □ for all 0 < 𝜆 < ∞, the second assertion is immediate by Corollary 3.29. Corollary 3.36 Suppose that 𝑏 >∫ 0. Then e𝑏𝑡 𝑙 𝑡 converges as 𝑡 → ∞ to 𝑞(∞)𝜋0 , ∞ which is non-trivial if and only if 1 𝑧 log 𝑧𝑚(d𝑧) < ∞. Theorem 3.37 Suppose that 𝑏 > 0. Then for 𝑟 ≥ 0 and 𝑥 > 0 the distribution of 𝑥(𝑡) under Q 𝑥 {·|𝑟 + 𝑡 < 𝜏0 } converges as 𝑡 → ∞ to a probability measure 𝜋𝑟 on (0, ∞) independent of 𝑥. Moreover, 𝜋0 is also the limit distribution of 𝑣¯ −1 𝑡 𝑙 𝑡 given by Theorem 3.34 and 𝜋0 𝑄 ◦𝑡 = e−𝑏𝑡 𝜋0 for all 𝑡 ≥ 0.

82

3 One-Dimensional Branching Processes

Proof For 𝑟 > 0 we can use calculations similar to those in the proof of Theorem 3.31 to see e−𝑥𝑣𝑡 (𝜆) − e−𝑥𝑣𝑡 (𝜆+𝑣¯𝑟 ) . 1 − e−𝑥 𝑣¯𝑟+𝑡

Q 𝑥 [e−𝜆𝑥 (𝑡) |𝑟 + 𝑡 < 𝜏0 ] =

(3.47)

By letting 𝑟 → 0 we obtain Q 𝑥 [e−𝜆𝑥 (𝑡) |𝑡 < 𝜏0 ] =

1 − e−𝑥𝑣𝑡 (𝜆) e−𝑥𝑣𝑡 (𝜆) − e−𝑥 𝑣¯𝑡 = 1 − . 1 − e−𝑥 𝑣¯𝑡 1 − e−𝑥 𝑣¯𝑡

Let 𝜋0 be the probability measure on (0, ∞) given by Theorem 3.34. It follows that lim Q 𝑥 [e−𝜆𝑥 (𝑡) |𝑡 < 𝜏0 ] = 1 − lim 𝑣¯ −1 𝑡 𝑣 𝑡 (𝜆) = 𝐿 𝜋0 (𝜆),

𝑡→∞

(3.48)

𝑡→∞

so the distribution of 𝑥(𝑡) under Q 𝑥 {·|𝑡 < 𝜏0 } converges to 𝜋0 . In view of (3.46) we have ∫ ∞ ∫ ∞
−𝑣𝑡 (𝜆)𝑢
1−e 𝜋0 (d𝑢) = 1 − e−𝜆𝑢 e−𝑏𝑡 𝜋0 (d𝑢), 0

0

and hence 𝜋0 𝑄 ◦𝑡 = e−𝑏𝑡 𝜋0 . On the other hand, as 𝑡 → ∞ the right-hand side of (3.47) is equivalent to −1 𝑣¯ 𝑟+𝑡 (𝑣 𝑡 (𝜆 + 𝑣¯ 𝑟 ) − 𝑣 𝑡 (𝜆)) = 𝑣 𝑡 ( 𝑣¯ 𝑟 ) −1 (𝑣 𝑡 (𝜆 + 𝑣¯ 𝑟 ) − 𝑣 𝑡 (𝜆)).

Using the canonical representation (3.15) we may write this as ∞

∫


1 − e−𝑣¯𝑟 𝑢 𝑙 𝑡 (d𝑢)

 −1 ∫

0

∞


e−𝜆𝑢 1 − e−𝑣¯𝑟 𝑢 𝑙 𝑡 (d𝑢),

0

which converges as 𝑡 → ∞ to ∫

 −1 ∫

∞

1−e

−𝑣¯𝑟 𝑢


𝜋0 (d𝑢)

0

∞


e−𝜆𝑢 1 − e−𝑣¯𝑟 𝑢 𝜋0 (d𝑢),

(3.49)

0

giving the Laplace transform of a probability 𝜋𝑟 on (0, ∞).

□

Corollary ∫ ∞ 3.38 Suppose that 𝑏 > 0. Let 𝜋0 be given by Theorem 3.34. Then we have 0 𝑢𝜋0 (d𝑢) = 𝑞(∞) −1 . Consequently, 𝜋0 has finite mean if and only if ∫∞ 𝑧 log 𝑧𝑚(d𝑧) < ∞. 1 Proof By Theorem 3.34 the measure 𝜋0 has Laplace transform 1 − 𝑔(𝜆). Letting 𝜆 → ∞ in (3.46) we have 𝑔( 𝑣¯ 𝑠 ) = e−𝑏𝑠 . By Theorem 3.11 and Corollary 3.13 we have lim𝑠→∞ 𝑣¯ 𝑠 = 0. It follows that −1 −𝑏𝑠 𝑔 ′ (0) = lim 𝑣¯ −1 = lim 𝑞 𝑠 (∞) −1 = 𝑞(∞) −1 , 𝑠 𝑔( 𝑣¯ 𝑠 ) = lim 𝑣¯ 𝑠 e 𝑠→∞

𝑠→∞

𝑠→∞

which together with Theorem 3.35 gives the desired conclusion.

□

3.4 More Conditional Limit Theorems

83

There are counterparts of the above conditional limit theorems in the supercritical case. In fact, there is a symmetry in the limit theorems between the subcritical and supercritical processes if we use suitable conditioning. In the strictly supercritical case, we have 𝑏 < 0 and Q 𝑥 [e−𝜆𝑥 (𝑡) |𝜏0 < ∞] = e−𝑥𝑤𝑡 (𝜆) ,

𝜆 ≥ 0,

(3.50)

where 𝑤 𝑡 (𝜆) = 𝑣 𝑡 (𝜆 + 𝑣¯ ) − 𝑣¯ . Setting 𝜓(𝜆) = 𝜙(𝜆 + 𝑣¯ ) one may see that 𝑤 𝑡 (𝜆) satisfies 𝜕 𝑤 𝑡 (𝜆) = −𝜓(𝑤 𝑡 (𝜆)), 𝜕𝑡

𝑤 0 (𝜆) = 𝜆.

(3.51)

Recall that 𝑣¯ > 0 is the largest root of 𝜙(𝑧) = 0. Then it is simple to check that 𝜓(𝜆) = 𝜙(𝜆 + 𝑣¯ ) − 𝜙( 𝑣¯ ) has the representation (3.1) with parameters 𝑏 𝜓 := 𝜙 ′ ( 𝑣¯ ) > ¯ 𝑚(d𝑧). Thus (3.50) implies that {𝑥(𝑡) : 𝑡 ≥ 0} 0, 𝑐 𝜓 := 𝑐 and 𝑚 𝜓 (d𝑧) := e−𝑣𝑧 conditioned on 𝜏0 < ∞ is a strictly subcritical CB-process with cumulant semigroup (𝑤 𝑡 )𝑡 ≥0 . By Corollary 3.16 we have 𝑣 𝑡 ( 𝑣¯ ) = 𝑣¯ , so 𝑤 𝑡 (𝜆) = 𝑣 𝑡 (𝜆 + 𝑣¯ ) − 𝑣 𝑡 ( 𝑣¯ ) has the ¯ 𝑙 (d𝑢) for 𝑡 > 0. representation (3.15) with canonical measure e−𝑣𝑢 𝑡 Theorem 3.39 Suppose that 𝑏 < 0. Then for any 𝑡 ≥ 0 and 𝑥 > 0 the distribution of 𝑥(𝑡) under Q 𝑥 {·|𝑟 + 𝑡 < 𝜏0 < ∞} converges as 𝑟 → ∞ to a probability measure 𝑄¯ 𝑡𝑏 (𝑥, ·) on (0, ∞) given by   ∫ ∞ ∫ 𝑡 −𝜆𝑦 ¯ 𝑏 ′ e 𝑄 𝑡 (𝑥, d𝑦) = exp − 𝑥𝑤 𝑡 (𝜆) − 𝜓0 (𝑤 𝑠 (𝜆))d𝑠 , 0

0

where 𝜓0′ (𝑧) = 𝜙 ′ (𝑧+ 𝑣¯ )−𝑏 𝜓 . Moreover, 𝑄¯ 𝑡𝑏 (𝑥, ·) converges as 𝑡 → ∞ to a probability measure 𝜂 on (0, ∞). Proof The first assertion follows from Theorem 3.31. Since Condition 3.9 is assumed, we have 𝜙 ′ (∞) = ∞. Then the second assertion is a consequence of Theorem 3.28 and Corollary 3.30. □ Similarly, from Theorem 3.37 we derive the following: Theorem 3.40 Suppose that 𝑏 < 0. Then for 𝑟 ≥ 0 and 𝑥 > 0 the distribution of 𝑥(𝑡) under Q 𝑥 {·|𝑟 + 𝑡 < 𝜏0 < ∞} converges as 𝑡 → ∞ to a probability measure 𝜋𝑟 on (0, ∞) which is independent of 𝑥. Moreover, 𝜋0 (d𝑢) is also the limit distribution ¯ 𝑙 (d𝑢). of ( 𝑣¯ 𝑡 − 𝑣¯ ) −1 e−𝑣𝑢 𝑡 We now consider the critical CB-process. In this case, we shall see that suitable conditioning of the process may lead to some universal limit laws independent of the explicit form of the branching mechanism.

84

3 One-Dimensional Branching Processes

Theorem 3.41 Suppose that 𝑏 = 0 and 𝜎 2 := 𝜙 ′′ (0) < ∞. Then as 𝑡 → ∞ we have   1 1 1 1 − → 𝜎2 𝑡 𝑣 𝑡 (𝜆) 𝜆 2 uniformly in 0 < 𝜆 ≤ ∞ with the convention 1/∞ = 0. In particular, we have 𝑡𝑣 𝑡 (𝜆) → 2/𝜎 2 for 0 < 𝜆 ≤ ∞ as 𝑡 → ∞. Proof Since Condition 3.9 holds, we have 𝜎 2 > 0. For 0 < 𝜆 ≤ ∞ and 𝑡 > 0, we may use the backward equation (3.5) to see that   ∫ ∫ 1 1 1 𝑡 1 𝜕 1 1 𝑡 𝜙(𝑣 𝑠 (𝜆)) − =− d𝑠. (3.52) 𝑣 𝑠 (𝜆)d𝑠 = 𝑡 𝑣 𝑡 (𝜆) 𝜆 𝑡 0 𝑣 𝑠 (𝜆) 2 𝜕𝑠 𝑡 0 𝑣 𝑠 (𝜆) 2 By l’Hôpital’s rule, lim 𝜙(𝑧)/𝑧2 = lim 𝜙 ′′ (𝑧)/2 = 𝜎 2 /2. 𝑧↓0

(3.53)

𝑧↓0

But by Theorem 3.11 and Corollary 3.13, we have lim𝑡→∞ 𝑣¯ 𝑡 = 0, and hence lim𝑡→∞ 𝑣 𝑡 (𝜆) = 0 uniformly on 0 < 𝜆 ≤ ∞. Then the assertion follows from (3.52) and (3.53). □ Corollary 3.42 Suppose that 𝑏 = 0 and 𝜎 2 := 𝜙 ′′ (0) < ∞. Then for any 𝜆 ≥ 0 we have lim 𝑣 ′ (𝜆/𝑡) 𝑡→∞ 𝑡

= (1 + 𝜎 2 𝜆/2) −2 .

Proof For 𝜆 = 0 the above limit relation holds trivially. For 𝜆 > 0 we can use (3.5) and (3.6) to get 𝑣 𝑡′ (𝜆/𝑡) = 𝜙(𝜆/𝑡) −1 𝜙(𝑣 𝑡 (𝜆/𝑡)). Then the result follows by Theorem 3.41. □ Theorem 3.43 Suppose that 𝑏 = 0 and 𝜎 2 := 𝜙 ′′ (0) < ∞. Let {𝑦(𝑡) : 𝑡 ≥ 0} be a Markov process with transition semigroup (𝑄 𝑡𝑏 )𝑡 ≥0 given by (3.40). Then the distribution of 𝑦(𝑡)/𝑡 converges as 𝑡 → ∞ to the one on (0, ∞) with density 2 4𝜎 −4 𝑥e−2𝑥/𝜎 . Proof In this critical case, we have 𝑞 𝑡′ (𝜆) = 𝑣 𝑡′ (𝜆). Since lim𝑡→∞ 𝑣 𝑡 (𝜆/𝑡) = 0, by (3.39) and Corollary 3.42 we see that ∫ 1 , lim e−𝜆𝑦/𝑡 𝑄 𝑡𝑏 (𝑥, d𝑦) = lim 𝑣 𝑡′ (𝜆/𝑡) = 𝑡→∞ [0,∞) 𝑡→∞ (1 + 𝜎 2 𝜆/2) 2 which is the Laplace transform of the desired limit distribution.

□

3.4 More Conditional Limit Theorems

85

Theorem 3.44 Suppose that 𝑏 = 0 and 𝜎 2 := 𝜙 ′′ (0) < ∞. Then for any fixed 𝑟 ≥ 0 and 𝑥 > 0 we have 2

lim Q 𝑥 {𝑥(𝑡)/𝑡 > 𝑧|𝑟 + 𝑡 < 𝜏0 } = e−2𝑧/𝜎 ,

𝑧 ≥ 0.

(3.54)

𝑡→∞

Proof For any 𝑡 > 0 we get from (3.47) that   e−𝑥𝑣𝑡 (𝜆/𝑡) − e−𝑥𝑣𝑡 (𝜆/𝑡+𝑣¯𝑟 ) Q 𝑥 e−𝜆𝑥 (𝑡)/𝑡 |𝑟 + 𝑡 < 𝜏0 = , 1 − e−𝑥 𝑣¯𝑟+𝑡 which is still correct for 𝑟 = 0 if we understand 𝑣¯ 0 = ∞. The right-hand side is equivalent to −1 𝑣¯ 𝑟+𝑡 (𝑣 𝑡 (𝜆/𝑡 + 𝑣¯ 𝑟 ) − 𝑣 𝑡 (𝜆/𝑡))

as 𝑡 → ∞. By Theorem 3.41 we have lim 𝑡 𝑣¯ 𝑟+𝑡 = 2/𝜎 2 and lim 𝑡𝑣 𝑡 (𝜆/𝑡) = (1/𝜆 + 𝜎 2 /2) −1 . 𝑡→∞

(3.55)

𝑡→∞

From the uniform convergence we get   1 1 𝜎2 1 1 − = . = lim 𝑡→∞ 𝑡𝑣 𝑡 (𝜆/𝑡 + 𝑣¯ 𝑟 ) 𝑡→∞ 𝑡 𝑣 𝑡 (𝜆/𝑡 + 𝑣¯ 𝑟 ) 𝜆/𝑡 + 𝑣¯ 𝑟 2 lim

Then it follows immediately that     𝜎2 2 1 1 − = . lim Q 𝑥 e−𝜆𝑥 (𝑡)/𝑡 |𝑟 + 𝑡 < 𝜏0 = 𝑡→∞ 2 𝜎 2 1/𝜆 + 𝜎 2 /2 1 + 𝜎 2 𝜆/2 The right-hand side gives the Laplace transform of the desired limit distribution. □ Theorem 3.45 Suppose that 𝑏 = 0 and 𝜎 2 := 𝜙 ′′ (0) < ∞. Then for any 𝑥 > 0 and 𝑎 ≥ 0 the distribution of 𝑥(𝑡)/𝑡 under Q 𝑥 {·|(1 + 𝑎)𝑡 < 𝜏0 } converges as 𝑡 → ∞ to the one on (0, ∞) with density 2

2

2𝜎 −2 (1 + 𝑎)e−2𝑥/𝜎 [1 − e−2𝑥/𝑎 𝜎 ]

(3.56)

with e−∞ = 0 by convention. Proof For any 𝑡 > 0 we use (3.47) to get   e−𝑥𝑣𝑡 (𝜆/𝑡) − e−𝑥𝑣𝑡 (𝜆/𝑡+𝑣¯𝑎𝑡 ) Q 𝑥 e−𝜆𝑥 (𝑡)/𝑡 |(1 + 𝑎)𝑡 < 𝜏0 = 1 − e−𝑥 𝑣¯ (1+𝑎) 𝑡 under the convention 𝑣¯ 0 = ∞. The right-hand side is equivalent to 𝑣¯ −1 (1+𝑎)𝑡 (𝑣 𝑡 (𝜆/𝑡 + 𝑣¯ 𝑎𝑡 ) − 𝑣 𝑡 (𝜆/𝑡)) as 𝑡 → ∞. By (3.55) and the uniform convergence stated in Theorem 3.41 we have

86

3 One-Dimensional Branching Processes



1 2 1 1 1 𝜎 = lim − 𝑡→∞ 𝑡 𝑣 𝑡 (𝜆/𝑡 + 𝑣¯ 𝑎𝑡 ) 2 𝜆/𝑡 + 𝑣¯ 𝑎𝑡   1 1 = lim − 𝑡→∞ 𝑡𝑣 𝑡 (𝜆/𝑡 + 𝑣¯ 𝑎𝑡 ) 𝜆 + 𝑡 𝑣¯ 𝑎𝑡 1 1 = lim − . 𝑡→∞ 𝑡𝑣 𝑡 (𝜆/𝑡 + 𝑣¯ 𝑎𝑡 ) 𝜆 + 2/𝑎𝜎 2



It follows that  lim 𝑡𝑣 𝑡 (𝜆/𝑡 + 𝑣¯ 𝑎𝑡 ) = 𝑡→∞

1 𝜎2 + 2 𝜆 + 2/𝑎𝜎 2

 −1 =

𝜎 2 (1

2 + 𝑎𝜎 2 𝜆 . + 𝑎 + 𝑎𝜎 2 𝜆/2)

Then one shows easily   lim Q 𝑥 e−𝜆𝑥 (𝑡)/𝑡 |(1 + 𝑎)𝑡 < 𝜏0 =

𝑡→∞

1+𝑎 , (1 + 𝜎 2 𝜆/2)(1 + 𝑎 + 𝑎𝜎 2 𝜆/2)

which is the Laplace transform of the distribution with density (3.56).

□

3.5 Scaling Limits of Discrete Processes Let 𝑔 and ℎ be two probability generating functions. Suppose that {𝜉 𝑛,𝑖 : 𝑛, 𝑖 = 1, 2, . . .} and {𝜂 𝑛 : 𝑛 = 1, 2, . . .} are independent families of positive integer-valued i.i.d. random variables with distributions given by 𝑔 and ℎ, respectively. Given another positive integer-valued random variable 𝑦(0) independent of {𝜉 𝑛,𝑖 } and {𝜂 𝑛 }, we define inductively 𝑦(𝑛) =

𝑦 (𝑛−1) ∑︁

𝜉 𝑛,𝑖 + 𝜂 𝑛 ,

𝑛 = 1, 2, . . . .

(3.57)

𝑖=1

It is easy to show that {𝑦(𝑛) : 𝑛 = 0, 1, 2, . . .} is a discrete-time positive integervalued Markov chain with transition matrix 𝑄(𝑖, 𝑗) determined by ∞ ∑︁

𝑄(𝑖, 𝑗)𝑧 𝑗 = 𝑔(𝑧) 𝑖 ℎ(𝑧),

|𝑧| ≤ 1.

(3.58)

𝑗=0

The random variable 𝑦(𝑛) can be thought of as the number of individuals in generation 𝑛 ≥ 0 of an evolving particle system. After one unit time, each of the 𝑦(𝑛) particles splits independently of others into a random number of offspring according to the distribution given by 𝑔 and a random number of immigrants are added to the system according to the probability law given by ℎ. The 𝑛-step transition matrix 𝑄 𝑛 (𝑖, 𝑗) of {𝑦(𝑛) : 𝑛 = 0, 1, 2, . . .} is given by

3.5 Scaling Limits of Discrete Processes ∞ ∑︁

𝑄 𝑛 (𝑖, 𝑗)𝑧 𝑗 = 𝑔 𝑛 (𝑧) 𝑖

𝑗=0

87 𝑛 Ö

ℎ(𝑔 𝑗−1 (𝑧)),

|𝑧| ≤ 1,

(3.59)

𝑗=1

where 𝑔 𝑛 (𝑧) is defined by 𝑔 𝑛 (𝑧) = 𝑔(𝑔 𝑛−1 (𝑧)) successively with 𝑔 0 (𝑧) = 𝑧. We call any positive integer-valued Markov chain with transition probabilities given by (3.58) or (3.59) a Galton–Watson branching process with immigration (GWI-process) with parameters (𝑔, ℎ). If 𝑔 ′ (1−) < ∞ and ℎ ′ (1−) < ∞, then the first-moment of the discrete probability distribution {𝑄 𝑛 (𝑖, 𝑗) : 𝑗 = 0, 1, 2, . . .} is given by ∞ ∑︁

𝑗𝑄 𝑛 (𝑖, 𝑗) = 𝑖𝑔 ′ (1−) 𝑛 +

𝑛 ∑︁

ℎ ′ (1−)𝑔 ′ (1−) 𝑗−1 ,

(3.60)

𝑗=1

𝑗=1

which can be obtained by differentiating both sides of (3.59). In the special case where ℎ(𝑧) ≡ 1, we simply call {𝑦(𝑛) : 𝑛 = 0, 1, 2, . . .} a Galton–Watson branching process (GW-process). Suppose that for each integer 𝑘 ≥ 1 we have a GWI-process {𝑦 𝑘 (𝑛) : 𝑛 ≥ 0} with parameters (𝑔 𝑘 , ℎ 𝑘 ). Let 𝑧 𝑘 (𝑛) = 𝑦 𝑘 (𝑛)/𝑘. Then {𝑧 𝑘 (𝑛) : 𝑛 ≥ 0} is a Markov chain with state space 𝐸 𝑘 := {0, 1/𝑘, 2/𝑘, . . .} and 𝑛-step transition probability 𝑄 𝑛𝑘 (𝑥, d𝑦) determined by ∫ 𝐸𝑘

e−𝜆𝑦 𝑄 𝑛𝑘 (𝑥, d𝑦) = 𝑔 𝑘𝑛 (e−𝜆/𝑘 ) 𝑘 𝑥

𝑛 Ö

ℎ(𝑔 𝑘 (e−𝜆/𝑘 )), 𝑗−1

𝜆 ≥ 0.

(3.61)

𝑗=1

Suppose that {𝛾 𝑘 } is a positive real sequence such that 𝛾 𝑘 → ∞ increasingly as 𝑘 → ∞. Let ⌊𝛾 𝑘 𝑡⌋ denote the integer part of 𝛾 𝑘 𝑡 ≥ 0. We are interested in the asymptotic behavior of the continuous-time process {𝑧 𝑘 ( ⌊𝛾 𝑘 𝑡⌋) : 𝑡 ≥ 0} as 𝑘 → ∞. For any 𝑧 ≥ 0 define 𝐻 𝑘 (𝑧) = 𝛾 𝑘 [1 − ℎ 𝑘 (e−𝑧/𝑘 )]

(3.62)

𝐺 𝑘 (𝑧) = 𝑘𝛾 𝑘 [𝑔 𝑘 (e−𝑧/𝑘 ) − e−𝑧/𝑘 ].

(3.63)

and

For convenience of statement of the results, we formulate the following conditions: Condition 3.46 There is a function 𝜓 on [0, ∞) such that 𝐻 𝑘 (𝑧) → 𝜓(𝑧) uniformly on [0, 𝑎] for every 𝑎 ≥ 0 as 𝑘 → ∞. Condition 3.47 The sequence {𝐺 𝑘 } is uniformly Lipschitz on [0, 𝑎] for every 𝑎 ≥ 0 and there is a function 𝜙 on [0, ∞) such that 𝐺 𝑘 (𝑧) → 𝜙(𝑧) uniformly on [0, 𝑎] for every 𝑎 ≥ 0 as 𝑘 → ∞. Proposition 3.48 If Condition 3.46 holds, the limit function 𝜓 has the representation (3.26). If Condition 3.47 holds, then 𝜙 has the representation (3.1).

88

3 One-Dimensional Branching Processes

Proof The representation (3.26) for 𝜓 follows by a modification of the proof of Theorem 1.46. Since 𝜙 is locally Lipschitz, by Proposition 1.48 it suffices to show the function has the representation (1.41). This could be done by modifying the proofs of Theorems 1.46 and 1.47. Here we give a derivation of the representation by considering the sequence 𝜙 𝑘 (𝑧) = 𝑘𝛾 𝑘 [𝑔 𝑘 (1 − 𝑧/𝑘) − (1 − 𝑧/𝑘)],

0 ≤ 𝑧 ≤ 𝑘.

Fix the constant 𝑎 ≥ 0. By the mean-value theorem, for 𝑘 ≥ 𝑎 and 0 ≤ 𝑧 ≤ 𝑎 we have 𝐺 𝑘 (𝑧) − 𝜙 𝑘 (𝑧) = 𝑘𝛾 𝑘 [𝑔 𝑘′ (𝜂 𝑘 ) − 1] (e−𝑧/𝑘 − 1 + 𝑧/𝑘),

(3.64)

where 1 − 𝑎/𝑘 ≤ 1 − 𝑧/𝑘 ≤ 𝜂 𝑘 ≤ e−𝑧/𝑘 ≤ 1. Choose 𝑘 0 ≥ 𝑎 so that e−2𝑎/𝑘0 ≤ 1− 𝑎/𝑘 0 . Then for 𝑘 ≥ 𝑘 0 we have e−2𝑎/𝑘 ≤ 1− 𝑎/𝑘 and hence 𝛾 𝑘 |𝑔 𝑘′ (𝜂 𝑘 ) − 1| ≤ sup 𝛾 𝑘 |𝑔 𝑘′ (e−𝜆/𝑘 ) − 1|. 0≤𝜆≤2𝑎

Since {𝐺 𝑘 } is uniformly Lipschitz on [0, 2𝑎], the sequence 𝐺 ′𝑘 (𝑧) = 𝛾 𝑘 e−𝑧/𝑘 [1 − 𝑔 𝑘′ (e−𝑧/𝑘 )] is uniformly bounded on [0, 2𝑎]. Thus {𝛾 𝑘 |𝑔 𝑘′ (𝜂 𝑘 ) − 1| : 𝑘 ≥ 𝑘 0 } is a bounded sequence and (3.64) implies 𝜙(𝑧) = lim 𝐺 𝑘 (𝑧) = lim 𝜙 𝑘 (𝑧). 𝑘→∞

𝑘→∞

Then we can use Theorem 1.47 to see 𝜙 has the representation (1.41).

□

We shall work with the Laplace transform of the process {𝑧 𝑘 ( ⌊𝛾 𝑘 𝑡⌋) : 𝑡 ≥ 0}. In ⌊𝛾 𝑡 ⌋ view of (3.61), given 𝑧 𝑘 (0) = 𝑥 the conditional distribution 𝑄 𝑘 𝑘 (𝑥, ·) of 𝑧 𝑘 ( ⌊𝛾 𝑘 𝑡⌋) on 𝐸 𝑘 is determined by ∫ ⌊𝛾 𝑡 ⌋ e−𝜆𝑦 𝑄 𝑘 𝑘 (𝑥, d𝑦) 𝐸𝑘



∫

= exp − 𝑥𝑣 𝑘 (𝑡, 𝜆) −

⌊𝛾𝑘 𝑡⌋ 𝛾𝑘

 ¯ 𝐻 𝑘 (𝑣 𝑘 (𝑠, 𝜆))d𝑠 ,

(3.65)

0

where ⌊𝛾𝑘 𝑡 ⌋

𝑣 𝑘 (𝑡, 𝜆) = −𝑘 log 𝑔 𝑘

(e−𝜆/𝑘 )

(3.66)

3.5 Scaling Limits of Discrete Processes

89

and 𝐻¯ 𝑘 (𝜆) = −𝛾 𝑘 log ℎ 𝑘 (e−𝜆/𝑘 ),

𝜆 ≥ 0.

Lemma 3.49 Suppose that the sequence {𝐺 𝑘 } defined by (3.63) is uniformly Lipschitz on [0, 1]. Then there are constants 𝐵 ≥ 0 and 𝑁 ≥ 1 such that 𝑣 𝑘 (𝑡, 𝜆) ≤ 𝜆e 𝐵𝑡 for every 𝑡, 𝜆 ≥ 0 and 𝑘 ≥ 𝑁. Proof Let 𝑏 𝑘 = 𝐺 ′𝑘 (0+) for 𝑘 ≥ 1. Since {𝐺 𝑘 } is uniformly Lipschitz on [0, 1], the sequence {𝑏 𝑘 } is bounded. Let 𝐵 ≥ 0 be a constant such that 2|𝑏 𝑘 | ≤ 𝐵 for all ⌊𝛾 𝑡 ⌋ 𝑘 ≥ 1. In view of (3.65), there is a probability kernel 𝑃 𝑘 𝑘 (𝑥, d𝑦) on 𝐸 𝑘 such that ∫ ⌊𝛾 𝑡 ⌋ e−𝜆𝑦 𝑃 𝑘 𝑘 (𝑥, d𝑦) = exp{−𝑥𝑣 𝑘 (𝑡, 𝜆)}, 𝜆 ≥ 0. (3.67) 𝐸𝑘

From (3.63) we have 𝑏 𝑘 = 𝛾 𝑘 [1 − 𝑔 𝑘′ (1−)]. It is not hard to obtain ∫  𝑏 𝑘  ⌊𝛾𝑘 𝑡 ⌋ ⌊𝛾 𝑡 ⌋ 𝑦𝑃 𝑘 𝑘 (𝑥, d𝑦) = 𝑥𝑔 𝑘′ (1−) ⌊𝛾𝑘 𝑡 ⌋ = 𝑥 1 − . 𝛾𝑘 𝐸𝑘 Since 𝛾 𝑘 → ∞ as 𝑘 → ∞, there is an 𝑁 ≥ 1 such that 𝛾 𝛾   𝐵  𝐵𝑘 𝑏 𝑘  𝐵𝑘 ≤ 1+ ≤ e, 0 ≤ 1− 𝛾𝑘 2𝛾 𝑘

𝑘 ≥ 𝑁.

It follows that, for 𝑡 ≥ 0 and 𝑘 ≥ 𝑁, ∫ o n𝐵 ⌊𝛾 𝑡 ⌋ ⌊𝛾 𝑘 𝑡⌋ ≤ 𝑥e 𝐵𝑡 . 𝑦𝑃 𝑘 𝑘 (𝑥, d𝑦) ≤ 𝑥 exp 𝛾𝑘 𝐸𝑘

(3.68)

Then the desired estimate follows from (3.67), (3.68) and Jensen’s inequality.

□

Theorem 3.50 Suppose that Condition 3.47 is satisfied. Let (𝑡, 𝜆) ↦→ 𝑣 𝑡 (𝜆) be the unique locally bounded positive solution of (3.3). Then for every 𝑎 ≥ 0 we have 𝑣 𝑘 (𝑡, 𝜆) → 𝑣 𝑡 (𝜆) uniformly on [0, 𝑎] 2 as 𝑘 → ∞. Proof It suffices to show 𝑣 𝑘 (𝑡, 𝜆) converges uniformly on [0, 𝑎] 2 for every 𝑎 ≥ 0 and the limit solves (3.3). Let   𝐺¯ 𝑘 (𝑧) = 𝑘𝛾 𝑘 log 𝑔 𝑘 (e−𝑧/𝑘 )e𝑧/𝑘 , 𝑧 ≥ 0. For any integer 𝑛 ≥ 0 we may write   log 𝑔 𝑘𝑛+1 (e−𝜆/𝑘 ) = log 𝑔 𝑘 (𝑔 𝑘𝑛 (e−𝜆/𝑘 ))𝑔 𝑘𝑛 (e−𝜆/𝑘 ) −1 + log 𝑔 𝑘𝑛 (e−𝜆/𝑘 )
= (𝑘𝛾 𝑘 ) −1 𝐺¯ 𝑘 − 𝑘 log 𝑔 𝑘𝑛 (e−𝜆/𝑘 ) + log 𝑔 𝑘𝑛 (e−𝜆/𝑘 ). From this and (3.66) it follows that 𝑣 𝑘 (𝑡 + 𝛾 𝑘−1 , 𝜆) = 𝑣 𝑘 (𝑡, 𝜆) − 𝛾 𝑘−1 𝐺¯ 𝑘 (𝑣 𝑘 (𝑡, 𝜆)),

𝑡 ≥ 0.

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3 One-Dimensional Branching Processes

By applying the above equation to 𝑡 = 0, 1/𝛾 𝑘 , . . . , ( ⌊𝛾 𝑘 𝑡⌋ − 1)/𝛾 𝑘 and adding the resulting equations we obtain 𝑣 𝑘 (𝑡, 𝜆) = 𝜆 −

⌊𝛾 𝑘𝑡 ⌋ ∑︁


𝛾 𝑘−1 𝐺¯ 𝑘 𝑣 𝑘 (𝛾 𝑘−1 (𝑖 − 1), 𝜆) .

𝑖=1

Then we have ∫

𝑡

𝑣 𝑘 (𝑡, 𝜆) = 𝜆 + 𝜀 𝑘 (𝑡, 𝜆) −

𝐺¯ 𝑘 (𝑣 𝑘 (𝑠, 𝜆))d𝑠,

(3.69)

0

where

𝜀 𝑘 (𝑡, 𝜆) = 𝑡 − 𝛾 𝑘−1 ⌊𝛾 𝑘 𝑡⌋ 𝐺¯ 𝑘 𝑣 𝑘 (𝛾 𝑘−1 ⌊𝛾 𝑘 𝑡⌋, 𝜆) . It is elementary to see   𝐺¯ 𝑘 (𝑧) = 𝑘𝛾 𝑘 log 1 + (𝑘𝛾 𝑘 ) −1 𝐺 𝑘 (𝑧)e𝑧/𝑘 . Let 𝐵 ≥ 0 and 𝑁 ≥ 1 be chosen as in Lemma 3.49. Under Condition 3.47 one can show 𝐺¯ 𝑘 (𝑧) → 𝜙(𝑧) uniformly on every bounded interval. Then for any 0 < 𝜀 ≤ 1 we can enlarge the constant 𝑁 ≥ 1 so that | 𝐺¯ 𝑘 (𝑧) − 𝜙(𝑧)| ≤ 𝜀,

0 ≤ 𝑧 ≤ 𝑎e 𝐵𝑎 , 𝑘 ≥ 𝑁.

(3.70)

0 ≤ 𝑡, 𝜆 ≤ 𝑎,

(3.71)

It follows that |𝜀 𝑘 (𝑡, 𝜆)| ≤ 𝛾 𝑘−1 𝑀, where 𝑀 =1+

|𝜙(𝑧)|.

sup 0≤𝑧 ≤𝑎e 𝐵𝑎

For 𝑛 ≥ 𝑘 ≥ 𝑁 let 𝐾 𝑘,𝑛 (𝑡, 𝜆) = sup |𝑣 𝑛 (𝑠, 𝜆) − 𝑣 𝑘 (𝑠, 𝜆)|. 0≤𝑠 ≤𝑡

By (3.69), (3.70) and (3.71) we obtain 𝐾 𝑘,𝑛 (𝑡, 𝜆) ≤ 2(𝛾 𝑘−1 𝑀 + 𝜀𝑎) + 𝐿

∫

𝑡

𝐾 𝑘,𝑛 (𝑠, 𝜆)d𝑠,

0 ≤ 𝑡, 𝜆 ≤ 𝑎,

0

where 𝐿 = sup0≤𝑧 ≤𝑎e𝐵𝑎 |𝜙 ′ (𝑧)|. By Gronwall’s inequality, 𝐾 𝑘,𝑛 (𝑡, 𝜆) ≤ 2(𝛾 𝑘−1 𝑀 + 𝜀𝑎)e 𝐿𝑡 ,

0 ≤ 𝑡, 𝜆 ≤ 𝑎.

Then 𝑣 𝑘 (𝑡, 𝜆) → some 𝑣 𝑡 (𝜆) uniformly on [0, 𝑎] 2 as 𝑘 → ∞. In view of (3.71) and □ (3.69) we have (3.3).

3.5 Scaling Limits of Discrete Processes

91

Let 𝐷 ( [0, ∞), R+ ) denote the space of càdlàg paths from [0, ∞) to R+ furnished with the Skorokhod topology. The main limit theorem of this section is the following: Theorem 3.51 Suppose that Conditions 3.46 and 3.47 are satisfied. Let {𝑦(𝑡) : 𝑡 ≥ 𝛾 0} be a càdlàg CBI-process with transition semigroup (𝑄 𝑡 )𝑡 ≥0 defined by (3.29). If 𝑧 𝑘 (0) converges to 𝑦(0) in distribution, then {𝑧 𝑘 ( ⌊𝛾 𝑘 𝑡⌋) : 𝑡 ≥ 0} converges to {𝑦(𝑡) : 𝑡 ≥ 0} in distribution on 𝐷 ( [0, ∞), R+ ). Proof For 𝜆 > 0 and 𝑥 ≥ 0 set 𝑒 𝜆 (𝑥) = e−𝜆𝑥 . We denote by 𝐷 1 the linear span of {𝑒 𝜆 : 𝜆 > 0}. It is easy to see that 𝐷 1 is an algebra strongly separating the points of R+ in the sense of Ethier and Kurtz (1986, pp. 112–113). Let 𝐶0 (R+ ) be the space of continuous functions on R+ vanishing at infinity. Then 𝐷 1 is uniformly dense in 𝐶0 (R+ ) by the Stone–Weierstrass theorem; see, e.g., Hewitt and Stromberg (1965, pp. 98–99). By Proposition 3.1 it is easy to see that the function 𝑡 ↦→ 𝑣 𝑡 (𝜆) is locally bounded away from zero. Under Condition 3.46 we have 𝐻¯ 𝑘 (𝑧) → 𝜓(𝑧) uniformly on every bounded interval. Then one can use (3.29), (3.65) and Theorem 3.50 to show ⌊𝛾 𝑡 ⌋ 𝛾 lim sup 𝑄 𝑘 𝑘 𝑒 𝜆 (𝑥) − 𝑄 𝑡 𝑒 𝜆 (𝑥) = 0 𝑘→∞ 𝑥 ∈𝐸𝑘

for every 𝑡 ≥ 0. It follows that ⌊𝛾 𝑡 ⌋ 𝛾 lim sup 𝑄 𝑘 𝑘 𝑓 (𝑥) − 𝑄 𝑡 𝑓 (𝑥) = 0

𝑘→∞ 𝑥 ∈𝐸𝑘

for every 𝑡 ≥ 0 and 𝑓 ∈ 𝐶0 (R+ ). By Ethier and Kurtz (1986, p. 226 and pp. 233–234) we conclude that {𝑧 𝑘 ( ⌊𝛾 𝑘 𝑡⌋) : 𝑡 ≥ 0} converges to the CBI-process {𝑦(𝑡) : 𝑡 ≥ 0} □ in distribution on 𝐷 ( [0, ∞), R+ ). The theorem above gives an interpretation of the CBI-process as the limit of a sequence of rescaled GWI-processes. The following examples describe some typical situations where Conditions 3.46 and 3.47 are satisfied. Example 3.4 Suppose that ℎ is a probability generating function such that 𝛽 := ℎ ′ (1−) < ∞. Let 𝛾 𝑘 = 𝑘 and ℎ 𝑘 (𝑧) = ℎ(𝑧). Then the sequence 𝐻 𝑘 (𝑧) defined by (3.62) converges to 𝛽𝑧 as 𝑘 → ∞. Example 3.5 For any 0 < 𝛼 ≤ 1 let 𝛾 𝑘 = 𝑘 𝛼 and ℎ 𝑘 (𝑧) = 1 − (1 − 𝑧) 𝛼 . Then the sequence 𝐻 𝑘 (𝑧) defined by (3.62) converges to 𝑧 𝛼 as 𝑘 → ∞. Example 3.6 Suppose that 𝑔 is a probability generating function such that 𝑔 ′ (1−) = 1 and 𝑐 := 𝑔 ′′ (1−)/2 < ∞. Let 𝛾 𝑘 = 𝑘 and 𝑔 𝑘 (𝑧) = 𝑔(𝑧). By Taylor’s expansion one sees that the sequence 𝐺 𝑘 (𝑧) defined by (3.63) converges to 𝑐𝑧2 as 𝑘 → ∞. Example 3.7 For any 1 ≤ 𝛼 ≤ 2 let 𝛾 𝑘 = 𝛼𝑘 𝛼−1 and 𝑔 𝑘 (𝑧) = 𝑧 + 𝛼−1 (1 − 𝑧) 𝛼 . Then the sequence 𝐺 𝑘 (𝑧) defined by (3.63) converges to 𝑧 𝛼 as 𝑘 → ∞.

92

3 One-Dimensional Branching Processes

We now give some applications of Theorem 3.51 to the characterizations of local times. Let 𝑋 0 = (Ω, ℱ, ℱ𝑡 , 𝑋𝑡0 , P 𝑥 ) be a standard one-dimensional Brownian motion. Given any constant 𝑎 ∈ R we define 𝑋𝑡𝑎 = 𝑋𝑡0 − 𝑎𝑡 for 𝑡 ≥ 0. Then 𝑋 𝑎 = (Ω, ℱ, ℱ𝑡 , 𝑋𝑡𝑎 , P 𝑥 ) is a Brownian motion with drift −𝑎. It is well known that there is a positive continuous two-parameter process {𝑙 𝑎 (𝑡, 𝑦) : 𝑡 ≥ 0, 𝑦 ∈ R} such that ∫ ∫ 𝑡 2𝑙 𝑎 (𝑡, 𝑦)d𝑦, 1 𝐴 (𝑋𝑠𝑎 )d𝑠 = 𝐴 ∈ ℬ(R). 0

𝐴

The process {2𝑙 𝑎 (𝑡, 𝑦) : 𝑡 ≥ 0, 𝑦 ∈ R} is the local time of {𝑋𝑡𝑎 : 𝑡 ≥ 0}. Let 𝜏𝑥𝑎 = inf{𝑡 > 0 : 𝑋𝑡𝑎 = 𝑥} denote the hitting time of 𝑥 ∈ R by the Brownian motion with drift. By a 𝛿-downcrossing of {𝑋𝑡𝑎 : 𝑡 ≥ 0} at 𝑦 ∈ R before time 𝑇 > 0 we mean an interval [𝑢, 𝑣] ⊂ [0, 𝑇) such that 𝑋𝑢𝑎 = 𝑦 + 𝛿, 𝑋𝑣𝑎 = 𝑦 and 𝑦 < 𝑋𝑡𝑎 < 𝑦 + 𝛿 for all 𝑢 < 𝑡 < 𝑣. We first consider the special case 𝑎 = 0. It is well known that P 𝑥 {𝑙 0 (𝑡, 𝑦) → ∞ as 𝑡 → ∞} = 1 for all 𝑥, 𝑦 ∈ R. Then for every 𝑢 ≥ 0 we have P 𝑥 -a.s. 𝜎 0 (𝑢) := inf{𝑡 ≥ 0 : 𝑙 0 (𝑡, 0) ≥ 𝑢} < ∞. The following theorem is the well-known Ray–Knight theorem on Brownian local times. Theorem 3.52 For any 𝑥 ≥ 0, under the probability law P 𝑥 we have: (1) {𝑙 0 (𝜎 0 (𝑢), −𝑡) : 𝑡 ≥ 0} and {𝑙 0 (𝜎 0 (𝑢), 𝑥 + 𝑡) : 𝑡 ≥ 0} are CB-processes with branching mechanism 𝜙(𝑧) = 𝑧 2 ; (2) {𝑙 0 (𝜎 0 (𝑢), 𝑡) : 0 ≤ 𝑡 ≤ 𝑥} is a CBI-process with branching mechanism 𝜙(𝑧) = 𝑧 2 and immigration mechanism 𝜓(𝑧) = 𝑧. 0 Proof (1) Let 𝜉 𝑘 denote the number of (1/𝑘)-downcrossings at 𝑥 before time 𝜏𝑥−1/𝑘 . 0 By the property of independent increments of {𝑋𝑡 : 𝑡 ≥ 0} we have

P0 [𝑧 𝜉𝑘 ] =

∞ ∑︁ 1 𝑧𝑖 = , 𝑖+1 2−𝑧 2 𝑖=0

|𝑧| ≤ 1.

For 𝑘 ≥ 1 and 𝑖 ≥ 0 let 𝑍 𝑘 (𝑖) denote the number of (1/𝑘)-downcrossings of {𝑋𝑡0 : 𝑡 ≥ 0} at 𝑥 𝑖 = 𝑥 + 𝑖/𝑘 before time 𝜎 0 (𝑢). It is easy to see that 𝑍 𝑘 (𝑖 + 1) is the sum of 𝑍 𝑘 (𝑖) independent copies of 𝜉 𝑘 . Thus {𝑍 𝑘 (𝑖) : 𝑖 = 0, 1, . . .} is a GW-process determined by the probability generating function 𝑔(𝑧) =

1 , 2−𝑧

|𝑧| ≤ 1.

(3.72)

By the approximation of the local time by downcrossing numbers, for every 𝑡 ≥ 0 we have 𝑍 𝑘 ( ⌊𝑘𝑡⌋)/𝑘 → 𝑙 0 (𝜎 0 (𝑢), 𝑥 + 𝑡) in probability as 𝑘 → ∞; see, e.g, Revuz and Yor (1999, p. 227). On the other hand, it is easy to show

3.5 Scaling Limits of Discrete Processes

93

𝜙(𝑧) := lim 𝑘 2 [𝑔(e−𝑧/𝑘 ) − e−𝑧/𝑘 ] = 𝑧 2 . 𝑘→∞

By Theorem 3.51 one can see {𝑙 0 (𝜎 0 (𝑢), 𝑥 + 𝑡) : 𝑡 ≥ 0} is a CB-process with branching mechanism 𝜙(𝑧) = 𝑧 2 . The result for {𝑙 0 (𝜎 0 (𝑢), −𝑡) : 𝑡 ≥ 0} follows in a similar way. (2) This is similar to the first part of the proof, so we only give a sketch. For 𝑘 ≥ 1 and 0 ≤ 𝑖 ≤ ⌊𝑘𝑥⌋ let 𝑌𝑘 (𝑖) denote the number of (1/𝑘)-downcrossings of {𝑋𝑡0 : 𝑡 ≥ 0} at 𝑧𝑖 = 𝑖/𝑘 before time 𝜎 0 (𝑢). One can see that 𝑌𝑘 (𝑖 +1) −1 is the sum of 𝑌𝑘 (𝑖) independent copies of 𝜉 𝑘 . Then {𝑌𝑘 (𝑖) : 𝑖 = 0, 1, . . . , ⌊𝑘𝑥⌋} is a GWI-process determined by the pair of generating functions (𝑔, ℎ), where 𝑔(𝑧) is given by (3.72) and ℎ(𝑧) = 𝑧. For any 0 ≤ 𝑡 ≤ 𝑥 we have 𝑌𝑘 ( ⌊𝑘𝑡⌋)/𝑘 → 𝑙 0 (𝜎 0 (𝑢), 𝑡) in probability □ as 𝑘 → ∞. Then the result follows by Theorem 3.51. Let {2𝑙 (𝑡, 𝑦) : 𝑡 ≥ 0, 𝑦 ≥ 0} denote the local time of the reflecting Brownian motion {|𝑋𝑡0 | : 𝑡 ≥ 0}. Then {𝑙 (𝑡, 𝑦) : 𝑡 ≥ 0, 𝑦 ≥ 0} is a positive continuous two-parameter process such that ∫ ∫ 𝑡 𝐴 ∈ ℬ(R+ ). 1 𝐴 (|𝑋𝑠0 |)d𝑠 = 2 𝑙 (𝑡, 𝑦)d𝑦, 0

𝐴

For any 𝑥 ≥ 0 and 𝑢 ≥ 0 we have P 𝑥 -a.s. 𝜎(𝑢) := inf{𝑡 ≥ 0 : 𝑙 (𝑡, 0) ≥ 𝑢} < ∞. By modifying the arguments in the proof of Theorem 3.52 one can show the following: Theorem 3.53 For any 𝑥 ≥ 0, under the probability law P 𝑥 we have: (1) {𝑙 (𝜎(𝑢), 𝑥 + 𝑡) : 𝑡 ≥ 0} is a CB-process with branching mechanism 𝜙(𝑧) = 𝑧2 ; (2) {𝑙 (𝜎(𝑢), 𝑡) : 0 ≤ 𝑡 ≤ 𝑥} is a CBI-process with branching mechanism 𝜙(𝑧) = 𝑧2 and immigration mechanism 𝜓(𝑧) = 𝑧. We next consider the case 𝑎 > 0. In this case we have P 𝑥 {𝜏0𝑎 < ∞} = 1 for every 𝑥 > 0. For 𝛿 > 0 and |𝑥| ≤ 𝛿 let 𝑢 𝛿 (𝑥) = P 𝑥 {𝜏−𝑎𝛿 < 𝜏𝛿𝑎 }. Then 𝑥 ↦→ 𝑢 𝛿 (𝑥) solves the differential equation 1 ′′ 𝑢 (𝑥) − 𝑎𝑢 ′ (𝑥) = 0, 2

|𝑥| ≤ 𝛿

with boundary conditions 𝑢(𝛿) = 0 and 𝑢(−𝛿) = 1. By solving the above boundary value problem we find 𝑢 𝛿 (𝑥) =

e2𝑎 𝛿 − e2𝑎𝑥 , e2𝑎 𝛿 − e−2𝑎 𝛿

|𝑥| ≤ 𝛿.

The following theorem slightly generalizes the Ray–Knight theorem.

(3.73)

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3 One-Dimensional Branching Processes

Theorem 3.54 Suppose that 𝑎 > 0 and 𝑥 > 0. Then under P 𝑥 we have: (1) {𝑙 𝑎 (𝜏0𝑎 , 𝑥+𝑡) : 𝑡 ≥ 0} is a CB-process with branching mechanism 𝜙(𝑧) = 𝑧 2 +2𝑎𝑧; (2) {𝑙 𝑎 (𝜏0𝑎 , 𝑡) : 0 ≤ 𝑡 ≤ 𝑥} is a CBI-process with branching mechanism 𝜙(𝑧) = 𝑧2 + 2𝑎𝑧 and immigration mechanism 𝜓(𝑧) = 𝑧. Proof The arguments are modifications of those in the proof of Theorem 3.52, so we only describe the difference. For 𝑘 ≥ 1 and 𝑖 ≥ 0 let 𝑍 𝑘 (𝑖) denote the number of (1/𝑘)-downcrossings of {𝑋𝑡𝑎 : 𝑡 ≥ 0} at 𝑥𝑖 = 𝑥 + 𝑖/𝑘 before time 𝜏0𝑎 . Then {𝑍 𝑘 (𝑖) : 𝑖 = 0, 1, . . .} is a GW-process corresponding to the generating function 𝑔 𝑘 (𝑧) =

∞ ∑︁

𝑝 𝑘 (𝑞 𝑘 𝑧) 𝑖 =

𝑖=0

𝑝𝑘 , 1 − 𝑞𝑘 𝑧

|𝑧| ≤ 1,

(3.74)

where 𝑝 𝑘 = 𝑢 1/𝑘 (0) and 𝑞 𝑘 = 1 − 𝑝 𝑘 . From (3.73) we get 𝑝𝑘 =

1 1 𝑎 e2𝑎/𝑘 − 1 = + + 𝑜 𝑘 e2𝑎/𝑘 − e−2𝑎/𝑘 2 2𝑘

𝑞𝑘 =

1 1 − e−2𝑎/𝑘 1 𝑎 = − + 𝑜 𝑘 e2𝑎/𝑘 − e−2𝑎/𝑘 2 2𝑘

and

as 𝑘 → ∞. Then we use (3.74) to see 𝜙(𝑧) := lim 𝑘 2 [𝑔 𝑘 (e−𝑧/𝑘 ) − e−𝑧/𝑘 ] 𝑘→∞

𝑘 2 [1 − e−𝑧/𝑘 − 𝑞 𝑘 (1 − e−2𝑧/𝑘 )] 𝑘→∞ 1 − 𝑞 𝑘 e−𝑧/𝑘 2 = 𝑧 + 2𝑎𝑧. = lim

For 𝑡 ≥ 0 we have 𝑍 𝑘 ( ⌊𝑘𝑡⌋)/𝑘 → 𝑙 𝑎 (𝜏0𝑎 , 𝑥 + 𝑡) in probability as 𝑘 → ∞. Thus the assertion (1) follows by Theorem 3.51. Similarly one obtains (2). □

3.6 Notes and Comments The convergence of rescaled GW-processes to diffusion processes was first studied by Feller (1951). Jiřina (1958) introduced CB-processes in both discrete and continuous times. Lamperti (1967a) showed that the continuous-time processes are weak limits of rescaled GW-processes; see also Aliev and Shchurenkov (1983) and Grimvall (1974). The convergence of rescaled GWI-processes to CBI-processes has been discussed by a number of authors, see, e.g., Aliev (1985), Kawazu and Watanabe (1971) and Li (2006) among others.

3.6 Notes and Comments

95

The “if” part of Theorem 3.14 was proved in Silverstein (1967/8). It seems the “only if” part is a new result. Most of the other results on extinction probabilities and growth rates in Sections 3.1 and 3.2 can be found in Grey (1974). It is simple to check that if {𝑋𝑡 : 𝑡 ≥ 0} is a Dawson–Watanabe superprocess with conservative underlying spatial motion and spatially constant branching mechanism, then the total mass {𝑋𝑡 (1) : 𝑡 ≥ 0} is a CB-process. The properties of local extinction and growth rate for superprocesses were studied in Engländer and Kyprianou (2004), Liu et al. (2009) and Pinsky (1995, 1996). A zero-one law on the local extinction for a super-Brownian motion was given in Zhou (2008). See also Engländer (2007) and the references therein. Theorem 3.24 and Corollary 3.25 were given in Pinsky (1972). A similar result for Ornstein–Uhlenbeck type processes was proved in Sato and Yamazato (1984). For finite-dimensional affine Markov processes, a sufficient condition for the ergodicity in weak convergence was proved by Jin et al. (2020), which covers partially the results of Pinsky (1972) and Sato and Yamazato (1984). The necessity of the condition of Jin et al. (2020) was still an open problem. The transformation of probability laws given in Theorem 3.32 is standard in the theory of Markov processes; see, e.g., Sharpe (1988, p. 296). Most other results in Sections 3.3 and 3.4 can be found in Li (2000). The result of Theorem 3.37 was already expected by Pakes (1988, p. 86); see also Pakes and Trajstman (1985). A number of conditional limit theorems for Galton–Watson processes were proved in Pakes (1999) by introducing some general conditioning events. Theorems 3.44 and 3.45 treat the two simplest special cases of the conditional events of Pakes (1999). Some of the results in Section 3.4 were proved in Lambert (2007) by different methods; see also Kyprianou and Pardo (2008). A conditional limit theorem for generalized diffusion processes was proved in Li et al. (2003). Let 𝑋 = {𝑋 (𝑡) : 𝑡 ≥ 0} be such a process with initial state 𝑋 (0) = 𝑥 > 0, hitting time 𝜏𝑋 (0) at the origin and speed measure 𝑚 regularly varying at infinity with exponent 1/𝛼−1 > 0. They proved that, for a suitable function 𝑢(𝑐), the probability law of {𝑢(𝑐) −1 𝑋 (𝑐𝑡) : 0 < 𝑡 ≤ 1} converges as 𝑐 → ∞ to the conditioned 2(1 − 𝛼)-dimensional Bessel excursion on natural scale and that the latter is equivalent to the 2(1 − 𝛼)-dimensional Bessel meander up to a scale transformation. In particular, the distribution of 𝑢(𝑐) −1 𝑋 (𝑐) converges to the Weibull distribution: (1 − 𝛼)𝑥 1/𝛼−1 exp{−𝛼(1 − 𝛼)𝑥 1/𝛼 }d𝑥,

𝑥 > 0.

From the conditional limit theorem they also derive a limit theorem for some regenerative processes associated with 𝑋. For the sake of simplicity, we have assumed the branching mechanism is given by (3.1). Proposition 3.48 and Theorem 3.51 suggest that the class of CBI-processes with transition semigroups given by (3.29) essentially includes all possible rescaling limits of GWI-processes with finite first-moments. One may consider a more general branching mechanism 𝜙 defined by (1.41) or (1.43). By the result of Silverstein (1967/8), in this case for every 𝜆 > 0 there is a unique strictly positive solution 𝑡 ↦→ 𝑣 𝑡 (𝜆) to (3.5); see also Kawazu and Watanabe (1971). Let 𝑣 𝑡 (0) = lim𝜆→0 𝑣 𝑡 (𝜆)

96

3 One-Dimensional Branching Processes 𝛾

for 𝑡 ≥ 0. Then one can also define the transition semigroups (𝑄 𝑡 )𝑡 ≥0 and (𝑄 𝑡 )𝑡 ≥0 by (3.2) and (3.29), respectively, but they are not necessarily conservative. For instance, we have 𝑄 𝑡 (𝑥, [0, ∞)) = exp{−𝑥𝑣 𝑡 (0)}. It was shown in Kawazu and Watanabe (1971) that 𝑣 𝑡 (0) = 0 for all 𝑡 ≥ 0 if and only if ∫ 1 d𝑧 = ∞. (3.75) 0+ 0 ∨ (−𝜙(𝑧)) Clearly, the above condition holds for 𝜙(𝜆) ≡ 𝜆 log 𝜆. A CB-process with this branching mechanism is known as Neveu’s CB-process, which was used by Neveu (1992) in the study of the generalized random energy models of Derrida (1985) and Ruelle (1987). Conditions 3.46 and 3.47 were essentially given in Aliev (1985) and Aliev and Shchurenkov (1983). Slightly different forms of the two conditions can be found in Li (2006). The proof of Theorem 3.50 follows Aliev and Shchurenkov (1983). The convergence in distribution on the path space 𝐷 ( [0, ∞), R+ ) of Theorem 3.51 was established in Li (2006) by proving the convergence of the generators of the rescaled GWI-processes; see also Ma (2009). Theorem 3.52 was originally proved by Knight (1963) and Ray (1963). There are many generalizations of the Ray–Knight theorem; see, e.g., Borodin and Salminen (1996). A kind of convolution among stochastic processes with state space [0, ∞) was introduced by Shiga and Watanabe (1973). Using this convolution, they defined the notion of an infinitely decomposable process. They showed that a Markov process is infinitely decomposable if and only if it is a CBI-process, which includes the squared Bessel diffusion as a typical example. This special case was investigated much further by Pitman and Yor (1982). In particular, they constructed a path-valued random field {𝑌𝑥𝑑 : 𝑥 ≥ 0, 𝑑 ≥ 0} with independent increments, where 𝑌𝑥𝑑 = {𝑌𝑥𝑑 (𝑡) : 𝑡 ≥ 0} is a squared Bessel diffusion process with initial value 𝑥 and with generator 𝐴 determined by 𝐴 𝑓 (𝑥) = 2𝑥

d2 d 𝑓 (𝑥) + 𝑑 𝑓 (𝑥), 2 d𝑥 d𝑥

𝑓 ∈ 𝐶 2 [0, ∞).

Their construction was given by some Poisson random measures based on excursion laws. See Revuz and Yor (1999) for a compact theory of squared Bessel diffusions. The genealogical structures of Galton–Watson branching processes are represented by Galton–Watson trees. Those trees can be coded by two kinds of discrete paths called height functions and contour functions. By the result of Aldous (1993), a sequence of suitably rescaled critical Galton–Watson trees converges to the socalled continuum random tree coded by a Brownian excursion. The basic idea of the Ray–Knight theorem is to code the genealogical structures of Feller’s branching diffusion by the Brownian paths. A discrete time–space counterpart of the Ray–Knight theorem was given by Dwass (1975), who characterized the crossing numbers at different levels by random walks in terms Galton–Watson processes. Le Gall and Le Jan (1998a) proposed an approach of coding the genealogy of a general subcritical branching CB-process using a spectrally positive Lévy process, which corresponds

3.6 Notes and Comments

97

to the reflecting Brownian motion in the case of Feller’s branching diffusion. A key contribution of Le Gall and Le Jan (1998a) is an explicit expression for the height process as a functional of the Lévy process whose Laplace exponent is precisely the branching mechanism. This suggests that many problems concerning the genealogies of CB-processes can be restated and solved in terms of spectrally positive Lévy processes, for which there is a rich literature; see, e.g., Bertoin (1996) and Sato (1999). In view of Theorem 3.51, one may want to look for limit theorems of branching models involving genealogical structures. Some limit theorems of this type were established in Duquesne and Le Gall (2002) in terms of height processes and contour processes. Pitman (2006) studied various combinatorial models of random partitions and trees, and the asymptotics of these models related to stochastic processes. See Aldous (1991a, 1991b, 1993) for the early work in the subject. The method of Gromov–Hausdorff distance was developed in Evans et al. (2006) and Evans and Winter (2006) to study the asymptotic behavior of random trees when the number of vertices goes to infinity. Evans (2008) and Winter (2007) gave surveys of the relevant backgrounds and applications; see also Le Gall (2005). The genealogical structures of catalytic branching models were studied in Greven et al. (2009). If (𝑣 𝑡 )𝑡 >0 admits the representation (3.15), then each 𝑙 𝑡 (d𝑢) is a diffuse measure on (0, ∞); see Bertoin and Le Gall (2000). CBI-processes were used by Bertoin and Le Gall (2000, 2006) in studying the coalescent processes with multiple collisions of Pitman (1999) and Sagitov (1999). See also Limic and Sturm (2006) and Schweinsberg (2000, 2003) for some related results. Using the results for self-similar CBI-processes, Patie (2009) gave a characterization of the density of the law of an exponential functional associated to some one-sided Lévy processes. A natural generalization of the CBI-process is described as follows. Let 𝑚 ≥ 0 and 𝑛 ≥ 0 be integers and define 𝐷 = R+𝑚 × R𝑛 and 𝑈 = C−𝑚 × (𝑖R) 𝑛 , where C− = {𝑎 + 𝑖𝑏 : 𝑎 ≤ 0, 𝑏 ∈ R} and 𝑖R = {𝑖𝑏 : 𝑏 ∈ R}. Let (·, ·) denote the duality between 𝐷 and 𝑈. A transition semigroup (𝑃𝑡 )𝑡 ≥0 on 𝐷 is called an affine semigroup if its characteristic function has the representation ∫ (3.76) e ( 𝑦,𝑢) 𝑃𝑡 (𝑥, d𝑦) = exp{(𝑥, 𝜓(𝑡, 𝑢)) + 𝜙(𝑡, 𝑢)}, 𝑢 ∈ 𝑈, 𝐷

where 𝑢 ↦→ 𝜓(𝑡, 𝑢) is a continuous mapping of 𝑈 into itself and 𝑢 ↦→ 𝜙(𝑡, 𝑢) is a continuous function on 𝑈 satisfying 𝜙(𝑡, 0) = 0. A Markov process in 𝐷 is called an affine process if it has affine transition semigroup. The process reduces to an 𝑚-dimensional CBI-process when 𝑛 = 0. The CIR-model was first used by Cox et al. (1985) to model the stochastic evolution of interest rates. General affine processes have been used widely in mathematical finance; see Duffie et al. (2003) and the references therein. The affine semigroup defined by (3.76) is called regular if it is stochastically continuous and the right derivatives 𝜓𝑡′ (0, 𝑢) and 𝜙𝑡′ (0, 𝑢) exist for all 𝑢 ∈ 𝑈 and are continuous at 𝑢 = 0. A number of characterizations of regular affine processes were given in Duffie et al. (2003). By a result of Kawazu and Watanabe (1971),

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3 One-Dimensional Branching Processes

a stochastically continuous CBI-process is automatically regular. The regularity of affine processes was studied in Dawson and Li (2006) under a moment assumption. The regularity problem was settled in Keller-Ressel et al. (2011), where it was proved that any stochastically continuous affine process is regular.

Chapter 4

Branching Particle Systems

Branching particle systems arise from applications in a number of subjects. Typical examples of those systems are biological populations in isolated regions, families of neutrons in nuclear reactions, cosmic-ray showers and so on. In this chapter, we show that suitable scaling limits of those particle systems lead to the Dawson–Watanabe superprocesses in finite-dimensional distributions, giving intuitive interpretations for the superprocesses. To show the ideas in a simple and clear way, we shall first develop the results in detail for local branching particle systems. After that we show how the argument can be modified to general non-local branching models.

4.1 Particle Systems with Local Branching In this section, we introduce a special class of branching particle systems, which can be regarded as the discrete-state counterpart of the local branching Dawson– Watanabe superprocesses. Let 𝐸 be a Lusin topological space and let 𝑁 (𝐸) denote the space of integer-valued finite measures on 𝐸. Let 𝜉 = (Ω, ℱ, ℱ𝑡 , 𝜉𝑡 , P 𝑥 ) be a Borel right process in 𝐸 with conservative transition semigroup (𝑃𝑡 )𝑡 ≥0 . We assume the sample path {𝜉𝑡 : 𝑡 ≥ 0} is right continuous in both the original and Ray topologies and has left limits {𝜉𝑡− : 𝑡 > 0} in the Ray–Knight completion 𝐸¯ of 𝐸. Let 𝛾 ≥ 0 be a constant. Suppose that 𝑔 ∈ 𝐵(𝐸 × [−1, 1]) and 𝑔(𝑥, ·) is a probability generating function for each 𝑥 ∈ 𝐸, that is, 𝑔(𝑥, 𝑧) =

∞ ∑︁

𝑝 𝑘 (𝑥)𝑧 𝑘 ,

|𝑧| ≤ 1,

𝑘=0

where 𝑝 𝑘 (𝑥) ≥ 0 and

Í∞ 𝑘=0

𝑝 𝑘 (𝑥) = 1. Moreover, we assume

sup 𝑔 𝑧′ (𝑥, 1−) = sup 𝑥 ∈𝐸

∞ ∑︁

𝑘 𝑝 𝑘 (𝑥) < ∞.

(4.1)

𝑥 ∈𝐸 𝑘=1

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_4

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4 Branching Particle Systems

100

Let 𝑔(𝑥, 𝑧) = 1 for 𝑥 ∈ 𝐸¯ \ 𝐸 and |𝑧| ≤ 1. We consider a particle system on 𝐸 characterized by the following properties: (1) The particles in 𝐸 move independently according to the law given by the transition probabilities of 𝜉. (2) For a particle which is alive at time 𝑟 ≥ 0 and follows the path {𝜉 𝑠 : 𝑠 ≥ 𝑟 }, the conditional probability of survival in the time interval [𝑟, 𝑡) is exp{−(𝑡 − 𝑟)𝛾}. (3) When a particle following the path {𝜉 𝑠 : 𝑠 ≥ 𝑟 } dies at time 𝑡 > 𝑟, it gives birth to a random number of offspring at 𝜉𝑡 ∈ 𝐸 according to the probability distribution given by the generating function 𝑔(𝜉𝑡− , ·). The offspring then start to move from their common birth site. In addition, we assume that the lifetimes and the branchings of different particles are independent. By a branching particle system with parameters (𝜉, 𝛾, 𝑔) we mean the measure-valued process {𝑋𝑡 : 𝑡 ≥ 0}, where 𝑋𝑡 (𝐵) denotes the number of particles in 𝐵 ∈ ℬ(𝐸) that are alive at time 𝑡 ≥ 0. A construction of the branching particle system with initial value 𝜎 ∈ 𝑁 (𝐸) is given as follows. Let 𝒜 be the set of all finite strings of the form 𝛼 = 𝑛0 𝑛1 · · · 𝑛𝑙 ( 𝛼) for integers 𝑙 (𝛼) ≥ 0 and 𝑛𝑖 ≥ 1. We provide 𝒜 with the arboreal ordering. Then 𝑚 0 𝑚 1 · · · 𝑚 𝑝 ≺ 𝑛0 𝑛1 · · · 𝑛𝑞 if and only if 𝑝 ≤ 𝑞 and 𝑚 0 = 𝑛0 , 𝑚 1 = 𝑛1 , . . . , 𝑚 𝑝 = 𝑛 𝑝 . The particles will be labeled by the strings in 𝛼 ∈ 𝒜. The integer 𝑙 (𝛼) is interpreted as the generation number of the particle with label 𝛼 ∈ 𝒜. Then this particle has exactly 𝑙 (𝛼) predecessors, which we denote respectively by 𝛼 \ 1, 𝛼 \ 2, . . ., 𝛼 \ 𝑙 (𝛼). For example, if 𝛼 = 12436, then 𝛼 \ 1 = 1243 and 𝛼 \ 3 = 12. Suppose we are given a probability space (𝑊, 𝒢, P) on which the following family of independent random elements are defined: {𝜉 𝛼 (𝑥), 𝑆 𝛼 , 𝜂 𝛼 (𝑥) : 𝛼 ∈ 𝒜, 𝑥 ∈ 𝐸 },

(4.2)

where each 𝜉 𝛼 (𝑥) = {𝜉 𝛼 (𝑥, 𝑡) : 𝑡 ≥ 0} is a Markov process with transition semigroup (𝑃𝑡 )𝑡 ≥0 and 𝜉 𝛼 (𝑥, 0) = 𝑥, each 𝑆 𝛼 is an exponential random variable with parameter 𝛾, and each 𝜂 𝛼 (𝑥) is an integer-valued random variable with distribution defined by the probability generating function 𝑔(𝑥, ·). Given a finite set {𝑥1 , . . . , 𝑥 𝑛 } ⊂ 𝐸, the branching particle system {𝑋𝑡𝜎 : 𝑡 ≥ 0} Í𝑛 with initial state 𝜎 = 𝑖=1 𝛿 𝑥𝑖 is constructed as follows. Let 𝜕 be a point that is not in 𝐸. For all labels 𝛼 ∈ 𝒜 we shall define the birth times 𝛽 𝛼 , birth places 𝑏 𝛼 , death times 𝜁 𝛼 and trajectories 𝜉 𝛼 = {𝜉 𝛼 (𝑡) : 𝑡 ≥ 0} of the corresponding particles in an inductive way. If 𝛼 = 𝑛0 ∈ 𝒜 has generation number 𝑙 (𝛼) = 0, we define the birth time and place by 𝛽𝑛0 =

n0 ∞

if 𝑛0 ≤ 𝑛, if 𝑛0 > 𝑛

and

𝑏 𝑛0 =

n𝑥

𝑛0

𝜕

if 𝑛0 ≤ 𝑛, if 𝑛0 > 𝑛.

The death time is defined by 𝜁 𝑛0 = 𝛽𝑛0 + 𝑆 𝑛0 and the trajectory by 𝜉 𝑛0 (𝑡) =

n 𝜉 (𝑏 , 𝑡 − 𝛽 ) 𝑛0 𝑛0 𝑛0 𝜕

if 𝛽𝑛0 ≤ 𝑡 < ∞, if 0 ≤ 𝑡 < 𝛽𝑛0 .

4.1 Particle Systems with Local Branching

101

Suppose now the birth times, birth places, death times and trajectories are already defined for the particles in the (𝑘 − 1)-th generation. For 𝛼 = 𝑛0 𝑛1 · · · 𝑛𝑙 ( 𝛼) ∈ 𝒜 with generation number 𝑙 (𝛼) = 𝑘 ≥ 1, we define the birth time and place of the particle by 𝛽𝛼 =

n𝜁

( 𝛼\1)

∞

if 𝑛 𝑘 ≤ 𝜂 ( 𝛼\1) (𝜉 ( 𝛼\1) (𝜁 ( 𝛼\1) −)), if 𝑛 𝑘 > 𝜂 ( 𝛼\1) (𝜉 ( 𝛼\1) (𝜁 ( 𝛼\1) −))

and 𝑏𝛼 =

n𝜉

( 𝛼\1) (𝜁 ( 𝛼\1) )

𝜕

if 𝑛 𝑘 ≤ 𝜂 ( 𝛼\1) (𝜉 ( 𝛼\1) (𝜁 ( 𝛼\1) −)), if 𝑛 𝑘 > 𝜂 ( 𝛼\1) (𝜉 ( 𝛼\1) (𝜁 ( 𝛼\1) −)).

The death time of the particle is defined by 𝜁 𝛼 = 𝛽 𝛼 + 𝑆 𝛼 and the trajectory by 𝜉 𝛼 (𝑡) =

n 𝜉 (𝑏 , 𝑡 − 𝛽 ) 𝛼 𝛼 𝛼 𝜕

if 𝛽 𝛼 ≤ 𝑡 < ∞, if 0 ≤ 𝑡 < 𝛽 𝛼 .

Now the branching particle system generated by the initial mass 𝜎 is constructed as ∑︁ (4.3) 𝑋𝑡𝜎 = 1 [𝛽 𝛼 ,𝜁 𝛼 ) (𝑡)𝛿 𝜉𝛼 (𝑡) , 𝑡 ≥ 0. 𝛼∈𝒜

At this moment, it is not clear if 𝑋𝑡𝜎 (𝐸) is a.s. finite, so we think of {𝑋𝑡𝜎 : 𝑡 ≥ 0} as a process taking values in 𝑁 (𝐸)∪{Δ}, where Δ denotes infinity. The independence of the family (4.2) implies P exp{−𝑋𝑡𝜎 ( 𝑓 )} = exp{−𝜎(𝑢 𝑡 )},

𝑓 ∈ 𝐵(𝐸) ++ ,

(4.4)

where 𝑢 𝑡 (𝑥) ≡ 𝑢 𝑡 (𝑥, 𝑓 ) = − log P exp{−𝑋𝑡𝑥 ( 𝑓 )} and {𝑋𝑡𝑥 : 𝑡 ≥ 0} is a system with 𝑋0𝑥 = 𝛿 𝑥 . Here and in the sequel, we make the convention that Δ( 𝑓 ) = ∞ for 𝑓 ∈ 𝐵(𝐸) ++ . Moreover, we have the following renewal equation: ∫ 𝑡   −𝛾𝑡− 𝑓 ( 𝜉𝑡 ) −𝑢𝑡 ( 𝑥) 𝛾e−𝛾𝑠 P 𝑥 𝑔(𝜉 𝑠 , e−𝑢𝑡−𝑠 ( 𝜉𝑠 ) ) d𝑠. = P 𝑥 [e ]+ (4.5) e 0

This follows as we think about the Laplace functional of the random measure 𝑋𝑡𝑥 produced by a single particle that starts moving from the point 𝑥 ∈ 𝐸. Suppose the particle is labeled by 𝛼 ∈ 𝒜 with 𝑙 (𝛼) = 0. Then it has birth time 𝛽 𝛼 = 0, birth place 𝑏 𝛼 = 𝑥, death time 𝜁 𝛼 = 𝑆 𝛼 and trajectory 𝜉 𝛼 = {𝜉 𝛼 (𝑥, 𝑡) : 𝑡 ≥ 0}. In view of (4.4), the Laplace functional of 𝑋𝑡𝑥 is given by the left-hand side of (4.5). By the independence of 𝜁 𝛼 and 𝜉 𝛼 we have

102

4 Branching Particle Systems

n

  o  𝑥 𝑥 P 1 {𝜁 𝛼 >𝑡 } e−𝑋𝑡 ( 𝑓 ) = P 1 {𝜁 𝛼 >𝑡 } P e−𝑋𝑡 ( 𝑓 ) 𝜁 𝛼 , 𝜉 𝛼 o n = P 1 {𝜁 𝛼 >𝑡 } e− 𝑓 ( 𝜉𝛼 ( 𝑥,𝑡)) = P 𝑥 [e−𝛾𝑡− 𝑓 ( 𝜉𝑡 ) ], where the expectations related to {𝑋𝑡𝑥 : 𝑡 ≥ 0} and {𝜉 𝛼 (𝑥, 𝑡) : 𝑡 ≥ 0} are taken on the probability space (𝑊, 𝒢, P) and the one related to {𝜉𝑡 : 𝑡 ≥ 0} is taken on (Ω, ℱ, P 𝑥 ). This gives the first term on the right-hand side of (4.5). By property (3), if it happens that 0 < 𝜁 𝛼 ≤ 𝑡, then the particle dies at 𝜉 𝛼 (𝑥, 𝜁 𝛼 −) and gives birth to a random number of offspring at 𝜉 𝛼 (𝑥, 𝜁 𝛼 ) according to the probability law given by the generating function 𝑔(𝜉 𝛼 (𝑥, 𝜁 𝛼 −), ·). With those considerations we compute   𝑥 P 1 {0 𝑘.

Since 𝑧 ↦→ 𝜙 𝑘 (𝑥, 𝑧) is a convex function, we have −𝜙 𝑘 (𝑥, 𝑧) ≤ 𝐵𝑧 for all 𝑧 ≥ 0. Then the convergence of 𝑣 𝑘 (𝑡, 𝑥, 𝑓 ) is true by Proposition 2.16. The convergence of 𝑢 𝑘 (𝑡, 𝑥, 𝑓 ) follows by the relation (4.14). □ Let 𝜇 ∈ 𝑀 (𝐸) and let Q (𝑘) denote the conditional law given that 𝑌𝑘 (0) = 𝑘 𝑋 𝑘 (0) ( 𝜇) is a Poisson random measure on 𝐸 with intensity 𝑘 𝜇. By (4.13) and Theorem 1.26 it is not hard to show that   Q (𝑘) exp − ⟨𝑋 𝑘 (𝑡), 𝑓 ⟩ = exp − ⟨𝜇, 𝑣 𝑘 (𝑡)⟩ . (4.20) ( 𝜇) By Theorem 2.21 the solution of (4.19) defines a cumulant semigroup (𝑉𝑡 )𝑡 ≥0 . Theorem 4.6 If Condition 4.2 is satisfied, then the finite-dimensional distributions of {𝑋 𝑘 (𝑡) : 𝑡 ≥ 0} under Q (𝑘) converge to those of the (𝜉, 𝜙)-superprocess {𝑋𝑡 : 𝑡 ≥ 0} ( 𝜇) with initial value 𝑋0 = 𝜇. Proof Let Q 𝜇 denote the conditional law of the (𝜉, 𝜙)-superprocess {𝑋𝑡 : 𝑡 ≥ 0} given 𝑋0 = 𝜇. To get the desired convergence of the finite-dimensional distributions of {𝑋 𝑘 (𝑡) : 𝑡 ≥ 0} it suffices to prove 𝑛 𝑛 n ∑︁ o n ∑︁ o lim Q (𝑘) exp − ⟨𝑋 (𝑡 ), 𝑓 ⟩ = Q exp − ⟨𝑋 , 𝑓 ⟩ 𝑘 𝑖 𝑖 𝜇 𝑡 𝑖 𝑖 ( 𝜇)

𝑘→∞

𝑖=1

𝑖=1

(4.21)

4.2 Scaling Limits of Local Branching Systems

107

for all {𝑡 1 < · · · < 𝑡 𝑛 } ⊂ [0, ∞) and { 𝑓1 , . . . , 𝑓𝑛 } ⊂ 𝐵(𝐸) + and use Theorem 1.25. For 𝑛 = 1 this follows by Proposition 4.5. Now suppose (4.21) holds when 𝑛 is replaced by 𝑛 − 1. For any 𝑛 ≥ 2 the Markov property of {𝑋 𝑘 (𝑡) : 𝑡 ≥ 0} implies that 𝑛 o n ∑︁ 𝑓 exp ⟨𝑋 ), Q (𝑘) (𝑡 − ⟩ 𝑖 𝑖 𝑘 ( 𝜇) 𝑖=1 𝑛−1 o n ∑︁ ⟨𝑋 𝑓 ), exp 𝑢 , (Δ𝑡 − (𝑡 ), )⟩ = Q ((𝑘) ⟩ − ⟨𝑋 (𝑡 𝑖 𝑛 𝑘 𝑛−1 𝑘 𝑖 𝑘 𝜇)

(4.22)

𝑖=1

where Δ𝑡 𝑛 = 𝑡 𝑛 − 𝑡 𝑛−1 . Let 𝐵 ≥ 0 and 𝑏 𝑘 ∈ 𝐵(𝐸) be defined as in the proof of Proposition 4.5. For 𝑓 ∈ 𝐵(𝐸) + one uses Proposition 4.1 and a property of the Poisson random measure to see Q (𝑘) [⟨𝑋 𝑘 (𝑡), 𝑓 ⟩] = Q ((𝑘) [⟨𝑌𝑘 (0), 𝑃𝑡𝑏𝑘 ( 𝑓 /𝑘)⟩] = ⟨𝜇, 𝑃𝑡𝑏𝑘 𝑓 ⟩ ≤ e 𝐵𝑡 ⟨𝜇, 𝑃𝑡 𝑓 ⟩. ( 𝜇) 𝜇) It then follows that 𝑛−1 o h n ∑︁  ⟩ ⟨𝑋 − exp ), Q (𝑘) (𝑡 𝑓 exp − ⟨𝑋 𝑘 (𝑡 𝑛−1 ), 𝑢 𝑘 (Δ𝑡 𝑛 )⟩ 𝑖 𝑘 𝑖 ( 𝜇) 𝑖=1  i − exp − ⟨𝑋 𝑘 (𝑡 𝑛−1 ), 𝑉Δ𝑡𝑛 𝑓𝑛 ⟩ Ei hD (𝑡 𝑉 |𝑢 ≤ Q (𝑘) (Δ𝑡 ), ) − 𝑋 𝑓 | Δ𝑡𝑛 𝑛 𝑛 𝑘 𝑘 𝑛−1 ( 𝜇) D E 𝐵𝑡𝑛−1 𝜇, 𝑃𝑡𝑛−1 |𝑢 𝑘 (Δ𝑡 𝑛 ) − 𝑉Δ𝑡𝑛 𝑓𝑛 | . ≤ e

By Proposition 4.5, the right-hand side goes to zero as 𝑘 → ∞. Since (4.21) holds when 𝑛 is replaced by 𝑛 − 1, we have 𝑛 n ∑︁ o − ⟨𝑋 𝑘 (𝑡𝑖 ), 𝑓𝑖 ⟩ lim Q (𝑘) exp ( 𝜇)

𝑘→∞

𝑖=1 𝑛−1 o n ∑︁ − exp ⟨𝑋 𝑘 (𝑡𝑖 ), 𝑓𝑖 ⟩ − ⟨𝑋 𝑘 (𝑡 𝑛−1 ), 𝑉Δ𝑡𝑛 𝑓𝑛 ⟩ = lim Q ((𝑘) 𝜇) 𝑘→∞

𝑖=1

n

𝑛−1 ∑︁

n

𝑛 ∑︁

= Q 𝜇 exp −

⟨𝑋𝑡𝑖 , 𝑓𝑖 ⟩ − ⟨𝑋𝑡𝑛−1 , 𝑉Δ𝑡𝑛 𝑓𝑛 ⟩

o

𝑖=1

= Q 𝜇 exp −

o ⟨𝑋𝑡𝑖 , 𝑓𝑖 ⟩ ,

𝑖=1

so (4.21) follows by induction on 𝑛 ≥ 1.

□

The above theorem gives the heuristical interpretations for the parameters of the superprocess. That is, the process 𝜉 gives the law of migration of the “particles” and 𝜙 arises from the branching rate and the generating function determining the distribution of the offspring production.

4 Branching Particle Systems

108

In the remainder of this section, we consider the special case where (𝐸, 𝑑) is a complete separable metric space. Let 𝐷 𝐸 := 𝐷 ([0, ∞), 𝐸) be the space of càdlàg paths from [0, ∞) to 𝐸. We fix a metric 𝑞 on 𝐷 𝐸 for the Skorokhod topology. By definition, a stopped path is a pair (𝑤, 𝑧), where 𝑧 ≥ 0 and 𝑤 ∈ 𝐷 𝐸 satisfies 𝑤(𝑡) = 𝑤(𝑡 ∧ 𝑧) for all 𝑡 ≥ 0. Let 𝑆 be the set of all stopped paths and let 𝜌 be the metric on 𝑆 defined by 𝜌((𝑤 1 , 𝑧1 ), (𝑤 2 , 𝑧2 )) = |𝑧1 − 𝑧 2 | + 𝑞(𝑤 1 , 𝑤 2 ). Then (𝑆, 𝜌) is a complete separable metric space. Let 𝜉 = (Ω, ℱ, ℱ𝑡 , 𝜉𝑡 , P 𝑥 ) be a càdlàg Borel right process in 𝐸 satisfying: Condition 4.7 For every 𝜀 > 0 we have o n lim sup P 𝑥 sup 𝑑 (𝑥, 𝜉 𝑠 ) ≥ 𝜀 = 0. 𝑡→0 𝑥 ∈𝐸

0≤𝑠 ≤𝑡

Let (𝑢, 𝑦) ∈ 𝑆 and let 𝑏 ≥ 𝑎 ≥ 0 be constants satisfying 𝑎 ≤ 𝑦. With those parameters we define a probability measure 𝑅 𝑎,𝑏 ((𝑢, 𝑦), d(𝑤, 𝑧)) on 𝑆 by the following prescriptions: (1) 𝑅 𝑎,𝑏 ((𝑢, 𝑦), d(𝑤, 𝑧))-a.s. 𝑧 = 𝑏 and 𝑤(𝑡) = 𝑢(𝑡) for 0 ≤ 𝑡 ≤ 𝑎; (2) the law of {𝑤(𝑎 + 𝑡) : 0 ≤ 𝑡 ≤ 𝑏 − 𝑎} under 𝑅 𝑎,𝑏 ((𝑢, 𝑦), d(𝑤, 𝑧)) coincides with that of {𝜉𝑡 : 0 ≤ 𝑡 ≤ 𝑏 − 𝑎} under P𝑢(𝑎) . For 𝑠 > 0 and 𝑦 ≥ 0 let 𝛽 = {𝛽𝑡 : 𝑡 ≥ 0} be a reflecting standard Brownian motion 𝑦 starting at 𝛽0 = 𝑦 and let 𝛾𝑠 (d𝑎, d𝑏) be the distribution of (inf 0≤𝑟 ≤𝑠 𝛽𝑟 , 𝛽𝑠 ) on R+2 . The reflection principle gives that   2(𝑦 + 𝑏 − 2𝑎) (𝑦 + 𝑏 − 2𝑎) 2 𝑦 𝛾𝑠 (d𝑎, d𝑏) = exp − 1 {0 𝑡 + 1/𝑘. By an observation of Le Gall (1993), the measure-valued process 𝑌𝑘 (𝑡) =

𝑛𝑡 ∑︁

𝛿 𝜂𝑎𝑖 (𝑡 ) (𝑡) ,

𝑡≥0

𝑖=1

is a branching particle system with parameters (𝜉, 𝑔, 2𝑘) in the sense of Section 4.1, where 𝑔(𝑧) = 1/2 + 𝑧 2 /2. Note that 𝑛0 is a Poisson random variable with mean 𝑘𝑢 by Itô’s excursion theory. Then the continuity in probability of the process 𝑠 ↦→ 𝜂 𝑠 and the approximation of the Brownian local time by upcrossing numbers imply 𝑘 −1𝑌𝑘 (𝑡) → 𝑋𝑡 as 𝑘 → ∞ for a random measure 𝑋𝑡 on 𝐸 defined by ∫

𝜎 (𝑢)

𝑓 (𝜂 𝑠 (𝑡))d𝑙 𝑠 (𝑡),

𝑋𝑡 ( 𝑓 ) =

𝑓 ∈ 𝐵(𝐸),

(4.24)

0

where d𝑙 𝑠 (𝑡) denotes the integration with respect to the increasing function 𝑠 ↦→ 𝑙 𝑠 (𝑡). By Theorem 4.6 we conclude {𝑋𝑡 : 𝑡 ≥ 0} is a Dawson–Watanabe superprocess with spatial motion 𝜉 and binary local branching mechanism 𝜙(𝑧) = 𝑧 2 . This gives a representation of the superprocess in terms of the 𝜉-Brownian snake. When 𝐸 is a singleton, the representation reduces to the first result of Theorem 3.53.

4.3 General Branching Particle Systems In this section, we consider a model of branching particle systems that generalizes the system introduced in Section 4.1. The high-density limits of these systems will lead to Dawson–Watanabe superprocesses with decomposable branching mechanisms. Let 𝐸 be a Lusin topological space and let 𝑁 (𝐸) denote the space of integer-valued finite measures on 𝐸. Let 𝜉 = (Ω, ℱ, ℱ𝑡 , 𝜉𝑡 , P 𝑥 ) be a Borel right process with state space 𝐸 and conservative transition semigroup (𝑃𝑡 )𝑡 ≥0 . We assume the sample path {𝜉𝑡 : 𝑡 ≥ 0} is right continuous in both the original and the Ray topologies and has

110

4 Branching Particle Systems

¯ Let {𝐾 (𝑡) : 𝑡 ≥ 0} be a left limits {𝜉𝑡− : 𝑡 > 0} in the Ray–Knight completion 𝐸. continuous admissible additive functional of 𝜉. Let 𝛼 ∈ 𝐵(𝐸) + and let 𝐹 (𝑥, d𝜈) be a Markov kernel from 𝐸 to 𝑁 (𝐸) such that ∫ (4.25) 𝜈(1)𝐹 (𝑥, d𝜈) < ∞. sup 𝑥 ∈𝐸

𝑁 (𝐸)

For 𝑥 ∈ 𝐸¯ \ 𝐸 let 𝐹 (𝑥, d𝜈) be the unit mass at 𝛿 𝑥 . A general branching particle system is characterized by the following properties: (1) The particles in 𝐸 move independently according to the law given by the transition probabilities of 𝜉. (2) For a particle which is alive at time 𝑟 ≥ 0 and follows the path {𝜉 𝑠 : 𝑠 ≥ 𝑟 }, ∫ 𝑡 the conditional probability of survival in the time interval [𝑟, 𝑡) is exp{− 𝑟 𝛼(𝜉 𝑠 )𝐾 (d𝑠)}. (3) When a particle following the path {𝜉 𝑠 : 𝑠 ≥ 𝑟 } dies at time 𝑡 > 𝑟, it gives birth to a random number of offspring in 𝐸 according to the probability kernel 𝐹 (𝜉𝑡− , d𝜈). The offspring then start to move from their birth places. We also assume that the lifetimes and the branchings of different particles are independent. Let 𝑋𝑡 (𝐵) denote the number of particles in 𝐵 ∈ ℬ(𝐸) that are alive at time 𝑡 ≥ 0. If we assume 𝑋0 (𝐸) < ∞, then {𝑋𝑡 : 𝑡 ≥ 0} is a Markov process with state space 𝑁 (𝐸), which will be referred to as the general branching particle system with parameters (𝜉, 𝐾, 𝛼, 𝐹). We are not going to give the rigorous construction of the general branching system, which involves the same ideas as the construction in Section 4.1 but is considerably more complicated. Let 𝜎 ∈ 𝑁 (𝐸) and let {𝑋𝑡𝜎 : 𝑡 ≥ 0} be a general branching particle system with parameters (𝜉, 𝐾, 𝛼, 𝐹) and initial state 𝑋0 = 𝜎. Suppose that the process is defined on the probability space (𝑊, 𝒢, P). The above properties imply P exp{−𝑋𝑡𝜎 ( 𝑓 )} = exp{−𝜎(𝑢 𝑡 )},

𝑓 ∈ 𝐵(𝐸) + ,

(4.26)

where 𝑢 𝑡 (𝑥) ≡ 𝑢 𝑡 (𝑥, 𝑓 ) is determined by the renewal equation ∫ 𝑡 o n −𝑢𝑡 ( 𝑥) e 𝛼(𝜉 𝑠 )𝐾 (d𝑠) = P 𝑥 exp − 𝑓 (𝜉𝑡 ) − 0 ∫ 𝑡 ∫  ∫ 𝑠 − 0 𝛼( 𝜉𝑟 ) 𝐾 (d𝑟) −𝜈 (𝑢𝑡−𝑠 ) 𝐹 (𝜉 𝑠 , d𝜈) . e e 𝛼(𝜉 𝑠 )𝐾 (d𝑠) + P𝑥 𝑁 (𝐸)

0

This equation is derived by arguments similar to those used for (4.5). By Proposition 2.9 the above equation implies ∫ 𝑡  − 𝑓 ( 𝜉𝑡 ) −𝑢𝑡−𝑠 ( 𝜉𝑠 ) −𝑢𝑡 ( 𝑥) e − P𝑥 = P𝑥 e 𝐾 (d𝑠) 𝛼(𝜉 𝑠 )e 0  ∫ 𝑡 ∫ e−𝜈 (𝑢𝑡−𝑠 ) 𝐹 (𝜉 𝑠 , d𝜈) . 𝛼(𝜉 𝑠 )𝐾 (d𝑠) + P𝑥 (4.27) 0

𝑁 (𝐸)

4.3 General Branching Particle Systems

111

For the general branching particle system, it is natural to treat separately from others the offspring that start their migration from the death sites of their parents. To this end, we need to introduce some additional parameters as follows. Let 𝑔 ∈ 𝐵(𝐸 × [−1, 1]) be such that for each 𝑥 ∈ 𝐸, 𝑔(𝑥, 𝑧) =

∞ ∑︁

𝑝 𝑖 (𝑥)𝑧𝑖 ,

|𝑧| ≤ 1,

𝑖=0

is a probability generating function with sup 𝑥 𝑔 𝑧′ (𝑥, 1−) < ∞. Recall that 𝑃(𝐸) denotes the space of probability measures on 𝐸. Let 𝐺 (𝑥, d𝜋) be a probability kernel from 𝐸 to 𝑃(𝐸) and let ℎ ∈ 𝐵(𝐸 × 𝑃(𝐸) × [−1, 1]) be such that for each (𝑥, 𝜋) ∈ 𝐸 × 𝑃(𝐸), ℎ(𝑥, 𝜋, 𝑧) =

∞ ∑︁

𝑞 𝑖 (𝑥, 𝜋)𝑧𝑖 ,

|𝑧| ≤ 1,

𝑖=0

is a probability generating function with sup 𝑥, 𝜋 ℎ ′𝑧 (𝑥, 𝜋, 1−) < ∞. Now we can define the probability kernels 𝐹0 (𝑥, d𝜈) and 𝐹1 (𝑥, d𝜈) from 𝐸 to 𝑁 (𝐸) by ∫

e−𝜈 ( 𝑓 ) 𝐹0 (𝑥, d𝜈) = 𝑁 (𝐸)

∞ ∑︁

𝑝 𝑖 (𝑥)e−𝑖 𝑓 ( 𝑥) = 𝑔 𝑥, e− 𝑓 ( 𝑥)




𝑖=0

and ∫

−𝜈 ( 𝑓 )

e

∫ 𝐹1 (𝑥, d𝜈) =

𝑁 (𝐸)


ℎ 𝑥, 𝜋, 𝜋(e− 𝑓 ) 𝐺 (𝑥, d𝜋).

𝑃 (𝐸)

Suppose we have the decomposition 𝛼(𝑥) = 𝛾(𝑥) + 𝜌(𝑥) for 𝛾, 𝜌 ∈ 𝐵(𝐸) + . Let 𝐹 (𝑥, d𝜈) =

 1  𝛾(𝑥)𝐹0 (𝑥, d𝜈) + 𝜌(𝑥)𝐹1 (𝑥, d𝜈) 𝛼(𝑥)

(4.28)

if 𝛼(𝑥) > 0 and let 𝐹 (𝑥, d𝜈) = 𝐹0 (𝑥, d𝜈) if 𝛼(𝑥) = 0. For the kernel 𝐹 (𝑥, d𝜈) given by (4.28), the general branching particle system is determined by the parameters (𝜉, 𝐾, 𝛾, 𝑔, 𝜌, ℎ, 𝐺). Intuitively, as a particle dies at 𝑥 ∈ 𝐸, the branching is of local type with probability 𝛾(𝑥)/𝛼(𝑥) and is of non-local type with probability 𝜌(𝑥)/𝛼(𝑥). If the local branching type is chosen, the particle gives birth to a number of offspring at its death site 𝑥 according to the distribution {𝑝 𝑖 (𝑥)}. If non-local branching occurs, an offspring-location-distribution 𝜋 ∈ 𝑃(𝐸) is first selected according to the probability kernel 𝐺 (𝑥, d𝜋), the particle then gives birth to a random number of offspring according to the distribution {𝑞 𝑖 (𝑥, 𝜋)}, and those offspring choose their locations in 𝐸 independently of each other according to the distribution 𝜋(d𝑦). Therefore the locations of non-locally displaced offspring involve two sources of randomness. From (4.27) and (4.28) we obtain

112

4 Branching Particle Systems −𝑢𝑡 ( 𝑥)

e

− 𝑓 ( 𝜉𝑡 )

∫

𝑡 −𝑢𝑡−𝑠 ( 𝜉𝑠 )



𝐾 (d𝑠) 𝛼(𝜉 𝑠 )e  ∫
𝜌(𝜉 𝑠 )𝐾 (d𝑠) ℎ 𝜉 𝑠 , 𝜋, 𝜋(e−𝑢𝑡−𝑠 ) 𝐺 (𝜉 𝑠 , d𝜋) + P𝑥 𝑃 (𝐸)  ∫0 𝑡  −𝑢𝑡−𝑠 ( 𝜉𝑠 ) 𝛾(𝜉 𝑠 )𝑔(𝜉 𝑠 , e + P𝑥 )𝐾 (d𝑠) .

= P𝑥 e

− P𝑥

0

∫

𝑡

0

Setting 𝑣(𝑡, 𝑥) = 1 − exp{−𝑢 𝑡 (𝑥)} we have ∫ 𝑡    𝑣(𝑡, 𝑥) = P 𝑥 1 − e− 𝑓 ( 𝜉𝑡 ) − P 𝑥 𝛼(𝜉 𝑠 )𝑣(𝑡 − 𝑠, 𝜉 𝑠 )𝐾 (d𝑠) 0  ∫ 𝑡 ∫   −𝑢𝑡−𝑠 𝜌(𝜉 𝑠 ) )) 𝐺 (𝜉 𝑠 , d𝜋)𝐾 (d𝑠) + P𝑥 1 − ℎ(𝜉 𝑠 , 𝜋, 𝜋(e 𝑃 (𝐸)  ∫0 𝑡    − P𝑥 𝛾(𝜉 𝑠 ) 𝑔(𝜉 𝑠 , e−𝑢𝑡−𝑠 ( 𝜉𝑠 ) ) − 1 𝐾 (d𝑠) . 0

Then (𝑡, 𝑥) ↦→ 𝑣(𝑡, 𝑥) solves the equation − 𝑓 ( 𝜉𝑡 )





∫

𝑡

𝑣(𝑡, 𝑥) = P 𝑥 1 − e 𝜌(𝜉 𝑠 )𝑣(𝑡 − 𝑠, 𝜉 𝑠 )𝐾 (d𝑠) − P𝑥 0  ∫ 𝑡 − P𝑥 𝜙(𝜉 𝑠 , 𝑣(𝑡 − 𝑠, 𝜉 𝑠 ))𝐾 (d𝑠)  ∫0 𝑡  + P𝑥 𝜌(𝜉 𝑠 )𝜓(𝜉 𝑠 , 𝑣(𝑡 − 𝑠))𝐾 (d𝑠) ,



(4.29)

0

where 𝜙(𝑥, 𝑧) = 𝛾(𝑥) [𝑔(𝑥, 1 − 𝑧) − (1 − 𝑧)] and ∫ [1 − ℎ(𝑥, 𝜋, 1 − 𝜋( 𝑓 ))]𝐺 (𝑥, d𝜋).

𝜓(𝑥, 𝑓 ) = 𝑃 (𝐸)

The uniqueness of the solution of (4.29) follows by a standard application of Gronwall’s inequality.

4.4 Scaling Limits of General Branching Systems In this section, we prove that a Dawson–Watanabe superprocess with local and non-local branching mechanisms arises as the small particle limit of a sequence of the general branching particle systems introduced in Section 4.3. The arguments are similar to those used in the local branching case. For each integer 𝑘 ≥ 1, let {𝑌𝑘 (𝑡) : 𝑡 ≥ 0} be a sequence of general branching particle systems determined by

4.4 Scaling Limits of General Branching Systems

113

the parameters (𝜉, 𝐾, 𝛾 𝑘 , 𝑔 𝑘 , 𝜌 𝑘 , ℎ 𝑘 , 𝐺). Recall that 𝑁 𝑘 (𝐸) = {𝑘 −1 𝜈 : 𝜈 ∈ 𝑁 (𝐸)}. Let 𝑋 𝑘 (𝑡) = 𝑘 −1𝑌𝑘 (𝑡) for 𝑡 ≥ 0. Then {𝑋 𝑘 (𝑡) : 𝑡 ≥ 0} is a Markov process in 𝑁 𝑘 (𝐸). For 0 ≤ 𝑧 ≤ 𝑘 let 𝜙 𝑘 (𝑥, 𝑧) = 𝑘𝛾 𝑘 (𝑥) [𝑔 𝑘 (𝑥, 1 − 𝑧/𝑘) − (1 − 𝑧/𝑘)]

(4.30)

𝜁 𝑘 (𝑥, 𝜋, 𝑧) = 𝑘 [1 − ℎ 𝑘 (𝑥, 𝜋, 1 − 𝑧/𝑘)].

(4.31)

and

For 𝑓 ∈ 𝐵(𝐸) + let ∫ 𝜁 𝑘 (𝑥, 𝜋, 𝜋( 𝑓 ))𝐺 (𝑥, d𝜋),

𝜓 𝑘 (𝑥, 𝑓 ) =

(4.32)

𝑃 (𝐸)

which makes sense when 𝑘 ≥ 1 is sufficiently large. Let Q𝜈(𝑘) denote the conditional law given 𝑋 𝑘 (0) = 𝜈 ∈ 𝑁 𝑘 (𝐸). By (4.26) for any 𝑓 ∈ 𝐵(𝐸) + we have   Q𝜈(𝑘) exp − ⟨𝑋 𝑘 (𝑡), 𝑓 ⟩ = exp − ⟨𝜈, 𝑢 𝑘 (𝑡)⟩ ,

(4.33)

where 𝑢 𝑘 (𝑡, 𝑥) is determined by 𝑣 𝑘 (𝑡, 𝑥) = 𝑘 [1 − exp{−𝑢 𝑘 (𝑡, 𝑥)/𝑘 }] and − 𝑓 ( 𝜉𝑡 )/𝑘





∫

𝑡



𝜌 𝑘 (𝜉 𝑠 )𝑣 𝑘 (𝑡 − 𝑠, 𝜉 𝑠 )𝐾 (d𝑠) 𝑣 𝑘 (𝑡, 𝑥) = P 𝑥 𝑘 1 − e − P𝑥 0 ∫ 𝑡  − P𝑥 𝜙 𝑘 (𝜉 𝑠 , 𝑣 𝑘 (𝑡 − 𝑠, 𝜉 𝑠 ))𝐾 (d𝑠)  ∫0 𝑡  𝜌 𝑘 (𝜉 𝑠 )𝜓 𝑘 (𝜉 𝑠 , 𝑣 𝑘 (𝑡 − 𝑠))𝐾 (d𝑠) . + P𝑥 (4.34) 0

For convenience of statement of the results in this section, we formulate the following conditions: Condition 4.8 For each 𝑎 ≥ 0 the sequence {𝜙 𝑘 (𝑥, 𝑧)} is Lipschitz with respect to 𝑧 uniformly on 𝐸 × [0, 𝑎] and 𝜙 𝑘 (𝑥, 𝑧) converges to some 𝜙(𝑥, 𝑧) uniformly on 𝐸 × [0, 𝑎] as 𝑘 → ∞. Condition 4.9 For each 𝑘 ≥ 1 we have (d/d𝑧)ℎ 𝑘 (𝑥, 𝜋, 1−) ≤ 1 uniformly on 𝐸 × 𝑃(𝐸). Condition 4.10 For each 𝑎 ≥ 0 the sequence 𝜁 𝑘 (𝑥, 𝜋, 𝑧) converges to some 𝜁 (𝑥, 𝜋, 𝑧) uniformly on 𝐸 × 𝑃(𝐸) × [0, 𝑎] as 𝑘 → ∞. Under Condition 4.8, one can see as in the proof of Proposition 4.3 that 𝜙(𝑥, 𝑧) has the representation (2.49).

114

4 Branching Particle Systems

Proposition 4.11 If Conditions 4.9 and 4.10 hold, then 𝜁 (𝑥, 𝜋, 𝑧) has the representation ∫ ∞ (4.35) (1 − e−𝑧𝑢 )𝑛(𝑥, 𝜋, d𝑢), 𝜁 (𝑥, 𝜋, 𝑧) = 𝛽(𝑥, 𝜋)𝑧 + 0

where 𝛽 ∈ 𝐵(𝐸 × 𝑃(𝐸)) + and 𝑢𝑛(𝑥, 𝜋, d𝑢) is a bounded kernel from 𝐸 × 𝑃(𝐸) to (0, ∞) satisfying ∫ ∞ (4.36) 𝑢𝑛(𝑥, 𝜋, d𝑢) ≤ 1, 𝑥 ∈ 𝐸, 𝜋 ∈ 𝑃(𝐸). 𝛽(𝑥, 𝜋) + 0

Proof We first apply Theorem 1.46 for fixed (𝑥, 𝜋) ∈ 𝐸 × 𝑃(𝐸) to get the representation (4.35), where 𝛽(𝑥, 𝜋) ≥ 0 is a constant and (1 ∧ 𝑢)𝑛(𝑥, 𝜋, d𝑢) is a finite measure on (0, ∞). From Condition 4.9 we have d d 𝜁 𝑘 (𝑥, 𝜋, 𝑧) = ℎ 𝑘 (𝑥, 𝜋, 1 − 𝑧/𝑘) ≤ 1, d𝑧 d𝑧 and hence 𝜁 𝑘 (𝑥, 𝜋, 𝑧) ≤ 𝑧 for 0 ≤ 𝑧 ≤ 𝑘. It then follows that 𝜁 (𝑥, 𝜋, 𝑧) ≤ 𝑧 for all 𝑧 ≥ 0. Therefore ∫ ∞ d 𝛽(𝑥, 𝜋) + 𝑢𝑛(𝑥, 𝜋, d𝑢) = 𝜁 (𝑥, 𝜋, 0+) ≤ 1 d𝑧 0 uniformly on 𝐸 ×𝑃(𝐸). As in the proof of Proposition 4.3 one sees 𝛽 ∈ 𝐵(𝐸 ×𝑃(𝐸)) + and 𝑛(𝑥, 𝜋, d𝑢) is a kernel from 𝐸 × 𝑃(𝐸) to (0, ∞). □ Proposition 4.12 To each function 𝜁 (𝑥, 𝜋, 𝑧) given by (4.35) and (4.36) there corresponds a sequence {𝜁 𝑘 (𝑥, 𝜋, 𝑧)} in the form (4.31) such that Conditions 4.9 and 4.10 are satisfied. Proof We can give a direct construction of the sequence as in the proof of Proposition 4.4. For (𝑥, 𝜋, 𝑧) ∈ 𝐸 × 𝑃(𝐸) × [−1, 1] and 𝑘 ≥ 1 set ∫ 1 ∞ 𝑘𝑢(𝑧−1) (e ℎ 𝑘 (𝑥, 𝜋, 𝑧) = 1 + 𝛽(𝑥, 𝜋) (𝑧 − 1) + − 1)𝑛(𝑥, 𝜋, d𝑢). 𝑘 0 Clearly, the function 𝑧 ↦→ ℎ 𝑘 (𝑥, 𝜋, 𝑧) is analytic in (−1, 1) and ℎ 𝑘 (𝑥, 𝜋, 1) = 1. Observe also that ∫ ∞ ℎ 𝑘 (𝑥, 𝜋, 0) ≥ 1 − 𝛽(𝑥, 𝜋) − 𝑢𝑛(𝑥, 𝜋, d𝑢) ≥ 0 0

and d𝑖 ℎ 𝑘 (𝑥, 𝜋, 0) ≥ 0, d𝑧𝑖

𝑖 = 1, 2, . . . .

4.4 Scaling Limits of General Branching Systems

115

Therefore ℎ 𝑘 (𝑥, 𝜋, ·) is a probability generating function. Let 𝜁 𝑘 (𝑥, 𝜋, 𝑧) be defined by (4.31). Then 𝜁 𝑘 (𝑥, 𝜋, 𝑧) = 𝜁 (𝑥, 𝜋, 𝑧) for 0 ≤ 𝑧 ≤ 𝑘. □ Proposition 4.13 Suppose 𝜁 (𝑥, 𝜋, 𝑧) is given by (4.35) and (4.36) and 𝐺 (𝑥, d𝜋) is a probability kernel from 𝐸 to 𝑃(𝐸). Let ∫ 𝜁 (𝑥, 𝜋, 𝜋( 𝑓 ))𝐺 (𝑥, d𝜋), 𝜓(𝑥, 𝑓 ) = 𝑥 ∈ 𝐸, 𝑓 ∈ 𝐵(𝐸) + . (4.37) 𝑃 (𝐸)

Then 𝜓(𝑥, 𝑓 ) admits the representation (2.21) with ∫ 𝜈(1)𝑅(𝑥, d𝜈) ≤ 1, 𝜌(𝑥, 1) +

𝑥 ∈ 𝐸.

(4.38)

𝑀 (𝐸) ◦

Proof We get (2.21) from (4.35) and (4.37) by changing the variables of integration. □ The boundedness (4.38) follows from (4.36). denote the conditional law given that 𝑌𝑘 (0) = 𝑘 𝑋 𝑘 (0) Let 𝜇 ∈ 𝑀 (𝐸) and let Q (𝑘) ( 𝜇) is a Poisson random measure on 𝐸 with intensity 𝑘 𝜇. From (4.33) and Theorem 1.26 we get   Q (𝑘) exp − ⟨𝑋 𝑘 (𝑡), 𝑓 ⟩ = exp − ⟨𝜇, 𝑣 𝑘 (𝑡)⟩ . ( 𝜇)

(4.39)

By modifications of the arguments in Section 4.2 we have: Proposition 4.14 Suppose that 𝜌 𝑘 → 𝜌 ∈ 𝐵(𝐸) + in the supremum norm and Conditions 4.8, 4.9 and 4.10 are satisfied. Then for each 𝑎 ≥ 0 both 𝑣 𝑘 (𝑡, 𝑥, 𝑓 ) and 𝑢 𝑘 (𝑡, 𝑥, 𝑓 ) converge uniformly on the set [0, 𝑎] × 𝐸 × 𝐵 𝑎 (𝐸) + of (𝑡, 𝑥, 𝑓 ) to the unique locally bounded positive solution (𝑡, 𝑥) ↦→ 𝑉𝑡 𝑓 (𝑥) of the evolution equation  ∫ 𝑡 𝑉𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) − P 𝑥 𝜌(𝜉 𝑠 )𝑉𝑡−𝑠 𝑓 (𝜉 𝑠 )𝐾 (d𝑠) 0 ∫ 𝑡  − P𝑥 𝜙(𝜉 𝑠 , 𝑉𝑡−𝑠 𝑓 (𝜉 𝑠 ))𝐾 (d𝑠)   ∫0 𝑡 (4.40) + P𝑥 𝜌(𝜉 𝑠 )𝜓(𝜉 𝑠 , 𝑉𝑡−𝑠 𝑓 )𝐾 (d𝑠) . 0

Moreover, the operators (𝑉𝑡 )𝑡 ≥0 constitute a cumulant semigroup. Theorem 4.15 Suppose that 𝜌 𝑘 → 𝜌 ∈ 𝐵(𝐸) + in the supremum norm and Conditions 4.8, 4.9 and 4.10 are satisfied. Then the finite-dimensional distributions of {𝑋 𝑘 (𝑡) : 𝑡 ≥ 0} under Q (𝑘) converge to those of a Dawson–Watanabe superprocess ( 𝜇) {𝑋𝑡 : 𝑡 ≥ 0} with initial state 𝑋0 = 𝜇 and with cumulant semigroup given by (4.40). The Dawson–Watanabe superprocess with cumulant semigroup given by (4.40) has local branching mechanism (𝑥, 𝑧) ↦→ 𝜌(𝑥)𝑧 + 𝜙(𝑥, 𝑧) and non-local branching mechanism (𝑥, 𝑓 ) ↦→ 𝜌(𝑥)𝜓(𝑥, 𝑓 ). Note that conditions (4.36) and (4.38) actually put no restriction on the non-local branching mechanism because of the multiplying

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4 Branching Particle Systems

factor 𝜌(𝑥). Heuristically, the local branching mechanism describes the death and birth of particles at 𝑥 ∈ 𝐸 and 𝜁 (𝑥, 𝜋, ·) describes the birth of particles at 𝑥 ∈ 𝐸 that are displaced into 𝐸 following the distribution 𝜋 ∈ 𝑃(𝐸), which is selected according to the kernel 𝐺 (𝑥, d𝜋). More specifically, we can explain the non-local branching mechanisms given by (2.51) and (2.52) as follows. In the first case, the non-locally displaced offspring born at 𝑥 ∈ 𝐸 are produced according to 𝜁 (𝑥, ·) and they choose their locations independently according to the distribution 𝜋(𝑥, d𝑦). In the second case, once a particle dies at 𝑥 ∈ 𝐸, a point 𝑦 ∈ 𝐸 is first chosen following the distribution 𝜋(𝑥, d𝑦) and then the non-locally displaced offspring are produced at this point according to the law given by 𝜁 (𝑥, 𝑦, ·). Example 4.2 If there is a 𝑐 ∈ 𝐵(𝐸) + such that 𝛾 𝑘 (𝑥) = 𝑘𝑐(𝑥) and 𝑔 𝑘 (𝑥, 𝑧) = (1 + 𝑧2 )/2, then from (4.30) we have 𝜙 𝑘 (𝑥, 𝑧) = 𝑐(𝑥)𝑧2 /2, which gives a binary local branching mechanism for the corresponding superprocess. The (𝜉, 𝐾, 𝜙)-superprocess with general branching mechanism given by (2.29) or (2.30) also arises as the high-density limits of branching particle systems in finite-dimensional distributions. We leave the considerations to the reader.

4.5 Notes and Comments Silverstein (1968) proved an existence theorem for branching particle systems as measure-valued processes. A different but equivalent formulation of branching systems was given in Ikeda et al. (1968a, 1968b, 1969). The construction given in Section 4.1 follows that of Walsh (1986). Watanabe (1968) established a rescaling limit theorem of discrete-time branching particle systems, which gave a super-Brownian motion as the limit process. See also Dawson (1975) and Ethier and Kurtz (1986, pp. 400–407). The main references for this chapter are Dawson et al. (2002c) and Dynkin (1993a), where non-local branching superprocesses were obtained. For a Hunt spatial motion process, one can prove the weak convergence of the rescaled branching systems in the space of càdlàg paths; see, e.g., Schied (1999). The construction of binary branching superprocesses using Brownian snakes was originally given by Le Gall (1991). The proof of (4.24) was given in Le Gall (1993) for a continuous spatial motion. The idea of the representation is to code the genealogical trees of the population by Brownian excursions and to construct the spatial motions by attaching to each individual in the trees a path of the spatial motion process in such a way that the paths of two individuals are the same up to the level corresponding to the generation of their last common ancestor. A Brownian snake representation for catalytic branching superprocesses was provided in Klenke (2003). The super-Brownian motion also arises in limit theorems of other rescaled interacting particle systems. For example, such limit theorems were proved for contact processes in Durrett and Perkins (1999), for high-dimensional percolation in Hara and Slade (2000) and van der Hofstad and Slade (2003), for voter models in Cox et

4.5 Notes and Comments

117

al. (2000), for interacting diffusions in Cox and Klenke (2003), for Lotka–Volterra models in Cox and Perkins (2005, 2008) and for epidemic models in Lalley (2009). See Slade (2002) for a nice introduction of the explorations in the subject. The Donsker invariance principle is deeply involved in those results. Generally speaking, if the relevant spatial transition kernel has finite variance, a super-Brownian motion would arise as the limit process. He (2011) considered the case where the transition kernel is in the contraction domain of a stable law and obtained a super-stable process in the limit. One motivation of the study is to use the limit theorems to investigate the asymptotic properties of the approximating systems; see, e.g., Cox and Perkins (2004, 2007). We refer the reader to the survey of Aldous (2010) on continuum limits of discrete random structures, which is illustrated by the theory of the Brownian continuum random tree as a limiting object of various models. For general references of interacting particle systems, one may see Chen (2004), Durrett (1995) and Liggett (1985, 1999).

Chapter 5

Basic Regularities of Superprocesses

In this chapter we prove some basic regularities of Dawson–Watanabe superprocesses. The theory is developed here in the Borel right setting, which is particularly suitable for the applications of various transformations. We shall see that if the underlying spatial motion 𝜉 is a Borel right process, the (𝜉, 𝜙)-superprocess is a Borel right process with quasi-left continuous natural filtration; and if 𝜉 is a Hunt process, so is the superprocess. We give characterizations of the so-called occupation times. Some useful upper and lower bounds for the cumulant semigroup are also proved.

5.1 Right Continuous Realizations Suppose that 𝐸 is a Lusin topological space and 𝑑 is a metric for its topology so that the 𝑑-completion of 𝐸 is compact. Let 𝜉 = (Ω, ℱ, ℱ𝑡 , 𝜉𝑡 , P 𝑥 ) be a conservative Borel right process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 and with resolvent (𝑈 𝛼 ) 𝛼>0 defined by (A.6). Let 𝜙 be a branching mechanism on 𝐸 given by (2.29) or (2.30) and let (𝑄 𝑡 )𝑡 ≥0 denote the transition semigroup on 𝑀 (𝐸) of the (𝜉, 𝜙)-superprocess defined by (2.35) and (2.36). Let 𝒟 be a countable and uniformly dense subset of 𝐶𝑢 (𝐸) ++ and assume 1 ∈ 𝒟. Let ℛ be the rational Ray cone for (𝑃𝑡 )𝑡 ≥0 constructed from 𝒟. We clearly have ℛ \ {0} ⊂ 𝐵(𝐸) ++ . Note also that for each 𝑓 ∈ ℛ there is a constant 𝛼 = 𝛼( 𝑓 ) > 0 such that 𝑓 is an 𝛼-excessive function relative to (𝑃𝑡 )𝑡 ≥0 . Let 𝐸¯ be the Ray–Knight ¯ completion of 𝐸 defined from ℛ. Then 𝐸¯ is a compact metric space and 𝐸 ∈ ℬ( 𝐸) by Proposition A.31. Let (𝑈¯ 𝛼 ) 𝛼>0 be the Ray extension of (𝑈 𝛼 ) 𝛼>0 . We denote the ¯ Ray extension of (𝑃𝑡 )𝑡 ≥0 by ( 𝑃¯𝑡 )𝑡 ≥0 , which is a Borel semigroup on 𝐸. ¯ Let 𝐸 𝜌 denote the set 𝐸 furnished with the Ray topology inherited from 𝐸. Then each 𝑓 ∈ ℛ is uniformly continuous on 𝐸 𝜌 and admits a unique continuous ¯ denote the class of those extensions. By Proposition A.28 ¯ Let ℛ extension 𝑓¯ to 𝐸. ¯ is uniformly dense in 𝐶 ( 𝐸). ¯ ¯ ¯ By the collection ℛ − ℛ := { 𝑓¯ − 𝑔¯ : 𝑓¯, 𝑔¯ ∈ ℛ} Proposition A.31 we have ℬ(𝐸 𝜌 ) = ℬ(𝐸). Let 𝐷 = {𝑥 ∈ 𝐸¯ : 𝑃¯0 (𝑥, ·) = 𝛿 𝑥 (·)} and 𝐵 = {𝑥 ∈ 𝐸¯ : 𝑃¯0 (𝑥, ·) ≠ 𝛿 𝑥 (·)} denote respectively the sets of non-branch © Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_5

119

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5 Basic Regularities of Superprocesses

¯ and 𝐷 ⊃ 𝐸. By points and branch points for ( 𝑃¯𝑡 )𝑡 ≥0 . Clearly, we have 𝐷 ∈ ℬ( 𝐸) ¯ Theorem A.24 the restriction of ( 𝑃𝑡 )𝑡 ≥0 to 𝐷 is a Borel right semigroup. We shall use the same notation for the restriction. Since 𝐸¯ is a compact metric space, the ¯ ¯ is locally compact, separable and metrizable. We fix a metric on 𝑀 ( 𝐸) space 𝑀 ( 𝐸) compatible with its topology and regard 𝑀 (𝐸 𝜌 ) and 𝑀 (𝐷) as topological subspaces ¯ comprising measures supported by 𝐸 𝜌 and 𝐷, respectively. of 𝑀 ( 𝐸) ¯ + to 𝐵( 𝐸) ¯ by ¯ 𝑓¯) from 𝐵( 𝐸) We extend 𝑓 ↦→ 𝜙(·, 𝑓 ) to an operator 𝑓¯ ↦→ 𝜙(·, ¯ ¯ ¯ ¯ ¯ setting 𝜙(𝑥, 𝑓 ) = 𝜙(𝑥, 𝑓 ) for 𝑥 ∈ 𝐸 and 𝜙(𝑥, 𝑓 ) = 0 for 𝑥 ∈ 𝐸 \ 𝐸, where 𝑓 = 𝑓¯| 𝐸 ¯ + . Then 𝜙¯ has the canonical representation (2.29) is the restriction to 𝐸 of 𝑓¯ ∈ 𝐵( 𝐸) ¯ ¯ ¯ 𝐻) defined in obvious ways. The construction of the with the parameters ( 𝑏, 𝑐, ¯ 𝜂, Dawson–Watanabe superprocess given by (2.35) and (2.36) certainly applies to the restrictions of ( 𝑃¯𝑡 )𝑡 ≥0 and 𝜙¯ to 𝐷. We can even extend the construction to the space ¯ More precisely, for every 𝑓¯ ∈ 𝐵( 𝐸) ¯ + there is a unique locally bounded positive 𝐸. ¯ ¯ solution 𝑡 ↦→ 𝑉𝑡 𝑓 to the equation ∫ 𝑡 ∫ ¯ ¯ ¯ ¯ ¯ ¯ 𝑉¯𝑠 𝑓¯) 𝑃¯𝑡−𝑠 (𝑥, d𝑦), 𝑡 ≥ 0, 𝑥 ∈ 𝐸, (5.1) 𝑉𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) − 𝜙(𝑦, d𝑠 0

𝐸¯

¯ In fact, which defines a cumulant semigroup (𝑉¯𝑡 )𝑡 ≥0 with underlying space 𝐸. ¯ by Proposition A.23 for every 𝑡 ≥ 0 and every 𝑥 ∈ 𝐸 the probability measure 𝑃¯𝑡 (𝑥, ·) is supported by 𝐷, so we can first solve the equation on 𝐷 and then define ¯ Thus the function 𝑉¯𝑡 𝑓¯ is actually independent of the 𝑉¯𝑡 𝑓¯(𝑥) = 𝑃¯0𝑉¯𝑡 𝑓¯(𝑥) for 𝑥 ∈ 𝐸. ¯ ¯ ¯ defined values of 𝑓 on 𝐸 \ 𝐷. Let (𝑄¯ 𝑡 )𝑡 ≥0 be the transition semigroup on 𝑀 ( 𝐸) ¯ ¯ by (2.3) from (𝑉𝑡 )𝑡 ≥0 . Then for every 𝑡 ≥ 0 and every 𝜇 ∈ 𝑀 ( 𝐸) the measure 𝑄¯ 𝑡 (𝜇, ·) = 𝑄¯ 𝑡 (𝜇 𝑃¯0 , ·) is carried by 𝑀 (𝐷). Since the restriction of ( 𝑃¯𝑡 )𝑡 ≥0 to 𝐷 is a Borel right semigroup, the restriction of ( 𝑄¯ 𝑡 )𝑡 ≥0 to 𝑀 (𝐷) is a normal Borel transition semigroup. We denote this restriction by the same notation (𝑄¯ 𝑡 )𝑡 ≥0 . Note also that the restriction of (𝑄¯ 𝑡 )𝑡 ≥0 to 𝑀 (𝐸) coincides with the transition semigroup (𝑄 𝑡 )𝑡 ≥0 of the (𝜉, 𝜙)-superprocess. Let 𝑊¯ denote the space of all right continuous paths from [0, ∞) into 𝑀 (𝐷) ¯ Let {𝑋𝑡 : 𝑡 ≥ 0} denote the coordinate process of 𝑊¯ with left limits in 𝑀 ( 𝐸). 0 0 ¯ ¯ and let (𝒢 , 𝒢𝑡 ) denote its natural 𝜎-algebras. The following theorem gives a right continuous realization of the semigroup (𝑄¯ 𝑡 )𝑡 ≥0 on 𝑀 (𝐷). ¯ 𝜇 on Theorem 5.1 For each 𝜇 ∈ 𝑀 (𝐷) there is a unique probability measure Q 0 ¯ ¯ ¯ (𝑊, 𝒢 ) such that Q 𝜇 {𝑋0 = 𝜇} = 1 and {𝑋𝑡 : 𝑡 ≥ 0} is a Markov process in 𝑀 (𝐷) ¯ 𝜇 ) with transition semigroup (𝑄¯ 𝑡 )𝑡 ≥0 . relative to (𝒢¯ 0 , 𝒢¯ 𝑡0 , Q Proof Let (Ω, 𝒜 0 , 𝒜𝑡0 , 𝑍𝑡 , P¯ 𝜇 ) be a Markov process with state space 𝑀 (𝐷) and ¯ is 𝛼-excessive relative to transition semigroup (𝑄¯ 𝑡 )𝑡 ≥0 . Recall that each 𝑓¯ ∈ ℛ ¯ ¯ ( 𝑃𝑡 )𝑡 ≥0 for some constant 𝛼 = 𝛼( 𝑓 ) > 0. By Corollary 2.34 there exists an 𝛼1 > 0 such that 𝑡 ↦→ e−𝛼1 𝑡 𝑍𝑡 ( 𝑓¯) is an (𝒜𝑡0 )-supermartingale. By Dellacherie and Meyer (1982, pp. 66–67), there is an Ω1 ∈ 𝒜 0 with P¯ 𝜇 (Ω1 ) = 1 such that {𝑍𝑡 (𝜔, 𝑓¯) : 𝑡 ≥ 0} ¯ For 𝑡 ≥ 0 possesses finite right and left limits along rationals for 𝜔 ∈ Ω1 and 𝑓¯ ∈ ℛ. ¯ let and 𝑓¯ ∈ ℛ

5.2 The Strong Markov Property

n lim rat.𝑟 ↓𝑡 𝑍𝑟 (𝜔, 𝑓¯) 𝑍𝑡+ (𝜔, 𝑓¯) = 0

121

if 𝜔 ∈ Ω1 , if 𝜔 ∈ Ω \ Ω1 .

Since ( 𝑃¯𝑡 )𝑡 ≥0 is a Borel right semigroup on 𝐷, by Theorem 2.22 the process {𝑍𝑡 : 𝑡 ≥ 0} is right continuous in probability under P¯ 𝜇 . Then we have ¯ = 1, P¯ 𝜇 {𝑍𝑡+ ( 𝑓¯) = 𝑍𝑡 ( 𝑓¯) for all 𝑓¯ ∈ ℛ}

𝑡 ≥ 0.

(5.2)

¯ is dense in 𝐶 ( 𝐸) ¯ −ℛ ¯ with the supremum norm. Thus for every 𝑡 ≥ 0 Recall that ℛ ¯ such that 𝑌𝑡 (𝜔, 𝑓¯) = 𝑍𝑡+ (𝜔, 𝑓¯) and 𝜔 ∈ Ω there is a unique measure 𝑌𝑡 (𝜔, ·) ∈ 𝑀 ( 𝐸) ¯ ¯ ¯ for every 𝜔 ∈ Ω. for all 𝑓 ∈ ℛ. It is easy to see that 𝑡 ↦→ 𝑌𝑡 (𝜔) is càdlàg in 𝑀 ( 𝐸) In view of (5.2), we have ¯ = 1, P¯ 𝜇 {𝑌𝑡 ( 𝑓¯) = 𝑍𝑡 ( 𝑓¯) for all 𝑓¯ ∈ ℛ}

𝑡 ≥ 0.

(5.3)

It follows that the random measure 𝑌𝑡 is P¯ 𝜇 -a.s. supported by 𝐷 and {𝑌𝑡 | 𝐷 : 𝑡 ≥ 0} is a Markov process in 𝑀 (𝐷) with transition semigroup ( 𝑄¯ 𝑡 )𝑡 ≥0 𝜇 relative to (𝒜 0 , 𝒜𝑡0 , P¯ 𝜇 ). Let (𝒜 𝜇 , 𝒜𝑡 ) be the augmentation of (𝒜 0 , 𝒜𝑡0 ) by P¯ 𝜇 . Then {𝑌𝑡 | 𝐷 : 𝑡 ≥ 0} is also a Markov process with semigroup (𝑄¯ 𝑡 )𝑡 ≥0 relative to 𝜇 (𝒜 𝜇 , 𝒜𝑡 , P¯ 𝜇 ). Recall that 𝐷 = {𝑥 ∈ 𝐸 : 𝑃¯0 (𝑥, ·) = 𝛿 𝑥 } and 𝐵 = 𝐸¯ \ 𝐷. Then (5.3) implies that ¯ = 1, P¯ 𝜇 {𝑌𝑡 ( 𝑓¯) = 𝑌𝑡 ( 𝑃¯0 𝑓¯) for all 𝑓¯ ∈ ℛ}

𝑡 ≥ 0.

(5.4)

¯ which is 𝛼-excessive for ( 𝑃¯𝑡 )𝑡 ≥0 let 𝑓¯𝑘 = 𝑘 ( 𝑓¯ − 𝑘 𝑈¯ 𝑘+𝛼 𝑓¯). Then 𝑓¯𝑘 ∈ For 𝑓¯ ∈ ℛ + ¯ ¯ + 𝐶 ( 𝐸) and 𝑈¯ 𝛼 𝑓¯𝑘 = 𝑘 𝑈¯ 𝑘+𝛼 𝑓¯ → 𝑃¯0 𝑓¯ increasingly as 𝑘 → ∞. Since 𝑈¯ 𝛼 𝑓¯𝑘 ∈ 𝐶 ( 𝐸) ¯ is 𝛼-excessive for ( 𝑃𝑡 )𝑡 ≥0 , by Corollary 2.34 there exists an 𝛼1 > 0 such that 𝜇 𝑡 ↦→ e−𝛼1 𝑡 𝑌𝑡 (𝑈¯ 𝛼 𝑓¯𝑘 ) is a right continuous (𝒜𝑡 )-supermartingale, and by Dellacherie 𝜇 𝜇 and Meyer (1982, p. 69) it is also an (𝒜𝑡+ )-supermartingale. Since (𝒜 𝜇 , 𝒜𝑡+ , P¯ 𝜇 ) satisfies the usual hypotheses, 𝑡 ↦→ e−𝛼1 𝑡 𝑌𝑡 ( 𝑃¯0 𝑓¯) and hence 𝑡 ↦→ 𝑌𝑡 ( 𝑃¯0 𝑓¯) is P¯ 𝜇 -a.s. right continuous; see Dellacherie and Meyer (1982, p. 79) or Sharpe (1988, p. 390). Then (5.4) and the right continuity of {𝑌𝑡 : 𝑡 ≥ 0} imply that ¯ = 1. P¯ 𝜇 {𝑌𝑡 ( 𝑓¯) = 𝑌𝑡 ( 𝑃¯0 𝑓¯) for all 𝑡 ≥ 0 and 𝑓¯ ∈ ℛ} ¯𝜇 Therefore we must have P¯ 𝜇 {𝑌𝑡 (𝐵) = 0 for all 𝑡 ≥ 0} = 1. Now we can simply let Q ¯ be the image of P 𝜇 under the mapping 𝜔 ↦→ {𝑌𝑡 (𝜔)| 𝐷 : 𝑡 ≥ 0}, and the theorem is proved. □

5.2 The Strong Markov Property ¯ 𝜇 ) be the Markov process in 𝑀 (𝐷) given by Theorem 5.1. ¯ 𝒢¯ 0 , 𝒢¯ 𝑡0 , 𝑋𝑡 , Q Let 𝑋¯ = (𝑊, 𝜌 ¯ at 𝑡 > 0. It is not hard to We write 𝑋𝑡− for the left limit of the process in 𝑀 ( 𝐸) ¯ 𝜇 (𝐺) is ℬ(𝑀 (𝐷))-measurable. Given show that for any 𝐺 ∈ b𝒢¯ 0 , the map 𝜇 ↦→ Q ¯ 𝐾 on a probability measure 𝐾 on 𝑀 (𝐷) we can define the probability measure Q

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5 Basic Regularities of Superprocesses

¯ 𝒢¯ 0 ) by (𝑊, ¯ 𝐾 (𝐺) = Q

∫

¯ 𝜇 (𝐺)𝐾 (d𝜇), Q

𝐺 ∈ b𝒢¯ 0 .

(5.5)

𝑀 (𝐷)

¯ 𝐾 -augmentation of (𝒢¯ 0 , 𝒢¯ 𝑡0 ), and simply write (𝒢¯ 𝜇 , 𝒢¯ 𝑡𝜇 ) in Let (𝒢¯ 𝐾 , 𝒢¯ 𝑡𝐾 ) be the Q the special case of 𝐾 = 𝛿 𝜇 for 𝜇 ∈ 𝑀 (𝐷). Let 𝒢¯ = ∩𝐾 𝒢¯ 𝐾 and 𝒢¯ 𝑡 = ∩𝐾 𝒢¯ 𝑡𝐾 , where the intersections are taken over all probability measures 𝐾 on 𝑀 (𝐷). Let 𝛾(𝑥, d𝑦) be the kernel on 𝐸 defined by (2.31) and let 𝑐 0 = sup [𝛾(𝑥, 1) − 𝑏(𝑥)]. 𝑥 ∈𝐸

¯ 𝐸¯ \ 𝐸) = 0 for 𝑥 ∈ 𝐸 and ¯ d𝑦) be the extension of 𝛾(𝑥, d𝑦) to 𝐸¯ so that 𝛾(𝑥, Let 𝛾(𝑥, ¯ = 0 for 𝑥 ∈ 𝐸¯ \ 𝐸. Let ( 𝜋¯ 𝑡 )𝑡 ≥0 be the semigroup of kernels on 𝐷 defined by ¯ 𝐸) 𝛾(𝑥, ¯ d𝑦). Let ( 𝑅¯ 𝛼 ) 𝛼>𝑐0 be defined by (A.48) from ( 𝜋¯ 𝑡 )𝑡 ≥0 . (2.38) from ( 𝑃¯𝑡 )𝑡 ≥0 and 𝛾(𝑥, ¯ By the formula 𝜋¯ 𝑡 = 𝑃¯0 𝜋¯ 𝑡 we can extend ( 𝜋¯ 𝑡 )𝑡 ≥0 to a semigroup of kernels on 𝐸. Let ( 𝑅¯ 𝛼 ) 𝛼>𝑐0 be extended accordingly. 𝜇 Proposition 5.2 Let 𝜇 ∈ 𝑀 (𝐷) and 𝑓¯ ∈ 𝐵(𝐷). For any bounded (𝒢¯ 𝑡+ )-stopping time 𝑇, we have   ¯ 𝜇 𝑋𝑇+𝑡 ( 𝑓¯)|𝒢¯ 𝜇 = 𝑋𝑇 ( 𝜋¯ 𝑡 𝑓¯), 𝑡 ≥ 0. (5.6) Q 𝑇+

If 𝑇 is predictable in addition, then   ¯ 𝜇 𝑋𝑇+𝑡 ( 𝑓¯)|𝒢¯ 𝜇 = 𝑋 𝜌 ( 𝜋¯ 𝑡 𝑓¯), Q 𝑇− 𝑇−

𝑡 ≥ 0.

(5.7)

¯ is 𝛼-excessive for ( 𝑃¯𝑡 )𝑡 ≥0 . By Corollary 2.34 and Proof Step 1. Suppose that 𝑔¯ ∈ ℛ 𝜇 ¯ is a the Markov property of (𝑋𝑡 , 𝒢¯ 𝑡 ), there exists an 𝛼1 > 0 such that 𝑡 ↦→ e−𝛼1 𝑡 𝑋𝑡 ( 𝑔) 𝜇 𝜇 right continuous (𝒢¯ 𝑡 )-supermartingale. Then it is a strong (𝒢¯ 𝑡+ )-supermartingale by 𝜇 Dellacherie and Meyer (1982, p. 69 and p. 74). In particular, for any (𝒢¯ 𝑡+ )-stopping time 𝑇 with upper bound 𝑈 > 0 we have ¯ 𝜇 [𝑋𝑇+𝑡 ( 𝑔)] Q ¯ ¯ ≤ e 𝛼1 (𝑈+𝑡) 𝜇( 𝑔).

(5.8)

Step 2. In view of (5.8), for any fixed 𝐺 ∈ pb𝒢¯ 𝑇+ we can define the finite measure ¯ let 𝑓¯ = 𝑈¯ 𝛽 𝑔¯ ∈ 𝐶 ( 𝐸) ¯ 𝜇 [𝐺 𝑋𝑇+𝑡 ] ∈ 𝑀 (𝐷). For 𝛽 > 𝑐 0 and 𝑔¯ ∈ ℛ ¯ +. 𝜇𝑡 = Q Theorem 2.36 implies that ∫ 𝑡 (5.9) 𝑀𝑡 ( 𝑓¯) := 𝑋𝑡 ( 𝑓¯) − 𝑋0 ( 𝑓¯) − 𝑋𝑠 (𝛽 𝑓¯ − 𝑔¯ + 𝛾¯ 𝑓¯ − 𝑏¯ 𝑓¯)d𝑠 𝜇

0 𝜇 (𝒢¯ 𝑡 )-martingale.

is a right continuous By Dellacherie and Meyer (1982, p. 69 and 𝜇 p. 74) we conclude that (5.9) is a strong (𝒢¯ 𝑡+ )-martingale. It follows that ∫ 𝑡 ¯ ¯ 𝜇𝑡 ( 𝑓 ) = 𝜇0 ( 𝑓 ) + 𝜇 𝑠 (𝛽 𝑓¯ − 𝑔¯ + 𝛾¯ 𝑓¯ − 𝑏¯ 𝑓¯)d𝑠. 0

5.2 The Strong Markov Property

Then 𝑡 ↦→ 𝜇𝑡 ( 𝑓¯) is continuous. Let 𝜇ˆ 𝛽 ( 𝑓¯) = implies that

123

∫∞ 0

e−𝛽𝑡 𝜇𝑡 ( 𝑓¯)d𝑡. The above equation

𝜇ˆ 𝛽 ( 𝑓¯) = 𝛽−1 𝜇0 ( 𝑓¯) + 𝛽−1 𝜇ˆ 𝛽 (𝛽 𝑓¯ − 𝑔¯ + 𝛾¯ 𝑓¯ − 𝑏¯ 𝑓¯), and so 𝜇ˆ 𝛽 ( 𝑔¯ − 𝛾¯ 𝑈¯ 𝛽 𝑔¯ + 𝑏¯ 𝑈¯ 𝛽 𝑔) ¯ = 𝜇0 (𝑈¯ 𝛽 𝑔). ¯

(5.10)

¯ is uniformly dense in 𝐶 ( 𝐸), ¯ −ℛ ¯ and ¯ we also have (5.10) for all 𝑔¯ ∈ 𝐶 ( 𝐸) Since ℛ hence for all 𝑔¯ ∈ 𝐵(𝐷). ¯ let 𝑓¯ = 𝑈¯ 𝛽 𝑔¯ and ℎ¯ = 𝑓¯ + ( 𝛾¯ − 𝑏) ¯ 𝑅¯ 𝛽 𝑓¯. By Step 3. For 𝛽 > 𝑐 0 and 𝑔¯ ∈ ℛ Proposition A.54 we have ¯ 𝑅¯ 𝛽 𝑓¯ = 𝑅¯ 𝛽 𝑓¯. 𝑈¯ 𝛽 ℎ¯ = 𝑈¯ 𝛽 𝑓¯ + 𝑈¯ 𝛽 ( 𝛾¯ − 𝑏) Then we can apply (5.10) to the function ℎ¯ ∈ 𝐵(𝐷) to see ¯ 𝑅¯ 𝛽 𝑓¯) = 𝜇ˆ 𝛽 ( ℎ¯ − ( 𝛾¯ − 𝑏) ¯ ¯ 𝑈¯ 𝛽 ℎ) 𝜇ˆ 𝛽 ( 𝑓¯) = 𝜇ˆ 𝛽 ( ℎ¯ − ( 𝛾¯ − 𝑏) 𝛽 𝛽 ¯ = 𝜇0 ( 𝑅¯ 𝑓¯). = 𝜇0 (𝑈¯ ℎ) By Proposition A.43 and the uniqueness of Laplace transforms we get 𝜇𝑡 ( 𝑓¯) = ¯ Then we also have 𝜇𝑡 ( 𝑔) 𝜇0 ( 𝜋¯ 𝑡 𝑓¯), and hence 𝜇𝑡 (𝛽𝑈¯ 𝛽 𝑔) ¯ = 𝜇0 (𝛽 𝜋¯ 𝑡 𝑈¯ 𝛽 𝑔). ¯ = 𝜇0 ( 𝜋¯ 𝑡 𝑔) ¯ 𝛽 ¯ was arbitrary, we get 𝜇𝑡 ( 𝑓¯) = 𝜇0 ( 𝜋¯ 𝑡 𝑓¯) ¯ because 𝛽𝑈 𝑔¯ → 𝑔¯ as 𝛽 → ∞. Since 𝑔¯ ∈ ℛ for all 𝑓 ∈ 𝐵(𝐷). This proves (5.6). ¯ we have 𝑓¯ := 𝑈¯ 𝛽 𝑔¯ ∈ 𝐶 ( 𝐸) ¯ + . Then the right Step 4. For 𝛽 > 𝑐 0 and 𝑔¯ ∈ ℛ ¯ continuous martingale {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0} defined by (5.9) has predictable projection {𝑀𝑡− ( 𝑓¯) : 𝑡 ≥ 0}; see Dellacherie and Meyer (1982, pp. 106–107). It follows that 𝜌 {𝑋𝑡 ( 𝑓¯) : 𝑡 ≥ 0} has predictable projection {𝑋𝑡− ( 𝑓¯) : 𝑡 ≥ 0}. From Dellacherie and 𝜇 𝜌 ¯ ¯ ¯ Meyer (1982, p. 103), we have Q 𝜇 [𝑋𝑇 ( 𝑓 )|𝒢𝑇− ] = 𝑋𝑇− ( 𝑓¯), and hence     ¯ 𝜇 𝑋𝑇 (𝑈¯ 𝛽 𝑔)𝐺 ¯ 𝜇 𝑋 𝜌 (𝑈¯ 𝛽 𝑔)𝐺 ¯ Q =Q ¯ 𝑇− → ¯ ¯ for every 𝑥 ∈ 𝐷 and 𝛽𝑈¯ 𝛽 𝑔(𝑥) ¯ for every 𝐺 ∈ pb𝒢¯ 𝑇− . Because 𝛽𝑈¯ 𝛽 𝑔(𝑥) → 𝑔(𝑥) for every 𝑥 ∈ 𝐸¯ as 𝛽 → ∞, the above equation implies 𝑃¯0 𝑔(𝑥) ¯ ¯ = 𝜋¯ 0 𝑔(𝑥)     ¯ 𝜇 𝑋𝑇 ( 𝑔)𝐺 ¯ 𝜇 𝑋 𝜌 ( 𝜋¯ 0 𝑔)𝐺 =Q Q . ¯ ¯ 𝑇− 𝜇

¯ was arbitrary, we obtain Since 𝑔¯ ∈ ℛ     ¯ 𝜇 𝑋 𝜌 ( 𝜋¯ 0 𝑓¯)𝐺 ¯ 𝜇 𝑋𝑇 ( 𝑓¯)𝐺 = Q Q 𝑇− for every 𝑓¯ ∈ 𝐵(𝐷). Then (5.7) holds for 𝑡 = 0. For 𝑡 ≥ 0 we may appeal to (5.6) to get

124

5 Basic Regularities of Superprocesses

¯ 𝜇 [𝑋𝑇+𝑡 ( 𝑓¯)|𝒢¯ 𝜇 ] |𝒢¯ 𝜇 } ¯ 𝜇 [𝑋𝑇+𝑡 ( 𝑓¯)|𝒢¯ 𝜇 ] = Q ¯ 𝜇 {Q Q 𝑇− 𝑇+ 𝑇− ¯ 𝜇 [𝑋𝑇 ( 𝜋¯ 𝑡 𝑓¯) 𝒢¯ 𝜇 ] = 𝑋 𝜌 ( 𝜋¯ 𝑡 𝑓¯). =Q 𝑇− 𝑇− This completes the proof of the result.

□

𝜇 Corollary 5.3 For any 𝑓¯ ∈ 𝐵(𝐷), the process 𝑡 ↦→ 𝑋𝑡 ( 𝑓¯) has (𝒢¯ 𝑡+ )-predictable 𝜌 ¯ ¯ 𝜌 ¯ projection 𝑡 ↦→ 𝑋𝑡− ( 𝜋¯ 0 𝑓 ) = 𝑋𝑡− ( 𝑃0 𝑓 ).

Proposition 5.4 Let 𝜇 ∈ 𝑀 (𝐷) and 𝑓¯ ∈ 𝐵(𝐷). For any bounded (𝒢¯ 𝑡+ )-stopping time 𝑇 satisfying 0 ≤ 𝑇 ≤ 𝑢, we have   ¯ 𝜇 𝑋𝑢 ( 𝑓¯)|𝒢¯ 𝜇 = 𝑋𝑇 ( 𝜋¯ 𝑢−𝑇 𝑓¯). (5.11) Q 𝑇+ 𝜇

If 𝑇 is predictable in addition, then   ¯ 𝜇 𝑋𝑢 ( 𝑓¯)|𝒢¯ 𝜇 = 𝑋 𝜌 ( 𝜋¯ 𝑢−𝑇 𝑓¯). Q 𝑇− 𝑇−

(5.12)

Proof Let {𝑆 𝑘 } be the sequence of random times defined by 𝑆 𝑘 (𝑤) = 𝑖/2 𝑘 for (𝑖 − 1)/2 𝑘 ≤ 𝑢 − 𝑇 (𝑤) < 𝑖/2 𝑘 . For any 𝑡 ≥ 0 we have {𝑇 + 𝑆 𝑘 ≤ 𝑡} =

∞ Ø

{𝑇 + 𝑖/2 𝑘 ≤ 𝑡} ∩ {(𝑖 − 1)/2 𝑘 ≤ 𝑢 − 𝑇 < 𝑖/2 𝑘 }

𝑖=1

=

∞ Ø

{𝑇 ≤ 𝑡 − 𝑖/2 𝑘 } ∩ {𝑢 − 𝑖/2 𝑘 < 𝑇 ≤ 𝑢 − (𝑖 − 1)/2 𝑘 },

𝑖=1

which belongs to 𝒢¯ 𝑡+ . Then 𝑇 + 𝑆 𝑘 is a (𝒢¯ 𝑡+ )-stopping time. Clearly, 𝑆 𝑘 → 𝑢 − 𝑇 ¯ there exists decreasingly as 𝑘 → ∞. By the proof of Proposition 5.2 for any 𝑔¯ ∈ ℛ 𝜇 an 𝛼1 > 0 such that 𝑡 ↦→ e−𝛼1 𝑡 𝑋𝑡 ( 𝑔) ¯ is a right continuous (𝒢¯ 𝑡+ )-supermartingale. Then the family 𝜇

𝜇

e−𝛼1 (𝑇+𝑆𝑘 ) 𝑋𝑇+𝑆𝑘 ( 𝑔), ¯

𝑘 = 1, 2, . . . ,

¯ 𝜇 -uniformly integrable; see Dellacherie and Meyer (1982, p. 24) or Sharpe (1988, is Q ¯ 𝜇 -uniformly p. 390). It follows that for each 𝑓¯ ∈ 𝐶 (𝐷) the sequence {𝑋𝑇+𝑆𝑘 ( 𝑓¯)} is Q 𝜇 𝜇 integrable. Since {𝑆 𝑘 = 𝑖/2 𝑘 } ∈ 𝒢¯ 𝑇+ , for any 𝐺 ∈ b𝒢¯ 𝑇+ we see by Proposition 5.2 that     ¯ 𝜇 𝑋𝑢 ( 𝑓¯)𝐺 = lim Q ¯ 𝜇 𝑋𝑇+𝑆𝑘 ( 𝑓¯)𝐺 Q 𝑘→∞ ∞ ∑︁   ¯ 𝜇 𝑋𝑇+𝑖/2𝑘 ( 𝑓¯)𝐺1 {𝑆 =𝑖/2𝑘 } Q = lim 𝑘 𝑘→∞

𝑖=1

= lim

∞ ∑︁

𝑘→∞ 𝑖=1

   ¯ 𝜇 𝑋𝑇+𝑖/2𝑘 ( 𝑓¯) 𝒢¯ 𝜇 𝐺1 {𝑆 =𝑖/2𝑘 } ¯𝜇 Q Q 𝑘 𝑇+

5.2 The Strong Markov Property

125 ∞ ∑︁

  ¯ 𝜇 𝑋𝑇 ( 𝜋¯ 𝑖/2𝑘 𝑓¯)𝐺1 {𝑆 =𝑖/2𝑘 } Q 𝑘 𝑘→∞ 𝑖=1   ¯ 𝜇 𝑋𝑇 ( 𝜋¯ 𝑆𝑘 𝑓¯)𝐺 = lim Q 𝑘→∞   ¯ 𝜇 𝑋𝑇 ( 𝜋¯ 𝑢−𝑇 𝑓¯)𝐺 , =Q = lim

where we have used (5.8), the pointwise right continuity of 𝑡 ↦→ 𝜋¯ 𝑡 𝑓¯ and the dominated convergence theorem for the last equality. This gives (5.11) for 𝑓¯ ∈ 𝐶 (𝐷), □ and the extension to 𝑓¯ ∈ 𝐵(𝐷) is trivial. The proof of (5.12) is similar. 𝜇 ¯ ¯ 𝒢¯ 𝜇 , 𝒢¯ 𝑡+ For any 𝜇 ∈ 𝑀 (𝐷) the space (𝑊, , Q 𝜇 ) clearly satisfies the usual hypotheses. If (𝑠, 𝑥) ↦→ ℎ 𝑠 (𝑥) is a bounded and uniformly continuous function on 𝜇 [0, ∞) × 𝐷, then 𝑠 ↦→ 𝑋𝑠 (ℎ 𝑠 ) is right continuous and hence (𝒢¯ 𝑡+ )-optional. Now 𝜇 ¯ Proposition A.1 implies that 𝑠 ↦→ 𝑋𝑠 (ℎ 𝑠 ) is also (𝒢𝑡+ )-optional for any bounded Borel function (𝑠, 𝑥) ↦→ ℎ 𝑠 (𝑥) on [0, ∞) × 𝐷. By Proposition 5.4 the process 𝜇 {𝑋𝑠 ( 𝜋¯ 𝑡−𝑠 𝑓¯) : 0 ≤ 𝑠 ≤ 𝑡} is a strong (𝒢¯ 𝑡+ )-martingale for every 𝑓¯ ∈ 𝐵(𝐷) + . ¯ ¯ Then {𝑋𝑠 ( 𝜋¯ 𝑡−𝑠 𝑓 ) : 0 ≤ 𝑠 ≤ 𝑡} is Q 𝜇 -a.s. right continuous; see Dellacherie and ¯ = 1𝐸 (𝑥)𝑏(𝑥) and Meyer (1982, p. 109) or Sharpe (1988, pp. 389–390). Let 𝑏(𝑥) ¯ ¯ ¯ ¯ ¯ ¯ 𝜙0 (𝑥, 𝑓 ) = 𝜙(𝑥, 𝑓 ) − 𝑏(𝑥) 𝑓 (𝑥). By Theorem 2.23 we can rewrite (5.1) as ∫ 𝑡 ¯ ¯ ¯ 𝑉𝑡 𝑓 (𝑥) = 𝜋¯ 𝑡 𝑓 (𝑥) − 𝑡 ≥ 0, 𝑥 ∈ 𝐷. 𝜋¯ 𝑡−𝑠 𝜙¯0 (·, 𝑉¯𝑠 𝑓¯) (𝑥)d𝑠, 0

¯ 𝜇 -a.s. Using the equation above it is easy to see that {𝑋𝑠 (𝑉¯𝑡−𝑠 𝑓¯) : 0 ≤ 𝑠 ≤ 𝑡} is Q right continuous. Theorem 5.5 For every 𝑡 ≥ 0, every initial law 𝐾 and every 𝐹 ∈ 𝐵(𝑀 (𝐷)), the 𝐾 )-martingale. ¯ 𝐾 -a.s. right continuous (𝒢¯ 𝑡+ process {𝑄¯ 𝑡−𝑠 𝐹 (𝑋𝑠 ) : 0 ≤ 𝑠 ≤ 𝑡} is a Q Proof Let us fix 𝑡 ≥ 0 and the initial law 𝐾 on 𝑀 (𝐷). By (5.5) and the above ¯ 𝐾 -a.s. right continuous if analysis, the process {𝑄¯ 𝑡−𝑠 𝐹 (𝑋𝑠 ) : 0 ≤ 𝑠 ≤ 𝑡} is Q ¯ ¯ The Markov property of {𝑋𝑡 : 𝑡 ≥ 0} implies that 𝐹 (𝜈) = e−𝜈 ( 𝑓 ) for some 𝑓¯ ∈ ℛ. {𝑄¯ 𝑡−𝑠 𝐹 (𝑋𝑠 ) : 0 ≤ 𝑠 ≤ 𝑡} is a (𝒢¯ 𝑡𝐾 )-martingale. Then {𝑄¯ 𝑡−𝑠 𝐹 (𝑋𝑠 ) : 0 ≤ 𝑠 ≤ 𝑡} 𝐾 )-martingale; see Dellacherie and Meyer (1982, p. 69). We choose a is also a (𝒢¯ 𝑡+ ¯ so that its completion coincides with its one-point comcompatible metric on 𝑀 ( 𝐸) ¯ pactification. By the Stone–Weierstrass theorem, the linear span of {𝜈 ↦→ e−𝜈 ( 𝑓 ) : ¯ is uniformly dense in 𝐶𝑢 (𝑀 ( 𝐸)). ¯ Since each 𝐹 ∈ 𝐶𝑢 (𝑀 (𝐷)) has 𝑓¯ ∈ ℛ} ¯ an extension in 𝐶𝑢 (𝑀 ( 𝐸)), we infer that {𝑄¯ 𝑡−𝑠 𝐹 (𝑋𝑠 ) : 0 ≤ 𝑠 ≤ 𝑡} is a 𝐾 )-martingale for every 𝐹 ∈ 𝐶 (𝑀 (𝐷)). By Propo¯ 𝐾 -a.s. right continuous (𝒢¯ 𝑡+ Q 𝑢 ¯ 𝐾 -a.s. right continuous sition A.1 we conclude that {𝑄¯ 𝑡−𝑠 𝐹 (𝑋𝑠 ) : 0 ≤ 𝑠 ≤ 𝑡} is a Q 𝐾 )-martingale for every 𝐹 ∈ 𝐵(𝑀 (𝐷)); see Dellacherie and Meyer (1982, p. 79) (𝒢¯ 𝑡+ or Sharpe (1988, p. 390). □

126

5 Basic Regularities of Superprocesses

Theorem 5.6 The filtrations (𝒢¯ 𝑡 ) and (𝒢¯ 𝑡𝐾 ) are right continuous and the process ¯ 𝒢¯ 𝑡 , 𝑋𝑡 , Q ¯ 𝐾 ) satisfies the strong Markov property, that is, for every 𝑡 ≥ 0, ¯ 𝒢, 𝑋¯ = (𝑊, ¯ every (𝒢𝑡 )-stopping time 𝑇, every initial law 𝐾 and every function 𝐹 ∈ 𝐵(𝑀 (𝐷)), we have   ¯ 𝐾 𝐹 (𝑋𝑇+𝑡 )1 {𝑇 0 and let 𝑇 = ∞ otherwise. Then {𝑇𝑛 } is an increasing sequence of stopping times and Proposition 5.26 implies Q𝑎 𝛿 𝑥 -a.s. 𝑇𝑛 → 𝑇. Moreover, we have Q𝑎 𝛿 𝑥 -a.s. lim 𝑋𝑡 (𝑔0 ) = 𝑥(𝑧)𝑔0 (0+) = 𝑥(𝑧) and 𝑋𝑧 (𝑔0 ) = 𝑥(𝑧)𝜇(𝑔0 ). 𝑡 ↑𝑧

Since 𝑥(𝑧) > 0 with strictly positive probability, by (5.46) we see 𝑡 ↦→ 𝑋𝑡 (𝑔0 ) cannot □ be quasi-left continuous at the stopping time 𝑇. By Corollary 5.27, any realization of this (𝜉, 𝜙)-superprocess cannot be quasileft continuous in 𝑀 (𝐸 𝜌 ), so the superprocess has no Hunt realization in 𝑀 (𝐸 𝜌 ). Then it seems the last assertion in Theorem 2.20 of Fitzsimmons (1988, p. 347) requires some additional condition. On the other hand, Theorem 5.8 implies that 𝑋 has a Hunt realization in its own Ray topology; see Sharpe (1988, p. 220). Thus the Ray topology of the (𝜉, 𝜙)-superprocess on 𝑀 (𝐸 1 ) is different from the topology of 𝑀 (𝐸 𝜌 ).

5.6 Bounds for the Cumulant Semigroup

139

5.6 Bounds for the Cumulant Semigroup In this section, we establish some useful upper and lower bounds for the cumulant semigroup of the (𝜉, 𝜙)-superprocess. Recall that the branching mechanism 𝜙 is given by (2.29) or (2.30). The discussions here are based on the following basic comparison theorem: Theorem 5.28 Let 𝑇 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + . Let 𝜙˜ be another branching mechanism ˜ Suppose and let (𝑡, 𝑥) ↦→ 𝑉˜𝑡 𝑓 (𝑥) be the solution of (2.36) with 𝜙 replaced by 𝜙. ˜ ˜ ˜ that 𝜙(𝑥, 𝑉𝑡 𝑓 ) ≥ 𝜙(𝑥, 𝑉𝑡 𝑓 ) for 0 ≤ 𝑡 ≤ 𝑇 and 𝑥 ∈ 𝐸. Then 𝑉𝑡 𝑓 (𝑥) ≤ 𝑉˜𝑡 𝑓 (𝑥) for 0 ≤ 𝑡 ≤ 𝑇 and 𝑥 ∈ 𝐸. Proof Fix 𝑡 ∈ [0, 𝑇] and let 𝑢𝑟 (𝑥) = 𝑉𝑡−𝑟 𝑓 (𝑥) and 𝑢˜ 𝑟 (𝑥) = 𝑉˜𝑡−𝑟 𝑓 (𝑥) for 0 ≤ 𝑟 ≤ 𝑡 and 𝑥 ∈ 𝐸. Then (𝑟, 𝑥) ↦→ 𝑢˜ 𝑟 (𝑥) is the unique bounded positive solution of ∫ 𝑡 ˜ 𝑠 , 𝑢˜ 𝑠 )]d𝑠 = P𝑟 , 𝑥 [ 𝑓 (𝜉𝑡 )]. P𝑟 , 𝑥 [ 𝜙(𝜉 𝑢˜ 𝑟 (𝑥) + 𝑟

From this we have ∫ ∫ 𝑡 P𝑟 , 𝑥 [𝜙(𝜉 𝑠 , 𝑢˜ 𝑠 )]d𝑠 = P𝑟 , 𝑥 [ 𝑓 (𝜉𝑡 )] + 𝑢˜ 𝑟 (𝑥) + 𝑟

𝑡

P𝑟 , 𝑥 [𝑔𝑠 (𝜉 𝑠 )]d𝑠,

𝑟

˜ 𝑢˜ 𝑠 ) is a bounded positive Borel function on [0, 𝑡] × 𝐸. where 𝑔𝑠 (𝑥) = 𝜙(𝑥, 𝑢˜ 𝑠 ) − 𝜙(𝑥, By Theorem 5.16 one can see that (𝑟, 𝑥) ↦→ 𝑢˜ 𝑟 (𝑥) appears in the characterization of the weighted occupation time of the (𝜉, 𝜙)-superprocess. Then 𝑢𝑟 (𝑥) ≤ 𝑢˜ 𝑟 (𝑥) for all 0 ≤ 𝑟 ≤ 𝑡 and 𝑥 ∈ 𝐸. This gives the desired result. □ Corollary 5.29 Let 𝑇 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + . Let 𝐶𝑇 ≥ 0 be a constant such that 𝑉𝑡 𝑓 (𝑥) ≤ 𝐶𝑇 for 0 ≤ 𝑡 ≤ 𝑇 and 𝑥 ∈ 𝐸. Let ∫ (1 − e−𝐶𝑇 𝜈 (1) )𝜈({𝑥})𝐻 (𝑥, d𝜈). 𝑏𝑇 (𝑥) = 𝑏(𝑥) + 𝐶𝑇 𝑐(𝑥) + 𝑀 (𝐸) ◦

Then we have 𝑉𝑡 𝑓 (𝑥) ≥ 𝑃𝑡𝑏𝑇 𝑓 (𝑥) for 0 ≤ 𝑡 ≤ 𝑇 and 𝑥 ∈ 𝐸, where (𝑃𝑡𝑏𝑇 )𝑡 ≥0 is the semigroup defined by (2.37) with 𝑏 replaced by 𝑏𝑇 . Proof Recall the elementary inequality e−𝑧 − 1 + 𝑧 ≤ (1 − e−𝑧 )𝑧 for 𝑧 ≥ 0. By (2.31), for 0 ≤ 𝑡 ≤ 𝑇 and 𝑥 ∈ 𝐸 we have 𝜙(𝑥, 𝑉𝑡 𝑓 ) ≤ 𝑏(𝑥)𝑉𝑡 𝑓 (𝑥) + 𝐶𝑇 𝑐(𝑥)𝑉𝑡 𝑓 (𝑥) − 𝛾(𝑥, 𝑉𝑡 𝑓 ) ∫ (1 − e−𝜈 (𝑉𝑡 𝑓 ) )𝜈(𝑉𝑡 𝑓 )𝐻 (𝑥, d𝜈) + 𝑀 (𝐸) ◦

≤ 𝑏(𝑥)𝑉𝑡 𝑓 (𝑥) + 𝐶𝑇 𝑐(𝑥)𝑉𝑡 𝑓 (𝑥) − 𝜂(𝑥, 𝑉𝑡 𝑓 ) ∫ − (1 − e−𝐶𝑇 𝜈 (1) )𝜈 𝑥 (𝑉𝑡 𝑓 )𝐻 (𝑥, d𝜈) 𝑀 (𝐸) ◦

140

5 Basic Regularities of Superprocesses

∫ +

(1 − e−𝐶𝑇 𝜈 (1) )𝜈(𝑉𝑡 𝑓 )𝐻 (𝑥, d𝜈)

𝑀 (𝐸) ◦

≤ 𝑏(𝑥)𝑉𝑡 𝑓 (𝑥) + 𝐶𝑇 𝑐(𝑥)𝑉𝑡 𝑓 (𝑥) − 𝜂(𝑥, 𝑉𝑡 𝑓 ) ∫ + (1 − e−𝐶𝑇 𝜈 (1) )𝜈({𝑥})𝑉𝑡 𝑓 (𝑥)𝐻 (𝑥, d𝜈) 𝑀 (𝐸) ◦

≤ 𝑏𝑇 (𝑥)𝑉𝑡 𝑓 (𝑥). Then we get the result by Theorem 5.28.

□ −𝑐 ∗ 𝑡

≤ 𝑉𝑡 1(𝑥) ≤ Corollary 5.30 If 𝑐 0 := sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)] ≤ 0, we have e e𝑐0 𝑡 ≤ 1 for all 𝑡 ≥ 0 and 𝑥 ∈ 𝐸, where   ∫ −𝜈 (1) ∗ )𝜈({𝑥})𝐻 (𝑥, d𝜈) . 𝑐 = sup 𝑏(𝑥) + 𝑐(𝑥) + (1 − e 𝑥 ∈𝐸

𝑀 (𝐸) ◦

Proof By Corollary 2.25 and Theorem A.53, we have 𝑉𝑡 1(𝑥) ≤ 𝜋𝑡 1(𝑥) ≤ e𝑐0 𝑡 ≤ 1. Then the result follows from Corollary 5.29. □ For the branching mechanism 𝜙 given by (2.29) or (2.30), its local projection is the function 𝜙1 on 𝐸 × [0, ∞) defined by ∫ 𝜙1 (𝑥, 𝜆) = [𝑏(𝑥) − 𝛾(𝑥, 1)]𝜆 + 𝑐(𝑥)𝜆2 + 𝐾 (𝜈, 𝜆1 {𝑥 } )𝐻 (𝑥, d𝜈). (5.47) 𝑀 (𝐸) ◦

Condition 5.31 The local projection 𝜙1 is bounded below by a spatially constant local branching mechanism 𝜙∗ , that is, 𝜙1 (𝑥, 𝜆) ≥ 𝜙∗ (𝜆) for every 𝑥 ∈ 𝐸 and 𝜆 ≥ 0. Theorem 5.32 Suppose that Condition 5.31 is satisfied. Let (𝑣 ∗𝑡 )𝑡 ≥0 denote the cumulant semigroup of the CB-process with branching mechanism 𝜙∗ . Then we have ∥𝑉𝑡 𝑓 ∥ ≤ 𝑣 ∗𝑡 (∥ 𝑓 ∥) for 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + . 𝛾

Proof We first consider the case where 𝜉 is conservative. Let (𝑃𝑡 )𝑡 ≥0 be the semigroup of kernels defined by (2.37) with 𝑏 replaced by 𝛾(·, 1). By Theorem A.44 we can define a conservative Borel right semigroup ( 𝑃˜𝑡 )𝑡 ≥0 on 𝐸 by ∫ 𝑡 ∫ 𝛾 𝛾 ˜ d𝑠 (5.48) 𝛾(𝑦, 𝑃˜ 𝑠 𝑓 )𝑃𝑡−𝑠 (𝑥, d𝑦). 𝑃𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) + 0

𝐸

Let 𝜙˜ be the branching mechanism defined by ˜ 𝜙(𝑥, 𝑓 ) = 𝜙1 (𝑥, 𝑓 (𝑥)) + 𝛾(𝑥, 1) 𝑓 (𝑥) − 𝛾(𝑥, 𝑓 ). ˜ 𝑓 ). Let (𝑉˜𝑡 )𝑡 ≥0 denote the cumulant semigroup It is easy to see that 𝜙(𝑥, 𝑓 ) ≥ 𝜙(𝑥, ˜ of the (𝜉, 𝜙)-superprocess. Then (𝑡, 𝑥) ↦→ 𝑉˜𝑡 𝑓 (𝑥) is the unique locally bounded positive solution to ∫ 𝑡 ∫ ˜ ˜ 𝑉˜𝑠 𝑓 )𝑃𝑡−𝑠 (𝑥, d𝑦). (5.49) 𝑉𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) − d𝑠 𝜙(𝑦, 0

𝐸

5.6 Bounds for the Cumulant Semigroup

141

By Theorem 5.28 we have 𝑉𝑡 𝑓 (𝑥) ≤ 𝑉˜𝑡 𝑓 (𝑥). By Proposition 2.9, we can rewrite (5.49) as ∫ 𝑡 ∫   𝛾 𝛾 𝜙1 (𝑦, 𝑉˜𝑠 𝑓 (𝑦)) − 𝛾(𝑦, 𝑉˜𝑠 𝑓 ) 𝑃𝑡−𝑠 (𝑥, d𝑦). 𝑉˜𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) − d𝑠 0

𝐸

Using (5.48) again we get ∫ 𝑡 ∫ 𝛾 𝑉˜𝑡 𝑓 (𝑥) = 𝑃˜𝑡 𝑓 (𝑥) − 𝑃𝑡−𝑠 𝜙1 (𝑉˜𝑠 𝑓 ) (𝑥)d𝑠 + 0

𝑡

𝑃𝑡−𝑠 𝛾(𝑉˜𝑠 𝑓 − 𝑃˜ 𝑠 𝑓 ) (𝑥)d𝑠. 𝛾

0

By using the above relation successively and arguing as in the proof of Theorem 2.23 we see (𝑡, 𝑥) ↦→ 𝑉˜𝑡 𝑓 (𝑥) is also the unique locally bounded positive solution to ∫ 𝑡 ∫ ˜ ˜ 𝜙1 (𝑦, 𝑉˜𝑠 𝑓 (𝑦)) 𝑃˜𝑡−𝑠 (𝑥, d𝑦). 𝑉𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) − d𝑠 0

𝐸

Then (𝑉˜𝑡 )𝑡 ≥0 is actually the cumulant semigroup of a Dawson–Watanabe superprocess with local branching mechanism 𝜙1 and underlying transition semigroup ( 𝑃˜𝑡 )𝑡 ≥0 . Since 𝜙1 (𝑥, 𝜆) ≥ 𝜙∗ (𝜆) for all 𝑥 ∈ 𝐸 and 𝜆 ≥ 0, using Theorem 5.28 again we see 𝑉˜𝑡 𝑓 (𝑥) ≤ 𝑉˜𝑡 ∥ 𝑓 ∥ (𝑥) ≤ 𝑣 ∗𝑡 (∥ 𝑓 ∥) for 𝑡 ≥ 0 and 𝑥 ∈ 𝐸. This gives the result when 𝜉 is conservative. In the general case, let 𝜉ˆ be the conservative extension of 𝜉 to the state space 𝐸ˆ := 𝐸 ∪ {𝜕} with 𝜕 being an isolated cemetery. Let 𝜙ˆ be the ˆ ˆ branching mechanism on 𝐸ˆ defined by 𝜙(𝜕, 𝑓 ) = 𝜙∗ ( 𝑓 (𝜕)) and 𝜙(𝑥, 𝑓 ) = 𝜙(𝑥, 𝑓 | 𝐸 ) + ˆ ˆ for 𝑥 ∈ 𝐸, where 𝑓 ∈ 𝐵( 𝐸) . Let (𝑉𝑡 )𝑡 ≥0 be the cumulant semigroup of the ˆ 𝜙)-superprocess. ˆ ( 𝜉, It is easy to show that 𝑉𝑡 𝑓 (𝑥) = 𝑉ˆ𝑡 ( 𝑓 1𝐸 ) (𝑥) for 𝑥 ∈ 𝐸 and + 𝑓 ∈ 𝐵(𝐸) . Then we still have the desired result. □ Corollary 5.33 Suppose that Condition 5.31 holds with 𝜙∗′ (∞) = ∞. Then we have 𝐸𝐶 = 𝐸. Proof By Theorem 5.32 we have 𝑉𝑡 𝑓 (𝑥) ≤ 𝑣 ∗𝑡 (∥ 𝑓 ∥) for 𝑥 ∈ 𝐸 and 𝑓 ∈ 𝐵(𝐸) + . Suppose that there exist 𝑡 > 0 and 𝑥 ∈ 𝐸 such that 𝑉𝑡 𝑓 (𝑥) is represented by (2.5) with 𝜆 𝑡 (𝑥, 1) > 0. By Theorem 3.14, we have the representation (3.15) for 𝑣 ∗𝑡 (𝜆) with 𝑡 > 0. Then (𝜕/𝜕𝜆)𝑣 ∗𝑡 (𝜆) → 0 as 𝜆 → ∞. It follows that 𝑉𝑡 𝜆(𝑥) ≥ 𝜆 𝑡 (𝑥, 1)𝜆 ≥ 𝑣 ∗𝑡 (𝜆) when 𝜆 ≥ 0 is sufficiently large, yielding a contradiction. This proves 𝜆 𝑡 (𝑥, 1) = 0 □ for all 𝑡 > 0 and 𝑥 ∈ 𝐸. Corollary 5.34 Suppose that Condition 5.31 holds with 𝜙∗ satisfying Grey’s condition. Then 𝐸𝐶 = 𝐸 and ∥ 𝑣¯ 𝑡 ∥ ≤ 𝑣¯ ∗𝑡 := lim𝜆→∞ 𝑣 ∗𝑡 (𝜆) < ∞ for 𝑡 > 0 and ∥ 𝑣¯ 𝑡 ∥ ≤ e𝑐0 (𝑡−𝑟) ∥ 𝑣¯ 𝑟 ∥ for 𝑡 ≥ 𝑟 > 0, where 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)]. Proof Since 𝜙∗ satisfies Grey’s condition, by Theorems 3.10 and 5.32, for any 𝑡 > 0 we have ∥ 𝑣¯ 𝑡 ∥ ≤ 𝑣¯ ∗𝑡 < ∞, which implies 𝐸𝐶 = 𝐸. By Theorem A.53 we have ∥𝜋𝑡 ∥ ≤ e𝑐0 𝑡 for 𝑡 ≥ 0. Then ∥ 𝑣¯ 𝑡 ∥ = ∥𝑉𝑡−𝑟 𝑣¯ 𝑟 ∥ ≤ ∥𝜋𝑡−𝑟 𝑣¯ 𝑟 ∥ ≤ e𝑐0 (𝑡−𝑟) ∥ 𝑣¯ 𝑟 ∥ for □ 𝑡 ≥ 𝑟 > 0.

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5 Basic Regularities of Superprocesses

5.7 Notes and Comments The proofs in the first three sections follow those of Fitzsimmons (1988, 1992), where local branching mechanisms were considered. Some different potential theoretical methods for the regularity of superprocesses were given in Beznea (2011). Starting from a local branching superprocess as the underlying spatial motion, a Markov branching systems taking values of configurations on the space of finite measures was constructed by Beznea and Lupaşcu (2016). Their proof of the sample path regularity relies on the analysis of some convenient superharmonic functions with compact level sets. For càdlàg spatial motions, the existence of right realizations of superprocesses was studied in Dynkin (1993b), Kuznetsov (1994), Leduc (2000) and Schied (1999). In particular, Leduc (2000) constructed a class of Hunt superprocesses under a second-moment condition on the kernel 𝐻 (𝑥, d𝜈) in the expression of the branching mechanism and proved that any Hunt MB-process satisfying certain assumptions has a version in his class. The weighted occupation times were first introduced by Iscoe (1986) for super-stable processes. They were then used in Iscoe (1988) to study supporting properties of super-Brownian motions. Theorem 5.15 was adopted from Dynkin (1993a). It generalizes the result of Iscoe (1986). Proposition 5.20 was proved in He and Li (2016). The results of Corollaries 5.21 and 5.22 can also be derived from the theory of Lévy processes; see, e.g. Corollaries 12.9 and 12.10 in Kyprianou (2014, p. 347). The present form of Theorem 5.32 was given in Li (2021); see also Dawson (1993, pp. 195–196) and Li (2001) for related discussions. We may think of (5.31) as a Feynman–Kac formula for the (𝜉, 𝜙)-superprocess 𝑋. The formula gives a characterization of the subprocess of 𝑋 obtained from the decreasing multiplicative functional  ∫ 𝑡  𝑡 ↦→ exp − 𝑋𝑠 (𝑔)d𝑠 . 0

In view of (2.36) and (5.32), this subprocess can also be regarded as a superprocess with branching mechanism 𝑓 ↦→ 𝜙(·, 𝑓 ) − 𝑔. This type of branching mechanism was considered in Dynkin (1994) under the technical condition ∫   lim sup 𝜈 𝑥 (1) + 𝜈({𝑥}) 2 𝐻 (𝑥, d𝜈) = 0, 𝜀→0 𝑥 ∈𝐸

{𝜈 (1) ≤ 𝜀 }

where 𝜈 𝑥 (d𝑦) denotes the restriction of 𝜈(d𝑦) to 𝐸 \ {𝑥}.

Chapter 6

Constructions by Transformations

In this chapter, we give the construction of several classes of superprocesses by transformations. In particular, we extend the state space of the superprocess to some 𝜎-finite measures. Other classes we shall construct include multitype superprocesses, age-structured superprocesses, conditioned superprocesses and timeinhomogeneous superprocesses. The constructions give not only the existence but also the regularity of those superprocesses. The setting of Borel right processes we have chosen is particularly suitable for the applications of those transformations.

6.1 Spaces of Tempered Measures In this section, we extend the state space of the Dawson–Watanabe superprocess to include some 𝜎-finite measures. Suppose that 𝐸 is a Lusin topological space. We fix a strictly positive function ℎ ∈ pℬ(𝐸). Recall that 𝑀ℎ (𝐸) is the space of tempered measures 𝜇 on 𝐸 satisfying 𝜇(ℎ) < ∞. Let 𝑀ℎ (𝐸) ◦ = 𝑀ℎ (𝐸) \ {0}. The topology on 𝑀ℎ (𝐸) is defined by the convention: 𝜇 𝑛 → 𝜇 if and only if 𝜇 𝑛 (ℎ 𝑓 ) → 𝜇(ℎ 𝑓 ) for all 𝑓 ∈ 𝐶 (𝐸). Suppose that 𝜉 = (Ω, ℱ, ℱ𝑡 , 𝜉𝑡 , P 𝑥 ) is a Borel right process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 . We here assume (ℱ, ℱ𝑡 ) are the augmentations of the natural 𝜎-algebras (ℱ 0 , ℱ𝑡0 ) generated by the sample path {𝜉𝑡 : 𝑡 ≥ 0}. Let (ℱ 𝑢 , ℱ𝑡𝑢 ) be the natural 𝜎-algebras on Ω generated by {𝜉𝑡 : 𝑡 ≥ 0} as random variables in 𝐸 furnished with the universal 𝜎-algebra ℬ𝑢 (𝐸). Let 𝑡 ↦→ 𝐾 (𝑡) be a continuous additive functional of 𝜉 and assume each 𝜔 ↦→ 𝐾𝑡 (𝜔) is measurable with respect to ℱ 0 . Let 𝜌 ∈ pℬ(𝐸) be a strictly positive function such that 𝜌 ≤ ℎ and define the continuous additive functional ∫ 𝑡 𝜌(𝜉 𝑠 )ℎ(𝜉 𝑠 ) −1 𝐾 (d𝑠), 𝑡 ≥ 0. 𝐽 (𝑡) = 0

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_6

143

144

6 Constructions by Transformations

Suppose that there is a constant 𝛼 ≥ 0 such that, as 𝑡 ↓ 0,   𝑥∈𝐸 P 𝑥 e−𝛼𝐽 (𝑡) ℎ(𝜉𝑡 ) ↑ ℎ(𝑥),

(6.1)

and −1

∫

sup ℎ(𝑥) P 𝑥 𝑥 ∈𝐸



𝑡

e

−𝛼𝐽 (𝑠)

𝜌(𝜉 𝑠 )𝐾 (d𝑠) → 0.

(6.2)

0

Let 𝑏 ∈ ℬ(𝐸) and 𝑐 ∈ pℬ(𝐸). Let 𝜂(𝑥, d𝑦) be a 𝜎-finite kernel on 𝐸 and let 𝐻 (𝑥, d𝜈) be a 𝜎-finite kernel from 𝐸 to 𝑀ℎ (𝐸) ◦ such that  −1 sup 𝜌(𝑥) |𝑏(𝑥)|ℎ(𝑥) + 𝑐(𝑥)ℎ(𝑥) 2 + 𝜂(𝑥, ℎ) 𝑥 ∈𝐸  ∫   + (6.3) 𝜈(ℎ) ∧ 𝜈(ℎ) 2 + 𝜈 𝑥 (ℎ) 𝐻 (𝑥, d𝜈) < ∞, 𝑀ℎ (𝐸) ◦

where 𝜈 𝑥 (d𝑦) denotes the restriction of 𝜈(d𝑦) to 𝐸 \ {𝑥}. Recall that 𝐵 ℎ (𝐸) is the set of functions 𝑓 ∈ ℬ(𝐸) satisfying | 𝑓 | ≤ const. · ℎ. Let 𝐵𝜌 (𝐸) be defined similarly with 𝜌 replacing ℎ. We consider the operator 𝑓 ↦→ 𝜙(·, 𝑓 ) from 𝐵 ℎ (𝐸) + to 𝐵𝜌 (𝐸) with the representation ∫ 2 𝑓 (𝑦)𝜂(𝑥, d𝑦) 𝜙(𝑥, 𝑓 ) = 𝑏(𝑥) 𝑓 (𝑥) + 𝑐(𝑥) 𝑓 (𝑥) − 𝐸 ∫   −𝜈 ( 𝑓 ) (6.4) + − 1 + 𝜈({𝑥}) 𝑓 (𝑥) 𝐻 (𝑥, d𝜈). e 𝑀ℎ (𝐸) ◦

Theorem 6.1 For each 𝑓 ∈ 𝐵 ℎ (𝐸) + there is a unique positive solution (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥, 𝑓 ) = 𝑉𝑡 𝑓 (𝑥) to the evolution equation  ∫ 𝑡 (6.5) 𝑣 𝑡 (𝑥) = P 𝑥 [ 𝑓 (𝜉𝑡 )] − P 𝑥 𝜙(𝜉 𝑠 , 𝑣 𝑡−𝑠 )𝐾 (d𝑠) , 𝑡 ≥ 0, 𝑥 ∈ 𝐸, 0

↦ ∥ℎ−1 𝑣 𝑡 (·, 𝑓 ) ∥ is bounded on each bounded interval [0, 𝑇]. Moreover, such that 𝑡 → ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈) = exp{−𝜇(𝑉𝑡 𝑓 )}, 𝑓 ∈ 𝐵 ℎ (𝐸) + , (6.6) 𝑀ℎ (𝐸)

defines a transition semigroup (𝑄 𝑡 )𝑡 ≥0 on 𝑀ℎ (𝐸). A realization of the transition semigroup (𝑄 𝑡 )𝑡 ≥0 defined by (6.6) is naturally called a (𝜉, 𝐾, 𝜙)-superprocess with state space 𝑀ℎ (𝐸). The proof of the above theorem is based on a number of transformations. Since (𝑃𝑡 )𝑡 ≥0 is Borel and each 𝜔 ↦→ 𝐽𝑡 (𝜔) is measurable with respect to the natural 𝜎-algebra ℱ 0 , we can define a Borel right semigroup (𝑃𝑡𝛼 )𝑡 ≥0 on 𝐸 by   𝑃𝑡𝛼 𝑓 (𝑥) = P 𝑥 e−𝛼𝐽 (𝑡) 𝑓 (𝜉𝑡 ) , 𝑥 ∈ 𝐸, 𝑓 ∈ 𝐵(𝐸).

6.1 Spaces of Tempered Measures

145

Let 𝜁 denote the lifetime of 𝜉. By the discussions in Sharpe (1988, pp. 286–287), for every initial law 𝜇 on 𝐸 there exists a probability measure P 𝜇𝛼 on (Ω, ℱ 𝑢 ) such that   P 𝜇𝛼 (𝐻1 {𝑡 0 such that 𝑑 ∑︁ 𝑖, 𝑗=1

𝑎 𝑖 𝑗 (𝑥)𝑢 𝑖 𝑢 𝑗 ≥ 𝜃 0

𝑑 ∑︁

𝑢 2𝑖 ,

𝑥 ∈ R𝑑 , 𝑢 𝑖 ∈ R, 𝑖 = 1, . . . , 𝑑.

𝑖=1

Fix 𝑝 > 0 and let ℎ(𝑥) = (1 + |𝑥| 2 ) − 𝑝/2 for 𝑥 ∈ R𝑑 , where | · | denotes the Euclidean norm. It is easy to find a constant 𝛼 > 0 such that | 𝐴ℎ(𝑥)| ≤ 𝛼ℎ(𝑥) for all 𝑥 ∈ R𝑑 .

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6 Constructions by Transformations

Example 6.2 Let 𝜉 be the standard one-dimensional Brownian motion killed at the origin. Then 𝜉 has state space R◦ := R \ {0}. Let (𝑃𝑡 )𝑡 ≥0 denote the transition semigroup of 𝜉. For any 𝑡 > 0 the sub-Markov kernel 𝑃𝑡 (𝑥, d𝑦) has density  𝑔 (𝑥 − 𝑦) − 𝑔𝑡 (𝑥 + 𝑦) if 𝑥𝑦 > 0, 𝑝 𝑡 (𝑥, 𝑦) = 𝑡 0 otherwise, where 𝑔𝑡 (𝑧) is given by (2.48). It is easy to show that ℎ(𝑥) ≡ |𝑥| is an invariant function for (𝑃𝑡 )𝑡 ≥0 . Let 𝜙(𝑥, 𝑓 ) = |𝑥| −𝜎 𝑓 (𝑥) 1+𝛽 for constants 0 < 𝛽 ≤ 1 and 𝛽 ≤ 𝜎 ≤ 1 + 𝛽. Then (6.3) is satisfied with 𝜌(𝑥) = |𝑥| 1+𝛽−𝜎 . Moreover, since 0 ≤ 1 + 𝛽 − 𝜎 ≤ 1, we have ∫ 𝑡 ∫ 𝑡 P 𝑥 [𝜌(𝜉 𝑠 )]d𝑠 ≤ P 𝑥 [1 + |𝜉 𝑠 |]d𝑠 0 ∫0 𝑡 ∫ |𝑥 | 2 d𝑠 ≤ e−𝑧 /2𝑠 d𝑧 + 𝑡|𝑥| √  0 √ 2𝜋𝑠  − |𝑥 | 2 2𝑡 ≤ √ + 𝑡 ℎ(𝑥). 𝜋 By Theorem 6.1 we can define a cumulant semigroup (𝑉𝑡 )𝑡 ≥0 on 𝐵 ℎ (R◦ ) + by ∫ 𝑡  𝑉𝑡 𝑓 (𝑥) = P 𝑥 [ 𝑓 (𝜉𝑡 )] − P 𝑥 |𝜉 𝑠 | −𝜎 𝑉𝑡−𝑠 𝑓 (𝜉 𝑠 ) 1+𝛽 d𝑠, 𝑡 ≥ 0, 𝑥 ∈ R◦ . 0

This gives a (𝜉, 𝜙)-superprocess 𝑋 in 𝑀ℎ (R◦ ). By Proposition 2.27 and the construction in the proof of Theorem 6.1 we have the moment formula Q 𝜇 [𝑋𝑡 ( 𝑓 )] = 𝜇(𝑃𝑡 𝑓 ),

𝑡 ≥ 0, 𝜇 ∈ 𝑀ℎ (R◦ ), 𝑓 ∈ 𝐵 ℎ (R◦ ).

Then we can also take 𝑀 (R◦ ) as the state space of 𝑋 and the above formula remains true for 𝜇 ∈ 𝑀 (R◦ ) and 𝑓 ∈ 𝐵(R◦ ). It is easy to see that {𝑋𝑡 (1) : 𝑡 ≥ 0} is a supermartingale but not a martingale unless 𝑋0 = 0. Example 6.3 For the local branching mechanism 𝜙 defined by (2.49), the assumption (6.3) means  −1 sup ℎ(𝑥) 𝜌(𝑥) |𝑏(𝑥)| + 𝑐(𝑥)ℎ(𝑥) 𝑥 ∈𝐸  ∫ ∞   2 + 𝑢 ∧ (𝑢 ℎ(𝑥)) 𝑚(𝑥, d𝑢) < ∞. 0

Clearly, the above condition is satisfied when 𝜌 = ℎ is a bounded function on 𝐸.

6.2 Multitype Superprocesses

149

6.2 Multitype Superprocesses In this section, we derive the existence of some multitype superprocesses from the non-local branching superprocess. For simplicity we only consider Lebesgue killing densities. Suppose that 𝐸 and 𝑇 are Lusin topological spaces. Let 𝜉 = {ℱ, ℱ𝑡 , (𝜉𝑡 , 𝛼𝑡 ), P ( 𝑥,𝑎) } be a Borel right process with state space 𝐸 × 𝑇. In general, the factors {𝜉𝑡 : 𝑡 ≥ 0} and {𝛼𝑡 : 𝑡 ≥ 0} are not necessarily independent. Let 𝜙 = 𝜙(𝑥, 𝑎, 𝑓 ) be a branching mechanism given by (2.29) or (2.30) with 𝐸 replaced by 𝐸 × 𝑇. By Theorem 5.13 we have the following: Theorem 6.4 There is a Borel right superprocess 𝑋 = (𝑊, 𝑋𝑡 , 𝒢, 𝒢𝑡 , Q 𝜇 ) with state space 𝑀 (𝐸 × 𝑇) and with the transition probabilities determined by Q 𝜇 exp{−𝑋𝑡 ( 𝑓 )} = exp{−𝜇(𝑉𝑡 𝑓 )},

𝑡 ≥ 0, 𝑓 ∈ 𝐵(𝐸 × 𝑇) + ,

(6.13)

where (𝑡, 𝑥, 𝑎) ↦→ 𝑉𝑡 𝑓 (𝑥, 𝑎) is the unique locally bounded positive solution of ∫ 𝑡    𝑉𝑡 𝑓 (𝑥, 𝑎) = P ( 𝑥,𝑎) 𝑓 (𝜉𝑡 , 𝛼𝑡 ) − P ( 𝑥,𝑎) 𝜙(𝜉 𝑠 , 𝛼𝑠 , 𝑉𝑡−𝑠 𝑓 )d𝑠 . (6.14) 0

The process 𝑋 defined by (6.13) and (6.14) is called a multitype superprocess with type space 𝑇. Heuristically, the process {𝜉𝑡 : 𝑡 ≥ 0} describes the migration of the particles and {𝛼𝑡 : 𝑡 ≥ 0} represents the mutation of their types. The two factors are treated equally in (6.14). We often apply Theorem 6.4 to a special form of the branching mechanism given as follows. For any 𝑓 ∈ 𝐵(𝐸 × 𝑇) + let 𝜓 = 𝜓(𝑥, 𝑎, 𝑓 ) and 𝜙 = 𝜙(𝑥, 𝑎, 𝑓 ) be given respectively by (2.21) and (2.29) with 𝑥 ∈ 𝐸 replaced by (𝑥, 𝑎) ∈ 𝐸 × 𝑇. Let 𝜋(𝑥, 𝑎, d𝛽) be a probability kernel from 𝐸 × 𝑇 to 𝑇. We can define (𝑡, 𝑥, 𝑎) ↦→ 𝑉𝑡 𝑓 (𝑥, 𝑎) by the unique locally bounded positive solution of ∫ 𝑡    𝑉𝑡 𝑓 (𝑥, 𝑎) = P ( 𝑥,𝑎) 𝑓 (𝜉𝑡 , 𝛼𝑡 ) − P ( 𝑥,𝑎) 𝜙(𝜉 𝑠 , 𝛼𝑠 , 𝑉𝑡−𝑠 𝑓 (·, 𝛼𝑠 ))d𝑠 0  ∫ 𝑡 ∫ (6.15) + P ( 𝑥,𝑎) d𝑠 𝜓(𝜉 𝑠 , 𝛽, 𝑉𝑡−𝑠 𝑓 (·, 𝛽))𝜋(𝜉 𝑠 , 𝛼𝑠 , d𝛽) . 0

𝑇

In this case, the functional 𝜙(𝑦, 𝑎, ·) describes the reproduction of a parent at site 𝑦 ∈ 𝐸 of type 𝑎 ∈ 𝑇 without mutation and 𝜓(𝑦, 𝛽, ·) describes the reproduction of a parent with mutated type 𝛽 ∈ 𝑇 chosen randomly according to the distribution 𝜋(𝑦, 𝑎, d𝛽). Example 6.4 The 𝑑-dimensional CB-process introduced in Example 2.5 is a special form of the multitype superprocess provided by Theorem 6.4 with 𝐸 being a singleton and with 𝑇 = {1, 2, . . . , 𝑑}. Example 6.5 Let us consider the case where 𝑇 = R+ and 𝛼𝑡 = 𝛼0 + 𝑡 for all 𝑡 ≥ 0. Suppose that 𝜉 = (Ω, ℱ, ℱ𝑡 , 𝜉𝑡 , P 𝑥 ) is a Borel right process with state space 𝐸. Let 𝜌 ∈ 𝐵(𝐸 × R+ ) and let 𝜁 = 𝜁 (𝑥, 𝑎, 𝜆) be given by (2.50) with 𝑥 ∈ 𝐸 replaced by

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6 Constructions by Transformations

(𝑥, 𝑎) ∈ 𝐸 × R+ . In addition, we assume sup 𝑥,𝑎 𝜁 𝑧′ (𝑥, 𝑎, 0+) ≤ 1. A special form of (6.15) is the equation ∫ 𝑡   𝑉𝑡 𝑓 (𝑥, 𝑎) = P 𝑥 [ 𝑓 (𝜉𝑡 , 𝑎 + 𝑡)] − P 𝑥 𝜌(𝜉 𝑠 , 𝑎 + 𝑠)𝑉𝑡−𝑠 𝑓 (𝜉 𝑠 , 𝑎 + 𝑠) d𝑠 0 ∫ 𝑡   P 𝑥 𝜌(𝜉 𝑠 , 𝑎 + 𝑠)𝜁 (𝜉 𝑠 , 𝑎 + 𝑠, 𝑉𝑡−𝑠 𝑓 (𝜉 𝑠 , 0)) d𝑠. + (6.16) 0

The corresponding multitype superprocess in 𝑀 (𝐸 × R+ ) is called an age-structured superprocess. Clearly, we can also get (6.16) as a special case of (4.40) with 𝜙 = 0. Here 𝜉𝑡 represents the location of a particle and 𝛼𝑡 represents its age. At its branching time a particle gives birth to a random number of offspring whose spatial motions start from the branching site and whose ages start from zero. See also the explanations following Theorem 4.15. In many cases, we only consider multitype superprocesses with finite or countable type spaces. Let 𝑇 = {1, 2, . . . , 𝑑} or {1, 2, . . . }. Suppose that for each 𝑖 ∈ 𝑇 we have: • a Borel right process 𝜉𝑖 in 𝐸 with transition semigroup (𝑃𝑖 (𝑡))𝑡 ≥0 ; • an operator 𝜙𝑖 = 𝜙𝑖 (𝑥, 𝑓 ) belonging to the class given by (2.29) or (2.30); • a discrete probability distribution 𝑝 𝑖 (𝑥) = {𝑝 𝑖 𝑗 (𝑥) : 𝑗 ∈ 𝑇 } on 𝑇 with 𝑝 𝑖 𝑗 ∈ 𝐵(𝐸) + ; • an operator 𝜓𝑖 = 𝜓𝑖 (𝑥, 𝑓 ) belonging to the class given by (2.21). The next theorem deals with a process with state space   ∑︁ 𝑇 𝑀𝑇 (𝐸) := 𝜇 = (𝜇𝑖 : 𝑖 ∈ 𝑇) ∈ 𝑀 (𝐸) : 𝜇𝑖 (𝐸) < ∞ . 𝑖 ∈𝑇

Of course, we have 𝑀𝑇 (𝐸) = 𝑀 (𝐸) 𝑇 if 𝑇 is a finite set. Write 𝑌𝑡 = (𝑌𝑖 (𝑡) : 𝑖 ∈ 𝐼) for 𝑡 ≥ 0. Theorem 6.5 There is a Borel right multitype superprocess 𝑌 = (𝑊, 𝑌𝑡 , 𝒢, 𝒢𝑡 , Q 𝜇 ) in 𝑀𝑇 (𝐸) with transition probabilities defined by n ∑︁ o n ∑︁ o Q 𝜇 exp − ⟨𝜇𝑖 , 𝑣 𝑖 (𝑡)⟩ , ⟨𝑌𝑖 (𝑡), 𝑓𝑖 ⟩ = exp − (6.17) 𝑖 ∈𝑇

𝑖 ∈𝑇

where 𝑓𝑖 ∈ 𝐵(𝐸) + and 𝑣 𝑖 (𝑡) = 𝑣 𝑖 (𝑡, 𝑥) is defined by the system of equations ∫ 𝑡 ∫ d𝑠 𝜙𝑖 (𝑦, 𝑣 𝑖 (𝑡 − 𝑠))𝑃𝑖 (𝑠, 𝑥, d𝑦) 𝑣 𝑖 (𝑡, 𝑥) = 𝑃𝑖 (𝑡) 𝑓𝑖 (𝑥) − 0 𝐸  ∫ 𝑡 ∫  ∑︁ 𝑝 𝑖 𝑗 (𝑦)𝜓 𝑗 (𝑦, 𝑣 𝑗 (𝑡 − 𝑠)) 𝑃𝑖 (𝑠, 𝑥, d𝑦). (6.18) d𝑠 + 0

𝐸

𝑗 ∈𝑇

6.2 Multitype Superprocesses

151

Proof Let 𝜉 be the Borel right process in the product space 𝐸 × 𝑇 with transition semigroup (𝑃𝑡 )𝑡 ≥0 defined by ∫ 𝑃𝑡 𝑓 (𝑥, 𝑖) = (𝑥, 𝑖) ∈ 𝐸 × 𝑇 . 𝑓 (𝑦, 𝑖)𝑃𝑖 (𝑡, 𝑥, d𝑦), 𝐸

Let 𝜙(𝑥, 𝑖, 𝑓 ) = 𝜙𝑖 (𝑥, 𝑓 ) and let 𝜋(𝑥, 𝑖, ·) be the Markov kernel from 𝐸 × 𝑇 to 𝑇 defined by ∑︁ 𝑝 𝑖 𝑗 (𝑥)𝛿 𝑗 (·), 𝜋(𝑥, 𝑖, ·) = 𝑗 ∈𝑇

where 𝛿 𝑗 (·) stands for the unit mass at 𝑗 ∈ 𝑇. Then we have a Borel right superprocess {𝑋𝑡 : 𝑡 ≥ 0} in 𝑀 (𝐸 ×𝑇) defined by (6.13) and (6.15). For 𝑖 ∈ 𝑇 and 𝜇 ∈ 𝑀 (𝐸 ×𝑇) we define 𝑈𝑖 𝜇 ∈ 𝑀 (𝐸) by 𝑈𝑖 𝜇(𝐵) = 𝜇(𝐵×{𝑖}) for 𝐵 ∈ ℬ(𝐸). Then 𝜇 ↦→ (𝑈𝑖 𝜇 : 𝑖 ∈ 𝑇) is a homeomorphism between 𝑀 (𝐸 ×𝑇) and 𝑀𝑇 (𝐸). Let 𝑌𝑖 (𝑡) = 𝑈𝑖 𝑋𝑡 . It is clear that {(𝑌𝑖 (𝑡) : 𝑖 ∈ 𝑇) : 𝑡 ≥ 0} is a Markov process in 𝑀𝑇 (𝐸) with transition probabilities defined by (6.17) and (6.18). By Theorems 6.4 and A.21 this process has a realization as a right process. □ The heuristical meaning of the process {(𝑌𝑖 (𝑡) : 𝑖 ∈ 𝑇) : 𝑡 ≥ 0} constructed in Theorem 6.5 is described as follows. The process 𝜉𝑖 gives the law of migration of the particles of type 𝑖 ∈ 𝑇. The functional 𝜙𝑖 (𝑦, ·) describes the reproduction of a parent at site 𝑦 ∈ 𝐸 of type 𝑖 ∈ 𝑇 without mutation and 𝜓 𝑗 (𝑦, ·) describes the reproduction of a parent with mutated type 𝑗 ∈ 𝑇 chosen randomly according to the discrete distribution 𝑝 𝑖 (𝑦) = {𝑝 𝑖 𝑗 (𝑦) : 𝑗 ∈ 𝑇 }. The process {𝑌𝑖 (𝑡) : 𝑡 ≥ 0} gives the mass distribution in 𝐸 of particles of type 𝑖 ∈ 𝑇. Since the multitype superprocesses are constructed from the single-type processes, the results we established before apply to the multitype case by simple modifications. In particular, by Corollary 5.17 we have: Corollary 6.6 In the setup of Theorem 6.5, for any 𝑡 ≥ 0 and 𝑓𝑖 , 𝑔𝑖 ∈ 𝐵(𝐸) + , 𝑖 ∈ 𝑇 we have  ∑︁   ∫ 𝑡 o n ∑︁ ⟨𝑌𝑖 (𝑡), 𝑓𝑖 ⟩ + Q 𝜇 exp − ⟨𝜇𝑖 , 𝑣 𝑖 (𝑡)⟩ , ⟨𝑌𝑖 (𝑠), 𝑔𝑖 ⟩d𝑠 = exp − 0

𝑖 ∈𝑇

𝑖 ∈𝑇

where 𝑣 𝑖 (𝑡) = 𝑣 𝑖 (𝑡, 𝑥) is defined by the system of equations  ∫ 𝑡 ∫  ∑︁ d𝑠 𝑝 𝑖 𝑗 (𝑦)𝜓 𝑗 (𝑦, 𝑣 𝑗 (𝑡 − 𝑠)) 𝑃𝑖 (𝑠, 𝑥, d𝑦) 𝑣 𝑖 (𝑡, 𝑥) = 𝑃𝑖 (𝑡) 𝑓𝑖 (𝑥) + 0

∫

𝐸

∫

𝑡

𝑃𝑖 (𝑠)𝑔𝑖 (𝑥)d𝑠 −

+ 0

𝑗 ∈𝑇 𝑡

∫ 𝜙𝑖 (𝑦, 𝑣 𝑖 (𝑡 − 𝑠))𝑃𝑖 (𝑠, 𝑥, d𝑦).

d𝑠 0

𝐸

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6 Constructions by Transformations

6.3 Two-Type Superprocesses Let 𝜉1 and 𝜉2 be Borel right processes in 𝐸 with transition semigroups (𝑃1 (𝑡))𝑡 ≥0 and (𝑃2 (𝑡))𝑡 ≥0 , respectively. Let 𝜙1 and 𝜙2 be operators belonging to the class given by (2.29) or (2.30). Let 𝜓 an operator of the form (2.21). By Theorem 6.5, we have the following: Theorem 6.7 There is a Borel right two-type superprocess 𝑌 = (𝑊, 𝑌𝑡 , 𝒢, 𝒢𝑡 , Q 𝜇 ) in 𝑀 (𝐸) 2 with transition probabilities defined by, for 𝑓1 , 𝑓2 ∈ 𝐵(𝐸) + ,  Q 𝜇 exp − ⟨𝑌1 (𝑡), 𝑓1 ⟩ − ⟨𝑌2 (𝑡), 𝑓2 ⟩  (6.19) = exp − ⟨𝜇1 , 𝑣 1 (𝑡)⟩ − ⟨𝜇2 , 𝑣 2 (𝑡)⟩ , where 𝑣 1 (𝑡, 𝑥) = 𝑣 1 (𝑡, 𝑥, 𝑓1 , 𝑓2 ) and 𝑣 2 (𝑡, 𝑥) = 𝑣 2 (𝑡, 𝑥, 𝑓2 ) are defined by the system of equations ∫ 𝑡 ∫ 𝑣 1 (𝑡, 𝑥) = 𝑃1 (𝑡) 𝑓1 (𝑥) − d𝑠 𝜙1 (𝑦, 𝑣 1 (𝑡 − 𝑠))𝑃1 (𝑠, 𝑥, d𝑦) 𝐸 0 ∫ 𝑡 ∫ d𝑠 (6.20) + 𝜓(𝑦, 𝑣 2 (𝑡 − 𝑠))𝑃1 (𝑠, 𝑥, d𝑦) 0

𝐸

and ∫

∫

𝑡

𝑣 2 (𝑡, 𝑥) = 𝑃2 (𝑡) 𝑓2 (𝑥) −

𝜙2 (𝑦, 𝑣 2 (𝑡 − 𝑠))𝑃2 (𝑠, 𝑥, d𝑦).

d𝑠 0

(6.21)

𝐸

In this model, particles of type one can produce the two types of offspring, but particles of type two can only produce offspring of their own type. Corollary 6.8 Let 𝑌 be the two-type superprocess defined in Theorem 6.7. Then (𝑊, 𝑌1 (𝑡), 𝒢, 𝒢𝑡 , Q ( 𝜇1 ,0) ) is a (𝜉1 , 𝜙1 )-superprocess. Proof By (6.20) we see (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥) = 𝑣 1 (𝑡, 𝑥, 𝑓1 , 0) is the unique locally bounded positive solution to ∫ 𝑡 ∫ 𝑣 𝑡 (𝑥) = 𝑃1 (𝑡) 𝑓1 (𝑥) − d𝑠 𝜙1 (𝑦, 𝑣 𝑡−𝑠 )𝑃1 (𝑠, 𝑥, d𝑦). 0

Then the result follows by Theorem A.21.

𝐸

□

Corollary 6.9 Let 𝑌 be the two-type superprocess defined in Theorem 6.7 with 𝜉1 = 𝜉2 and 𝜙2 = 𝜙1 + 𝜓. Then (𝑊, 𝑌1 (𝑡) + 𝑌2 (𝑡), 𝒢, 𝒢𝑡 , Q ( 𝜇1 ,0) ) is a (𝜉2 , 𝜙2 )superprocess. Proof From (6.20) and (6.21)it is easy to see that 𝑣 1 (𝑡, 𝑥, 𝑓 , 𝑓 ) = 𝑣 2 (𝑡, 𝑥, 𝑓 ). Then we get the result by Theorem A.21. □

6.4 A Change of the Probability Measure

153

Theorem 6.10 Let (𝑉𝑡 )𝑡 ≥0 be the cumulant semigroup of the (𝜉, 𝜙)-superprocess with branching mechanism 𝜙 given by (2.29) or (2.30). Let 𝜓 an operator of the form (2.21). Then for any 𝛾 ∈ 𝑀 (𝐸) there is a Borel right process in 𝑀 (𝐸) with 𝛾 semigroup (𝑄 𝑡 )𝑡 ≥0 defined by, for 𝜇 ∈ 𝑀 (𝐸) and 𝑓 ∈ 𝐵(𝐸) + ,   ∫ ∫ 𝑡 − ⟨𝜈, 𝑓 ⟩ 𝛾 𝑄 𝑡 (𝜇, d𝜈) = exp − ⟨𝜇, 𝑉𝑡 𝑓 ⟩ − e ⟨𝛾, 𝜓(·, 𝑉𝑠 𝑓 ))⟩d𝑠 . 𝑀 (𝐸)

0

Proof Let 𝑌 be the two-type superprocess defined in Theorem 6.7 such that 𝜉1 (𝑡) ≡ 𝜉1 (0), 𝜙1 = 0, 𝜉2 = 𝜉 and 𝜙2 = 𝜙. From (6.20) and (6.21) it follows that 𝑣 2 (𝑡, 𝑥, 𝑓 ) = 𝑉𝑡 𝑓 (𝑥) and ∫ 𝑡 𝜓(𝑥, 𝑉𝑠 𝑓 )d𝑠, 𝑡 ≥ 0, 𝑥 ∈ 𝐸 . 𝑣 1 (𝑡, 𝑥, 0, 𝑓 ) = 0 𝛾

Let Q 𝜇 = Q (𝛾, 𝜇) for 𝜇 ∈ 𝑀 (𝐸). Then one can use Theorem A.21 to see that 𝛾 (𝑊, 𝑌2 (𝑡), 𝒢, 𝒢𝑡 , Q 𝜇 ) is a Borel right process in 𝑀 (𝐸) with transition semigroup 𝛾 (𝑄 𝑡 )𝑡 ≥0 . □

6.4 A Change of the Probability Measure Let 𝐸 be a Lusin topological space and let 𝜉 be a conservative Borel right process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 . For simplicity we consider a local branching mechanism (𝑥, 𝜆) ↦→ 𝜙(𝑥, 𝜆) given by (2.49) with constant function 𝑏(𝑥) ≡ 𝑏 ≥ 0. Let (𝑄 𝑡 )𝑡 ≥0 denote the transition semigroup of the (𝜉, 𝜙)-superprocess defined by (2.35) and (2.36). By Corollary 2.28, ∫ 𝜈(1)𝑄 𝑡 (𝜇, d𝜈) = e−𝑏𝑡 𝜇(1) 𝑀 (𝐸)

for 𝑡 ≥ 0 and 𝜇 ∈ 𝑀 (𝐸). Then we can define a Borel transition semigroup ( 𝑄˜ 𝑡 )𝑡 ≥0 on 𝑀 (𝐸) ◦ by 𝑄˜ 𝑡 (𝜇, d𝜈) = e𝑏𝑡 𝜇(1) −1 𝜈(1)𝑄 𝑡 (𝜇, d𝜈).

(6.22)

This formula is a simple variation of the ℎ-transform of Doob; see, e.g., Sharpe (1988, p. 298). A realization of ( 𝑄˜ 𝑡 )𝑡 ≥0 can be obtained by a change of the probability measure in the (𝜉, 𝜙)-superprocess. Let 𝑊 be the space of paths 𝑤 : [0, ∞) → 𝑀 (𝐸) that are right continuous in ¯ where 𝐸¯ is a Ray–Knight both 𝑀 (𝐸) and 𝑀 (𝐸 𝜌 ) and have left limits in 𝑀 ( 𝐸), completion of 𝐸 with respect to 𝜉 and 𝐸 𝜌 denotes the set 𝐸 with the Ray topology ¯ Let 𝑊0 be the set of paths 𝑤 ∈ 𝑊 that have zero as a trap. inherited from 𝐸. Let 𝑋 = (𝑊0 , 𝒢, 𝒢𝑡 , 𝑋𝑡 , Q 𝜇 ) be the canonical Borel right realization of the (𝜉, 𝜙)superprocess. Let 𝜏0 = inf{𝑡 ≥ 0 : 𝑋𝑡 (1) = 0} denote the extinction time of 𝑋. It is

154

6 Constructions by Transformations

easy to show that 𝑚 𝑡 := e𝑏𝑡 𝑋0 (1) −1 𝑋𝑡 (1),

𝑡 ≥ 0,

(6.23)

defines a positive martingale multiplicative functional of the restriction of 𝑋 on 𝑀 (𝐸) ◦ . Let (𝒢𝑢 , 𝒢𝑡𝑢 ) be the natural 𝜎-algebras on 𝑊0 generated by {𝑋𝑡 : 𝑡 ≥ 0} as random variables on 𝑀 (𝐸) furnished with the universal 𝜎-algebra ℬ𝑢 (𝑀 (𝐸)). By the results in Sharpe (1988, p. 296), for each 𝜇 ∈ 𝑀 (𝐸) ◦ there is a unique probability ˜ 𝜇 on (𝑊0 , 𝒢𝑢 ) such that {𝑋𝑡 : 𝑡 ≥ 0} is a Markov process with transition measure Q ˜ 𝜇 {𝑋0 = 𝜇} = 1. In addition, we have semigroup ( 𝑄˜ 𝑡 )𝑡 ≥0 and Q   ˜ 𝜇 (𝐻1 {𝑡 0 be the resolvent of (𝑃𝑡 )𝑡 ≥0 defined by (A.6). Let 𝜙 be a branching mechanism given by (2.29) or (2.30). We assume the following conditions: © Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_7

165

166

7 Martingale Problems of Superprocesses

Condition 7.1 𝑏 ∈ 𝐶 (𝐸), 𝑐 ∈ 𝐶 (𝐸) + and the operator 𝑓 ↦→ 𝛾(·, 𝑓 ) preserves 𝐶0 (𝐸) + . Condition 7.2 𝑥 ↦→ [𝜈(1) ∧ 𝜈(1) 2 ]𝐻 (𝑥, d𝜈) is continuous by weak convergence on 𝑀 (𝐸) ◦ and 𝐶0 (𝐸) + is preserved by the operator ∫   𝜈( 𝑓 ) ∧ 𝜈( 𝑓 ) 2 𝐻 (𝑥, d𝜈). 𝑓 ↦→ 𝑀 (𝐸) ◦

We are going to prove some analytic properties of the cumulant semigroup (𝑉𝑡 )𝑡 ≥0 of the (𝜉, 𝜙)-superprocess. Recall that the cumulant semigroup is defined by the nonlinear integral evolution equation, for 𝑓 ∈ 𝐵(𝐸) + , ∫ 𝑡 ∫ d𝑠 𝜙(𝑦, 𝑉𝑠 𝑓 )𝑃𝑡−𝑠 (𝑥, d𝑦) = 𝑃𝑡 𝑓 (𝑥), 𝑡 ≥ 0, 𝑥 ∈ 𝐸 . (7.3) 𝑉𝑡 𝑓 (𝑥) + 0

𝐸

Proposition 7.3 For any 𝛼 > 0 we have 𝐷 0 ( 𝐴) = 𝑈 𝛼 𝐶0 (𝐸). Moreover, if 𝑓 = 𝑈 𝛼 𝑔 ∈ 𝐷 0 ( 𝐴) for 𝑔 ∈ 𝐶0 (𝐸), then 𝐴 𝑓 = 𝛼 𝑓 − 𝑔. Proof We first prove 𝐷 0 ( 𝐴) ⊂ 𝑈 𝛼 𝐶0 (𝐸) for 𝛼 > 0. Let 𝑓 ∈ 𝐷 0 ( 𝐴). For any 𝑡 ≥ 0 we have ∫ ∞ ∫ ∞ −𝛼𝑠 𝛼𝑡 𝛼 e−𝛼𝑠 𝑃𝑠 𝑓 d𝑠. 𝑈 𝑃𝑡 𝑓 = e 𝑃𝑠+𝑡 𝑓 d𝑠 = 𝑒 0

𝑡

By differentiating the equality at 𝑡 = 0 and applying the dominated convergence theorem we get 𝑈 𝛼 𝐴 𝑓 = 𝛼𝑈 𝛼 𝑓 − 𝑓 and so 𝑓 = 𝑈 𝛼 (𝛼− 𝐴) 𝑓 ∈ 𝑈 𝛼 𝐶0 (𝐸). This shows 𝐷 0 ( 𝐴) ⊂ 𝑈 𝛼 𝐶0 (𝐸). We next assume 𝑓 = 𝑈 𝛼 𝑔 for some 𝛼 > 0 and 𝑔 ∈ 𝐶0 (𝐸). For 𝑡 ≥ 0 we have ∫ ∞ ∫ ∞ −𝛼𝑠 𝛼𝑡 e 𝑃𝑡+𝑠 𝑔d𝑠 = e e−𝛼𝑠 𝑃𝑠 𝑔d𝑠. 𝑃𝑡 𝑓 = 0

𝑡

Then we can differentiate the equality at 𝑡 = 0 to see that 𝑓 ∈ 𝐷 0 ( 𝐴) and 𝐴 𝑓 = 𝛼 𝑓 − 𝑔. □ Let 𝜙 𝑛 (𝑥, 𝑓 ) be defined by (2.32). Then Conditions 7.1 and 7.1 imply that both 𝑓 ↦→ 𝜙(·, 𝑓 ) and 𝑓 ↦→ 𝜙 𝑛 (·, 𝑓 ) map 𝐶0 (𝐸) + into 𝐶0 (𝐸). By Theorem A.53 we have ∥𝜋𝑡 ∥ ≤ e𝑐0 𝑡 for 𝑡 ≥ 0, where 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)]. Lemma 7.4 Let 𝑓 ∈ 𝐶0 (𝐸) + and let 𝑡 ↦→ 𝜋𝑡 𝑓 be defined by (2.38). Then as 𝑛 → ∞ we have 𝜙 𝑛 (𝑥, 𝜋𝑡 𝑓 ) → 𝜙(𝑥, 𝜋𝑡 𝑓 ) uniformly and increasingly on the set [0, 𝑇] × 𝐸 for each 𝑇 ≥ 0. Proof Clearly, 𝜙 𝑛 (𝑥, 𝑓 ) → 𝜙(𝑥, 𝑓 ) increasingly for 𝑓 ∈ 𝐶0 (𝐸) + . Let 𝑏 ∗ = ∥𝑏 − ∥ and let 𝑡 ↦→ 𝜋𝑡∗ 𝑓 be defined by (2.38) with 𝑏 replaced by −𝑏 ∗ . By Theorem 5.28 we have 𝜋𝑡 𝑓 ≤ 𝜋𝑡∗ 𝑓 for 𝑡 ≥ 0. From (2.40) it is easy to see the operators (𝜋𝑡∗ )𝑡 ≥0 preserve ∗ 𝐶0 (𝐸) + and 𝑡 ↦→ 𝜋𝑡∗ 𝑓 is strongly continuous for each 𝑓 ∈ 𝐶0 (𝐸) + . Then ∥𝜋𝑡∗ ∥ ≤ e𝑐 𝑡 for 𝑡 ≥ 0, where 𝑐∗ = ∥𝛾(·, 1) + 𝑏 ∗ ∥. Observe that

7.1 The Differential Evolution Equation

167

𝜙(𝑥, 𝑓 ) = 𝜙 𝑛 (𝑥, 𝑓 ) + 𝑐(𝑥) 𝑓 (𝑥) 2 − 2𝑛2 𝑐(𝑥)𝐾 (𝛿 𝑥 , 𝑓 /𝑛) ∫ + 𝐾 (𝜈, 𝑓 ) [1 − 𝑛𝜈(1)]𝐻 (𝑥, d𝜈).

(7.4)

{𝑛𝜈 (1) 0 we take 𝑁 = 𝑁𝑡 (𝑥, 𝜀) ≥ 1 so that ∫ 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) < 𝜀. {𝜈 (1) >𝑁 }

If 𝜈(1) ≤ 𝑁, we have −𝜂 ( 𝑓 ) e 𝑠 − e−𝜈 ( 𝑓 ) ≤ 𝜈(𝑉𝑠 𝑓 ) − 𝜈( 𝑓 ) ≤ 𝑁 ∥𝑉𝑠 𝑓 − 𝑓 ∥.

7.1 The Differential Evolution Equation

171

It follows that 1 [𝑉𝑡+𝑠 𝑓 (𝑥) − 𝑉𝑡 𝑓 (𝑥)] − 𝐵𝑡 𝑓 (𝑥) 𝑠

1

≤ (𝑉𝑠 𝑓 − 𝑓 ) − 𝐵0 𝑓 𝜋𝑡 1(𝑥) + 𝜀∥𝐵0 𝑓 ∥ 𝑠 ∫ + 𝑁 ∥𝐵0 𝑓 ∥ ∥𝑉𝑠 𝑓 − 𝑓 ∥ 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) 𝑀 (𝐸) ◦

 1 

≤ (𝑉𝑠 𝑓 − 𝑓 ) − 𝐵0 𝑓 + 𝑁 ∥𝐵0 𝑓 ∥ ∥𝑉𝑠 𝑓 − 𝑓 ∥ 𝜋𝑡 1(𝑥) + 𝜀∥𝐵0 𝑓 ∥. 𝑠 Consequently,

1

lim (𝑉𝑡+𝑠 𝑓 − 𝑉𝑡 𝑓 ) − 𝐵𝑡 𝑓 = 0. 𝑠↓0 𝑠 In particular, for any 𝑥 ∈ 𝐸 the function 𝑡 ↦→ 𝑉𝑡 𝑓 (𝑥) has continuous right derivative 𝑡 ↦→ 𝐵𝑡 𝑓 (𝑥), and thus 𝑡 ↦→ 𝑉𝑡 𝑓 (𝑥) is continuously differentiable. This implies ∫ 𝑡 𝑉𝑡 𝑓 (𝑥) = 𝑓 (𝑥) + 𝐵𝑠 𝑓 (𝑥)d𝑠, 𝑡 ≥ 0, 𝑥 ∈ 𝐸 . 0

Then one can use the strong continuity of 𝑡 ↦→ 𝐵𝑡 𝑓 to see (7.10) holds in the supremum norm. □ Theorem 7.11 For 𝑓 ∈ 𝐷 0 ( 𝐴) + the unique locally bounded positive solution 𝑡 ↦→ 𝑉𝑡 𝑓 of the integral equation (7.3) also solves the differential equation (7.7). Proof Recall that 𝑉𝑡+𝑟 𝑓 = 𝑉𝑟 𝑉𝑡 𝑓 for 𝑡, 𝑢 ≥ 0. Then from (7.3) it follows that ∫ 𝑟 𝑃𝑟 𝑉𝑡 𝑓 − 𝑉𝑡 𝑓 = 𝑉𝑡+𝑟 𝑓 − 𝑉𝑡 𝑓 + 𝑃𝑟−𝑠 𝜙(𝑉𝑠+𝑡 𝑓 )d𝑠. 0

By Corollary 7.6 and Lemma 7.10 we see 𝑉𝑡 𝑓 ∈ 𝐷 0 ( 𝐴) + and 𝐴𝑉𝑡 𝑓 = 𝐵𝑡 𝑓 + 𝜙(𝑉𝑡 𝑓 ). This gives (7.7). □ Corollary 7.12 Let (𝜋𝑡 )𝑡 ≥0 be defined by (2.38). Then for 𝑡 ≥ 0 and 𝑓 ∈ 𝐷 0 ( 𝐴) we have 𝜋𝑡 𝑓 ∈ 𝐷 0 ( 𝐴) and d 𝜋𝑡 𝑓 (𝑥) = 𝜋𝑡 ( 𝐴 + 𝛾 − 𝑏) 𝑓 (𝑥) = ( 𝐴 + 𝛾 − 𝑏)𝜋𝑡 𝑓 (𝑥), d𝑡

𝑥 ∈ 𝐸,

(7.11)

where the derivative is taken in the supremum norm. Proof The first equality in (7.11) is a consequence of Theorem A.59. If 𝑓 ∈ 𝐷 0 ( 𝐴) + , the second equality follows from Theorem 7.11. For an arbitrary 𝑓 ∈ 𝐷 0 ( 𝐴), we may assume 𝑓 = 𝑈 𝛼 𝑔 for some 𝛼 > 0 and 𝑔 ∈ 𝐶0 (𝐸) by Proposition 7.3. Let 𝑔 + = 0 ∨ 𝑔 ∈ 𝐶0 (𝐸) + and 𝑔 − = 0 ∨ (−𝑔) ∈ 𝐶0 (𝐸) + . Then the second equality holds

172

7 Martingale Problems of Superprocesses

for 𝑓+ := 𝑈 𝛼 𝑔 + ∈ 𝐷 0 ( 𝐴) + and 𝑓− := 𝑈 𝛼 𝑔 − ∈ 𝐷 0 ( 𝐴) + .

(7.12)

By linearity the equality also holds for 𝑓 = 𝑓+ − 𝑓− .

□

By a combination of Theorems 7.8 and 7.11 we obtain: Theorem 7.13 For any 𝑓 ∈ 𝐷 0 ( 𝐴) + , the integral equation (7.3) and the differential equation (7.7) for (𝑡, 𝑥) ↦→ 𝑉𝑡 𝑓 (𝑥) are equivalent. By modifications of the arguments given above one can prove the following: Theorem 7.14 For any 𝑓 ∈ 𝐷 0 ( 𝐴) + and 𝑔 ∈ 𝐶0 (𝐸) + , the integral equation (5.32) is equivalent to the differential evolution equation   d   𝑣 𝑡 (𝑥) = 𝐴𝑣 𝑡 (𝑥) − 𝜙(𝑥, 𝑣 𝑡 ) + 𝑔(𝑥), d𝑡   𝑣 0 (𝑥) = 𝑓 (𝑥), 

𝑡 ≥ 0, 𝑥 ∈ 𝐸,

(7.13)

𝑥 ∈ 𝐸.

Suppose that 𝑋 = (𝑊, 𝒢, 𝒢𝑡 , 𝑋𝑡 , Q 𝜇 ) is a right continuous realization of the (𝜉, 𝜙)-superprocess. Since any function in 𝐶 (𝐸) + is the increasing limit of a sequence of functions from 𝐶0 (𝐸) + , we can define the transition semigroup (𝑄 𝑡 )𝑡 ≥0 of 𝑋 by ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈) = exp{−𝜇(𝑉𝑡 𝑓 )}, 𝑓 ∈ 𝐶0 (𝐸) + , (7.14) 𝑀 (𝐸)

where 𝑡 ↦→ 𝑉𝑡 𝑓 is the unique locally bounded positive solution of (7.3). Since 𝛼𝑈 𝛼 𝑓 → 𝑓 uniformly as 𝛼 → ∞ for any 𝑓 ∈ 𝐶0 (𝐸), by Proposition 7.3 we see that 𝐷 0 ( 𝐴) + is uniformly dense in 𝐶0 (𝐸) + . It follows that the operators (𝑉𝑡 )𝑡 ≥0 are uniquely determined by their restrictions to 𝐷 0 ( 𝐴) + . Then the transition semigroup (𝑄 𝑡 )𝑡 ≥0 can be defined by (7.14) for 𝑓 ∈ 𝐷 0 ( 𝐴) + with 𝑡 ↦→ 𝑉𝑡 𝑓 being the unique positive solution of the differential equation (7.7). Similarly, the joint distribution ∫𝑡 of 𝑋𝑡 and 0 𝑋𝑠 d𝑠 can also be determined by (5.31) and (7.13). In applications we may also consider (7.7) in a smaller class of functions, as shown in the following example. Example 7.1 Let 𝐶02 (R𝑑 ) denote the set of twice continuously differentiable functions on R𝑑 that together with all their partial derivatives up to the second order vanish at infinity. If 𝜉 is a 𝑑-dimensional diffusion process with generator 𝐴 specified in Example 6.1, then for any 𝑓 ∈ 𝐶02 (R𝑑 ) + we can also define (𝑡, 𝑥) ↦→ 𝑉𝑡 𝑓 (𝑥) by the nonlinear partial differential equation ( d 𝑉𝑡 𝑓 (𝑥) = 𝐴𝑉𝑡 𝑓 (𝑥) − 𝜙(𝑥, 𝑉𝑡 𝑓 ), d𝑡 𝑉0 𝑓 (𝑥) = 𝑓 (𝑥),

𝑡 ≥ 0, 𝑥 ∈ R𝑑 ,

(7.15)

𝑥 ∈ R𝑑 .

The operators (𝑉𝑡 )𝑡 ≥0 are uniquely determined by their restrictions on 𝐶02 (R𝑑 ) + . This follows from the fact that any function in 𝐶0 (R𝑑 ) + is the limit of a sequence of functions from 𝐶02 (R𝑑 ) + in the supremum norm.

7.2 Generators and Martingale Problems

173

7.2 Generators and Martingale Problems Suppose that 𝐸 is a locally compact separable metric space. Let 𝜉 be a Hunt process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 and let 𝜙 be a branching mechanism given by (2.29) or (2.30). We assume that (𝑃𝑡 )𝑡 ≥0 and 𝜙 satisfy the conditions specified at the beginning of Section 7.1. Let (𝑄 𝑡 )𝑡 ≥0 and (𝑉𝑡 )𝑡 ≥0 denote respectively the transition semigroup and the cumulant semigroup of the (𝜉, 𝜙)-superprocess. By Theorem 5.13, the process has a càdlàg realization in 𝑀 (𝐸). Proposition 7.15 If {𝑋𝑡 : 𝑡 ≥ 0} is a càdlàg Markov process in 𝑀 (𝐸) relative to a filtration (ℱ𝑡 )𝑡 ≥0 with transition semigroup (𝑄 𝑡 )𝑡 ≥0 , then {𝑋𝑡 : 𝑡 ≥ 0} is also a ¯ 𝑡+ )𝑡 ≥0 with Markov process relative to the augmented right continuous filtration ( ℱ the same transition semigroup. Proof Under the condition of the proposition, one easily sees that {𝑋𝑡 : 𝑡 ≥ 0} ¯ 𝑡 )𝑡 ≥0 with transition is a Markov process relative to the augmented filtration ( ℱ semigroup (𝑄 𝑡 )𝑡 ≥0 . Let 𝑡 > 𝑟 ≥ 0 and let {𝑟 𝑛 } ⊂ (𝑟, 𝑡] be a decreasing sequence such that lim𝑛→∞ 𝑟 𝑛 = 𝑟. For any 𝑓 ∈ 𝐶0 (𝐸) + we have   ¯ 𝑟𝑛 = exp{−𝑋𝑟𝑛 (𝑉𝑡−𝑟𝑛 𝑓 )}. P e−𝑌𝑡 ( 𝑓 ) | ℱ Then we can let 𝑛 → ∞ and use Corollary 7.6 to get   ¯ 𝑟+ = exp{−𝑋𝑟 (𝑉𝑡−𝑟 𝑓 )}. P e−𝑋𝑡 ( 𝑓 ) ℱ ¯ 𝑡+ )𝑡 ≥0 with the same This gives the Markov property of {𝑋𝑡 : 𝑡 ≥ 0} relative to ( ℱ transition semigroup. □ We shall give several equivalent formulations of the (𝜉, 𝜙)-superprocess in terms of martingale problems and discuss some consequences. Let 𝒟0 be the class of functions on 𝑀 (𝐸) of the form 𝐹 (𝜇) = 𝐺 (𝜇( 𝑓1 ), . . . , 𝜇( 𝑓𝑛 )),

(7.16)

where 𝐺 ∈ 𝐶 2 (R𝑛 ) and { 𝑓1 , . . . , 𝑓𝑛 } ⊂ 𝐷 0 ( 𝐴). For 𝐹 ∈ 𝒟0 define ∫  ′  𝐿 0 𝐹 (𝜇) = 𝐴𝐹 (𝜇; 𝑥) + 𝛾(𝑥, 𝐹 ′ (𝜇)) − 𝑏(𝑥)𝐹 ′ (𝜇; 𝑥) 𝜇(d𝑥) 𝐸 ∫ ∫ ∫  𝜇(d𝑥) 𝐹 (𝜇 + 𝜈) 𝑐(𝑥)𝐹 ′′ (𝜇; 𝑥)𝜇(d𝑥) + + 𝐸 𝐸 𝑀 (𝐸) ◦  − 𝐹 (𝜇) − 𝜈(𝐹 ′ (𝜇)) 𝐻 (𝑥, d𝜈), (7.17) where 𝐹 ′ (𝜇; 𝑥) = lim 𝜀↓0

 1 𝐹 (𝜇 + 𝜀𝛿 𝑥 ) − 𝐹 (𝜇) 𝜀

174

7 Martingale Problems of Superprocesses

and 𝐹 ′′ (𝜇; 𝑥) is defined by the limit with 𝐹 (·) replaced by 𝐹 ′ (·; 𝑥). In particular, if 𝜙 is the local branching mechanism given by (2.49), the operator 𝐿 0 is given by ∫   ′ 𝐴𝐹 (𝜇; 𝑥) − 𝑏(𝑥)𝐹 ′ (𝜇; 𝑥) 𝜇(d𝑥) 𝐿 0 𝐹 (𝜇) = 𝐸 ∫ ∫ ∞ ∫  + 𝑐(𝑥)𝐹 ′′ (𝜇; 𝑥)𝜇(d𝑥) + 𝐹 (𝜇 + 𝑢𝛿 𝑥 ) 𝜇(d𝑥) 𝐸 𝐸 0  − 𝐹 (𝜇) − 𝑢𝐹 ′ (𝜇; 𝑥) 𝑚(𝑥, d𝑢). (7.18) Suppose that (Ω, 𝒢, 𝒢𝑡 , P) is a filtered probability space satisfying the usual hypotheses and {𝑋𝑡 : 𝑡 ≥ 0} is a càdlàg process in 𝑀 (𝐸) that is adapted to (𝒢𝑡 )𝑡 ≥0 and satisfies P[𝑋0 (1)] < ∞. Let us consider the following properties: (1) For every 𝑇 ≥ 0 and 𝑓 ∈ 𝐶0 (𝐸) + , exp{−𝑋𝑡 (𝑉𝑇−𝑡 𝑓 )}, is a martingale. (2) For every 𝑓 ∈ 𝐷 0 ( 𝐴) + ,  ∫ 𝐻𝑡 ( 𝑓 ) := exp − 𝑋𝑡 ( 𝑓 ) +

0 ≤ 𝑡 ≤ 𝑇,



𝑡

𝑋𝑠 ( 𝐴 𝑓 − 𝜙( 𝑓 ))d𝑠 ,

𝑡 ≥ 0,

0

is a local martingale. (3) The process {𝑋𝑡 : 𝑡 ≥ 0} has no negative jumps. Moreover, we have: (a) Let 𝑁 (d𝑠, d𝜈) be the optional random measure on [0, ∞) × 𝑀 (𝐸) ◦ defined by ∑︁ 1 {Δ𝑋𝑠 ≠0} 𝛿 (𝑠,Δ𝑋𝑠 ) (d𝑠, d𝜈), 𝑁 (d𝑠, d𝜈) = (7.19) 𝑠>0

where Δ𝑋𝑠 = 𝑋𝑠 −𝑋𝑠− . Then 𝑁 (d𝑠, d𝜈) has predictable compensator 𝑁ˆ (d𝑠, d𝜈) = d𝑠𝐾 (𝑋𝑠− , d𝜈), where ∫ 𝜇(d𝑥)𝐻 (𝑥, d𝜈). 𝐾 (𝜇, d𝜈) = 𝐸

(b) Let 𝑁˜ (d𝑠, d𝜈) = 𝑁 (d𝑠, d𝜈) − 𝑁ˆ (d𝑠, d𝜈) be the compensated random measure. Then for any 𝑓 ∈ 𝐷 0 ( 𝐴) we have ∫ 𝑡 𝑐 𝑑 𝑋𝑡 ( 𝑓 ) = 𝑋0 ( 𝑓 ) + 𝑀𝑡 ( 𝑓 ) + 𝑀𝑡 ( 𝑓 ) + 𝑋𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠, (7.20) 0

where 𝑡 ↦→ 2𝑋𝑡 (𝑐 𝑓 2 )d𝑡 and

𝑀𝑡𝑐 ( 𝑓 )

is a continuous local martingale with quadratic variation ∫

𝑡

∫

𝑡 ↦→ 𝑀𝑡𝑑 ( 𝑓 ) = 0

𝑀 (𝐸) ◦

𝜈( 𝑓 ) 𝑁˜ (d𝑠, d𝜈)

(7.21)

7.2 Generators and Martingale Problems

175

is a purely discontinuous local martingale. (4) For every 𝐹 ∈ 𝒟0 we have ∫ 𝑡 𝐿 0 𝐹 (𝑋𝑠 )d𝑠 + local mart. 𝐹 (𝑋𝑡 ) = 𝐹 (𝑋0 ) +

(7.22)

0

(5) For every 𝐺 ∈ 𝐶 2 (R) and 𝑓 ∈ 𝐷 0 ( 𝐴) we have ∫ 𝑡 𝐺 (𝑋𝑡 ( 𝑓 )) = 𝐺 (𝑋0 ( 𝑓 )) + 𝐺 ′ (𝑋𝑠 ( 𝑓 )) 𝑋𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠 0 ∫ 𝑡 𝐺 ′′ (𝑋𝑠 ( 𝑓 )) 𝑋𝑠 (𝑐 𝑓 2 )d𝑠 + local mart. + 0 ∫ 𝑡 ∫ ∫  𝑋𝑠 (d𝑥) + 𝐺 (𝑋𝑠 ( 𝑓 ) + 𝜈( 𝑓 )) d𝑠 𝐸 0 𝑀 (𝐸) ◦  (7.23) − 𝐺 (𝑋𝑠 ( 𝑓 )) − 𝜈( 𝑓 )𝐺 ′ (𝑋𝑠 ( 𝑓 )) 𝐻 (𝑥, d𝜈). Theorem 7.16 The above properties (1), (2), (3), (4) and (5) are equivalent to each other. Those properties hold if and only if {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a (𝜉, 𝜙)-superprocess with transition semigroup (𝑄 𝑡 )𝑡 ≥0 . Proof Clearly, (1) holds if and only if {𝑋𝑡 : 𝑡 ≥ 0} is a Markov process relative to (𝒢𝑡 )𝑡 ≥0 with transition semigroup (𝑄 𝑡 )𝑡 ≥0 defined by (7.14). Then we only need to prove the equivalence of the five properties. (1)⇒(2): If (1) holds, then {𝑋𝑡 : 𝑡 ≥ 0} is a (𝜉, 𝜙)-superprocess, so Corollary 2.28 implies P[𝑋𝑡 ( 𝑓 )] = P[𝑋0 (𝜋𝑡 𝑓 )],

𝑡 ≥ 0, 𝑓 ∈ 𝐵(𝐸),

(7.24)

where (𝜋𝑡 )𝑡 ≥0 is defined by (2.38). Now we fix 𝑟 ≥ 0 and 𝐵 ∈ 𝒢𝑟 and define 𝐽𝑡 ( 𝑓 ) = P[1 𝐵 e−𝑋𝑡 ( 𝑓 ) ] = P[1 𝐵 e−𝑋𝑟 (𝑉𝑡−𝑟 𝑓 ) ] for 𝑡 ≥ 𝑟 and 𝑓 ∈ 𝐷 0 ( 𝐴) + . In view of (7.24), we can use Theorem 7.11 and the dominated convergence theorem to show that 𝐽𝑡 ( 𝑓 ) is continuously differentiable in 𝑡 ≥ 𝑟. By calculating the right derivative, we have   d d 𝐽𝑡 ( 𝑓 ) = P[1 𝐵 e−𝑋𝑡 (𝑉𝑠 𝑓 ) ] = −P 1 𝐵 𝑋𝑡 ( 𝐴 𝑓 − 𝜙( 𝑓 ))e−𝑋𝑡 ( 𝑓 ) . 𝑠=0 d𝑡 d𝑠 It follows that 𝑌𝑡 ( 𝑓 ) := e

−𝑋𝑡 ( 𝑓 )

∫ +

𝑡

𝑋𝑠 ( 𝐴 𝑓 − 𝜙( 𝑓 ))e−𝑋𝑠 ( 𝑓 ) d𝑠,

𝑡≥0

0

is a martingale. By integration by parts applied to ∫ 𝑍𝑡 ( 𝑓 ) := e−𝑋𝑡 ( 𝑓 ) and 𝑊𝑡 ( 𝑓 ) := exp 0



𝑡

𝑋𝑠 ( 𝐴 𝑓 − 𝜙( 𝑓 ))d𝑠

(7.25)

176

7 Martingale Problems of Superprocesses

we obtain d𝐻𝑡 ( 𝑓 ) = e−𝑋𝑡− ( 𝑓 ) d𝑊𝑡 ( 𝑓 ) + 𝑊𝑡 ( 𝑓 )de−𝑋𝑡 ( 𝑓 ) = 𝑊𝑡 ( 𝑓 )d𝑌𝑡 ( 𝑓 ). Then {𝐻𝑡 ( 𝑓 )} is a local martingale.

(2)⇒(3): For 𝑓 ∈ 𝐷 0 ( 𝐴) + define 𝑍𝑡 ( 𝑓 ) and 𝑊𝑡 ( 𝑓 ) by (7.25). We have 𝑍𝑡 ( 𝑓 ) = 𝐻𝑡 ( 𝑓 )𝑊𝑡 ( 𝑓 ) −1 and so d𝑍𝑡 ( 𝑓 ) = 𝑊𝑡 ( 𝑓 ) −1 d𝐻𝑡 ( 𝑓 ) − 𝑍𝑡− ( 𝑓 ) 𝑋𝑡− ( 𝐴 𝑓 − 𝜙( 𝑓 ))d𝑡

(7.26)

by integration by parts. Then {𝑍𝑡 ( 𝑓 )} is a special semi-martingale; see, e.g., Dellacherie and Meyer (1982, p. 213). By Itô’s formula we find that {𝑋𝑡 ( 𝑓 )} is a semi-martingale. Let 𝑆(𝐸) denote the space of finite Borel signed measures on 𝐸 endowed with the 𝜎-algebra generated by the mappings 𝜇 ↦→ 𝜇(𝐵) for all 𝐵 ∈ ℬ(𝐸). Let 𝑆(𝐸) ◦ = 𝑆(𝐸) \ {0}. We define the optional random measure 𝑁 (d𝑠, d𝜈) on [0, ∞) × 𝑆(𝐸) ◦ by ∑︁ 𝑁 (d𝑠, d𝜈) = 1 {Δ𝑋𝑠 ≠0} 𝛿 (𝑠,Δ𝑋𝑠 ) (d𝑠, d𝜈), 𝑠>0

where Δ𝑋𝑠 = 𝑋𝑠 − 𝑋𝑠− ∈ 𝑆(𝐸). Let 𝑁ˆ (d𝑠, d𝜈) denote the predictable compensator of 𝑁 (d𝑠, d𝜈) and let 𝑁˜ (d𝑠, d𝜈) denote the compensated random measure; see Dellacherie and Meyer (1982, pp. 371–374). Then there is a càdlàg process {𝑈𝑡 ( 𝑓 )} with locally bounded variations such that 𝑋𝑡 ( 𝑓 ) = 𝑋0 ( 𝑓 ) + 𝑈𝑡 ( 𝑓 ) + 𝑀𝑡𝑐 ( 𝑓 ) + 𝑀𝑡𝑑 ( 𝑓 ), where {𝑀𝑡𝑐 ( 𝑓 )} is a continuous local martingale and ∫ 𝑡∫ 𝑑 𝜈( 𝑓 ) 𝑁˜ (d𝑠, d𝜈), 𝑀𝑡 ( 𝑓 ) = 0

𝑡 ≥ 0,

(7.27)

(7.28)

𝑆 (𝐸) ◦

is a purely discontinuous local martingale; see Dellacherie and Meyer (1982, p. 353 and p. 376) or Jacod and Shiryaev (2003, p. 84). Let {𝐶𝑡 ( 𝑓 )} denote the quadratic variation process of {𝑀𝑡𝑐 ( 𝑓 )}. By Itô’s formula, ∫ ∫ 𝑡 1 𝑡 𝑍 𝑠− ( 𝑓 )d𝐶𝑠 ( 𝑓 ) 𝑍 𝑠− ( 𝑓 )d𝑈𝑠 ( 𝑓 ) + 𝑍𝑡 ( 𝑓 ) = 𝑍0 ( 𝑓 ) − 2 0 ∫ 𝑡∫ 0 + 𝑍 𝑠− ( 𝑓 )𝐾 (𝜈, 𝑓 ) 𝑁ˆ (d𝑠, d𝜈) + local mart., (7.29) 0

𝑆 (𝐸) ◦

where 𝐾 (𝜈, 𝑓 ) = e−𝜈 ( 𝑓 ) − 1 + 𝜈( 𝑓 ). In view of (7.26) and (7.29) we get ∫ 1 d𝑈𝑡 ( 𝑓 ) = d𝐶𝑡 ( 𝑓 ) + 𝑋𝑡− ( 𝐴 𝑓 − 𝜙( 𝑓 ))d𝑡 + 𝐾 (𝜈, 𝑓 ) 𝑁ˆ (d𝑡, d𝜈) 2 𝑆 (𝐸) ◦

7.2 Generators and Martingale Problems

177

by the uniqueness of canonical decomposition of the special semi-martingale; see Dellacherie and Meyer (1982, p. 213). By substituting the representation (2.30) of 𝜙 into the above equation and comparing both sides it is easy to show that (3.a) and (3.b) hold for 𝑓 ∈ 𝐷 0 ( 𝐴) + . By linearity we see that (3.b) also holds for an arbitrary 𝑓 = 𝑓+ − 𝑓− ∈ 𝐷 0 ( 𝐴), where 𝑓+ ∈ 𝐷 0 ( 𝐴) + and 𝑓− ∈ 𝐷 0 ( 𝐴) + are defined by (7.12). (3)⇒(4): For the function 𝐹 ∈ 𝒟0 given by (7.16), it is easy to show that 𝐹 ′ (𝜇; 𝑥) =

𝑛 ∑︁

𝑓𝑖 (𝑥)𝐺 𝑖′ (𝜇( 𝑓1 ), . . . , 𝜇( 𝑓𝑛 ))

𝑖=1

and 𝐹 ′′ (𝜇; 𝑥) =

𝑛 ∑︁

𝑓𝑖 (𝑥) 𝑓 𝑗 (𝑥)𝐺 𝑖′′𝑗 (𝜇( 𝑓1 ), . . . , 𝜇( 𝑓𝑛 )).

𝑖, 𝑗=1

Consequently, we have 𝐿0 𝐹 ( 𝜇) =

𝑛 ∑︁

𝐺𝑖′ ( 𝜇 ( 𝑓1 ), . . . , 𝜇 ( 𝑓 𝑛 )) 𝜇 ( 𝐴 𝑓𝑖 + 𝛾 𝑓𝑖 − 𝑏 𝑓𝑖 ) ∫ h 𝜇 (d𝑥) + 𝐺 ( 𝜇 ( 𝑓1 ) + 𝜈 ( 𝑓1 ) , . . . , 𝜇 ( 𝑓 𝑛 ) + 𝜈 ( 𝑓 𝑛 )) 𝑖=1 ∫

𝑀 (𝐸) ◦

𝐸

− 𝐺 ( 𝜇 ( 𝑓1 ) , . . . , 𝜇 ( 𝑓 𝑛 )) −

𝑛 ∑︁

i 𝜈 ( 𝑓𝑖 )𝐺𝑖′ ( 𝜇 ( 𝑓1 ) , . . . , 𝜇 ( 𝑓 𝑛 )) 𝐻 ( 𝑥, d𝜈)

𝑖=1 𝑛 ∑︁

+

𝐺𝑖′′𝑗 ( 𝜇 ( 𝑓1 ) ,

. . . , 𝜇 ( 𝑓 𝑛 )) 𝜇 (𝑐 𝑓𝑖 𝑓 𝑗 ).

(7.30)

𝑖, 𝑗=1

Clearly, the continuous local martingales 𝑡 ↦→ 𝑀𝑡𝑐 ( 𝑓𝑖 ) and 𝑡 ↦→ 𝑀𝑡𝑐 ( 𝑓 𝑗 ) have quadratic covariation 2𝑋𝑡 (𝑐 𝑓𝑖 𝑓 𝑗 )d𝑡. Then by Itô’s formula, 𝐹 (𝑋𝑡 ) = 𝐹 (𝑋0 ) +

𝑛 ∫ ∑︁ 𝑖=1

∫

𝑡

∫

𝑡

𝐺 𝑖′ (𝑋𝑠− ( 𝑓1 ), . . . , 𝑋𝑠− ( 𝑓𝑛 ))d𝑀𝑠𝑐 ( 𝑓𝑖 )

0 𝑛 ∑︁

𝐺 𝑖′ (𝑋𝑠− ( 𝑓1 ), . . . , 𝑋𝑠− ( 𝑓𝑛 ))𝜈( 𝑓𝑖 ) 𝑁˜ (d𝑠, d𝜈) 𝑀 (𝐸) ◦ 𝑖=1 ∫ 𝑡 ∑︁ 𝑛 𝐺 𝑖′ (𝑋𝑠− ( 𝑓1 ), . . . , 𝑋𝑠− ( 𝑓𝑛 )) 𝑋𝑠− ( 𝐴 𝑓𝑖 + 𝛾 𝑓𝑖 − 𝑏 𝑓𝑖 )d𝑠 + 0 𝑖=1 ∫ 𝑡 ∑︁ 𝑛 + 𝐺 𝑖′′𝑗 (𝑋𝑠− ( 𝑓1 ), . . . , 𝑋𝑠− ( 𝑓𝑛 )) 𝑋𝑠− (𝑐 𝑓𝑖 𝑓 𝑗 )d𝑠 0 𝑖, 𝑗=1 ∫ 𝑡∫  +

0

+

0

𝑀 (𝐸) ◦

𝐺 (𝑋𝑠− ( 𝑓1 ) + 𝜈( 𝑓1 ), . . . , 𝑋𝑠− ( 𝑓𝑛 ) + 𝜈( 𝑓𝑛 ))

− 𝐺 (𝑋𝑠− ( 𝑓1 ), . . . , 𝑋𝑠− ( 𝑓𝑛 )) 𝑛 ∑︁  − 𝜈( 𝑓𝑖 )𝐺 𝑖′ (𝑋𝑠− ( 𝑓1 ), . . . , 𝑋𝑠− ( 𝑓𝑛 )) 𝑁 (d𝑠, d𝜈) 𝑖=1

178

7 Martingale Problems of Superprocesses

∫

𝑡

= 𝐹 (𝑋0 ) +

𝐿 0 𝐹 (𝑋𝑠 )d𝑠 + 𝑀𝑡 (𝐹), 0

where 𝑀𝑡 (𝐹) =

𝑛 ∫ ∑︁

𝑡

𝐺 𝑖′ (𝑋𝑠− ( 𝑓1 ), . . . , 𝑋𝑠− ( 𝑓𝑛 ))d𝑀𝑠𝑐 ( 𝑓𝑖 )

0 𝑖=1 ∫ 𝑡∫

+ 0



𝐺 (𝑋𝑠− ( 𝑓1 ) + 𝜈( 𝑓1 ), . . . , 𝑋𝑠− ( 𝑓𝑛 ) + 𝜈( 𝑓𝑛 ))  (7.31) − 𝐺 (𝑋𝑠− ( 𝑓1 ), . . . , 𝑋𝑠− ( 𝑓𝑛 )) 𝑁˜ (d𝑠, d𝜈). 𝑀 (𝐸) ◦

For 𝑛 ≥ 1 define the stopping time 𝑇𝑛 = {𝑡 ≥ 0 : 𝑋𝑡 (1) ≥ 𝑛}. Then {𝑀𝑡 (𝐹) : 𝑡 ≥ 0} is a local martingale with localization sequence {𝑇𝑛 }. This proves (4). (4)⇒(5): Let 𝐹 (𝜇) = 𝐺 (𝜇( 𝑓 )) for 𝐺 ∈ 𝐶 2 (R) and 𝑓 ∈ 𝐷 0 ( 𝐴). As a special case of (7.30), we have 𝐿 0 𝐹 (𝜇) = 𝐺 ′ (𝜇( 𝑓 ))𝜇( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 ) + 𝐺 ′′ (𝜇( 𝑓 ))𝜇(𝑐 𝑓 2 ) ∫ ∫ h 𝜇(d𝑥) + 𝐺 (𝜇( 𝑓 ) + 𝜈( 𝑓 )) − 𝐺 (𝜇( 𝑓 )) ◦ 𝑀 (𝐸) 𝐸 i − 𝜈( 𝑓 )𝐺 ′ (𝜇( 𝑓 )) 𝐻 (𝑥, d𝜈).

(7.32)

Then (5) follows from (4). (5)⇒(1): Let 𝐺 ∈ 𝐶 2 (R) and let 𝑡 ↦→ 𝑓𝑡 be a mapping from [0, 𝑇] to 𝐷 0 ( 𝐴) + such that 𝑡 ↦→ 𝑓𝑡 is continuously differentiable and 𝑡 ↦→ 𝐴 𝑓𝑡 is continuous by the supremum norm. For 0 ≤ 𝑡 ≤ 𝑇 and 𝑘 ≥ 1 we have 𝐺 (𝑋𝑡 ( 𝑓𝑡 )) = 𝐺 (𝑋0 ( 𝑓0 )) +

∞ ∑︁ 

𝐺 (𝑋𝑡∧( 𝑗+1)/𝑘 ( 𝑓𝑡∧ 𝑗/𝑘 )) − 𝐺 (𝑋𝑡∧ 𝑗/𝑘 ( 𝑓𝑡∧ 𝑗/𝑘 ))



𝑗=0

+

∞ ∑︁ 

 𝐺 (𝑋𝑡∧( 𝑗+1)/𝑘 ( 𝑓𝑡∧( 𝑗+1)/𝑘 )) − 𝐺 (𝑋𝑡∧( 𝑗+1)/𝑘 ( 𝑓𝑡∧ 𝑗/𝑘 )) ,

𝑗=0

where the summations only consist of finitely many non-trivial terms. By applying (5) term by term we obtain 𝐺 (𝑋𝑡 ( 𝑓𝑡 )) = 𝐺 (𝑋0 ( 𝑓0 )) +

∞ ∫ ∑︁ 𝑗=0

𝑡∧( 𝑗+1)/𝑘



𝐺 ′ (𝑋𝑠 ( 𝑓𝑡∧ 𝑗/𝑘 )) 𝑋𝑠 (( 𝐴 + 𝛾) 𝑓𝑡∧ 𝑗/𝑘 )

𝑡∧ 𝑗/𝑘

2 − 𝐺 (𝑋𝑠 ( 𝑓𝑡∧ 𝑗/𝑘 )) 𝑋𝑠 (𝑏 𝑓𝑡∧ 𝑗/𝑘 ) + 𝐺 ′′ (𝑋𝑠 ( 𝑓𝑡∧ 𝑗/𝑘 )) 𝑋𝑠 (𝑐 𝑓𝑡∧ 𝑗/𝑘 ) ∫ ∫ h 𝐺 (𝑋𝑠 ( 𝑓𝑡∧ 𝑗/𝑘 ) + 𝜈( 𝑓𝑡∧ 𝑗/𝑘 )) + 𝑋𝑠 (d𝑥) 𝐸 𝑀 (𝐸) ◦  i − 𝐺 (𝑋𝑠 ( 𝑓𝑡∧ 𝑗/𝑘 )) − 𝜈( 𝑓𝑡∧ 𝑗/𝑘 )𝐺 ′ (𝑋𝑠 ( 𝑓𝑡∧ 𝑗/𝑘 )) 𝐻 (𝑥, d𝜈) d𝑠 ′

7.2 Generators and Martingale Problems

+

∞ ∫ ∑︁ 𝑗=0

179

𝑡∧( 𝑗+1)/𝑘

𝐺 ′ (𝑋𝑡∧( 𝑗+1)/𝑘 ( 𝑓𝑠 )) 𝑋𝑡∧( 𝑗+1)/𝑘 ( 𝑓𝑠′)d𝑠 + 𝑀𝑘 (𝑡),

𝑡∧ 𝑗/𝑘

where {𝑀𝑘 (𝑡)} is a local martingale. Since {𝑋𝑡 } is a càdlàg process, letting 𝑘 → ∞ in the equation above gives ∫ 𝑡 𝐺 (𝑋𝑡 ( 𝑓𝑡 )) = 𝐺 (𝑋0 ( 𝑓0 )) + 𝐺 ′ (𝑋𝑠 ( 𝑓𝑠 )) 𝑋𝑠 ( 𝐴 𝑓𝑠 + 𝛾 𝑓𝑠 − 𝑏 𝑓𝑠 + 𝑓𝑠′) 0 ∫ ∫ h 𝑋𝑠 (d𝑥) + 𝐺 ′′ (𝑋𝑠 ( 𝑓𝑠 )) 𝑋𝑠 (𝑐 𝑓𝑠2 ) + 𝐺 (𝑋𝑠 ( 𝑓𝑠 ) + 𝜈( 𝑓𝑠 )) 𝑀 (𝐸) ◦ 𝐸 i − 𝐺 (𝑋𝑠 ( 𝑓𝑠 )) − 𝜈( 𝑓𝑠 )𝐺 ′ (𝑋𝑠 ( 𝑓𝑠 )) 𝐻 (𝑥, d𝜈) d𝑠 + 𝑀 (𝑡), where {𝑀 (𝑡)} is a local martingale. For any 𝑓 ∈ 𝐷 0 ( 𝐴) + we may apply the above to 𝐺 (𝑧) = e−𝑧 and 𝑓𝑡 = 𝑉𝑇−𝑡 𝑓 to see 𝑡 ↦→ exp{−𝑋𝑡 (𝑉𝑇−𝑡 𝑓 )} is a local martingale. □ Then the assertion of (1) follows by the dominated convergence theorem.

Corollary 7.17 Suppose that {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a càdlàg (𝜉, 𝜙)-superprocess satisfying P[𝑋0 (1)] < ∞. Then the local martingales in the above properties (3), (4) and (5) are martingales. Proof For any 𝑓 ∈ 𝐷 0 ( 𝐴) let 𝑀𝑡𝑐 ( 𝑓 ) and 𝑀𝑡𝑑 ( 𝑓 ) be defined by (7.20) and (7.21). By Corollary 2.28 it is easy to see that h∫ 𝑡 i h∫ 𝑡 i P[𝑀𝑡𝑐 ( 𝑓 ) 2 ] = 2P 𝑋𝑠 (𝑐 𝑓 2 )d𝑠 ≤ 2∥𝑐 𝑓 2 ∥P 𝑋𝑠 (1)d𝑠 < ∞. 0

0

Then {𝑀𝑡𝑐 ( 𝑓 ) : 𝑡 ≥ 0} is a square-integrable martingale. Moreover, we have i2o nh ∫ 𝑡 ∫ P 𝜈( 𝑓 ) 𝑁˜ (d𝑠, d𝜈) 0 {𝜈 (1) ≤1} ∫ i h∫ 𝑡 ∫ d𝑠 𝑋𝑠 (d𝑥) 𝜈( 𝑓 ) 2 𝐻 (𝑥, d𝜈) =P 0 {𝜈 (1)∫≤1} 𝐸 i h∫ 𝑡 ∫ 2 d𝑠 𝑋𝑠 (d𝑥) 𝜈(1) 2 𝐻 (𝑥, d𝜈) < ∞ ≤ ∥𝑓∥ P 0

𝐸

{𝜈 (1) ≤1}

and P

h∫

𝑡

∫

i 𝜈( 𝑓 )𝑁 (d𝑠, d𝜈) 0 {𝜈∫(1) >1} ∫ ∫ h 𝑡 i =P 𝜈( 𝑓 )𝐻 (𝑥, d𝜈) d𝑠 𝑋𝑠 (d𝑥) 0 ∫ 𝐸 ∫ {𝜈 (1) ∫ >1} h 𝑡 i 𝑋𝑠 (d𝑥) ≤ ∥ 𝑓 ∥P 𝜈(1)𝐻 (𝑥, d𝜈) < ∞. d𝑠 0

𝐸

{𝜈 (1) >1}

180

7 Martingale Problems of Superprocesses

Then {𝑀𝑡𝑑 ( 𝑓 ) : 𝑡 ≥ 0} is a martingale. For 𝐹 ∈ 𝒟0 with representation (7.16), the local martingale in (7.22) is expressed explicitly by (7.31). Using this expression and the above estimates one can show that {𝑀𝑡 (𝐹)} is a martingale. □ Corollary 7.18 Suppose that {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a càdlàg (𝜉, 𝜙)-superprocess satisfying P[𝑋0 (1)] < ∞. Then for every 𝑇 ≥ 0 and 𝑓 ∈ 𝐷 0 ( 𝐴) there is a constant 𝐶 (𝑇, 𝑓 ) ≥ 0 such that n i h o √︁ P sup |𝑋𝑡 ( 𝑓 )| ≤ 𝐶 (𝑇, 𝑓 ) P[𝑋0 (1)] + P[𝑋0 (1)] . 0≤𝑡 ≤𝑇

Proof By the above property (3.b) and Doob’s martingale inequality we have h i h i P sup |𝑋𝑡 ( 𝑓 )| ≤ P[|𝑋0 ( 𝑓 )|] + P sup |𝑀𝑡𝑐 ( 𝑓 )| 0≤𝑡 ≤𝑇 0≤𝑡 ≤𝑇 ∫ 𝑡 ∫ i h + P sup 𝜈( 𝑓 )1 {𝜈 (1) ≤1} 𝑁˜ (d𝑠, d𝜈) ◦ 0≤𝑡 ≤𝑇 ∫0 ∫𝑀 (𝐸) 𝑡 i h + P sup 𝜈( 𝑓 )1 {𝜈 (1) >1} 𝑁˜ (d𝑠, d𝜈) 0≤𝑡 ≤𝑇 0 𝑀 (𝐸) ◦ h∫ 𝑇 i +P |𝑋𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )|d𝑠 0 i o 1/2 √ n h∫ 𝑇 𝑋𝑠 (𝑐 𝑓 2 )d𝑠 ≤ P[|𝑋0 ( 𝑓 )|] + 2 2 P 0 ∫ n h∫ 𝑇 ∫ i o 1/2 +2 P d𝑠 𝑋𝑠 (d𝑥) 𝜈( 𝑓 ) 2 𝐻 (𝑥, d𝜈) 0 𝐸 ∫ {𝜈 (1) ≤1} h∫ 𝑇 ∫ i + 2P d𝑠 𝑋𝑠 (d𝑥) 𝜈(| 𝑓 |)𝐻 (𝑥, d𝜈) 0 𝐸 {𝜈 (1) >1} h∫ 𝑇 i +P 𝑋𝑠 (| 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 |)d𝑠 . 0

Then the desired inequality follows by simple estimates based on Corollary 2.28. □ Corollary 7.19 Suppose that 𝜈(1) 2 𝐻 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ and {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a càdlàg (𝜉, 𝜙)-superprocess satisfying P[𝑋0 (1)] < ∞. Then for every 𝑓 ∈ 𝐷 0 ( 𝐴), ∫ 𝑡 𝑀𝑡 ( 𝑓 ) = 𝑋𝑡 ( 𝑓 ) − 𝑋0 ( 𝑓 ) − 𝑋𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠 (7.33) 0

is a square integrable (𝒢𝑡 )-martingale with quadratic variation process ∫ 𝑡 ∫ ⟨𝑀 ( 𝑓 )⟩𝑡 = d𝑠 𝑞(𝑥, 𝑓 ) 𝑋𝑠 (d𝑥), 0

where 𝑞(𝑥, 𝑓 ) is defined by (2.64).

𝐸

(7.34)

7.2 Generators and Martingale Problems

181

Proof Since {𝑋𝑡 : 𝑡 ≥ 0} is a (𝜉, 𝜙)-superprocess, it satisfies the properties (1)–(5). In particular, from (3) one sees that (7.33) defines a local martingale with quadratic variation process (7.34). From (7.24) we see that 𝑡 ↦→ P[𝑋𝑡 (1)] is locally bounded, so {𝑀𝑡 ( 𝑓 )} is actually a square integral martingale. □ Corollary 7.20 Suppose that the branching mechanism 𝜙 has the special form with 𝐻 (𝑥, 𝑀 (𝐸) ◦ ) = 0 for all 𝑥 ∈ 𝐸. Then any càdlàg (𝜉, 𝜙)-superprocess {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a.s. continuous. Conversely, if {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a continuous process in 𝑀 (𝐸) and if for every 𝑓 ∈ 𝐷 0 ( 𝐴) the process {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0} defined by (7.33) is a (𝒢𝑡 )-local martingale with quadratic variation process ∫ 𝑡 ∫ 𝑐(𝑥) 𝑓 (𝑥) 2 𝑋𝑠 (d𝑥), (7.35) d𝑠 ⟨𝑀 ( 𝑓 )⟩𝑡 = 2 0

𝐸

then {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a (𝜉, 𝜙)-superprocess. Proof If {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a càdlàg (𝜉, 𝜙)-superprocess, by Theorem 7.16 it has property (3) with 𝐾 (𝜇, 𝑀 (𝐸) ◦ ) = 0. Then {𝑋𝑡 : 𝑡 ≥ 0} is a.s. continuous. This gives the first assertion. Conversely, if {𝑋𝑡 : 𝑡 ≥ 0} is continuous in 𝑀 (𝐸) and if for every 𝑓 ∈ 𝐷 0 ( 𝐴) the process in (7.33) is a local martingale with quadratic variation process given by (7.35), we can use Itô’s formula to obtain ∫ 𝑡 𝐺 (𝑋𝑡 ( 𝑓 )) = 𝐺 (𝑋0 ( 𝑓 )) + 𝐺 ′ (𝑋𝑠 ( 𝑓 )) 𝑋𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠 0 ∫ 𝑡 𝐺 ′′ (𝑋𝑠 ( 𝑓 )) 𝑋𝑠 (𝑐 𝑓 2 )d𝑠 + local mart. + 0

Then another application of Theorem 7.16 gives the second assertion.

□

The above property (4) means that the (𝜉, 𝜙)-superprocess is a solution of the martingale problem for the operator (𝐿 0 , 𝒟0 ). Then the generator of the (𝜉, 𝜙)-superprocess is the closure of (𝐿 0 , 𝒟0 ); see, e.g., Theorem 4.1 in Ethier and Kurtz (1986, p. 182). Under suitable assumptions, we can replace 𝐷 0 ( 𝐴) by a larger function class in Theorem 7.16 and its corollaries. In particular, if 𝑃𝑡 1 ∈ 𝐶 (𝐸) for every 𝑡 ≥ 0 and there exists a function 𝐴1 ∈ 𝐶 (𝐸) such that  1 𝑃𝑡 1(𝑥) − 1 = 𝐴1(𝑥), 𝑡→0 𝑡

lim

𝑥 ∈ 𝐸,

(7.36)

where the convergence is uniform, we can extend the operator 𝐴 to the linear span 𝐷 ( 𝐴) of 𝐷 0 ( 𝐴) and the constant functions. In this case, the results of Theorem 7.16 and its corollaries remain true with 𝐷 0 ( 𝐴) replaced by 𝐷 ( 𝐴). Of course, we have 𝐷 ( 𝐴) = 𝐷 0 ( 𝐴) if 𝐸 is a compact metric space. The martingale problems for the (𝜉, 𝜙)-superprocess can also be reformulated on the state space of tempered measures. Let ℎ ∈ 𝐷 0 ( 𝐴) be a strictly positive function satisfying 𝐴ℎ ≤ 𝛼ℎ for some constant 𝛼 > 0. From (6.12) we see that ℎ is an 𝛼excessive function for (𝑃𝑡 )𝑡 ≥0 . Let 𝐶ℎ (𝐸) be the set of continuous functions 𝑓 on 𝐸 satisfying | 𝑓 | ≤ const.·ℎ and let 𝐷 ℎ ( 𝐴) = { 𝑓 ∈ 𝐷 0 ( 𝐴)∩𝐶ℎ (𝐸) : 𝐴 𝑓 ∈ 𝐶ℎ (𝐸)}. Let

182

7 Martingale Problems of Superprocesses

𝑓 ↦→ 𝜙(·, 𝑓 ) be a branching mechanism given as in Section 6.1 with 𝜌 = ℎ. Suppose that 𝑓 ↦→ ℎ−1 𝜙(·, ℎ 𝑓 ) − 𝛼 𝑓 satisfies the conditions for the branching mechanism specified at the beginning of Section 7.1. Let 𝑀ℎ (𝐸) be the space of measures 𝜇 on 𝐸 satisfying 𝜇(ℎ) < ∞ and let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup on 𝑀ℎ (𝐸) defined by (6.6) and (6.11). The proof of the following theorem is similar to that of Theorem 7.16. Theorem 7.21 Let (Ω, 𝒢, 𝒢𝑡 , P) be a filtered probability space satisfying the usual hypotheses and let {𝑋𝑡 : 𝑡 ≥ 0} be a càdlàg process in 𝑀ℎ (𝐸) that is adapted to (𝒢𝑡 )𝑡 ≥0 and satisfies P[𝑋0 (ℎ)] < ∞. Then Theorem 7.16 still holds when 𝑀 (𝐸), 𝐶0 (𝐸) and 𝐷 0 ( 𝐴) are replaced by 𝑀ℎ (𝐸), 𝐶ℎ (𝐸) and 𝐷 ℎ ( 𝐴), respectively. If the semigroup ( 𝑃˜𝑡 )𝑡 ≥0 given by (6.10) has a Hunt realization, the (𝜉, 𝜙)-superprocess has a càdlàg realization in 𝑀ℎ (𝐸) by Theorem 6.3. The following theorem describes another way to construct a càdlàg realization of the (𝜉, 𝜙)-superprocess in 𝑀ℎ (𝐸). Theorem 7.22 For every 𝜇 ∈ 𝑀ℎ (𝐸) the (𝜉, 𝜙)-superprocess has a càdlàg realization {𝑋𝑡 : 𝑡 ≥ 0} in 𝑀ℎ (𝐸) with initial value 𝑋0 = 𝜇. the process by a series of càdlàg proProof This is based on a construction of Í∞ 𝜇𝑖 for a sequence of finite measures cesses. Given 𝜇 ∈ 𝑀ℎ (𝐸) we write 𝜇 = 𝑖=1 {𝜇𝑖 : 𝑖 = 1, 2, . . .} ⊂ 𝑀 (𝐸). Let {𝑋𝑖 (𝑡) : 𝑡 ≥ 0}, 𝑖 = 1, 2, . . ., be a sequence of independent càdlàg (𝜉, 𝜙)-superprocesses in 𝑀 (𝐸) with 𝑋𝑖 (0) = 𝜇𝑖 , 𝑖 = 1, 2, . . .. For 𝑛 ≥ 𝑘 ≥ 1 it is easy to see that 𝑍 𝑘,𝑛 (𝑡) =

𝑛 ∑︁

𝑋𝑖 (𝑡),

𝑡≥0

𝑖=𝑘

is a càdlàg realization of the (𝜉, 𝜙)-superprocess in 𝑀 (𝐸) with initial state 𝜇 𝑘,𝑛 := Í𝑛 −1 𝑖=𝑘 𝜇𝑖 . The result of Theorem 6.3 implies 𝑡 ↦→ ∥ℎ 𝜋 𝑡 ℎ∥ is a locally bounded function. Then, by a modification of the proof of Corollary 7.18, we have h i i h P sup ⟨𝑍 𝑘,𝑛 (𝑠), ℎ⟩ ≤ 𝐶 (𝑡, ℎ) ⟨𝜇 𝑘,𝑛 , ℎ⟩ + ⟨𝜇 𝑘,𝑛 , ℎ⟩ 1/2 , 0≤𝑠 ≤𝑡

where 𝑡 ↦→ 𝐶 (𝑡, ℎ) is a locally bounded function. The right-hand side tends to zero as 𝑘, 𝑛 → ∞. Then 𝑋𝑡 =

∞ ∑︁

𝑋𝑖 (𝑡),

𝑡≥0

𝑖=1

defines a càdlàg process in 𝑀ℎ (𝐸). This process is clearly a realization of the (𝜉, 𝜙)-superprocess with 𝑋0 = 𝜇. □ Recall that 𝐶 2 (R+ ) denotes the set of bounded continuous real functions on R+ with bounded continuous derivatives up to the second order. From Theorem 7.16 we derive immediately the following characterization of a CB-process.

7.3 Worthy Martingale Measures

183

Theorem 7.23 Suppose that {(𝑥(𝑡), 𝒢𝑡 ) : 𝑡 ≥ 0} is a positive càdlàg process such that P[𝑥(0)] < ∞. Then {(𝑥(𝑡), 𝒢𝑡 ) : 𝑡 ≥ 0} is a CB-process with branching mechanism given by (3.1) if and only if for every 𝑓 ∈ 𝐶 2 (R+ ) we have ∫ 𝑡 𝑓 (𝑥(𝑡)) = 𝑓 (𝑥(0)) + 𝐿 0 𝑓 (𝑥(𝑠))d𝑠 + local mart., (7.37) 0

where 𝐿 0 𝑓 (𝑥) = 𝑐𝑥 𝑓 ′′ (𝑥) − 𝑏𝑥 𝑓 ′ (𝑥) +

∫

∞

𝑥 [ 𝑓 (𝑥 + 𝑧) − 𝑓 (𝑥) − 𝑧 𝑓 ′ (𝑥)]𝑚(d𝑧).

0

By Corollary 7.17, if {(𝑥(𝑡), 𝒢𝑡 ) : 𝑡 ≥ 0} is a CB-process with branching mechanism given by (3.1), the local martingale in (7.37) is actually a martingale.

7.3 Worthy Martingale Measures In this section, we assume 𝐸 is a Lusin topological space. However, the results obtained here can obviously be modified to the case of a Lusin measurable space. Given a signed measure 𝐾 (d𝑠, d𝑥, d𝑦) on ℬ((0, ∞) × 𝐸 2 ) with the total variation |𝐾 |(d𝑠, d𝑥, d𝑦) satisfying |𝐾 |((0, 𝑇] × 𝐸 2 ) < ∞ for all 𝑇 ≥ 0, we define the bilinear form ∫ 𝑇∫ 𝑓 (𝑠, 𝑥)𝑔(𝑠, 𝑦)𝐾 (d𝑠, d𝑥, d𝑦), (7.38) ( 𝑓 , 𝑔) 𝐾 ,𝑇 = 0

𝐸2

where 𝑓 , 𝑔 ∈ 𝐵((0, 𝑇] × 𝐸). We say the signed measure is symmetric if ( 𝑓 , 𝑔) 𝐾 ,𝑇 = (𝑔, 𝑓 ) 𝐾 ,𝑇 for all 𝑇 ≥ 0 and 𝑓 , 𝑔 ∈ 𝐵((0, 𝑇] × 𝐸), and say 𝐾 (d𝑠, d𝑥, d𝑦) is positive definite if ( 𝑓 , 𝑓 ) 𝐾 ,𝑇 ≥ 0 for all 𝑇 ≥ 0 and 𝑓 ∈ 𝐵((0, 𝑇] × 𝐸). For a symmetric and positive definite signed measure 𝐾 (d𝑠, d𝑥, d𝑦), one shows by a standard argument the following Schwarz’s inequality 2 ( 𝑓 , 𝑔) 𝐾 ,𝑇 ≤ ( 𝑓 , 𝑓 ) 𝐾 ,𝑇 (𝑔, 𝑔) 𝐾 ,𝑇 .

(7.39)

In particular, these apply to random (positive) measures 𝐾 (d𝑠, d𝑥, d𝑦). Now let (Ω, 𝒢, 𝒢𝑡 , P) be a filtered probability space satisfying the usual hypotheses. Suppose that for each 𝐵 ∈ ℬ(𝐸) there exists a square-integrable càdlàg (𝒢𝑡 )-martingale {𝑀𝑡 (𝐵) : 𝑡 ≥ 0} satisfying 𝑀0 (𝐵) = 0. The system {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} is called a martingale measure on 𝐸 if for every 𝑡 ≥ 0 and every disjoint sequence {𝐵1 , 𝐵2 , . . .} ⊂ ℬ(𝐸) we have 𝑀𝑡

∞ Ø 𝑘=1

∞  ∑︁ 𝑀𝑡 (𝐵 𝑘 ) 𝐵𝑘 = 𝑘=1

184

7 Martingale Problems of Superprocesses

by the convergence in 𝐿 2 (Ω, P). A martingale measure {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} is said to be worthy if there is a random measure 𝐾 (d𝑠, d𝑥, d𝑦) on ℬ((0, ∞) × 𝐸 2 ) such that: (1) 𝐾 (d𝑠, d𝑥, d𝑦) is symmetric and positive definite; (2) 𝑡 ↦→ 𝐾 ((0, 𝑡] × 𝐴 × 𝐵) is predictable for all 𝐴, 𝐵 ∈ ℬ(𝐸) and   P 𝐾 ((0, 𝑡] × 𝐸 2 ) < ∞, 𝑡 ≥ 0;

(7.40)

(3) for every 𝑡 ≥ 𝑠 ≥ 0 and 𝐴, 𝐵 ∈ ℬ(𝐸) we have |⟨𝑀 ( 𝐴), 𝑀 (𝐵)⟩𝑡 − ⟨𝑀 ( 𝐴), 𝑀 (𝐵)⟩𝑠 | ≤ 𝐾 ((𝑠, 𝑡] × 𝐴 × 𝐵).

(7.41)

In this case, we call 𝐾 (d𝑠, d𝑥, d𝑦) the dominating measure of {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)}. Let ℛ denote the semi-algebra consisting of rectangles on (0, ∞) × 𝐸 2 of the form (𝑠, 𝑡] × 𝐴 × 𝐵 for 𝑡 ≥ 𝑠 ≥ 0 and 𝐴, 𝐵 ∈ ℬ(𝐸). Given a worthy martingale measure {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} with dominating measure 𝐾 (d𝑠, d𝑥, d𝑦), we define a random set function 𝜂(·) on ℛ by 𝜂((𝑠, 𝑡] × 𝐴 × 𝐵) = ⟨𝑀 ( 𝐴), 𝑀 (𝐵)⟩𝑡 − ⟨𝑀 ( 𝐴), 𝑀 (𝐵)⟩𝑠 . Since the 𝜎-algebra ℬ(𝐸) is separable, we can extend 𝜂(·) to a random signed measure 𝜂(d𝑠, d𝑥, d𝑦) on ℬ((0, ∞)×𝐸 2 ) with total variation dominated by 𝐾 (d𝑠, d𝑥, d𝑦). It is easy to see that 𝜂(d𝑠, d𝑥, d𝑦) is symmetric and positive definite. We refer to 𝜂(d𝑠, d𝑥, d𝑦) as the covariance measure of {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)}. We say a martingale measure {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} is orthogonal if its covariance measure 𝜂(d𝑠, d𝑥, d𝑦) is a.s. carried by [0, ∞) × Δ(𝐸), where Δ(𝐸) = {(𝑥, 𝑥) : 𝑥 ∈ 𝐸 }. In this case, we let 𝜂(d𝑠, d𝑥) denote the image of 𝜂(d𝑠, d𝑥, d𝑦) induced by the mapping (𝑠, 𝑥, 𝑦) ↦→ (𝑠, 𝑥) and also call 𝜂(d𝑠, d𝑥) the covariance measure or intensity of {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)}. An orthogonal martingale measure is called a time–space Gaussian white noise if the one-dimensional process {𝑀𝑡 (𝐵) : 𝑡 ≥ 0} has Gaussian and independent increments for every 𝐵 ∈ ℬ(𝐸). Proposition 7.24 A worthy martingale measure {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} is orthogonal if and only if {𝑀𝑡 ( 𝐴) : 𝑡 ≥ 0} and {𝑀𝑡 (𝐵) : 𝑡 ≥ 0} are orthogonal martingales whenever 𝐴 and 𝐵 ∈ ℬ(𝐸) are disjoint. Proof Suppose that {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} is orthogonal and 𝐴, 𝐵 ∈ ℬ(𝐸) are disjoint sets. Then ⟨𝑀 ( 𝐴), 𝑀 (𝐵)⟩𝑡 = 𝜂((0, 𝑡] × 𝐴 × 𝐵) vanishes, so {𝑀𝑡 ( 𝐴) : 𝑡 ≥ 0} and {𝑀𝑡 (𝐵) : 𝑡 ≥ 0} are orthogonal martingales. Conversely, suppose that {𝑀𝑡 ( 𝐴) : 𝑡 ≥ 0} and {𝑀𝑡 (𝐵) : 𝑡 ≥ 0} are orthogonal whenever 𝐴 and 𝐵 ∈ ℬ(𝐸) are disjoint. Then 𝜂((0, 𝑡] × 𝐴 × 𝐵) = ⟨𝑀 ( 𝐴), 𝑀 (𝐵)⟩𝑡 vanishes when 𝐴 and 𝐵 ∈ ℬ(𝐸) are disjoint, and hence 𝜂(d𝑠, d𝑥, d𝑦) is carried by [0, ∞) × Δ(𝐸). □ Let ℒ be a linear space of Borel functions on 𝐸. Suppose that for each 𝑓 ∈ ℒ there is a square-integrable càdlàg (𝒢𝑡 )-martingale {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0} satisfying 𝑀0 ( 𝑓 ) = 0. The family {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ ℒ} is called a martingale functional if for every 𝑡 ≥ 0 the following properties hold:

7.3 Worthy Martingale Measures

185

(1) For each 𝑐 ∈ R and each 𝑓 ∈ ℒ we have a.s. 𝑀𝑡 (𝑐 𝑓 ) = 𝑐𝑀𝑡 ( 𝑓 ). Í (2) If 𝑓 , 𝑓1 , 𝑓2 , . . . ∈ ℒ and 𝑓 = ∞ 𝑘=1 𝑓 𝑘 by bounded pointwise convergence, then 𝑀𝑡 ( 𝑓 ) =

∞ ∑︁

𝑀𝑡 ( 𝑓 𝑘 )

𝑘=1

by the convergence in 𝐿 2 (Ω, P).

Proposition 7.25 For each worthy martingale measure {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} there is a martingale functional {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ 𝐵(𝐸)} such that 𝑀𝑡 (1 𝐵 ) = 𝑀𝑡 (𝐵) a.s. for every 𝑡 ≥ 0 and every 𝐵 ∈ ℬ(𝐸). Moreover, for any 𝑓 ∈ 𝐵(𝐸) the (𝒢𝑡 )-martingale {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0} has quadratic variation process ∫ 𝑡∫ (7.42) ( 𝑓 , 𝑓 ) 𝜂,𝑡 = 𝑓 (𝑥) 𝑓 (𝑦)𝜂(d𝑠, d𝑥, d𝑦). 𝐸2

0

Proof We shall give an explicit construction of the martingale functional {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ 𝐵(𝐸)}. If 𝑓 ∈ 𝐵(𝐸) is a simple function given by 𝑓 (𝑥) =

𝑛 ∑︁

𝑏 𝑖 1 𝐵𝑖 (𝑥),

𝑥 ∈ 𝐸,

𝑖=1

where 𝑏 𝑖 ∈ R and 𝐵𝑖 ∈ ℬ(𝐸) for 𝑖 = 1, . . . , 𝑛, we define 𝑀𝑡 ( 𝑓 ) =

𝑛 ∑︁

𝑏 𝑖 𝑀𝑡 (𝐵𝑖 ),

𝑡 ≥ 0.

𝑖=1

It is easy to see that {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0} is a càdlàg martingale with quadratic variation process given by (7.42). For a general function 𝑓 ∈ 𝐵(𝐸) let { 𝑓 𝑘 } be a sequence of simple functions on 𝐸 such that ∥ 𝑓 𝑘 − 𝑓 ∥ → 0 as 𝑘 → ∞. By Doob’s martingale inequality and the definition of the worthy martingale measure, i h   P sup |𝑀𝑠 ( 𝑓 𝑘 ) − 𝑀𝑠 ( 𝑓 𝑗 )| 2 ≤ 4P ( 𝑓 𝑘 − 𝑓 𝑗 , 𝑓 𝑘 − 𝑓 𝑗 ) 𝜂,𝑡 0≤𝑠 ≤𝑡   ≤ 4∥ 𝑓 𝑘 − 𝑓 𝑗 ∥ 2 P 𝐾 ( [0, 𝑡] × 𝐸 2 ) for any 𝑘 ≥ 𝑗 ≥ 1. Then there is a square-integrable càdlàg (𝒢𝑡 )-martingale {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0} independent of the choice of { 𝑓 𝑘 } such that h i lim P sup |𝑀𝑠 ( 𝑓 𝑘 ) − 𝑀𝑠 ( 𝑓 )| 2 = 0, 𝑡 ≥ 0. 𝑘→∞

0≤𝑠 ≤𝑡

Since (7.42) holds when 𝑓 is replaced by 𝑓 𝑘 , for any 𝑡 ≥ 𝑟 ≥ 0 we have   E 𝑀𝑡 ( 𝑓 𝑘 ) 2 − 𝑀𝑟 ( 𝑓 𝑘 ) 2 − ( 𝑓 𝑘 , 𝑓 𝑘 ) 𝜂,𝑡 + ( 𝑓 𝑘 , 𝑓 𝑘 ) 𝜂,𝑟 = 0.

186

7 Martingale Problems of Superprocesses

Then we can let 𝑘 → ∞ to see {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0} has quadratic variation process (7.42). It is easy to show that {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ 𝐵(𝐸)} satisfies the two properties in the definition of a martingale functional. □ Now suppose we are given a worthy martingale measure {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} on 𝐸 with covariance and dominating measures 𝜂(d𝑠, d𝑥, d𝑦) and 𝐾 (d𝑠, d𝑥, d𝑦), respectively. The martingale functional {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ 𝐵(𝐸)} given in Proposition 7.25 is clearly unique in the following sense: If {𝑍𝑡 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ 𝐵(𝐸)} is also a martingale functional such that 𝑍𝑡 (1 𝐵 ) = 𝑀𝑡 (𝐵) a.s. for every 𝑡 ≥ 0 and every 𝐵 ∈ ℬ(𝐸), then 𝑀𝑡 ( 𝑓 ) = 𝑍𝑡 ( 𝑓 ) a.s. for every 𝑡 ≥ 0 and every 𝑓 ∈ 𝐵(𝐸). A real-valued two-parameter process {ℎ 𝑠 (𝑥) : 𝑠 ≥ 0, 𝑥 ∈ 𝐸 } is said to be progressive if for every 𝑡 ≥ 0 the mapping (𝜔, 𝑠, 𝑥) ↦→ ℎ 𝑠 (𝜔, 𝑥) restricted to Ω × [0, 𝑡] × 𝐸 is measurable relative to 𝒢𝑡 × ℬ( [0, 𝑡] × 𝐸). Let 𝒫 = 𝒫(𝒢𝑡 ) denote the 𝜎-algebra on Ω × [0, ∞) generated by all real-valued left continuous processes adapted to (𝒢𝑡 ). A two-parameter process {ℎ 𝑠 (𝑥) : 𝑠 ≥ 0, 𝑥 ∈ 𝐸 } is said to be predictable if the mapping (𝜔, 𝑠, 𝑥) ↦→ ℎ 𝑠 (𝜔, 𝑥) is (𝒫×ℬ(𝐸))-measurable. Let ℒ𝐾2 (𝐸) be the space of two-parameter predictable processes ℎ = {ℎ 𝑠 (𝑥) : 𝑠 ≥ 0, 𝑥 ∈ 𝐸 } satisfying  1/2   < ∞, ∥ℎ∥ 𝐾 ,𝑇 := P (|ℎ|, |ℎ|) 𝐾 ,𝑇

𝑇 ≥ 0.

(7.43)

It is easy to show that each ∥ · ∥ 𝐾 ,𝑇 is a seminorm on ℒ𝐾2 (𝐸). We identify ℎ1 and ℎ2 ∈ ℒ𝐾2 (𝐸) if ∥ℎ1 − ℎ2 ∥ 𝐾 ,𝑇 = 0 for every 𝑇 ≥ 1. Then 𝑑2 (ℎ1 , ℎ2 ) =

∞ ∑︁ 1 (1 ∧ ∥ℎ1 − ℎ2 ∥ 𝐾 ,𝑛 ) 2𝑛 𝑛=1

(7.44)

defines a metric on ℒ𝐾2 (𝐸). We call {𝑞 𝑠 (𝑥) : 𝑠 ≥ 0, 𝑥 ∈ 𝐸 } a step process if it is of the form 𝑞 𝑠 (𝑥) = 𝑔0 (𝑥)1 {0} (𝑠) +

∞ ∑︁

𝑔𝑖 (𝑥)1 (𝑟𝑖 ,𝑟𝑖+1 ] (𝑠),

(7.45)

𝑖=0

where each (𝜔, 𝑥) ↦→ 𝑔𝑖 (𝜔, 𝑥) is a (𝒢𝑟𝑖 × ℬ(𝐸))-measurable function and {0 = 𝑟 0 < 𝑟 1 < 𝑟 2 < · · · } is a sequence increasing to infinity. Clearly, a step process is predictable and 𝒫 × ℬ(𝐸) is generated by the collection of step processes. Let ℒ𝐾0 (𝐸) be the set of step processes in ℒ𝐾2 (𝐸). Proposition 7.26 The metric space (ℒ𝐾2 (𝐸), 𝑑2 ) is complete and ℒ𝐾0 (𝐸) is a dense subset of ℒ𝐾2 (𝐸). Proof Suppose that {ℎ 𝑘 } is a Cauchy sequence in ℒ𝐾2 (𝐸). Then for any fixed 𝑛 ≥ 1 the restrictions of {ℎ 𝑘 } to Ω × [0, 𝑛] × 𝐸 form a Cauchy sequence with respect to the seminorm ∥ · ∥ 𝐾 ,𝑛 defined by (7.43). It is easily seen that Q𝑛 (d𝜔, d𝑠, d𝑥, d𝑦) = P(d𝜔)𝐾 (𝜔, d𝑠, d𝑥, d𝑦)

7.3 Worthy Martingale Measures

187

defines a finite measure on 𝒢 × ℬ((0, 𝑛] × 𝐸 2 ). For any 𝜀 > 0 and 𝑗, 𝑘 ≥ 1 we have Q𝑛 {(𝜔, 𝑠, 𝑥, 𝑦) : |ℎ 𝑗 (𝜔, 𝑠, 𝑥) − ℎ 𝑘 (𝜔, 𝑠, 𝑥)| ≥ 𝜀}  ∫ 𝑛∫ 1 ≤ P |ℎ 𝑗 (𝑠, 𝑥) − ℎ 𝑘 (𝑠, 𝑥)|𝐾 (d𝑠, d𝑥, d𝑦) 𝜀 0 𝐸2 h
1/2
1/2 i 1 ≤ P |ℎ 𝑗 − ℎ 𝑘 |, |ℎ 𝑗 − ℎ 𝑘 | 𝐾 ,𝑛 𝐾 (0, 𝑛] × 𝐸 2 𝜀 
 1/2 1 ≤ ∥ℎ 𝑗 − ℎ 𝑘 ∥ 𝐾 ,𝑛 P 𝐾 (0, 𝑛] × 𝐸 2 , 𝜀 where the second inequality follows from (7.39). Then for each 𝑖 ≥ 1 we can choose 𝑙𝑖 ≥ 1 so that  Q𝑛 (𝜔, 𝑠, 𝑥, 𝑦) : |ℎ 𝑗 (𝜔, 𝑠, 𝑥) − ℎ 𝑘 (𝜔, 𝑠, 𝑥)| ≥ 1/2𝑖 < 1/2𝑖 for all 𝑗, 𝑘 ≥ 𝑙𝑖 . In addition, we can assume 𝑙𝑖 → ∞ increasingly as 𝑖 → ∞. Let  𝐹𝑖 = (𝜔, 𝑠, 𝑥, 𝑦) : |ℎ𝑙𝑖 (𝜔, 𝑠, 𝑥) − ℎ𝑙𝑖+1 (𝜔, 𝑠, 𝑥)| ≥ 1/2𝑖 ∞ and 𝑁1 = ∩∞ 𝑚=1 ∪𝑖=𝑚 𝐹𝑖 . We have

Q𝑛

∞ Ø

∞  ∑︁ 1 1 = 𝑚−1 𝐹𝑖 ≤ 𝑖 2 2 𝑖=𝑚 𝑖=𝑚

and hence Q𝑛 (𝑁1 ) = 0. For any (𝜔, 𝑠, 𝑥, 𝑦) ∈ 𝑁1𝑐 there is some 𝑚 ≥ 1 such that |ℎ𝑙𝑖 (𝜔, 𝑠, 𝑥) − ℎ𝑙𝑖+1 (𝜔, 𝑠, 𝑥)| < 1/2𝑖 for all 𝑖 ≥ 𝑚, and hence {ℎ𝑙𝑖 (𝜔, 𝑠, 𝑥)} is a Cauchy sequence. Now define the predictable process ℎ(𝜔, 𝑠, 𝑥) = lim sup ℎ𝑙𝑖 (𝜔, 𝑠, 𝑥),

𝜔 ∈ Ω, 𝑠 ≥ 0, 𝑥 ∈ 𝐸 .

𝑖→∞

We have ℎ𝑙𝑖 (𝜔, 𝑠, 𝑥) → ℎ(𝜔, 𝑠, 𝑥) for all (𝜔, 𝑠, 𝑥, 𝑦) ∈ 𝑁1𝑐 . Let  𝑁 = (𝜔, 𝑠, 𝑥, 𝑦) ∈ Ω × (0, 𝑛] × 𝐸 2 : (𝜔, 𝑠, 𝑥, 𝑦) or (𝜔, 𝑠, 𝑦, 𝑥) ∈ 𝑁1 . Then Q𝑛 (𝑁) = 0 by the symmetry of 𝐾 (d𝑠, d𝑥, d𝑦). Moreover, ℎ𝑙𝑖 (𝜔, 𝑠, 𝑥) → ℎ(𝜔, 𝑠, 𝑥) and ℎ𝑙𝑖 (𝜔, 𝑠, 𝑦) → ℎ(𝜔, 𝑠, 𝑦) for all (𝜔, 𝑠, 𝑥, 𝑦) ∈ 𝑁 𝑐 . For any 𝜀 > 0 let 𝑚(𝜀) ≥ 1 be such that ∥ℎ 𝑘 − ℎ 𝑗 ∥ 𝐾 ,𝑛 ≤ 𝜀 for 𝑗, 𝑘 ≥ 𝑚(𝜀). Letting 𝑗 → ∞ along the sequence {𝑙𝑖 } and applying Fatou’s lemma we see ∥ℎ 𝑘 − ℎ∥ 𝐾 ,𝑛 ≤ 𝜀 for 𝑘 ≥ 𝑚(𝜀). Thus ∥ℎ 𝑘 − ℎ∥ 𝐾 ,𝑛 → 0 as 𝑘 → ∞. Since 𝑛 ≥ 1 was arbitrary, it is easy to define a process ℎ ∈ ℒ𝐾2 (𝐸) so that ℎ 𝑘 → ℎ relative to the metric defined by (7.44). This gives the first assertion of the proposition. We next prove the second assertion. Let ℒ = {ℎ ∈ ℒ𝐾2 (𝐸) : there exists {𝑞 𝑘 } ⊂ ℒ𝐾0 (𝐸) such that 𝑑2 (ℎ, 𝑞 𝑘 ) → 0 as 𝑘 → ∞}. It is easy to see that ℒ is a vector space. Suppose that {ℎ 𝑘 } is a bounded and increasing sequence of positive elements of ℒ such that ℎ 𝑘 → ℎ pointwise as 𝑘 → ∞. Then ℎ is a bounded positive two-parameter predictable process. In view of (7.43) and (7.44), we can use the dominated convergence theorem to see that 𝑑2 (ℎ, ℎ 𝑘 ) → 0 as 𝑘 → ∞. For each 𝑘 ≥ 1 choose 𝑞 𝑘 ∈ ℒ𝐾0 (𝐸) such that

188

7 Martingale Problems of Superprocesses

𝑑2 (ℎ 𝑘 , 𝑞 𝑘 ) ≤ 1/𝑘. Then 𝑑2 (ℎ, 𝑞 𝑘 ) → 0 as 𝑘 → ∞, and so ℎ ∈ ℒ. This shows ℒ is a monotone vector space. Since ℒ ⊃ ℒ𝐾0 (𝐸) and 𝜎(ℒ𝐾0 (𝐸)) = 𝒫 × ℬ(𝐸), by Proposition A.1 we have ℒ ⊃ b(𝒫×ℬ(𝐸)), and so ℒ𝐾0 (𝐸) is dense in b(𝒫×ℬ(𝐸)). Now consider ℎ ∈ ℒ𝐾2 (𝐸). For any 𝑘 ≥ 1 define ℎ 𝑘 ∈ b(𝒫×ℬ(𝐸)) by ℎ 𝑘 (𝜔, 𝑠, 𝑥) = ℎ(𝜔, 𝑠, 𝑥)1 { |ℎ( 𝜔,𝑠, 𝑥) | ≤𝑘 } . Clearly, we have  ∫ 𝑛∫ |ℎ(𝑠, 𝑥)ℎ(𝑠, 𝑦)|1 { |ℎ(𝑠, 𝑥) |>𝑘, |ℎ(𝑠,𝑦) |>𝑘 } 𝐾 (d𝑠, d𝑥, d𝑦) , ∥ℎ 𝑘 − ℎ∥ 2𝐾 ,𝑛 = P 0

𝐸2

which tends to zero as 𝑘 → ∞. Then b(𝒫 × ℬ(𝐸)) is dense in ℒ𝐾2 (𝐸), and so ℒ𝐾0 (𝐸) is dense in ℒ𝐾2 (𝐸). □ We are now ready to define the stochastic integrals of processes in ℒ𝐾2 (𝐸) with respect to the martingale measure {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)}. For a step process 𝑞 ∈ ℒ𝐾0 (𝐸) given by (7.45), each 𝑔𝑖 is a deterministic Borel function on 𝐸 under the conditional probability P{·|𝒢𝑟𝑖 }. Then we can use the martingale functional induced by {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} to define the process {𝑀𝑟𝑖+1 ∧𝑡 (𝑔𝑖 ) − 𝑀𝑟𝑖 ∧𝑡 (𝑔𝑖 ) : 𝑡 ≥ 0}, which is a square-integrable càdlàg (𝒢𝑡 )-martingale first under P{·|𝒢𝑟𝑖 } and then under P. It follows that 𝑀𝑡 (𝑞 𝑡 ) =

∞ ∑︁ 

 𝑀𝑟𝑖+1 ∧𝑡 (𝑔𝑖 ) − 𝑀𝑟𝑖 ∧𝑡 (𝑔𝑖 ) ,

𝑡≥0

𝑖=0

is a square-integrable càdlàg (𝒢𝑡 )-martingale. The quadratic variation process of {𝑀𝑡 (𝑞 𝑡 ) : 𝑡 ≥ 0} is clearly given by ∫ 𝑡∫ ⟨𝑀 (𝑞)⟩𝑡 = 𝑞 𝑠 (𝑥)𝑞 𝑠 (𝑦)𝜂(d𝑠, d𝑥, d𝑦). 0

𝐸2

For a general process ℎ ∈ ℒ𝐾2 (𝐸), choose a sequence {𝑞 𝑘 } ⊂ ℒ𝐾0 (𝐸) such that 𝑑2 (𝑞 𝑘 , ℎ) → 0 as 𝑘 → ∞. By Doob’s martingale inequality, h i   P sup 𝑀𝑠 (𝑞 𝑘 (𝑠) − 𝑞 𝑗 (𝑠)) 2 ≤ 4P (𝑞 𝑘 − 𝑞 𝑗 , 𝑞 𝑘 − 𝑞 𝑗 ) 𝜂,𝑛 0≤𝑠 ≤𝑛

≤ 4∥𝑞 𝑘 − 𝑞 𝑗 ∥ 2𝐾 ,𝑛 , which tends to zero as 𝑗, 𝑘 → ∞. Then there is a square-integrable càdlàg (𝒢𝑡 )-martingale {𝑀𝑡 (ℎ𝑡 ) : 𝑡 ≥ 0} such that h 2 i 𝑛 ≥ 1. lim P sup 𝑀𝑠 (𝑞 𝑘 (𝑠)) − 𝑀𝑠 (ℎ 𝑠 ) = 0, 𝑘→∞

0≤𝑠 ≤𝑛

It is easy to see that {𝑀𝑡 (ℎ𝑡 ) : 𝑡 ≥ 0} has quadratic variation process ∫ 𝑡∫ ℎ 𝑠 (𝑥)ℎ 𝑠 (𝑦)𝜂(d𝑠, d𝑥, d𝑦). (ℎ, ℎ) 𝜂,𝑡 = 0

𝐸2

(7.46)

7.3 Worthy Martingale Measures

189

We shall write ∫

𝑡

∫

𝑀𝑡 (ℎ𝑡 ) =

ℎ 𝑠 (𝑥) 𝑀 (d𝑠, d𝑥) 0

𝐸

and call it the stochastic integral of ℎ ∈ ℒ𝐾2 (𝐸) with respect to {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)}. We next prove an important property of stochastic integrals with respect to the martingale measure. Suppose that 𝐹 is another Lusin topological space and 𝜆 is a finite Borel measure on 𝐹. Let {ℎ(𝑠, 𝑥, 𝑧) : 𝑠 ≥ 0, 𝑥 ∈ 𝐸, 𝑧 ∈ 𝐹} be a predictable process satisfying ∫  ∫ 𝑛∫ 𝜆(d𝑧) (7.47) P |ℎ(𝑠, 𝑥, 𝑧)ℎ(𝑠, 𝑦, 𝑧)|𝐾 (d𝑠, d𝑥, d𝑦) < ∞ 𝐸2

0

𝐹

for every 𝑛 ≥ 1. The above condition implies ∫ 𝑛∫  P |ℎ(𝑠, 𝑥, 𝑧)ℎ(𝑠, 𝑦, 𝑧)|𝐾 (d𝑠, d𝑥, d𝑦) < ∞ 𝐸2

0

for 𝜆-a.e. 𝑧 ∈ 𝐹. Then the stochastic integral ∫ 𝑡∫ 𝑀𝑡 (𝑧) := ℎ(𝑠, 𝑥, 𝑧) 𝑀 (d𝑠, d𝑥) 0

(7.48)

𝐸

is well-defined for 𝜆-a.e. 𝑧 ∈ 𝐹. On the other hand, using (7.39) we have  ∫
P |ℎ(𝑧1 )|, |ℎ(𝑧 2 )| 𝐾 ,𝑛 𝜆(d𝑧1 )𝜆(d𝑧2 ) 𝐹2 ∫ h
1/2 i
1/2 P |ℎ(𝑧 1 )|, |ℎ(𝑧1 )| 𝐾 ,𝑛 |ℎ(𝑧 2 )|, |ℎ(𝑧2 )| 𝐾 ,𝑛 𝜆(d𝑧 1 )𝜆(d𝑧 2 ) ≤ ∫𝐹 2n 
 o 1/2 ≤ P |ℎ(𝑧1 )|, |ℎ(𝑧 1 )| 𝐾 ,𝑛 𝜆(d𝑧 1 ) 𝐹 ∫ n 
 o 1/2 · P |ℎ(𝑧2 )|, ℎ(𝑧2 )| 𝐾 ,𝑛 𝜆(d𝑧2 ) 𝐹 ∫ 
 ≤ 𝜆(1) (7.49) P |ℎ(𝑧)|, |ℎ(𝑧)| 𝐾 ,𝑛 𝜆(d𝑧). 𝐹

The right-hand side is finite by (7.47). It follows that ∫ 𝐻 (𝑠, 𝑥) := ℎ(𝑠, 𝑥, 𝑧)𝜆(d𝑧), 𝑠 ≥ 0, 𝑥 ∈ 𝐸 𝐹

defines a predictable process 𝐻 ∈ ℒ𝐾2 (𝐸). Therefore ∫

𝑡

∫ 𝐻 (𝑠, 𝑥)𝑀 (d𝑠, d𝑥)

𝑡 ↦→ 0

𝐸

(7.50)

190

7 Martingale Problems of Superprocesses

is well-defined as a square-integrable càdlàg martingale. From (7.50) and (7.48) it is natural to expect ∫ ∫ 𝑡∫ (7.51) 𝑀𝑡 (𝑧)𝜆(d𝑧) = 𝐻 (𝑠, 𝑥)𝑀 (d𝑠, d𝑥). 0

𝐹

𝐸

The above formula is called a stochastic Fubini’s theorem for the martingale measure, which means that (𝜔, 𝑧) ↦→ 𝑀𝑡 (𝜔, 𝑧) has a (𝒢𝑡 × ℬ(𝐹))-measurable version and the equality holds with probability one. To establish the formula rigorously we first prove the following: Lemma 7.27 Let ℎ and ℎ 𝑘 be predictable processes satisfying condition (7.47). Suppose that (7.51) holds for every ℎ 𝑘 and ∫ 
 P |ℎ 𝑘 (𝑧) − ℎ(𝑧)|, |ℎ 𝑘 (𝑧) − ℎ(𝑧)| 𝐾 ,𝑛 𝜆(d𝑧) → 0 (7.52) 𝐹

as 𝑘 → ∞ for every 𝑛 ≥ 1. Then (7.51) also holds for the process ℎ. Proof Step 1. Let 𝑀𝑘 (𝑡, 𝑧) be defined by the right-hand side of (7.48) with ℎ replaced by ℎ 𝑘 . Since (7.51) holds for ℎ 𝑘 , the function (𝜔, 𝑧) ↦→ 𝑀𝑘 (𝜔, 𝑡, 𝑧) has a (𝒢𝑡 × ℬ(𝐹))-measurable version. By (7.52) it is easy to show that ∫  2 P |𝑀𝑘 (𝑡, 𝑧) − 𝑀 𝑗 (𝑡, 𝑧)| 𝜆(d𝑧) → 0 𝐹

as 𝑗, 𝑘 → ∞. Then there is a (𝒢𝑡 × ℬ(𝐹))-measurable function (𝜔, 𝑧) ↦→ 𝑁𝑡 (𝜔, 𝑧) such that ∫  2 (7.53) |𝑀𝑘 (𝑡, 𝑧) − 𝑁𝑡 (𝑧)| 𝜆(d𝑧) → 0, P 𝐹

and hence ∫

∫ 𝑀𝑘 (𝑡, 𝑧)𝜆(d𝑧) → 𝐹

𝑁𝑡 (𝑧)𝜆(d𝑧)

(7.54)

𝐹

in 𝐿 2 (Ω, P) because 𝜆 is a finite measure. By (7.53) we can choose a sequence {𝑘 𝑖 } such that 𝑀𝑘𝑖 (𝑡, 𝑧) → 𝑁𝑡 (𝑧) in 𝐿 2 (Ω, P) for 𝜆-a.e. 𝑧 ∈ 𝐹. On the other hand, by (7.52) there is a subsequence {𝑘 𝑖′ } ⊂ {𝑘 𝑖 } such that ∥ℎ 𝑘𝑖′ (·, ·, 𝑧) − ℎ(·, ·, 𝑧) ∥ 𝐾 ,𝑛 → 0 for every 𝑛 ≥ 1 and 𝜆-a.e. 𝑧 ∈ 𝐹. Then 𝑀𝑘𝑖′ (𝑡, 𝑧) → 𝑀𝑡 (𝑧) in 𝐿 2 (Ω, P) for 𝜆-a.e. 𝑧 ∈ 𝐹. It follows that 𝑀𝑡 (𝑧) = 𝑁𝑡 (𝑧) a.s. for 𝜆-a.e. 𝑧 ∈ 𝐹. Step 2. Let 𝐻 𝑘 be defined by the right-hand side of (7.50) with ℎ replaced by ℎ 𝑘 . From (7.52) and the calculations in (7.49) we have ∥𝐻 𝑘 − 𝐻 ∥ 𝐾 ,𝑛 → 0 as 𝑘 → ∞ for every 𝑛 ≥ 1. It follows that ∫ 𝑡∫ ∫ 𝑡∫ (7.55) 𝐻 𝑘 (𝑠, 𝑥)𝑀 (d𝑠, d𝑥) → 𝐻 (𝑠, 𝑥) 𝑀 (d𝑠, d𝑥) 0

𝐸

0

𝐸

7.3 Worthy Martingale Measures

191

in 𝐿 2 (Ω, P) as 𝑘 → ∞. By the assumption of the lemma, ∫ ∫ 𝑡∫ 𝐻 𝑘 (𝑠, 𝑥)𝑀 (d𝑠, d𝑥). 𝑀𝑘 (𝑡, 𝑧)𝜆(d𝑧) = 0

𝐹

𝐸

This together with (7.54) and (7.55) yields a.s. ∫ 𝑡∫ ∫ 𝐻 (𝑠, 𝑥)𝑀 (d𝑠, d𝑥), 𝑁𝑡 (𝑧)𝜆(d𝑧) = 0

𝐹

which is just what (7.51) means.

𝐸

□

Theorem 7.28 Let {ℎ(𝑠, 𝑥, 𝑧) : 𝑠 ≥ 0, 𝑥 ∈ 𝐸, 𝑧 ∈ 𝐹} be a predictable process satisfying (7.47). Let {𝑀𝑡 (𝑧) : 𝑡 ≥ 0, 𝑧 ∈ 𝐹} and {𝐻 (𝑠, 𝑥) : 𝑠 ≥ 0, 𝑥 ∈ 𝐸 } be defined by (7.48) and (7.50), respectively. Then (7.51) holds a.s. for every 𝑡 ≥ 0.

Proof Let ℋ be the class of bounded predictable process ℎ = {ℎ(𝑠, 𝑥, 𝑧) : 𝑠 ≥ 0, 𝑥 ∈ 𝐸, 𝑧 ∈ 𝐹} for which the theorem holds. Then Lemma 7.27 implies that ℋ is closed under bounded pointwise convergence. If ℎ(𝜔, 𝑠, 𝑥, 𝑧) = 𝑞(𝜔, 𝑠, 𝑥) 𝑓 (𝑧) for a bounded (𝒫 × ℬ(𝐸))-measurable function 𝑞 on Ω × [0, ∞) × 𝐸 and a bounded ℬ(𝐹)-measurable function 𝑓 on 𝐹, then (7.51) holds clearly. By Proposition A.1, the class ℋ contains all bounded (𝒫 × ℬ(𝐸) × ℬ(𝐹))-measurable processes. For a general predictable process ℎ = {ℎ(𝑠, 𝑥, 𝑧) : 𝑠 ≥ 0, 𝑥 ∈ 𝐸, 𝑧 ∈ 𝐹} satisfying (7.47), the result follows from Lemma 7.27 by an approximation using the sequence defined □ by ℎ 𝑘 = ℎ1 { |ℎ | ≤𝑘 } . The worthy martingale measure defined above is finite. Let ℎ ∈ pℬ(𝐸) be a strictly positive function. Recall that 𝐵 ℎ (𝐸) denotes the set of functions 𝑓 ∈ ℬ(𝐸) such that | 𝑓 | ≤ const. · ℎ. Instead of (7.40) one can also consider a dominating measure 𝐾 (d𝑠, d𝑥, d𝑦) satisfying ∫ 𝑡 ∫  ℎ(𝑥)ℎ(𝑦)𝐾 (d𝑠, d𝑥, d𝑦) < ∞, 𝑡 ≥ 0. P 0

𝐸2

Let 𝐸 𝑛 = {𝑥 ∈ 𝐸 : ℎ(𝑥) ≥ 1/𝑛} and let ℬℎ (𝐸) = {𝐵 ∈ ℬ(𝐸) : 𝐵 ⊂ 𝐸 𝑛 for some 𝑛 ≥ 1}. A family of square-integrable càdlàg (𝒢𝑡 )-martingales {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬℎ (𝐸)} is called a 𝜎-finite worthy martingale measure on 𝐸 if (7.41) holds for all 𝑡 ≥ 𝑠 ≥ 0 and 𝐴, 𝐵 ∈ ℬℎ (𝐸). Following the proof of Proposition 7.25, we can extend the 𝜎-finite worthy martingale measure to a martingale functional {𝑀𝑡 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ 𝐵 ℎ (𝐸)}. All the results in this section can be reformulated for 𝜎-finite worthy martingale measures.

192

7 Martingale Problems of Superprocesses

7.4 A Stochastic Convolution Formula Suppose that 𝐸 is a locally compact separable metric space. Let 𝜉 be a Hunt process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 and let 𝜙 be a branching mechanism given by (2.29) or (2.30). We assume that (𝑃𝑡 )𝑡 ≥0 and 𝜙 satisfy the conditions specified at the beginning of Section 7.1. Suppose that {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a càdlàg realization of the (𝜉, 𝜙)-superprocess with P[𝑋0 (1)] < ∞, where the filtration satisfies the usual hypotheses. Proposition 7.29 The continuous martingale functional {𝑀𝑡𝑐 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ 𝐷 0 ( 𝐴)} defined by (7.20) induces a continuous orthogonal martingale measure with covariance measure 𝜂 𝑐 (d𝑠, d𝑥) = 2𝑐(𝑥)d𝑠𝑋𝑠 (d𝑥). Proof We first note that (7.20) indeed defines a continuous martingale functional {𝑀𝑡𝑐 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ 𝐷 0 ( 𝐴)}. For each 𝑛 ≥ 1 define the measure 𝜇 𝑛 ∈ 𝑀 (𝐸) by ∫ 𝑛 ∫  𝜇 𝑛 ( 𝑓 ) = 2P d𝑠 𝑐(𝑥) 𝑓 (𝑥) 𝑋𝑠 (d𝑥) , 𝑓 ∈ 𝐵(𝐸). 0

𝐸

It is well known that 𝐶0 (𝐸) is dense in 𝐿 2 (𝜇 𝑛 ); see, e.g., Hewitt and Stromberg (1965, p. 197). Since 𝐷 0 ( 𝐴) is dense in 𝐶0 (𝐸) in the supremum norm, it is also dense in 𝐿 2 (𝜇 𝑛 ). Consequently, for any 𝑓 ∈ 𝐵(𝐸) there is a sequence { 𝑓 𝑘 } ⊂ 𝐷 0 ( 𝐴) such that lim 𝑘→∞ 𝜇 𝑛 (| 𝑓 𝑘 − 𝑓 | 2 ) = 0 for every 𝑛 ≥ 1. By Theorem 7.16 and Doob’s martingale inequality,  ∫ 𝑛 ∫ i h P sup 𝑀𝑠𝑐 ( 𝑓 𝑘 − 𝑓 𝑗 ) 2 ≤ 4P 2𝑐(𝑥)| 𝑓 𝑘 (𝑥) − 𝑓 𝑗 (𝑥)| 2 𝑋𝑠 (d𝑥) d𝑠 0≤𝑠 ≤𝑛

0

𝐸 2

≤ 4𝜇 𝑛 (| 𝑓 𝑘 − 𝑓 𝑗 | ). The right-hand side goes to zero as 𝑗, 𝑘 → ∞. Then there is a square-integrable continuous martingale {𝑀𝑡𝑐 ( 𝑓 ) : 𝑡 ≥ 0} such that h i lim P sup |𝑀𝑠𝑐 ( 𝑓 𝑘 ) − 𝑀𝑠𝑐 ( 𝑓 )| 2 = 0, 𝑡 ≥ 0. 𝑘→∞

0≤𝑠 ≤𝑡

It is easy to see that {𝑀𝑡𝑐 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ 𝐵(𝐸)} is a martingale functional. Let 𝑀𝑡𝑐 (𝐵) = 𝑀𝑡𝑐 (1 𝐵 ) for 𝑡 ≥ 0 and 𝐵 ∈ ℬ(𝐸). Then {𝑀𝑡𝑐 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} is a continuous orthogonal martingale measure with covariance measure 𝜂 𝑐 (d𝑠, d𝑥). □ Proposition 7.30 Let 𝑁 (d𝑠, d𝜈) be given by (7.19). Then for any 𝑎 > 0 we can define a càdlàg worthy martingale measure ∫ 𝑡∫ (7.56) 𝑍𝑡𝑎 (𝐵) = 𝜈(𝐵) 𝑁˜ (d𝑠, d𝜈), 𝑡 ≥ 0, 𝐵 ∈ ℬ(𝐸), 0

{𝜈 (1) ≤𝑎 }

7.4 A Stochastic Convolution Formula

193

which has covariance measure ∫ ∫ 𝑎 𝑋𝑠 (d𝑧) 𝜂 (d𝑠, d𝑥, d𝑦) = d𝑠

𝜈(d𝑥)𝜈(d𝑦)𝐻 (𝑧, d𝜈).

(7.57)

{𝜈 (1) ≤𝑎 }

𝐸

Proof It is easy to see that (7.56) defines a square-integrable càdlàg martingale {𝑍𝑡𝑎 (𝐵) : 𝑡 ≥ 0}. The expression (7.57) is a consequence of the form of the pre□ dictable compensator 𝑁ˆ (d𝑠, d𝜈).

Corollary 7.31 Suppose that 𝜈(1) 2 𝐻 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ . Then we can define a càdlàg worthy martingale measure ∫ 𝑡∫ 𝜈(𝐵) 𝑁˜ (d𝑠, d𝜈), 𝑡 ≥ 0, 𝐵 ∈ ℬ(𝐸), 𝑍𝑡 (𝐵) = 0

𝑀 (𝐸) ◦

which has covariance measure ∫ ∫ 𝜂(d𝑠, d𝑥, d𝑦) = d𝑠 𝑋𝑠 (d𝑧)

𝜈(d𝑥)𝜈(d𝑦)𝐻 (𝑧, d𝜈).

𝑀 (𝐸) ◦

𝐸

The framework of stochastic integrals developed in Section 7.3 applies to the martingale measure {𝑍𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} defined in Corollary 7.31 under the second moment assumption. In the general case, we can define some integrals with respect to the compensated optional random measure 𝑁˜ (d𝑠, d𝜈) by considering separately the sets {𝜈 ∈ 𝑀 (𝐸) ◦ : 𝜈(1) ≤ 1} and {𝜈 ∈ 𝑀 (𝐸) ◦ : 𝜈(1) > 1}. More precisely, for a predictable two-parameter process 𝑓 = { 𝑓𝑠 (𝑥) : 𝑠 ≥ 0, 𝑥 ∈ 𝐸 } satisfying ∫ 𝑡 ∫ ∫  𝜈(| 𝑓𝑠 |) 2 1 {𝜈 (1) ≤1} d𝑠 𝑋𝑠 (d𝑧) P ◦ 𝐸 𝑀 (𝐸) 0   + 𝜈(| 𝑓𝑠 |)1 {𝜈 (1) >1} 𝐻 (d𝑧, d𝜈) < ∞, (7.58) we define ∫ 𝑡∫ 0

𝜈( 𝑓𝑠 ) 𝑁˜ (d𝑠, d𝜈) =

∫

𝑀 (𝐸) ◦

𝑡

0∫

∫ 𝑡

𝜈( 𝑓𝑠 )1 {𝜈 (1) >1} 𝑁˜ (d𝑠, d𝜈)

◦ 𝑀 ∫ (𝐸)

𝑓𝑠 (𝑥)𝑍 1 (d𝑠, d𝑥),

+ 0

(7.59)

𝐸

where the first term stands for the integral with respect to the random signed-measure 1 {𝜈 (1) >1} 𝑁˜ (d𝑠, d𝜈) and the second term stands for the integral with respect to the worthy martingale measure {𝑍𝑡1 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} defined in Proposition 7.30 with 𝑎 = 1. The next theorem gives a representation of the (𝜉, 𝜙)-superprocess by a formula involving stochastic convolution integrals.

194

7 Martingale Problems of Superprocesses

Theorem 7.32 Let (𝜋𝑡 )𝑡 ≥0 be the semigroup defined by (2.38). Then for any 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) we have a.s. ∫ 𝑡∫ 𝑋𝑡 ( 𝑓 ) = 𝑋0 (𝜋𝑡 𝑓 ) + 𝜋𝑡−𝑠 𝑓 (𝑥) 𝑀 𝑐 (d𝑠, d𝑥) 𝐸 0 ∫ 𝑡∫ (7.60) 𝜈(𝜋𝑡−𝑠 𝑓 ) 𝑁˜ (d𝑠, d𝜈). + 𝑀 (𝐸) ◦

0

Proof Since (𝑠, 𝑥) ↦→ 1 {𝑠 ≤𝑡 } 𝜋𝑡−𝑠 𝑓 (𝑥) is a deterministic measurable function on [0, ∞) × 𝐸, the stochastic integrals on the right-hand side of (7.60) are well-defined. Let Δ = {0 = 𝑡 0 < 𝑡 1 < · · · < 𝑡 𝑛 = 𝑡} be a partition of [0, 𝑡] and let |Δ| = max1≤𝑖 ≤𝑛 |𝑡𝑖 − 𝑡𝑖−1 |. For any 𝑓 ∈ 𝐷 0 ( 𝐴), by (7.20), (7.21) and Corollary 7.12 we have 𝑋𝑡 ( 𝑓 ) = 𝑋0 (𝜋𝑡 𝑓 ) +

𝑛 ∑︁

𝑋𝑡𝑖 (𝜋𝑡−𝑡𝑖 𝑓 − 𝜋𝑡−𝑡𝑖−1 𝑓 )

𝑖=1

+

𝑛 ∑︁ 

𝑋𝑡𝑖 (𝜋𝑡−𝑡𝑖−1 𝑓 ) − 𝑋𝑡𝑖−1 (𝜋𝑡−𝑡𝑖−1 𝑓 )



𝑖=1

= 𝑋0 (𝜋𝑡 𝑓 ) −

𝑛 ∫ ∑︁ 𝑖=1

+

𝑛 ∑︁ 

𝑡−𝑡𝑖−1

𝑋𝑡𝑖 (( 𝐴 + 𝛾 − 𝑏)𝜋 𝑠 𝑓 )d𝑠

𝑡−𝑡𝑖

𝑀𝑡𝑐𝑖 (𝜋𝑡−𝑡𝑖−1 𝑓 ) − 𝑀𝑡𝑐𝑖−1 (𝜋𝑡−𝑡𝑖−1 𝑓 )



𝑖=1

+ +

𝑛 ∫ ∑︁

𝑡𝑖

𝑖=1 𝑡𝑖−1 𝑛 ∫ 𝑡𝑖 ∑︁ 𝑖=1

∫

𝜈(𝜋𝑡−𝑡𝑖−1 𝑓 ) 𝑁˜ (d𝑠, d𝜈)

𝑀 (𝐸) ◦

𝑋𝑠 (( 𝐴 + 𝛾 − 𝑏)𝜋𝑡−𝑡𝑖−1 𝑓 )d𝑠.

𝑡𝑖−1

By letting |Δ| → 0 and using the right continuity of 𝑠 ↦→ 𝑋𝑠 and the strong continuity of 𝑠 ↦→ 𝜋 𝑠 𝑓 we obtain (7.60). Since 𝐷 0 ( 𝐴) is dense in 𝐶0 (𝐸) in the supremum norm, we also have (7.60) for an arbitrary 𝑓 ∈ 𝐶0 (𝐸). The result for a general function □ 𝑓 ∈ 𝐵(𝐸) follows by Proposition A.1. As an application of the representation (7.60), we prove a structural property of the (𝜉, 𝜙)-superprocess. For this purpose let us consider the following condition: Condition 7.33 (i) There exists a 𝜎-finite measure 𝜆 on 𝐸 and a Borel function (𝑡, 𝑥, 𝑦) ↦→ 𝑝 𝑡 (𝑥, 𝑦) on (0, ∞) × 𝐸 2 such that 𝑃𝑡 (𝑥, d𝑦) = 𝑝 𝑡 (𝑥, 𝑦)𝜆(d𝑦),

𝑡 > 0, 𝑥, 𝑦 ∈ 𝐸 .

(ii) There exists a constant 0 < 𝛼 < 1 and an increasing function 𝑡 ↦→ 𝐶 (𝑡) on [0, ∞) such that 𝑝 𝑡 (𝑥, 𝑦) ≤ 𝑡 −𝛼 𝐶 (𝑡),

𝑡 > 0, 𝑥, 𝑦 ∈ 𝐸 .

7.4 A Stochastic Convolution Formula

195

Theorem 7.34 Suppose that Condition 7.33 holds. Then for every 𝑡 > 0 we have P{𝑋𝑡 is absolutely continuous with respect to 𝜆} = 1.

Proof By the expression (2.40) one can show that for any 𝑡 > 0 the kernel 𝜋𝑡 (𝑥, d𝑦) is absolutely continuous with respect to 𝜆(d𝑦) with density 𝑞 𝑡 (𝑥, 𝑦) satisfying 𝑞 𝑡 (𝑥, 𝑦) ≤ 𝑔(𝑡) := 𝑡 −𝛼 e ∥𝑏 ∥𝑡 𝐶 (𝑡) + 𝐾 (𝑡), where 𝐾 (𝑡) = e ∥𝑏 ∥𝑡 𝐶 (𝑡)

∞ ∑︁ 𝑛=1

∫ ∥𝛾∥ 𝑛

∫

𝑡

0

∫

𝑠1

d𝑠2 · · ·

d𝑠1 0

𝑠𝑛−1

𝑠−𝛼 𝑛 d𝑠 𝑛 .

0

For 𝑓 ∈ 𝐶0 (𝐸) + satisfying 𝜆( 𝑓 ) < ∞ define 𝜆 𝑓 (d𝑦) = 𝑓 (𝑦)𝜆(d𝑦). Let 𝑞 𝑠 (𝑥, 𝑧) = 0 for 𝑠 ≤ 0. For any 𝑛 ≥ 1 we have  ∫ ∫ 𝑛∫ 2 2𝑐(𝑥)𝑞 𝑡−𝑠 (𝑥, 𝑧) 𝑋𝑠 (d𝑥) P 𝜆 𝑓 (d𝑧) 𝐸 0∫ 𝑡 𝐸  ∫ 𝜋𝑡−𝑠 𝑓 (𝑥) 𝑋𝑠 (d𝑥) 𝑔(𝑡 − 𝑠)d𝑠 ≤ 2∥𝑐∥P 0 ∫ 𝑡 𝐸 ≤ 2∥𝑐∥P[𝑋0 (𝜋𝑡 𝑓 )] 𝑔(𝑡 − 𝑠)d𝑠 < ∞. 0

Let 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)] and let 𝑐+0 = 0 ∨ 𝑐 0 . By Theorem A.53 we have + ∥𝜋𝑡 ∥ ≤ e𝑐0 𝑡 ≤ e𝑐0 𝑡 for all 𝑡 ≥ 0. It follows that  ∫ ∫ ∫ 𝑛 ∫ 2 d𝑠 𝜆 𝑓 (d𝑦) P ⟨𝜈, 𝑞 𝑡−𝑠 (·, 𝑦)⟩ 𝐻 (𝑧, d𝜈) 𝑋𝑠 (d𝑧) {𝜈 (1) 𝐸 ∫ 𝐸  ∫ 𝑡 0 ∫ ≤1} 𝜈(𝜋𝑡−𝑠 𝑓 )𝜈(1)𝐻 (𝑧, d𝜈) ≤ P 𝑋𝑠 (d𝑧) 𝑔(𝑡 − 𝑠)d𝑠 0 𝐸  ∫ 𝑡 ∫ {𝜈 (1) ≤1}∫ + 2 𝑐0 𝑡 𝑋𝑠 (d𝑧) ≤ e ∥ 𝑓 ∥P 𝑔(𝑡 − 𝑠)d𝑠 𝜈(1) 𝐻 (𝑧, d𝜈) {𝜈 (1) ≤1} 𝐸 ∫ 𝑡0 + ≤ e𝑐0 𝑡 ∥ 𝑓 ∥ 𝑔(𝑡 − 𝑠)P[𝑋0 (𝜋 𝑠 ℎ0 )]d𝑠 < ∞ 0

and  ∫ 𝑛 ∫ ∫ 𝑋𝑠 (d𝑧) ⟨𝜈, 𝑞 𝑡−𝑠 (·, 𝑦)⟩𝐻 (𝑧, d𝜈) 𝜆 𝑓 (d𝑦) d𝑠 𝐸 ∫ 𝑡 0 ∫ 𝐸  ∫ {𝜈 (1) >1} 𝜈(𝜋𝑡−𝑠 𝑓 )𝐻 (𝑧, d𝜈) ≤ P 𝑋𝑠 (d𝑧) d𝑠 0   ∫𝐸 𝑡 ∫ {𝜈 (1) >1}∫ + ≤ e𝑐0 𝑡 ∥ 𝑓 ∥P d𝑠 𝑋𝑠 (d𝑧) 𝜈(1)𝐻 (𝑧, d𝜈) {𝜈 (1) >1} 𝐸 ∫ 𝑡0 𝑐0+ 𝑡 ≤ e ∥𝑓∥ P[𝑋0 (𝜋 𝑠 ℎ1 )]d𝑠 < ∞,

∫ P

0

196

7 Martingale Problems of Superprocesses

where ℎ0 , ℎ1 ∈ 𝐵(𝐸) + are defined by ∫ ℎ0 (𝑧) = 𝜈(1) 2 𝐻 (𝑧, d𝜈),

∫ 𝜈(1)𝐻 (𝑧, d𝜈).

ℎ1 (𝑧) =

{𝜈 (1) ≤1}

{𝜈 (1) >1}

Then using (7.59) and Theorems 7.28 and 7.32 we get ∫ ∫ 𝑋𝑡 ( 𝑓 ) = 𝑌𝑡 (𝑦)𝜆 𝑓 (d𝑦) = 𝑓 (𝑧)𝑌𝑡 (𝑦)𝜆(d𝑦), 𝐸

𝐸

where ∫ 𝑌𝑡 (𝑦) =

∫ 𝑡∫ 𝑞 𝑡 (𝑥, 𝑦) 𝑋0 (d𝑥) + 𝑞 𝑡−𝑠 (𝑥, 𝑦)𝑀 𝑐 (d𝑠, d𝑥) 𝐸 0 𝐸 ∫ 𝑡∫ + ⟨𝜈, 𝑞 𝑡−𝑠 (·, 𝑦)⟩ 𝑁˜ (d𝑠, d𝜈). 0

𝑀 (𝐸) ◦

By considering a sequence { 𝑓𝑛 } dense in 𝐶0 (𝐸) + we obtain the desired result.

□

Example 7.2 If 𝜉 is a Brownian motion in R, then Condition 7.33 holds with 𝜆 being the Lebesgue measure. Thus for the super-Brownian motion the random measures {𝑋𝑡 : 𝑡 > 0} are absolutely continuous with respect to the Lebesgue measure.

7.5 Transforms by Martingales In this section, we assume 𝐸 is a locally compact separable metric space and 𝜉 is a Hunt process with Feller transition semigroup (𝑃𝑡 )𝑡 ≥0 . Let us consider a local branching mechanism 𝜙 given by (2.49) with constant function 𝑏(𝑥) ≡ 𝑏 ≥ 0. Suppose in addition that 𝑐 ∈ 𝐶 (𝐸) + and 𝑥 ↦→ (𝑢 ∧ 𝑢 2 )𝑚(𝑥, d𝑢) is continuous by ˜ 𝒢˜ 𝑡 , 𝑋𝑡 , Q ˜ 𝜇 ) be the subprocess of the weak convergence on (0, ∞). Let 𝑋 = (𝑊0 , 𝒢, (𝜉, 𝜙)-superprocess generated by the multiplicative functional {𝑚 𝑡 : 𝑡 ≥ 0} given by (6.23). Recall that 𝜇ˆ = 𝜇(1) −1 𝜇 for 𝜇 ∈ 𝑀 (𝐸) ◦ . Let 𝐿 0 be defined by (7.18). ˜ 𝜇 the process {𝑋𝑡 : 𝑡 ≥ 0} solves the martingale problem: Theorem 7.35 Under Q For any 𝐹 ∈ 𝒟0 given by (7.16), ∫ 𝑡 ∫ ∫ 𝑡 d𝑠 𝑐(𝑥)𝐹 ′ (𝑋𝑠 ; 𝑥) 𝑋ˆ 𝑠 (d𝑥) + local mart. 𝐿 0 𝐹 (𝑋𝑠 )d𝑠 + 2 𝐹 (𝑋𝑡 ) = 𝐸 0∫ 0 ∫ ∫ ∞ 𝑡 d𝑠 𝑋ˆ 𝑠 (d𝑥) + [𝐹 (𝑋𝑠 + 𝑢𝛿 𝑥 ) − 𝐹 (𝑋𝑠 )]𝑢𝑚(𝑥, d𝑢). (7.61) 0

𝐸

0

Proof Let 𝐻 (𝜇) = 𝜇(1)𝐹 (𝜇) for 𝜇 ∈ 𝑀 (𝐸). The operator 𝐿 0 can still be applied to 𝐻 although the function is not necessarily in 𝒟0 . In fact, it is easy to see that 𝐻 ′ (𝜇; 𝑥) = 𝐹 (𝜇) + 𝜇(1)𝐹 ′ (𝜇; 𝑥)

7.5 Transforms by Martingales

197

and 𝐻 ′′ (𝜇; 𝑥) = 2𝐹 ′ (𝜇; 𝑥) + 𝜇(1)𝐹 ′′ (𝜇; 𝑥). Then we have ∫

𝑐(𝑥)𝐹 ′ (𝜇; 𝑥)𝜇(d𝑥) 𝐿 0 𝐻 (𝜇) = 𝜇(1) (𝐿 0 − 𝑏)𝐹 (𝜇) + 2 𝐸 ∫ ∫ ∞ + 𝜇(d𝑥) [𝐹 (𝜇 + 𝑢𝛿 𝑥 ) − 𝐹 (𝜇)]𝑢𝑚(𝑥, d𝑢). 0

𝐸

By extending Theorem 7.16 slightly and applying it to suitable truncations of 𝐻 we get, under Q 𝜇 , ∫ 𝐻 (𝑋𝑡 ) =

𝑡

𝑋𝑠 (1) (𝐿 0 − 𝑏)𝐹 (𝑋𝑠 )d𝑠 + local mart. ∫ 𝑡 ∫ +2 d𝑠 𝑐(𝑥)𝐹 ′ (𝑋𝑠 ; 𝑥) 𝑋𝑠 (d𝑥) ∫ ∞ ∫ 𝑡0 ∫ 𝐸 [𝐹 (𝑋𝑠 + 𝑢𝛿 𝑥 ) − 𝐹 (𝑋𝑠 )]𝑢𝑚(𝑥, d𝑢). + 𝑋𝑠 (d𝑥) d𝑠 0

0

0

𝐸

By integration by parts, ∫ 𝑡 e𝑏𝑡 𝐻 (𝑋𝑡 ) = e𝑏𝑠 𝑋𝑠 (1)𝐿 0 𝐹 (𝑋𝑠 )d𝑠 + local mart. 0 ∫ ∫ 𝑡 𝑏𝑠 e d𝑠 +2 𝑐(𝑥)𝐹 ′ (𝑋𝑠 ; 𝑥) 𝑋𝑠 (d𝑥) ∫ 𝑡0 ∫ ∞ ∫ 𝐸 𝑏𝑠 𝑋𝑠 (d𝑥) e d𝑠 + [𝐹 (𝑋𝑠 + 𝑢𝛿 𝑥 ) − 𝐹 (𝑋𝑠 )]𝑢𝑚(𝑥, d𝑢) 0 𝐸 ∫ 𝑡0 = (7.62) e𝑏𝑠 𝑋𝑠 (1)𝐽 (𝑋𝑠 )d𝑠 + local mart., 0

where ∫

𝐽 (𝑋𝑠 ) = 𝐿 0 𝐹 (𝑋𝑠 ) + 2 𝑐(𝑥)𝐹 ′ (𝑋𝑠 ; 𝑥) 𝑋ˆ 𝑠 (d𝑥) ∫ ∫ 𝐸∞ [𝐹 (𝑋𝑠 + 𝑢𝛿 𝑥 ) − 𝐹 (𝑋𝑠 )]𝑢𝑚(𝑥, d𝑢). + 𝑋ˆ 𝑠 (d𝑥) 𝐸

0

Since 𝑡 ↦→ e𝑏𝑡 𝑋𝑡 (1) is a martingale under Q 𝜇 , we can use integration by parts again on the right-hand side of (7.62) to see ∫ 𝑡 𝑏𝑡 𝑏𝑡 𝑏𝑡 e 𝑋𝑡 (1)𝐹 (𝑋𝑡 ) = e 𝐻 (𝑋𝑡 ) = e 𝑋𝑡 (1) 𝐽 (𝑋𝑠 )d𝑠 + local mart. 0

˜ 𝜇 by a simple calculation. Then we have (7.61) under Q

□

198

7 Martingale Problems of Superprocesses

Now suppose that (Ω, 𝒢, 𝒢𝑡 , P) is a probability space satisfying the usual hypothesis. Let {𝑋𝑡 : 𝑡 ≥ 0} be a continuous 𝑀 (𝐸)-valued adapted process satisfying P[⟨𝑋0 , 1⟩] < ∞. For 𝑏 ∈ 𝐶 (𝑀 (𝐸) × 𝐸) and 𝑐 ∈ 𝐶 (𝐸) + we consider the following martingale problem: For every 𝑓 ∈ 𝐷 0 ( 𝐴) the process ∫ 𝑡 𝑀𝑡 ( 𝑓 ) = ⟨𝑋𝑡 , 𝑓 ⟩ − ⟨𝑋0 , 𝑓 ⟩ − ⟨𝑋𝑠 , 𝐴 𝑓 − 𝑏(𝑋𝑠 ) 𝑓 ⟩d𝑠 (7.63) 0

is a square-integrable (𝒢𝑡 )-martingale with quadratic variation process ∫ 𝑡 ∫ 2𝑐(𝑥) 𝑓 (𝑥) 2 𝑋𝑠 (d𝑥). d𝑠 ⟨𝑀 ( 𝑓 )⟩𝑡 = 0

(7.64)

𝐸

This should be compared with the martingale problem given by (7.33) and (7.35). If {𝑋𝑡 : 𝑡 ≥ 0} is a solution of the martingale problem above, we can follow the arguments in Section 7.3 to show that there is a continuous (𝒢𝑡 )-martingale measure {𝑀𝑡 (𝐵) : 𝑡 ≥ 0; 𝐵 ∈ ℬ(𝐸)} satisfying ∫ 𝑡∫ 𝑓 (𝑥)𝑀 (d𝑠, d𝑥), 𝑀𝑡 ( 𝑓 ) = (7.65) 𝑡 ≥ 0, 𝑓 ∈ 𝐷 0 ( 𝐴) 0

𝐸

and having covariance measure ∫ 2𝑐(𝑧)𝛿 𝑧 (d𝑥)𝛿 𝑧 (d𝑦) 𝑋𝑠 (d𝑧).

𝜂(d𝑠, d𝑥, d𝑦) = d𝑠

(7.66)

𝐸

Then for any function 𝛽 ∈ 𝐶 (𝑀 (𝐸) × 𝐸) we can define the continuous and strictly positive local martingale {𝑍𝑡 : 𝑡 ≥ 0} by  ∫ 𝑡 ∫ ∫ 𝑡 ∫ 2 𝛽(𝑋𝑠 , 𝑥)𝑀 (d𝑠, d𝑥) − 𝑐(𝑥) 𝛽(𝑋𝑠 , 𝑥) 𝑋𝑠 (d𝑥) . d𝑠 𝑍𝑡 = exp 0

0

𝐸

𝐸

Lemma 7.36 Suppose that {𝑋𝑡 : 𝑡 ≥ 0} is a solution of the martingale problem given by (7.63) and (7.64). Then {𝑍𝑡 : 𝑡 ≥ 0} is actually a (𝒢𝑡 )-martingale. Proof It suffices to prove P[𝑍𝑡 ] = 1 for every 𝑡 ≥ 0; see, e.g., Ikeda and Watanabe (1989, p. 152). For each 𝑛 ≥ 1 let 𝜏𝑛 = inf{𝑡 ≥ 0 : ⟨𝑋𝑡 , 1⟩ ≥ 𝑛}. It is easy to see that 𝜏𝑛 → ∞ as 𝑛 → ∞. Observe also that   ∫ 𝑡∧𝜏𝑛 ∫ 2 𝑐(𝑥) 𝛽(𝑋𝑠 , 𝑥) 𝑋𝑠 (d𝑥) < ∞. P exp d𝑠 0

𝐸

Then {𝑍𝑡∧𝜏𝑛 : 𝑡 ≥ 0} is a continuous and strictly positive (𝒢𝑡 )-martingale. It follows that 1 = P[𝑍𝑡∧𝜏𝑛 ] = P[𝑍𝑡∧𝜏𝑛 1 {𝜏𝑛 ≤𝑡 } ] + P[𝑍𝑡∧𝜏𝑛 1 {𝜏𝑛 >𝑡 } ].

(7.67)

For fixed 𝑛 ≥ 1 and 𝑡 ≥ 0 we define the new probability measure P𝑡 on 𝒢𝑡 by P𝑡 (d𝜔) = 𝑍𝑡∧𝜏𝑛 (𝜔)P(d𝜔). Under the measure P𝑡 , for each 𝑓 ∈ 𝐷 0 ( 𝐴),

7.5 Transforms by Martingales

199

∫

𝑢∧𝜏𝑛

𝑁𝑢 ( 𝑓 ) := 𝑀𝑢∧𝜏𝑛 ( 𝑓 ) − 2

⟨𝑋𝑠 , 𝑐𝛽(𝑋𝑠 ) 𝑓 ⟩d𝑠,

0 ≤ 𝑢 ≤ 𝑡,

(7.68)

0

is a square-integrable martingale with quadratic variation process given by ∫ 𝑢∧𝜏𝑛 ∫ ⟨𝑁 ( 𝑓 )⟩𝑢 = 2 𝑐(𝑥) 𝑓 (𝑥) 2 𝑋𝑠 (d𝑥); d𝑠 0

𝐸

see, e.g., Ikeda and Watanabe (1989, p. 191). Since (𝑃𝑡 )𝑡 ≥0 is conservative, it is easy to extend the martingale problems to the constant function 𝑓 = 1 with 𝐴1 = 0. Consequently, for any 0 ≤ 𝑢 ≤ 𝑡 we have ∫ 𝑢 𝑡 𝑡 P𝑡 [⟨𝑋𝑠∧𝜏𝑛 , 1⟩]d𝑠, P [⟨𝑋𝑢∧𝜏𝑛 , 1⟩] ≤ P [⟨𝑋0 , 1⟩] + ∥2𝑐𝛽 − 𝑏∥ 0

where P𝑡 [⟨𝑋0 , 1⟩] = P[⟨𝑋0 , 1⟩𝑍𝑡∧𝜏𝑛 ] = P[⟨𝑋0 , 1⟩]. Then Gronwall’s inequality implies P𝑡 [⟨𝑋𝑢∧𝜏𝑛 , 1⟩] ≤ P[⟨𝑋0 , 1⟩] exp{∥2𝑐𝛽 − 𝑏∥𝑢}.

(7.69)

From (7.63) and (7.68) it follows that n o P𝑡 sup ⟨𝑋𝑠∧𝜏𝑛 , 1⟩ ≥ 𝑛 0≤𝑠 ≤𝑡

 𝑡



𝑡

≤ P {⟨𝑋0 , 1⟩ ≥ 𝑛/3} + P sup |𝑁 𝑠 (1)| ≥ 𝑛/3 0≤𝑠 ≤𝑡 ∫ 𝑡  𝑡 + ∥2𝑐𝛽 − 𝑏∥P ⟨𝑋𝑠∧𝜏𝑛 , 1⟩d𝑠 ≥ 𝑛/3 . 0

By Doob’s martingale inequality, P𝑡

n

o 18∥𝑐∥ ∫ 𝑡 P𝑡 [⟨𝑋𝑠∧𝜏𝑛 , 1⟩]d𝑠. sup |𝑁 𝑠 (1)| ≥ 𝑛/3 ≤ 𝑛2 0≤𝑠 ≤𝑡 0

In view of (7.69), we can use Chebyshev’s inequality to see n o lim P[𝑍𝑡∧𝜏𝑛 1 {𝜏𝑛 ≤𝑡 } ] = lim P𝑡 sup ⟨𝑋𝑠∧𝜏𝑛 , 1⟩ ≥ 𝑛 = 0. 𝑛→∞

𝑛→∞

0≤𝑠 ≤𝑡

Then letting 𝑛 → ∞ in (7.67) we get P[𝑍𝑡 ] = 1.

□

𝐶 (𝐸) +

Theorem 7.37 Suppose that 𝑐 ∈ is bounded away from zero. Then there is a unique solution to the martingale problem given by (7.63) and (7.64). Proof If {𝑋𝑡 : 𝑡 ≥ 0} is a solution of the martingale problem given by (7.63) and (7.64) under P and if P 𝑍 is the probability measure on (Ω, 𝒢) such that P 𝑍 (d𝜔) = 𝑍𝑡 (𝜔)P(d𝜔) on 𝒢𝑡 for every 𝑡 ≥ 0, then for each 𝑓 ∈ 𝐷 0 ( 𝐴),

200

7 Martingale Problems of Superprocesses

∫ 𝑀𝑡 ( 𝑓 ) = ⟨𝑋𝑡 , 𝑓 ⟩ − ⟨𝑋0 , 𝑓 ⟩ −

𝑡

⟨𝑋𝑠 , 𝐴 𝑓 − 𝑏(𝑋𝑠 ) 𝑓 + 2𝑐𝛽(𝑋𝑠 ) 𝑓 ⟩d𝑠 0

is a square-integrable (𝒢𝑡 )-martingale with quadratic variation process (7.64) under P 𝑍 . Here we may assume (Ω, 𝒢, 𝒢𝑡 ) is the P-augmentation of the canonical space consisting of continuous paths from [0, ∞) to 𝑀 (𝐸), which is a standard measurable space, so that the measure P 𝑍 described as above is well-defined; see, e.g., Ikeda and Watanabe (1989, p. 190). By Corollary 7.20 the existence and uniqueness of the martingale problem given by (7.63) and (7.64) hold for 𝑏(𝑥) ≡ 0. Since 𝑐 ∈ 𝐶 (𝐸) + is bounded away from zero, using changes of the probability measures as above one can see that the existence and uniqueness also hold for a general 𝑏 ∈ 𝐶 (𝑀 (𝐸) × 𝐸).□ Here we may interpret {𝑋𝑡 : 𝑡 ≥ 0} as a superprocess with interactive growth rate given by the function 𝑏(𝜇, 𝑥). The transformation based on the strictly positive martingale {𝑍𝑡 : 𝑡 ≥ 0} used in the above proof is known as Dawson’s Girsanov transform.

7.6 Notes and Comments A systematic treatment of martingale problems for diffusions was given in Stroock and Varadhan (1979). Those for Markov processes with abstract state spaces were discussed in Ethier and Kurtz (1986). Nonlinear functional integral and differential evolution equations were discussed in Pazy (1983). Our approach in Section 7.1 is different from that of Pazy (1983) and uses heavily the special structures of the cumulant semigroup. The approach of martingale problems plays an important role in the study of measure-valued processes. Martingale problems for Dawson–Watanabe superprocesses with Feller spatial motion and binary branching mechanism were studied in Roelly (1986). The treatment in Section 7.2 follows El Karoui and Roelly (1991). Fitzsimmons (1988, 1992) studied martingale problems of superprocesses in the Borel right setting. Our main references for worthy martingale measures are El Karoui and Méléard (1990) and Walsh (1986). Dawson (1978) first used the Girsanov type transform to derive superprocesses with interactive branching structures. Martingale problems of the type given by (7.63) and (7.64) were considered in Etheridge (2004) and Fournier and Méléard (2004) in the study of locally regulated population models; see also Méléard and Roelly (1993). Martingale problems for superprocesses with general killing rates were studied in Leduc (2006). Champagnat and Roelly (2008) gave a martingale problem characterization for a continuous multitype superprocess conditioned on non-extinction, and proved several results on the long-time behavior of the conditioned superprocess. The study of stochastic partial differential equations has attracted a lot of attention. The basic theory of the subject was developed in Walsh (1986) on the basis of martingale measures. Mueller (2009) gave a survey of the tools and results for stochastic parabolic equations with emphasis on the techniques from Dawson–

7.6 Notes and Comments

201

Watanabe superprocesses and interacting particle systems; see also Krylov (1997). The approaches of Hilbert spaces and Sobolev spaces for stochastic partial differential equations were developed in Da Prato and Zabczyk (1992) and Krylov (1996). A series of recent results on the well-posedness of singular parabolic stochastic partial differential equations were presented in Hairer (2014). A prototype of those is the Kardar–Parisi–Zhang (KPZ) equation arising in interface propagation, which was solved in Hairer (2013). We refer the reader to Zambotti (2021) for a brief history of the subject. Let 𝑏 ∈ 𝐶 (R) and 𝑐 ∈ 𝐶 (R) + . The super-Brownian motion {𝑋𝑡 : 𝑡 ≥ 0} on R with local branching mechanism 𝜙(𝑥, 𝜆) ≡ 𝑏(𝑥)𝜆 + 𝑐(𝑥)𝜆2 has a continuous realization. It was proved in Konno and Shiga (1988) that {𝑋𝑡 : 𝑡 > 0} has a continuous density field {𝑢 𝑡 (𝑥) : 𝑡 > 0, 𝑥 ∈ R} with respect to the Lebesgue measure. The density field solves the stochastic integral equation, for any 𝑓 ∈ 𝐶 2 (R), ∫ ∫ 𝑡∫ ∫ √︁ 𝑓 (𝑥) 𝑋0 (d𝑥) + 𝑓 (𝑥)𝑢 𝑡 (𝑥)d𝑥 = 𝑓 (𝑥) 2𝑐(𝑥)𝑢 𝑠 (𝑥)𝐵(d𝑠, d𝑥) R∫ R 0 R ∫ h i 𝑡 1 ′′ 𝑓 (𝑥) − 𝑏(𝑥) 𝑓 (𝑥) 𝑢 𝑠 (𝑥)d𝑥, + d𝑠 0 R 2 where {𝐵(d𝑠, d𝑥) : 𝑡 ≥ 0, 𝑥 ∈ R} is a time–space Gaussian white noise based on the Lebesgue measure. The above equation is usually abbreviated to the stochastic partial differential equation: √︁ 𝜕 ¤ 𝑥) + 1 Δ𝑢 𝑡 (𝑥) − 𝑏(𝑥)𝑢 𝑡 (𝑥), 𝑢 𝑡 (𝑥) = 2𝑐(𝑥)𝑢 𝑡 (𝑥) 𝐵(𝑡, 𝜕𝑡 2

(7.70)

where the “dot” denotes the derivative in the distribution sense. A special case of (7.70) was established independently in Reimers (1989). The uniqueness in distribution of the solution to (7.70) follows from Corollary 7.20. The pathwise uniqueness problem for (7.70) remains open. The main difficulty comes from the unbounded operator Δ and the non-Lipschitz diffusion coefficient. Let {𝑁 (d𝑡, d𝑥, d𝑧) : 𝑡 ≥ 0, 𝑥 ∈ R𝑑 , 𝑧 > 0} be a Poisson random measure with intensity 𝑐𝑧−1−𝛼 d𝑡d𝑥d𝑧, where 𝑐 > 0 and 1 < 𝛼 < 2. Then a one-sided 𝛼-stable noise on [0, ∞) × R𝑑 is defined by ∫ 𝐿 (d𝑡, d𝑥) = 𝑧 𝑁˜ (d𝑡, d𝑥, d𝑧). {0 0 the random measure 𝑋𝑡 has support with Hausdorff dimension two and distributes its mass over the support in a deterministic manner; see, e.g., Perkins (2002, p. 209 and p. 212). For 𝑑 ≥ 2 it was proved in Tribe (1994) that 𝑋𝑡 can be approximated by suitably normalized restrictions of the Lebesgue measure to the 𝜀-neighborhoods of support of the random measure. The analogous result for the more difficult case 𝑑 = 2 was established in Kallenberg (2008), which leads to a simple derivation of the property of deterministic mass distribution. The key assumption of a Dawson–Watanabe superprocess is the independence of different particles in the approximating system. When dependence is introduced into the branching or migrating mechanisms, the characterization of the limiting measurevalued process usually becomes very difficult. The method of dual processes plays an important role in the analysis of the uniqueness of martingale problems for measurevalued processes. A general theory of duality was developed in Ethier and Kurtz (1986). The reader may refer to Dawson (1993) for systematic applications of this

7.6 Notes and Comments

203

method to measure-valued processes. The approach of filtered martingale problems introduced by Kurtz (1998) and Kurtz and Ocone (1988) is another important tool in handling the uniqueness of martingale problems. A superprocess with dependent spatial motion over the real line R was constructed in Dawson et al. (2001), generalizing the model of Wang (1997a, 1998a). Let 𝑐 ∈ 𝐶 2 (R) and 𝜎 ∈ 𝐶 2 (R) + . Let ℎ ∈ 𝐶 1 (R) and assume both ℎ and ℎ ′ are squareintegrable. Let ∫ (7.73) 𝜌(𝑥) = ℎ(𝑦 − 𝑥)ℎ(𝑦)d𝑦 and 𝑎(𝑥) = 𝑐(𝑥) 2 + 𝜌(0), 𝑥 ∈ R. R

The superprocess with dependent spatial motion is a diffusion process {𝑋𝑡 : 𝑡 ≥ 0} in 𝑀 (R) characterized by the following martingale problem: For each 𝑓 ∈ 𝐶 2 (R), ∫ 1 𝑡 𝑀𝑡 ( 𝑓 ) = ⟨𝑋𝑡 , 𝑓 ⟩ − ⟨𝑋0 , 𝑓 ⟩ − ⟨𝑋𝑠 , 𝑎 𝑓 ′′⟩d𝑠, (7.74) 2 0 is a continuous martingale with quadratic variation process ∫ 𝑡 ∫ 𝑡 ∫ ⟨𝑀 ( 𝑓 )⟩𝑡 = ⟨𝑋𝑠 , 𝜎 𝑓 2 ⟩d𝑠 + d𝑠 ⟨𝑋𝑠 , ℎ(𝑧 − ·) 𝑓 ′⟩ 2 d𝑧. 0

0

(7.75)

R

The process {𝑋𝑡 : 𝑡 ≥ 0} arises as a weak limit of critical branching particle systems with dependent spatial motion. Consider a family of independent standard Brownian motions {𝐵𝑖 (𝑡) : 𝑡 ≥ 0, 𝑖 = 1, 2, . . .} and a time–space Gaussian white noise {𝑊 (d𝑡, d𝑦) : 𝑡 ≥ 0, 𝑦 ∈ R} based on the Lebesgue measure. Suppose that {𝐵𝑖 (𝑡) : 𝑡 ≥ 0, 𝑖 = 1, 2, . . .} and {𝑊 (d𝑡, d𝑦) : 𝑡 ≥ 0, 𝑦 ∈ R} are independent. The migration of the particle with label 𝑖 ≥ 1 in the approximating system is defined by the stochastic differential equation ∫ d𝑥 𝑖 (𝑡) = 𝑐(𝑥𝑖 (𝑡))d𝐵𝑖 (𝑡) + ℎ(𝑦 − 𝑥𝑖 (𝑡))𝑊 (d𝑡, d𝑦). R

The uniqueness of solution of the martingale problem given by (7.74) and (7.75) was established in Dawson et al. (2001) by considering a function-valued dual process. Clearly, the superprocess with dependent spatial motion reduces to a usual critical branching Dawson–Watanabe superprocess if ℎ ≡ 0. On the other hand, when 𝜎 ≡ 0, branching does not occur and the total mass of the process remains unchanged as time passes. By considering a stochastic equation driven by a time– space Gaussian white noise and the path process of a Brownian motion, Gill (2009) unified the approaches of Dawson et al. (2001) and Perkins (1995, 2002) and gave a new class of measure-valued diffusions. Ren et al. (2009) introduced a superprocess with dependent spatial motion in a bounded domain in R𝑑 with killing boundary. A discontinuous superprocess with dependent spatial motion and general branching mechanism was constructed in He (2009). Some probability-valued Markov processes arising from consistent particle systems were studied in Ma and Xiang (2001) and Xiang (2009).

Chapter 8

Entrance Laws and Kuznetsov Measures

The main purpose of this chapter is to investigate the structures of entrance laws for MB-processes. In particular, we establish a one-to-one correspondence between the minimal probability entrance laws for a Dawson–Watanabe superprocess and the entrance laws for its spatial motion. Based on the correspondence, a complete characterization is given for infinitely divisible probability entrance laws of the superprocess. We also prove some properties of the Kuznetsov measures determined by canonical entrance rules. Cluster representations for the MB-process are given by summing up measure-valued paths selected by Poisson random measures. We briefly discuss the special case where the spatial motion process is an absorbingbarrier Brownian motion in a domain. Some of the results presented here will be used in the study of immigration superprocesses.

8.1 Some Simple Properties Suppose that 𝐸 is a Lusin topological space. Let (𝑄 𝑡 )𝑡 ≥0 and (𝑉𝑡 )𝑡 ≥0 denote respectively the transition semigroup and the cumulant semigroup of an MB-process with state space 𝑀 (𝐸). Recall that (𝑉𝑡 )𝑡 ≥0 always has the representation (2.5) and 𝐸𝐶 is the set of points 𝑥 ∈ 𝐸 such that (2.11) holds. Let (𝑄 ◦𝑡 )𝑡 ≥0 denote the restriction of (𝑄 𝑡 )𝑡 ≥0 to 𝑀 (𝐸) ◦ . Theorem 8.1 There is a one-to-one correspondence between bounded entrance laws (𝐾𝑡◦ )𝑡 >0 for (𝑄 ◦𝑡 )𝑡 ≥0 and bounded entrance laws (𝐾𝑡 )𝑡 >0 for (𝑄 𝑡 )𝑡 ≥0 satisfying lim𝑡→0 𝐾𝑡 ({0}) = 0, which is given by 𝐾𝑡◦ = 𝐾𝑡 | 𝑀 (𝐸) ◦ and 𝐾𝑡 ({0}) = ↑lim 𝐾𝑠◦ (1) − 𝐾𝑡◦ (1).

(8.1)

𝑠↓0

Proof Suppose that (𝐾𝑡◦ )𝑡 >0 is a bounded entrance law for (𝑄 ◦𝑡 )𝑡 ≥0 . Since the null measure is a trap for (𝑄 𝑡 )𝑡 ≥0 , one can see that 𝑡 ↦→ 𝐾𝑡◦ (1) is decreasing. Let 𝐾𝑡 be the extension of 𝐾𝑡◦ to 𝑀 (𝐸) defined by (8.1). Then lim𝑡→0 𝐾𝑡 ({0}) = 0. For © Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_8

205

206

8 Entrance Laws and Kuznetsov Measures

𝑡 > 𝑟 > 0 and 𝐹 ∈ 𝐵(𝑀 (𝐸)) we have ∫ 𝐹 (𝜇)𝐾𝑡◦ (d𝜇) + 𝐹 (0)𝐾𝑡 ({0}) 𝐾𝑡 (𝐹) = ◦ (𝐸) 𝑀 ∫ h i 𝑄 ◦𝑡−𝑟 𝐹 (𝜇)𝐾𝑟◦ (d𝜇) + 𝐹 (0) ↑lim 𝐾𝑠◦ (1) − 𝐾𝑡◦ (1) = 𝑠↓0 ∫𝑀 (𝐸) ◦ ∫ 𝑄 𝑡−𝑟 𝐹 (𝜇)𝐾𝑟◦ (d𝜇) − 𝑄 𝑡−𝑟 (𝜇, {0})𝐹 (0)𝐾𝑟◦ (d𝜇) = 𝑀 (𝐸) ◦ ∫ 𝑀 (𝐸) ◦ + 𝐹 (0) ↑lim 𝐾𝑠◦ (1) − 𝐹 (0) 𝑄 ◦𝑡−𝑟 (𝜇, 𝑀 (𝐸) ◦ )𝐾𝑟◦ (d𝜇) ◦ 𝑠↓0 𝑀 (𝐸) ∫ h i = 𝑄 𝑡−𝑟 𝐹 (𝜇)𝐾𝑟◦ (d𝜇) + 𝐹 (0) ↑lim 𝐾𝑠◦ (1) − 𝐾𝑟◦ (1) 𝑠↓0 ∫𝑀 (𝐸) ◦ ◦ 𝑄 𝑡−𝑟 𝐹 (𝜇)𝐾𝑟 (d𝜇) + 𝑄 𝑡−𝑟 𝐹 (0)𝐾𝑟 ({0}) = ∫𝑀 (𝐸) ◦ 𝑄 𝑡−𝑟 𝐹 (𝜇)𝐾𝑟 (d𝜇). = 𝑀 (𝐸)

Thus (𝐾𝑡 )𝑡 >0 is a bounded entrance law for (𝑄 𝑡 )𝑡 ≥0 . Conversely, suppose that (𝐾𝑡 )𝑡 >0 is a bounded entrance law for (𝑄 𝑡 )𝑡 ≥0 satisfying lim𝑡→0 𝐾𝑡 ({0}) = 0. It is easy to see that 𝐾𝑡◦ = 𝐾𝑡 | 𝑀 (𝐸) ◦ defines a bounded entrance law (𝐾𝑡◦ )𝑡 >0 for (𝑄 ◦𝑡 )𝑡 ≥0 . Moreover, we have 𝐾𝑡 ({0}) = 𝐾𝑡 (1) − 𝐾𝑡 (𝑀 (𝐸) ◦ ) = ↑lim 𝐾𝑠◦ (1) − 𝐾𝑡◦ (1). 𝑠↓0

This proves the one-to-one correspondence.

□

Theorem 8.2 Let 𝐾 = (𝐾𝑡 )𝑡 >0 be a family of infinitely divisible probability measures on 𝑀 (𝐸) with 𝐼𝑡 := − log 𝐿 𝐾𝑡 ∈ ℐ(𝐸) given by ∫
𝐼𝑡 ( 𝑓 ) = 𝜂𝑡 ( 𝑓 ) + 1 − e−𝜈 ( 𝑓 ) 𝐻𝑡 (d𝜈), 𝑓 ∈ 𝐵(𝐸) + , (8.2) 𝑀 (𝐸) ◦

where 𝜂𝑡 ∈ 𝑀 (𝐸) and 𝐻𝑡 (d𝜈) is a 𝜎-finite measure on 𝑀 (𝐸) ◦ satisfying ∫ [1 ∧ 𝜈(1)]𝐻𝑡 (d𝜈) < ∞.

(8.3)

𝑀 (𝐸) ◦

Then 𝐾 is an entrance law for (𝑄 𝑡 )𝑡 ≥0 if and only if, for all 𝑟, 𝑡 > 0, ∫ ∫ 𝜂𝑟+𝑡 = 𝜂𝑟 (d𝑦)𝜆 𝑡 (𝑦, ·), 𝐻𝑟+𝑡 = 𝜂𝑟 (d𝑦)𝐿 𝑡 (𝑦, ·) + 𝐻𝑟 𝑄 ◦𝑡 . 𝐸

(8.4)

𝐸

Proof By Theorem 1.36 the family of infinitely divisible probability measures (𝐾𝑡 )𝑡 >0 on 𝑀 (𝐸) can be represented by (8.2). By Proposition 2.6 one can see (8.4) gives an alternative expression for the relation 𝐾𝑟+𝑡 = 𝐾𝑟 𝑄 𝑡 for 𝑟, 𝑡 > 0. □

8.1 Some Simple Properties

207

Corollary 8.3 Suppose that 𝐾 = (𝐾𝑡 )𝑡 >0 is an infinitely divisible probability entrance law for (𝑄 𝑡 )𝑡 ≥0 given by (8.2). Then the family 𝐻 = (𝐻𝑡 )𝑡 >0 is an entrance rule for the restricted semigroup (𝑄 ◦𝑡 )𝑡 ≥0 . Corollary 8.4 Suppose that 𝐻 = (𝐻𝑡 )𝑡 >0 is a 𝜎-finite entrance law for (𝑄 ◦𝑡 )𝑡 ≥0 satisfying (8.3). Then  ∫  ∫
(8.5) 1 − e−𝜈 ( 𝑓 ) 𝐻𝑡 (d𝜈) e−𝜈 ( 𝑓 ) 𝐾𝑡 (d𝜈) = exp − 𝑀 (𝐸) ◦

𝑀 (𝐸)

defines an infinitely divisible probability entrance law 𝐾 = (𝐾𝑡 )𝑡 >0 for (𝑄 𝑡 )𝑡 ≥0 . Corollary 8.5 If 𝐸𝐶 = 𝐸, then (8.5) establishes a one-to-one correspondence between infinitely divisible probability entrance laws 𝐾 for (𝑄 𝑡 )𝑡 ≥0 and 𝜎-finite entrance laws 𝐻 for (𝑄 ◦𝑡 )𝑡 ≥0 satisfying (8.3). In the situation of Corollary 8.3, we call 𝐻 = (𝐻𝑡 )𝑡 >0 the canonical entrance rule defined by 𝐾 = (𝐾𝑡 )𝑡 >0 . We next turn to the special case of a (𝜉, 𝜙)-superprocess 𝑋, where 𝜉 is a Borel right process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 and 𝜙 is a branching mechanism given by (2.29) or (2.30). The transition semigroup (𝑄 𝑡 )𝑡 ≥0 of the (𝜉, 𝜙)-superprocess is defined by (2.35) and (2.36). Let 𝛾(𝑥, d𝑦) be the kernel on 𝐸 defined by (2.31) and let (𝜋𝑡 )𝑡 ≥0 be the semigroup of kernels defined by (2.38). Let 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − + 𝑏(𝑥)] and let 𝑐+0 = 0∨ 𝑐 0 . By Theorem A.53 we have ∥𝜋𝑡 ∥ ≤ e𝑐0 𝑡 ≤ e𝑐0 𝑡 for 𝑡 ≥ 0. To study the structures of entrance laws for the (𝜉, 𝜙)-superprocess, we need to clarify some connections between entrance laws for the underlying semigroup (𝑃𝑡 )𝑡 ≥0 and those for (𝜋𝑡 )𝑡 ≥0 . Let 𝒦(𝑃) be the set of entrance laws 𝜅 = (𝜅 𝑡 )𝑡 >0 for (𝑃𝑡 )𝑡 ≥0 satisfying ∫

1

𝜅 𝑠 (1)d𝑠 < ∞

(8.6)

0

and let 𝒦(𝜋) be the set of entrance laws for (𝜋𝑡 )𝑡 ≥0 satisfying the above integral condition. We remark that the measures (𝜅 𝑡 )𝑡 >0 are finite for any 𝜅 ∈ 𝒦(𝑃) or 𝒦(𝜋). Indeed, by (8.6) for any 𝑡 > 0 there is an 𝑟 ∈ (0, 𝑡] such that 𝜅𝑟 (1) < ∞. Then 𝜅 𝑡 (1) = 𝜅𝑟 (𝑃𝑡−𝑟 1) ≤ 𝜅𝑟 (1) < ∞ if 𝜅 ∈ 𝒦(𝑃) and 𝜅 𝑡 (1) = 𝜅𝑟 (𝜋𝑡−𝑟 1) ≤ e𝑐0 (𝑡−𝑟) 𝜅𝑟 (1) < ∞ if 𝜅 ∈ 𝒦(𝜋). In particular, if (𝑃𝑡 )𝑡 ≥0 is a conservative semigroup, then 𝒦(𝑃) coincides with the space of bounded entrance laws for (𝑃𝑡 )𝑡 ≥0 . Proposition 8.6 There is a one-to-one correspondence between 𝜅 ∈ 𝒦(𝑃) and 𝜂 ∈ 𝒦(𝜋) given by, for 𝑡 > 0 and 𝑓 ∈ 𝐵(𝐸), 𝜂𝑡 ( 𝑓 ) = lim 𝜅𝑟 (𝜋𝑡−𝑟 𝑓 ) and 𝜅 𝑡 ( 𝑓 ) = lim 𝜂𝑟 (𝑃𝑡−𝑟 𝑓 ). 𝑟→0

(8.7)

𝑟→0

Moreover, if the two entrance laws are related by (8.7), we have ∫ 𝑡 𝜅 𝑡−𝑠 ((𝛾 − 𝑏)𝜋 𝑠 𝑓 )d𝑠. 𝜂𝑡 ( 𝑓 ) = 𝜅 𝑡 ( 𝑓 ) + 0

(8.8)

208

8 Entrance Laws and Kuznetsov Measures

Proof Suppose that 𝜅 ∈ 𝒦(𝑃). For 𝑡 > 𝑟 > 0 and 𝑓 ∈ 𝐵(𝐸) we can use (2.38) and the entrance law property of 𝜅 = (𝜅 𝑡 )𝑡 >0 to see ∫ 𝑡−𝑟 𝜅𝑟 (𝜋𝑡−𝑟 𝑓 ) = 𝜅 𝑡 ( 𝑓 ) + 𝜅 𝑡−𝑠 ((𝛾 − 𝑏)𝜋 𝑠 𝑓 )d𝑠. (8.9) 0

Then the first limit in (8.7) exists and is given by (8.8). Clearly, the family 𝜂 = (𝜂𝑡 )𝑡 >0 constitute an entrance law for (𝜋𝑡 )𝑡 ≥0 . Moreover, we have ∫ 𝑡 ∫ 𝑡 + 𝜅 𝑡−𝑠 (𝜋 𝑠 1)d𝑠 ≤ 𝜅 𝑡 (1) + 𝑐+0 e𝑐0 𝑡 𝜂𝑡 (1) ≤ 𝜅 𝑡 (1) + 𝑐+0 𝜅 𝑠 (1)d𝑠, 0

0

and hence 𝜂 ∈ 𝒦(𝜋). From (8.8) and the entrance law property of (𝜅 𝑡 )𝑡 >0 it follows that ∫ 𝑟 𝜅𝑟−𝑠 ((𝛾 − 𝑏)𝜋 𝑠 𝑃𝑡−𝑟 𝑓 )d𝑠. 𝜂𝑟 (𝑃𝑡−𝑟 𝑓 ) = 𝜅 𝑡 ( 𝑓 ) + 0

By letting 𝑟 → 0 we obtain the second equality in (8.7). Conversely, suppose that 𝜂 ∈ 𝒦(𝜋). For 𝑓 ∈ 𝐵(𝐸) + we get from (2.37) and (2.39) that e− ∥𝑏

+ ∥𝑡

𝑃𝑡 𝑓 (𝑥) ≤ 𝑃𝑡𝑏 𝑓 (𝑥) ≤ 𝜋𝑡 𝑓 (𝑥).

(8.10)

Then for any 𝑡 > 𝑠 > 𝑟 > 0 we have e ∥𝑏

+ ∥𝑟

𝜂𝑟 (𝑃𝑡−𝑟 𝑓 ) ≤ e ∥𝑏

+ ∥𝑠

𝜂𝑟 (𝜋 𝑠−𝑟 𝑃𝑡−𝑠 𝑓 ) = e ∥𝑏

+ ∥𝑠

𝜂 𝑠 (𝑃𝑡−𝑠 𝑓 ).

Consequently, we can define an entrance law 𝜅 = (𝜅 𝑡 )𝑡 >0 for (𝑃𝑡 )𝑡 ≥0 by 𝜅 𝑡 ( 𝑓 ) = lim e ∥𝑏

+ ∥𝑟

𝜂𝑟 (𝑃𝑡−𝑟 𝑓 ) = lim 𝜂𝑟 (𝑃𝑡−𝑟 𝑓 ).

(8.11)

𝑟→0

𝑟→0

Clearly, the above relation also holds for all 𝑓 ∈ 𝐵(𝐸). In view of (8.10) and (8.11), we have 𝜅 𝑡 (1) = lim 𝜂𝑟 (𝑃𝑡−𝑟 1) ≤ lim e ∥𝑏 𝑟→0

+ ∥𝑡

𝜂𝑟 (𝜋𝑡−𝑟 1) ≤ e ∥𝑏

+ ∥𝑡

𝜂𝑡 (1),

𝑟→0

and hence 𝜅 ∈ 𝒦(𝑃). Then we use (2.38) and the entrance law property of (𝜂𝑡 )𝑡 >0 to see ∫ 𝑡−𝑟 𝜂𝑡 ( 𝑓 ) = 𝜂𝑟 (𝑃𝑡−𝑟 𝑓 ) + 𝜂𝑟 (𝑃𝑡−𝑟−𝑠 (𝛾 − 𝑏)𝜋 𝑠 𝑓 )d𝑠. 0

By letting 𝑟 → 0 in both sides we get (8.8). The first equality in (8.7) follows from □ (8.9).

8.2 Minimal Probability Entrance Laws

209

The above proof also gives the following: Corollary 8.7 If 𝜅 ∈ 𝒦(𝑃) and 𝜂 ∈ 𝒦(𝜋) are related by (8.7) and (8.8), then for every 𝑡 > 0 we have ∫ 𝑡 − ∥𝑏+ ∥𝑡 + 𝑐0+ 𝑡 𝜅 𝑠 (1)d𝑠. 𝜅 𝑡 (1) ≤ 𝜂𝑡 (1) ≤ 𝜅 𝑡 (1) + 𝑐 0 e e (8.12) 0

Let (𝑄 𝑡 )𝑡 ≥0 denote the transition semigroup of the (𝜉, 𝜙)-superprocess defined by (2.35) and (2.36). Let 𝒦(𝑄) be the set of 𝜎-finite entrance laws 𝐾 = (𝐾𝑡 )𝑡 >0 for the semigroup (𝑄 𝑡 )𝑡 ≥0 satisfying ∫

1

∫ 𝜈(1)𝐾𝑠 (d𝜈) < ∞

d𝑠 0

(8.13)

𝑀 (𝐸) ◦

and let 𝒦(𝑄 ◦ ) be the set of entrance laws for the restricted semigroup (𝑄 ◦𝑡 )𝑡 ≥0 satisfying the above integral condition. By Corollary 2.28 we have (2.53) with (𝜋𝑡 )𝑡 ≥0 defined by (2.38). By Corollary 8.7 it is simple to check that for any 𝐾 ∈ 𝒦(𝑄) or 𝒦(𝑄 ◦ ) we can define 𝜂 := 𝜋𝐾 ∈ 𝒦(𝜋) and 𝜅 := 𝑝𝐾 ∈ 𝒦(𝑃) by, for 𝑡 > 0 and 𝑓 ∈ 𝐵(𝐸), ∫ 𝜈( 𝑓 )𝐾𝑡 (d𝜈) (8.14) 𝜂𝑡 ( 𝑓 ) = 𝑀 (𝐸) ◦

and ∫ 𝜈(𝑃𝑡−𝑟 𝑓 )𝐾𝑟 (d𝜈).

𝜅 𝑡 ( 𝑓 ) = lim 𝑟→0

(8.15)

𝑀 (𝐸) ◦

8.2 Minimal Probability Entrance Laws Suppose that 𝜉 is a Borel right process in the Lusin topological space 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 and 𝜙 is a branching mechanism given by (2.29) or (2.30). Let (𝑄 𝑡 )𝑡 ≥0 and (𝑉𝑡 )𝑡 ≥0 denote respectively the transition semigroup and the cumulant semigroup of the (𝜉, 𝜙)-superprocess. Given 𝜅 ∈ 𝒦(𝑃) we set ∫ 𝑡 ∫ 𝜙(𝑦, 𝑉𝑠 𝑓 )𝜅 𝑡−𝑠 (d𝑦) 𝑆𝑡 (𝜅, 𝑓 ) = 𝜅 𝑡 ( 𝑓 ) − (8.16) d𝑠 0

𝐵(𝐸) + .

for 𝑡 > 0 and 𝑓 ∈ have 𝑆𝑡 (𝜅, 𝑓 ) = 𝜇(𝑉𝑡 𝑓 ).

𝐸

In particular, if 𝜅 ∈ 𝒦(𝑃) is closed by 𝜇 ∈ 𝑀 (𝐸), we

Lemma 8.8 If 𝜅 ∈ 𝒦(𝑃) and 𝜂 ∈ 𝒦(𝜋) are related by (8.7), then for any 𝑡 > 0 and 𝑓 ∈ 𝐵(𝐸) + we have 𝑆𝑡 (𝜅, 𝑓 ) = lim 𝜅𝑟 (𝑉𝑡−𝑟 𝑓 ) = ↓lim 𝜂𝑟 (𝑉𝑡−𝑟 𝑓 ). 𝑟→0

𝑟→0

(8.17)

210

8 Entrance Laws and Kuznetsov Measures

Proof By (2.36) for 𝑡 > 𝑟 > 0 and 𝑓 ∈ 𝐵(𝐸) + we have ∫ 𝑡−𝑟 ∫ 𝜅𝑟 (𝑉𝑡−𝑟 𝑓 ) = 𝜅 𝑡 ( 𝑓 ) − d𝑠 𝜙(𝑦, 𝑉𝑠 𝑓 )𝜅 𝑡−𝑠 (d𝑦). 0

𝐸

Then the first equality in (8.17) holds. The second equality follows similarly from □ (2.41) and (8.8). Lemma 8.9 The entrance law 𝜅 ∈ 𝒦(𝑃) is non-trivial if and only if we have lim𝑡→0 lim 𝜃→∞ 𝑆𝑡 (𝜅, 𝜃) = ∞. Proof From (8.17) we see 𝑓 ↦→ 𝑆𝑡 (𝜅, 𝑓 ) is an increasing functional, so the limit lim 𝜃→∞ 𝑆𝑡 (𝜅, 𝜃) exists in [0, ∞]. By (8.16) for any 𝜃 0 ≥ 0 we have lim inf lim 𝑆𝑡 (𝜅, 𝜃) ≥ lim 𝑆𝑡 (𝜅, 𝜃 0 ) = lim 𝜅 𝑡 (𝜃 0 ) = 𝜃 0 lim 𝜅 𝑡 (1). 𝑡→0

𝜃→∞

𝑡→0

𝑡→0

𝑡→0

If 𝜅 ∈ 𝒦(𝑃) is non-trivial, then lim𝑡→0 𝜅 𝑡 (1) > 0 and hence lim lim 𝑆𝑡 (𝜅, 𝜃) = ∞. 𝑡→0 𝜃→∞

If 𝜅 ∈ 𝒦(𝑃) is trivial, then 𝑆𝑡 (𝜅, 𝜃) = 0 for all 𝑡 > 0 and 𝜃 ≥ 0.

□

For 0 < 𝑎 ≤ ∞ write 𝐾 ∈ 𝒦 𝑎 (𝑄) if 𝐾 ∈ 𝒦(𝑄) and 𝐾𝑡 (1) = 𝑎 for all 𝑡 > 0. Similarly, we write 𝐾 ∈ 𝒦 𝑎 (𝑄 ◦ ) if 𝐾 ∈ 𝒦(𝑄 ◦ ) and lim𝑡→0 𝐾𝑡 (1) = 𝑎. Let 𝒦𝑚𝑎 (𝑄) and 𝒦𝑚𝑎 (𝑄 ◦ ) denote the sets of minimal elements of 𝒦 𝑎 (𝑄) and 𝒦 𝑎 (𝑄 ◦ ), respectively. Let 𝒦(𝑃) ◦ = 𝒦(𝑃) \{0}, where 0 is the trivial entrance law of (𝑃𝑡 )𝑡 ≥0 . We refer the reader to Dynkin (1978, Section 10) and Sharpe (1988, Section 40) for discussions of the structures of entrance laws for Markov processes. Theorem 8.10 To each 𝜅 ∈ 𝒦(𝑃) there corresponds an entrance law 𝐾 := 𝑙𝜅 ∈ 𝒦𝑚1 (𝑄) given by ∫ e−𝜈 ( 𝑓 ) 𝐾𝑡 (d𝜈) = exp{−𝑆𝑡 (𝜅, 𝑓 )}, 𝑡 > 0, 𝑓 ∈ 𝐵(𝐸) + . (8.18) 𝑀 (𝐸)

Moreover, (8.15) and (8.18) give a one-to-one correspondence between 𝒦(𝑃) and 𝒦𝑚1 (𝑄). Proof Step 1. Suppose that 𝜅 ∈ 𝒦(𝑃) and 𝜂 ∈ 𝒦(𝜋) are related by (8.7). By Lemma 8.8 we have exp{−𝑆𝑡 (𝜅, 𝑓 )} = lim exp{−𝜅𝑟 (𝑉𝑡−𝑟 𝑓 )} 𝑟→0 ∫ = lim e−𝜈 ( 𝑓 ) 𝑄 𝑡−𝑟 (𝜅𝑟 , d𝜈). 𝑟→0

(8.19)

𝑀 (𝐸)

Then an application of Theorem 1.21 shows that (8.18) really defines a family of probability measures 𝐾 = (𝐾𝑡 )𝑡 >0 on 𝑀 (𝐸). By (2.36) and (8.16) it is easy to show that 𝑆𝑟+𝑡 (𝜅, 𝑓 ) = 𝑆𝑟 (𝜅, 𝑉𝑡 𝑓 ), so 𝐾 is an entrance law for (𝑄)𝑡 ≥0 . In view of

8.2 Minimal Probability Entrance Laws

211

(8.16) and (8.8) we have (d/d𝜃)𝑆𝑡 (𝜅, 𝜃 𝑓 )| 𝜃=0+ = 𝜂𝑡 ( 𝑓 ) and hence (8.14) holds. In particular, we have 𝐾 ∈ 𝒦 1 (𝑄). Write 𝐾 = 𝑙𝜅 = 𝜆𝜂. By Proposition 8.6 we have 𝜋𝐾 = 𝜂 and 𝑝𝐾 = 𝜅. Therefore 𝑝𝑙𝜅 = 𝜅 for 𝜅 ∈ 𝒦(𝑃) and 𝜋𝜆𝜂 = 𝜂 for 𝜂 ∈ 𝒦(𝜋). Step 2. We claim 𝐾 = 𝜆𝜋𝐾 = 𝑙 𝑝𝐾 for every 𝐾 ∈ 𝒦𝑚1 (𝑄). To see this let Q𝐾 be the probability measure on 𝑀 (𝐸) (0,∞) under which the coordinate process {𝑤 𝑡 : 𝑡 > 0} is a Markov process with one-dimensional distributions (𝐾𝑡 )𝑡 >0 and semigroup (𝑄 𝑡 )𝑡 ≥0 . Since 𝐾 is minimal, by Dynkin (1978, p. 724) or Sharpe (1988, p. 199) we have Q𝐾 -a.s. ∫ e−𝜈 ( 𝑓 ) 𝐾𝑡 (d𝜈) = lim exp{−𝑤 𝑟𝑛 (𝑉𝑡−𝑟𝑛 𝑓 )} (8.20) 𝑛→∞

𝑀 (𝐸)

for any sequence 𝑟 𝑛 → 0. By (2.53),   𝑤 𝑟𝑛 (𝑉𝑡−𝑟𝑛 𝑓 ) ≤ 𝑤 𝑟𝑛 (𝜋𝑡−𝑟𝑛 𝑓 ) = Q𝐾 𝑤 𝑡 ( 𝑓 ) 𝑤 𝑠 : 0 < 𝑠 ≤ 𝑟 𝑛 . Then the family of random variables {𝑤 𝑟𝑛 (𝑉𝑡−𝑟𝑛 𝑓 ) : 0 < 𝑟 𝑛 ≤ 𝑡} is uniformly Q𝐾 -integrable. By (8.20) and dominated convergence we have ∫   − log e−𝜈 ( 𝑓 ) 𝐾𝑡 (d𝜈) = lim Q𝐾 𝑤 𝑟𝑛 (𝑉𝑡−𝑟𝑛 𝑓 ) 𝑀 (𝐸)

𝑛→∞

= lim 𝜋𝐾𝑟𝑛 (𝑉𝑡−𝑟𝑛 𝑓 ) = 𝑆𝑡 ( 𝑝𝐾, 𝑓 ), 𝑛→∞

where the last equality follows by Lemma 8.8. This proves 𝐾 = 𝑙 𝑝𝐾. Then the results in the first step imply 𝐾 = 𝜆𝜋𝐾. Step 3. Now it suffices to show 𝑙𝜅 ∈ 𝒦𝑚1 (𝑄) for all 𝜅 ∈ 𝒦(𝑃). By Dynkin (1978, p. 723) there is a probability measure 𝐹 on 𝒦𝑚1 (𝑄) such that ∫ 𝑙𝜅 𝑡 = 𝐻𝑡 𝐹 (d𝐻). 1 (𝑄) 𝒦𝑚

Let 𝐺 be the image of 𝐹 under the mapping 𝑝 : 𝒦𝑚1 (𝑄) → 𝒦(𝑃). By the results proved in the first two steps it follows that ∫ exp{−𝑆𝑡 (𝜇, 𝑓 )}𝐺 (d𝜇). exp{−𝑆𝑡 (𝜅, 𝑓 )} = 𝒦 ( 𝑃)

Since 𝑢 ↦→ e−𝑢 is a strictly convex function, 𝐺 must be the unit mass concentrated at 𝜅. Then 𝐹 is the unit mass at 𝑙𝜅, yielding 𝑙𝜅 ∈ 𝒦𝑚1 (𝑄). □ Corollary 8.11 For any 𝜅 ∈ 𝒦(𝑃) the entrance law 𝐾 ∈ 𝒦𝑚1 (𝑄) given by (8.18) is infinitely divisible. Proof In view of (8.19) we have 𝐾𝑡 = lim𝑟→0 𝑄 𝑡−𝑟 (𝜅𝑟 , ·). Then the infinite divisibility of 𝐾𝑡 follows from that of 𝑄 𝑡−𝑟 (𝜅𝑟 , ·). □

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8 Entrance Laws and Kuznetsov Measures

Corollary 8.12 There is a one-to-one correspondence between 𝜅 ∈ 𝒦(𝑃) ◦ and 𝐾 ◦ ∈ 𝒦𝑚1 (𝑄 ◦ ) given by (8.15) and ∫ (8.21) (1 − e−𝜈 ( 𝑓 ) )𝐾𝑡◦ (d𝜈) = 1 − exp{−𝑆𝑡 (𝜅, 𝑓 )}, 𝑀 (𝐸) ◦

where 𝑡 > 0 and 𝑓 ∈ 𝐵(𝐸) + . Proof For the entrance laws 𝐾 ∈ 𝒦 1 (𝑄) and 𝐾 ◦ ∈ 𝒦 1 (𝑄 ◦ ) related by (8.1) one can see that 𝐾 ∈ 𝒦𝑚1 (𝑄) if and only if 𝐾 ◦ ∈ 𝒦𝑚1 (𝑄 ◦ ). On the other hand, for the entrance laws 𝜅 ∈ 𝒦(𝑃) and 𝐾 ∈ 𝒦𝑚1 (𝑄) related by (8.18) we have ∫ e−𝜈 ( 𝜃) 𝐾𝑡 (d𝜈) = lim exp{−𝑆𝑡 (𝜅, 𝜃)}. 𝐾𝑡 ({0}) = lim 𝜃→∞

𝑀 (𝐸)

𝜃→∞

By Lemma 8.9, we have lim𝑡→0 𝐾𝑡 ({0}) = 0 if and only if 𝜅 ∈ 𝒦(𝑃) is non-trivial. □ Then the result follows from Theorem 8.10. Theorem 8.13 For any 𝑥 ∈ 𝐸 the canonical entrance rule 𝐿 (𝑥) = {𝐿 𝑡 (𝑥, ·) : 𝑡 > 0} defined by (2.5) is regular. Proof Step 1. We first consider the case where (𝑃𝑡 )𝑡 ≥0 is a conservative semigroup. For 𝑓 , ℎ ∈ 𝐵(𝐸) + and a functional 𝑈 on 𝐵(𝐸) + let Δℎ𝑈 ( 𝑓 ) = 𝑈 ( 𝑓 + ℎ) − 𝑈 ( 𝑓 ). Write Δ2ℎ = Δℎ Δℎ . Then Δ2ℎ𝑈 ( 𝑓 ) = 𝑈 ( 𝑓 + 2ℎ) − 2𝑈 ( 𝑓 + ℎ) + 𝑈 ( 𝑓 ). From (2.5) it is easy to see that ∫ Δ2ℎ𝑉𝑡 ( 𝑓 ) (𝑥) =

(1 − e−𝜈 ( 𝑓 ) ) (1 − e−𝜈 (ℎ) ) 2 𝐿 𝑡 (𝑥, d𝜈).

(8.22)

(8.23)

𝑀 (𝐸) ◦

Since (𝑃𝑡 )𝑡 ≥0 is conservative, by (2.36) we have lim𝑡 ↓0 ∥𝑉𝑡 𝜆−𝜆∥ = 0 for any constant 𝜆 ≥ 0. By Corollary 2.25, for 𝑡 > 𝑟 ≥ 0, ∥𝑉𝑡 𝜆 − 𝑉𝑟 𝜆∥ = ∥𝑉𝑟 𝑉𝑡−𝑟 𝜆 − 𝑉𝑟 𝜆∥ ≤ ∥𝜋𝑟 ∥ ∥𝑉𝑡−𝑟 𝜆 − 𝜆∥, where ∥𝜋𝑟 ∥ ≤ e𝑐0 𝑟 by Theorem A.53. Then 𝑡 ↦→ 𝑉𝑡 𝜆 is continuous in the supremum norm. In view of (8.22), we have ∥Δ2𝜆𝑉𝑟 (𝑉𝑡−𝑟 1) − Δ𝜆2 𝑉𝑡 (1) ∥ ≤ ∥Δ2𝜆𝑉𝑟 (𝑉𝑡−𝑟 1) − Δ𝜆2 𝑉𝑟 (1) ∥ + ∥Δ2𝜆𝑉𝑟 (1) − Δ𝜆2 𝑉𝑡 (1) ∥ ≤ ∥𝑉𝑟 (𝑉𝑡−𝑟 1 + 2𝜆) − 2𝑉𝑟 (𝑉𝑡−𝑟 1 + 𝜆) + 𝑉𝑟 (𝑉𝑡−𝑟 1) − 𝑉𝑟 (1 + 2𝜆) + 2𝑉𝑟 (1 + 𝜆) − 𝑉𝑟 (1) ∥ + ∥𝑉𝑟 (1 + 2𝜆) − 2𝑉𝑟 (1 + 𝜆) + 𝑉𝑟 (1) − 𝑉𝑡 (1 + 2𝜆) + 2𝑉𝑡 (1 + 𝜆) − 𝑉𝑡 (1) ∥

8.2 Minimal Probability Entrance Laws

213

≤ 4∥𝜋𝑟 ∥ ∥𝑉𝑡−𝑟 1 − 1∥ + ∥𝑉𝑟 (1 + 2𝜆) − 𝑉𝑡 (1 + 2𝜆) ∥ + 2∥𝑉𝑟 (1 + 𝜆) − 𝑉𝑡 (1 + 𝜆) ∥ + ∥𝑉𝑟 (1) − 𝑉𝑡 (1) ∥. The right-hand side vanishes as 𝑟 ↑ 𝑡. It follows that, in the supremum norm, lim Δ𝜆2 𝑉𝑟 (𝑉𝑡−𝑟 1) = lim Δ𝜆2 𝑉𝑟 (1) = Δ2𝜆𝑉𝑡 (1). 𝑟 ↑𝑡

𝑟 ↑𝑡

Then (8.23) implies ∫

(1 − e−𝜇 (𝑉𝑡−𝑟 1) )𝐿 𝑟 (𝑥, d𝜇)

lim inf 𝑟 ↑𝑡

𝑀 (𝐸) ∫◦

≥ lim ∫𝑟 ↑𝑡 =

(1 − e−𝜇 (𝑉𝑡−𝑟 1) ) (1 − e−𝜆𝜇 (1) ) 2 𝐿 𝑟 (𝑥, d𝜇)

𝑀 (𝐸) ◦

(1 − e−𝜇 (1) ) (1 − e−𝜆𝜇 (1) ) 2 𝐿 𝑡 (𝑥, d𝜇).

𝑀 (𝐸) ◦

Since 𝜆 ≥ 0 was arbitrary, it follows that ∫ ∫ lim inf 𝐿 𝑟 (𝑥, d𝜇) (1 − e−𝜈 (1) )𝑄 ◦𝑡−𝑟 (𝜇, d𝜈) 𝑟 ↑𝑡 𝑀 (𝐸) ◦∫ 𝑀 (𝐸) ◦ (1 − e−𝜇 (𝑉𝑡−𝑟 1) )𝐿 𝑟 (𝑥, d𝜇) = lim inf ◦ 𝑟 ↑𝑡 𝑀 (𝐸) ∫ (1 − e−𝜇 (1) )𝐿 𝑡 (𝑥, d𝜇), ≥ 𝑀 (𝐸) ◦

which implies the regularity of 𝐿(𝑥).

Step 2. In the general case where (𝑃𝑡 )𝑡 ≥0 is not conservative, we can extend it to a conservative Borel right semigroup ( 𝑃˜𝑡 )𝑡 ≥0 on the Lusin topological space ˜ + let 𝜙(𝜕, ˜ 𝐸˜ := 𝐸 ∪ {𝜕} with 𝜕 being an isolated cemetery. For 𝑓˜ ∈ 𝐵( 𝐸) 𝑓˜) = 0 ˜ and let 𝜙(𝑥, 𝑓˜) = 𝜙(𝑥, 𝑓˜| 𝐸 ) if 𝑥 ∈ 𝐸. Let (𝑉˜𝑡 )𝑡 ≥0 and ( 𝑄˜ 𝑡 )𝑡 ≥0 be defined as in the proof of Theorem 5.13. Let 𝜆˜ 𝑡 (𝑥, ·) and 𝐿˜ 𝑡 (𝑥, ·) be defined by (2.5) from (𝑉˜𝑡 )𝑡 ≥0 . Then 𝐿˜ (𝑥) = { 𝐿˜ 𝑡 (𝑥, ·) : 𝑡 > 0} is a regular entrance rule for (𝑄˜ ◦𝑡 )𝑡 ≥0 by the first step. Since 𝜕 is a cemetery, we have 𝑉˜𝑡 𝑓˜(𝜕) = 𝑃˜𝑡 𝑓˜(𝜕) = 0 if 𝑓˜(𝜕) = 0. For any 𝑓 ∈ 𝐵(𝐸) + we extend its definition to 𝐸˜ by setting 𝑓 (𝜕) = 0. Then 𝑉˜𝑡 𝑓 (𝑥) = 𝑉𝑡 𝑓 (𝑥) ˜ and 𝑓 ∈ 𝐵(𝐸) + , for 𝑡 ≥ 0 and 𝑥 ∈ 𝐸. It follows that, for 𝜇 ∈ 𝑀 ( 𝐸) ∫ ∫ −𝜈 ( 𝑓 ) ˜ e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇| 𝐸 , d𝜈). e 𝑄 𝑡 (𝜇, d𝜈) = ˜ 𝑀 ( 𝐸)

𝑀 (𝐸)

For any 𝑥 ∈ 𝐸 and 𝑓 ∈ 𝐵(𝐸) + we have ∫ 𝑉˜𝑡 𝑓 (𝑥) = 𝜆˜ 𝑡 (𝑥, 𝑓 ) + ˜ ◦ 𝑀 ( 𝐸)

(1 − e−𝜈 ( 𝑓 ) ) 𝐿˜ 𝑡 (𝑥, d𝜈).

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8 Entrance Laws and Kuznetsov Measures

By the uniqueness of the canonical representation we have ∫ ∫

1 − e−𝜈 ( 𝑓 ) 𝐿˜ 𝑡 (𝑥, d𝜈). 1 − e−𝜈 ( 𝑓 ) 𝐿 𝑡 (𝑥, d𝜈) = ˜ ◦ 𝑀 ( 𝐸)

𝑀 (𝐸) ◦

It follows that, for 𝑡 > 𝑟 > 0, ∫ ∫ (1 − e−𝜈 (1) )𝑄 ◦𝑡−𝑟 (𝜇, d𝜈) 𝐿 𝑟 (𝑥, d𝜇) 𝑀 (𝐸) ◦∫ 𝑀 (𝐸) ◦∫ = 𝐿˜ 𝑟 (𝑥, d𝜇) (1 − e−𝜈 (1𝐸 ) ) 𝑄˜ ◦𝑡−𝑟 (𝜇, d𝜈). ˜ ◦ 𝑀 ( 𝐸)

˜ ◦ 𝑀 ( 𝐸)

The right-hand side converges as 𝑟 ↑ 𝑡 to ∫ ∫ (1 − e−𝜈 (1𝐸 ) ) 𝐿˜ 𝑡 (𝑥, d𝜈) = ˜ ◦ 𝑀 ( 𝐸)

(1 − e−𝜈 (1) )𝐿 𝑡 (𝑥, d𝜈).

𝑀 (𝐸) ◦

Then 𝐿 (𝑥) is a regular entrance rule for (𝑄 ◦𝑡 )𝑡 ≥0 .

□

Corollary 8.14 For any 𝜇 ∈ 𝑀 (𝐸) the entrance rule 𝜇𝐿 = (𝜇𝐿 𝑡 )𝑡 >0 is regular. In the special case where the underlying semigroup (𝑃𝑡 )𝑡 ≥0 is conservative, let 𝐸¯ be a Ray–Knight completion of 𝐸 with respect to this semigroup. Let ( 𝑃¯𝑡 )𝑡 ≥0 be ¯ Let 𝐸 𝐷 ⊂ 𝐸¯ be the entrance space of (𝑃𝑡 )𝑡 ≥0 . the Ray extension of (𝑃𝑡 )𝑡 ≥0 to 𝐸. In this chapter, we only need the restriction of ( 𝑃¯𝑡 )𝑡 ≥0 to 𝐸 𝐷 , which is also a Borel ¯ 𝑓¯) from 𝐵(𝐸 𝐷 ) + right semigroup. We extend 𝑓 ↦→ 𝜙(·, 𝑓 ) to an operator 𝑓¯ ↦→ 𝜙(·, ¯ ¯ 𝑓¯) = 0 for 𝑥 ∈ 𝐸 𝐷 \ 𝐸, 𝑓¯) = 𝜙(𝑥, 𝑓 ) for 𝑥 ∈ 𝐸 and 𝜙(𝑥, to 𝐵(𝐸 𝐷 ) by setting 𝜙(𝑥, where 𝑓 = 𝑓¯| 𝐸 is the restriction to 𝐸 of 𝑓¯ ∈ 𝐵(𝐸 𝐷 ) + . Then for every 𝑓¯ ∈ 𝐵(𝐸 𝐷 ) + there is a unique locally bounded positive solution 𝑡 ↦→ 𝑉¯𝑡 𝑓¯ to the equation ∫ 𝑡 ∫ ¯ 𝑉¯𝑠 𝑓¯(𝑦)) 𝑃¯𝑡−𝑠 (𝑥, d𝑦), d𝑠 𝑉¯𝑡 𝑓¯(𝑥) = 𝑃¯𝑡 𝑓¯(𝑥) − (8.24) 𝜙(𝑦, 0

𝐸𝐷

where 𝑡 ≥ 0 and 𝑥 ∈ 𝐸 𝐷 . That defines a cumulant semigroup (𝑉¯𝑡 )𝑡 ≥0 with underlying space 𝐸 𝐷 . By Proposition A.37 for 𝑡 > 0 and 𝑥 ∈ 𝐸 𝐷 the probability measure 𝑃¯𝑡 (𝑥, ·) is carried by 𝐸. Then we can also regard (𝑉¯𝑡 )𝑡 >0 as operators from 𝐵(𝐸) + to 𝐵(𝐸 𝐷 ) + . Indeed, for 𝑓 ∈ 𝐵(𝐸) + we have ∫ 𝑡 ∫ ¯ ¯ d𝑠 𝜙(𝑦, 𝑉𝑠 𝑓 ) 𝑃¯𝑡−𝑠 (𝑥, d𝑦), (8.25) 𝑉𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) − 0

𝐸

where 𝑡 > 0 and 𝑥 ∈ 𝐸 𝐷 . Theorem 8.15 If (𝑃𝑡 )𝑡 ≥0 is a conservative semigroup, there is a one-to-one correspondence between 𝐾 ∈ 𝒦𝑚1 (𝑄) and 𝜇 ∈ 𝑀 (𝐸 𝐷 ) given by ∫ (8.26) e−𝜈 ( 𝑓 ) 𝐾𝑡 (d𝜈) = exp{−𝜇(𝑉¯𝑡 𝑓 )}, 𝑡 > 0, 𝑓 ∈ 𝐵(𝐸) + . 𝑀 (𝐸)

8.3 Infinitely Divisible Probability Entrance Laws

215

Proof Since (𝑃𝑡 )𝑡 ≥0 is conservative, every 𝜅 ∈ 𝒦(𝑃) is finite. By Theorem A.38 the relation 𝜅 𝑡 = 𝜇 𝑃¯𝑡 gives a one-to-one correspondence between 𝜅 ∈ 𝒦(𝑃) and 𝜇 ∈ 𝑀 (𝐸 𝐷 ). Using Lemma 8.8 it is easy to show that 𝑆𝑡 (𝜅, 𝑓 ) = 𝜇(𝑉¯𝑡 𝑓 ) for 𝑡 > 0 and 𝑓 ∈ 𝐵(𝐸) + . Then (8.26) follows from (8.18). □ Corollary 8.16 If (𝑃𝑡 )𝑡 ≥0 is a conservative semigroup, there is a one-to-one correspondence between 𝐾 ◦ ∈ 𝒦𝑚1 (𝑄 ◦ ) and 𝜇 ∈ 𝑀 (𝐸 𝐷 ) ◦ given by ∫ (1 − e−𝜈 ( 𝑓 ) )𝐾𝑡◦ (d𝜈) = 1 − exp{−𝜇(𝑉¯𝑡 𝑓 )}, 𝑡 > 0, 𝑓 ∈ 𝐵(𝐸) + . 𝑀 (𝐸) ◦

8.3 Infinitely Divisible Probability Entrance Laws In this section, we study the structures of infinitely divisible probability entrance laws for the (𝜉, 𝜙)-superprocess. Suppose that 𝜉 is a Borel right process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 and 𝜙 is a branching mechanism given by (2.29) or (2.30). Let 𝑏 + = 0 ∨ 𝑏 and 𝑏 − = 0 ∨ (−𝑏). Let 𝛾(𝑥, d𝑦) be the kernel on 𝐸 defined by (2.31) and let 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)]. Let (𝑄 𝑡 )𝑡 ≥0 denote the transition semigroup of the (𝜉, 𝜙)-superprocess defined by (2.35) and (2.36). We first consider the case where (𝑃𝑡 )𝑡 ≥0 is conservative. Let 𝐸 𝐷 be the entrance space of 𝜉 and let (𝑉¯𝑡 )𝑡 ≥0 be the extension of (𝑉𝑡 )𝑡 ≥0 on 𝐵(𝐸 𝐷 ) + defined by (8.24). ¯ 𝐸 𝐷 \ 𝐸) = 0 for 𝑥 ∈ 𝐸 Let 𝛾(𝑥, ¯ d𝑦) be the extension of 𝛾(𝑥, d𝑦) to 𝐸 𝐷 such that 𝛾(𝑥, and 𝛾(𝑥, ¯ 𝐸 𝐷 ) = 0 for 𝑥 ∈ 𝐸 𝐷 \ 𝐸. Let ( 𝜋¯ 𝑡 )𝑡 ≥0 be the semigroup of kernels on 𝐸 𝐷 ¯ d𝑦). Since (𝑉¯𝑡 )𝑡 ≥0 is a cumulant semigroup, defined by (2.38) from ( 𝑃¯𝑡 )𝑡 ≥0 and 𝛾(𝑥, it can be represented in the form of (2.5). However, in view of (8.25), for 𝑡 > 0 and 𝑥 ∈ 𝐸 𝐷 we can write ∫
𝑉¯𝑡 𝑓 (𝑥) = 𝜆 𝑡 (𝑥, 𝑓 ) + 1 − e−𝜈 ( 𝑓 ) 𝐿 𝑡 (𝑥, d𝜈), 𝑓 ∈ 𝐵(𝐸) + , (8.27) 𝑀 (𝐸) ◦

where 𝜆 𝑡 (𝑥, d𝑦) is a bounded kernel from 𝐸 𝐷 to 𝐸 and 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 𝐷 to 𝑀 (𝐸) ◦ . Let 𝐸 𝐷𝐶 be the set of points 𝑥 ∈ 𝐸 𝐷 such that 𝜆 𝑡 (𝑥, 𝐸) = 0 for all 𝑡 > 0. Theorem 8.17 Suppose that (𝑃𝑡 )𝑡 ≥0 is a conservative semigroup. Then 𝐾 ∈ 𝒦 1 (𝑄) is an infinitely divisible entrance law if and only if it has the representation, for 𝑡 > 0 and 𝑓 ∈ 𝐵(𝐸) ∗ ,   ∫ ¯ (1 − e−𝜈 ( 𝑉𝑡 𝑓 ) )𝐺 𝐷 (d𝜈) , 𝐿 𝐾𝑡 ( 𝑓 ) = exp − 𝛾 𝐷 (𝑉¯𝑡 𝑓 ) − (8.28) 𝑀 (𝐸𝐷 ) ◦

where 𝛾 𝐷 ∈ 𝑀 (𝐸 𝐷 ) and 𝐺 𝐷 (d𝜈) is a 𝜎-finite measure on 𝑀 (𝐸 𝐷 ) ◦ satisfying ∫ (8.29) 𝜈(1)𝐺 𝐷 (d𝜈) < ∞. 𝑀 (𝐸𝐷 ) ◦

216

8 Entrance Laws and Kuznetsov Measures

Proof By Theorem 8.15 it is easy to see that (8.28) defines an infinitely divisible entrance law 𝐾 ∈ 𝒦 1 (𝑄). By letting 𝑓 ≡ 𝜃 ≥ 0 and differentiating both sides at 𝜃 = 0 we obtain ∫ ∫ 𝜈(1)𝐾𝑡 (d𝜈) = 𝛾 𝐷 ( 𝜋¯ 𝑡 1) − 𝜈( 𝜋¯ 𝑡 1)𝐺 𝐷 (d𝜈). 𝑀 (𝐸𝐷 ) ◦

𝑀 (𝐸) ◦

Applying (8.10) and Theorem A.53 to ( 𝜋¯ 𝑡 )𝑡 ≥0 gives e− ∥𝑏

+ ∥𝑡

≤ 𝜋¯ 𝑡 1(𝑥) ≤ e𝑐0 𝑡 ,

𝑡 ≥ 0, 𝑥 ∈ 𝐸 𝐷 .

Then (8.13) is equivalent to (8.29). On the other hand, since 𝒦 1 (𝑄) is a simplex, if 𝐾 ∈ 𝒦 1 (𝑄) is an infinitely divisible entrance law, by Theorem 8.15 there is a probability measure 𝐹𝐷 (d𝜈) on 𝑀 (𝐸 𝐷 ) such that ∫ ∫ ¯ e−𝜈 ( 𝑓 ) 𝐾𝑡 (d𝜈) = e−𝜇 ( 𝑉𝑡 𝑓 ) 𝐹𝐷 (d𝜇), 𝑡 > 0, 𝑓 ∈ 𝐵(𝐸) + . 𝑀 (𝐸)

𝑀 (𝐸𝐷 )

Since (𝑉¯𝑡 )𝑡 ≥0 corresponds to a Borel right semigroup (𝑄¯ 𝑡 )𝑡 ≥0 on 𝑀 (𝐸 𝐷 ), we have 𝐹𝐷 = lim𝑡→0 𝐾𝑡 by the weak convergence of probability measures on 𝑀 (𝐸 𝐷 ). Then 𝐹𝐷 is infinitely divisible and the representation (8.28) follows. □ Corollary 8.18 Suppose that (𝑃𝑡 )𝑡 ≥0 is conservative. Then 𝐻 ∈ 𝒦(𝑄 ◦ ) if and only if it is given by ∫ (1 − e−𝜈 ( 𝑓 ) )𝐻𝑡 (d𝜈) 𝑀 (𝐸) ◦ ∫ ¯ = 𝛾 𝐷 (𝑉¯𝑡 𝑓 ) + (1 − e−𝜈 ( 𝑉𝑡 𝑓 ) )𝐺 𝐷 (d𝜈), (8.30) 𝑀 (𝐸𝐷 ) ◦

where 𝛾 𝐷 ∈ 𝑀 (𝐸 𝐷𝐶 ) and 𝐺 𝐷 (d𝜈) is a 𝜎-finite measure on 𝑀 (𝐸 𝐷 ) ◦ satisfying (8.29). Proof It is easy to see that (8.30) defines an entrance law 𝐻 ∈ 𝒦(𝑄 ◦ ). Conversely, suppose that 𝐻 ∈ 𝒦(𝑄 ◦ ). By Corollary 8.4 an infinitely divisible probability entrance law 𝐾 = (𝐾𝑡 )𝑡 >0 ∈ 𝒦 1 (𝑄) is defined by (8.5). By Theorem 8.17, we can represent 𝐻 = (𝐻𝑡 )𝑡 >0 by formula (8.30) for 𝛾 𝐷 ∈ 𝑀 (𝐸 𝐷 ) and a 𝜎-finite measure 𝐺 𝐷 (d𝜈) on 𝑀 (𝐸 𝐷 ) ◦ satisfying (8.29). Since the formula defines a family of 𝜎-finite measures (𝐻𝑡 )𝑡 >0 on 𝑀 (𝐸) ◦ , the measure 𝛾 𝐷 ∈ 𝑀 (𝐸 𝐷 ) must be carried by 𝐸 𝐷𝐶 .□ Corollary 8.19 Suppose that (𝑃𝑡 )𝑡 ≥0 is conservative. Then 𝐻 ∈ 𝒦𝑚∞ (𝑄 ◦ ) if and only if there exist 𝑞 > 0 and 𝑥 ∈ 𝐸 𝐷𝐶 such that 𝐻𝑡 (d𝜈) = 𝑞𝐿 𝑡 (𝑥, d𝜈),

𝑡 > 0, 𝜈 ∈ 𝑀 (𝐸) ◦ .

8.3 Infinitely Divisible Probability Entrance Laws

217

Now let us turn to a general underlying semigroup (𝑃𝑡 )𝑡 ≥0 , not necessarily conservative. Let (𝜋𝑡 )𝑡 ≥0 be the semigroup defined by (2.38). We can define a strictly positive function ℎ ∈ 𝐵(𝐸) + by ∫

1

𝑥 ∈ 𝐸.

𝜋 𝑠 1(𝑥)d𝑠,

ℎ(𝑥) =

(8.31)

0

Proposition 8.20 Let 𝑏 0 = 𝑐 0 + ∥𝑏 + ∥. Then for 𝑡 ≥ 0 and 𝑥 ∈ 𝐸 we have e−𝑏0 𝑡 𝑃𝑡 ℎ(𝑥) ≤ e−𝑐0 𝑡 𝜋𝑡 ℎ(𝑥) ≤ ℎ(𝑥).

(8.32)

Moreover, the function ℎ is 𝑏 0 -excessive for (𝑃𝑡 )𝑡 ≥0 . Proof By Theorem A.53 we have ∥𝜋𝑡 ∥ ≤ e𝑐0 𝑡 for 𝑡 ≥ 0. Then by (8.10) we have e−𝑏0 𝑡 𝑃𝑡 ℎ(𝑥) ≤ e−𝑐0 𝑡 𝜋𝑡 ℎ(𝑥) =

∫

1

e−𝑐0 𝑡 𝜋 𝑠 𝜋𝑡 1(𝑥)d𝑠 ≤ ℎ(𝑥).

0

This proves (8.32). On the other hand, from (8.31) we get ∫

1+𝑡

∫ 𝜋 𝑠 1(𝑥)d𝑠 −

𝜋𝑡 ℎ(𝑥) =

𝑡

𝜋 𝑠 1(𝑥)d𝑠. 0

0

Then 𝑡 ↦→ 𝜋𝑡 ℎ(𝑥) is right continuous. By Proposition A.43 we see 𝑡 ↦→ 𝑃𝑡 ℎ(𝑥) is □ also right continuous. Therefore ℎ is a 𝑏 0 -excessive function for (𝑃𝑡 )𝑡 ≥0 . To investigate the structures of infinitely divisible entrance laws 𝐾 ∈ 𝒦 1 (𝑄) for a general underlying semigroup (𝑃𝑡 )𝑡 ≥0 , we introduce some transformations based on the result of Proposition 8.20. We first define a Borel right semigroup (𝑇𝑡 )𝑡 ≥0 on 𝐸 by 𝑇𝑡 𝑓 (𝑥) = ℎ(𝑥) −1 𝑃𝑡𝑏0 (ℎ 𝑓 ) (𝑥),

𝑡 ≥ 0, 𝑥 ∈ 𝐸, 𝑓 ∈ 𝐵(𝐸);

(8.33)

see, e.g., Sharpe (1988, pp. 298–299). Moreover, by (2.5) it is easy to show that 𝑈𝑡 𝑓 (𝑥) = ℎ(𝑥) −1𝑉𝑡 (ℎ 𝑓 ) (𝑥),

𝑡 ≥ 0, 𝑥 ∈ 𝐸, 𝑓 ∈ 𝐵(𝐸) +

defines a cumulant semigroup on 𝐸. Indeed, by Proposition 8.20 we have 𝑈𝑡 𝑓 (𝑥) ≤ 𝜋𝑡ℎ 𝑓 (𝑥) := ℎ(𝑥) −1 𝜋𝑡 (ℎ 𝑓 ) (𝑥) ≤ ∥ 𝑓 ∥e𝑐0 𝑡 . By Proposition 2.9 we can rewrite (2.36) as ∫ 𝑡 ∫   𝑏0 𝑏0 𝑏 0𝑉𝑠 𝑓 (𝑦) − 𝜙(𝑦, 𝑉𝑠 𝑓 ) 𝑃𝑡−𝑠 𝑉𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) + (𝑥, d𝑦). d𝑠 0

𝐸

(8.34)

218

8 Entrance Laws and Kuznetsov Measures

Then (𝑡, 𝑥) ↦→ 𝑈𝑡 𝑓 (𝑥) satisfies ∫

𝑡

∫

𝛾0 (𝑦, 𝑈𝑠 𝑓 )𝑇𝑡−𝑠 (𝑥, d𝑦) d𝑠 𝑈𝑡 𝑓 (𝑥) = 𝑇𝑡 𝑓 (𝑥) + 𝐸 ∫ 𝑡 ∫0 + d𝑠 [𝑏 0 − 𝑏(𝑦)]𝑈𝑠 𝑓 (𝑦)𝑇𝑡−𝑠 (𝑥, d𝑦) ∫0 𝑡 ∫𝐸 d𝑠 − 𝜓0 (𝑦, 𝑈𝑠 𝑓 )𝑇𝑡−𝑠 (𝑥, d𝑦), 0

(8.35)

𝐸

where 𝛾0 (𝑦, 𝑓 ) = ℎ(𝑦) −1 𝛾(𝑦, ℎ 𝑓 ) and 2

𝜓0 (𝑦, 𝑓 ) = 𝑐(𝑦)ℎ(𝑦) 𝑓 (𝑦) + ℎ(𝑦)

−1

∫ 𝐾 (𝜈, ℎ 𝑓 )𝐻 (𝑦, d𝜈). 𝑀 (𝐸) ◦

Note that although 𝑓 ↦→ 𝛾0 (·, 𝑓 ) and 𝑓 ↦→ 𝜓0 (·, 𝑓 ) are not necessarily bounded operators on 𝐵(𝐸) + , all the terms in (8.35) are bounded by ∥ 𝑓 ∥e𝑐0 𝑡 . Indeed, by (2.39) and Proposition 2.9 we have ∫ 𝑡 ∫ 𝑏0 𝑏0 𝜋𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) + d𝑠 𝛾(𝑦, 𝜋 𝑠 𝑓 )𝑃𝑡−𝑠 (𝑥, d𝑦) 𝐸 ∫ 𝑡 ∫ 0 𝑏0 + d𝑠 [𝑏 0 − 𝑏(𝑦)]𝜋 𝑠 𝑓 (𝑦)𝑃𝑡−𝑠 (𝑥, d𝑦). 0

𝐸

Then (8.34) yields ∫ 𝑡 ∫ d𝑠 𝜋𝑡ℎ 𝑓 (𝑥) = 𝑇𝑡 𝑓 (𝑥) + 𝛾0 (𝑦, 𝜋 𝑠ℎ 𝑓 )𝑇𝑡−𝑠 (𝑥, d𝑦) 0 𝐸 ∫ 𝑡 ∫ + d𝑠 [𝑏 0 − 𝑏(𝑦)]𝜋 𝑠ℎ 𝑓 (𝑦)𝑇𝑡−𝑠 (𝑥, d𝑦) 0 ∫𝐸 𝑡 ∫ ≥ 𝑇𝑡 𝑓 (𝑥) + d𝑠 𝛾0 (𝑦, 𝑈𝑠 𝑓 )𝑇𝑡−𝑠 (𝑥, d𝑦) 𝐸 ∫ 𝑡 ∫0 + d𝑠 [𝑏 0 − 𝑏(𝑦)]𝑈𝑠 𝑓 (𝑦)𝑇𝑡−𝑠 (𝑥, d𝑦). 0

𝐸

From this we see each term in (8.35) is bounded by ∥ 𝑓 ∥e𝑐0 𝑡 .

Now let (𝑇𝑡𝜕 )𝑡 ≥0 be the conservative extension of (𝑇𝑡 )𝑡 ≥0 to 𝐸 𝜕 := 𝐸 ∪ {𝜕} with 𝜕 being an isolated point. Let (𝑇¯𝑡𝜕 )𝑡 ≥0 be the Ray extension of (𝑇𝑡𝜕 )𝑡 ≥0 to its 𝜕,𝑇 𝑇 = 𝐸 𝜕,𝑇 \ {𝜕} and let (𝑇¯ ) with the Ray topology. Let 𝐸 𝐷 entrance space 𝐸 𝐷 𝑡 𝑡 ≥0 𝐷 𝑇 . Then 𝐸 𝑇 is Lusin and (𝑇¯ ) be the restriction of (𝑇¯𝑡𝜕 )𝑡 ≥0 to 𝐸 𝐷 𝑡 𝑡 ≥0 is a Borel right 𝐷 𝑇 the measure 𝑇¯ (𝑥, ·) is semigroup. It is known that for any 𝑡 > 0 and 𝑥 ∈ 𝐸 𝐷 𝑡 𝑇 ¯ supported by 𝐸; see Proposition A.37. Given 𝑓 ∈ 𝐵(𝐸 𝐷 ) + let 𝑓 = 𝑓¯| 𝐸 . By (8.35) it is easy to show that the limit 𝑈¯ 𝑡 𝑓¯(𝑥) := lim𝑟→0 𝑇¯𝑟 𝑈𝑡−𝑟 𝑓 (𝑥) exists for all 𝑡 > 0 𝑇 . Let 𝑈 ¯ 0 𝑓¯(𝑥) = 𝑓¯(𝑥) for 𝑥 ∈ 𝐸 𝑇 . Then (𝑈¯ 𝑡 )𝑡 ≥0 constitute a cumulant and 𝑥 ∈ 𝐸 𝐷 𝐷 𝑇 . Moreover, we have semigroup on 𝐸 𝐷

8.3 Infinitely Divisible Probability Entrance Laws

∫

219

∫

𝑡

𝛾0 (𝑦, 𝑈𝑠 𝑓 )𝑇¯𝑡−𝑠 (𝑥, d𝑦) d𝑠 𝑈¯ 𝑡 𝑓¯(𝑥) = 𝑇¯𝑡 𝑓¯(𝑥) + 𝐸 0 ∫ 𝑡 ∫ d𝑠 [𝑏 0 − 𝑏(𝑦)]𝑈𝑠 𝑓 (𝑦)𝑇¯𝑡−𝑠 (𝑥, d𝑦) + 0 ∫ 𝑡 ∫𝐸 − d𝑠 𝜓0 (𝑦, 𝑈𝑠 𝑓 )𝑇¯𝑡−𝑠 (𝑥, d𝑦) 0

(8.36)

𝐸

𝑇 . By the observations in the last paragraph, each term in (8.36) for 𝑡 ≥ 0 and 𝑥 ∈ 𝐸 𝐷 is bounded by ∥ 𝑓¯∥e𝑐0 𝑡 . Obviously, we can also regard (𝑈¯ 𝑡 )𝑡 >0 as operators from 𝑇 )+. 𝐵(𝐸) + to 𝐵(𝐸 𝐷 𝑇 ) and 𝜅 ∈ Lemma 8.21 There is a one-to-one correspondence between 𝜇 ∈ 𝑀 (𝐸 𝐷 𝒦(𝑃) determined by

𝜅 𝑡 ( 𝑓 ) = e𝑏0 𝑡 𝜇(𝑇¯𝑡 (ℎ−1 𝑓 )),

𝑡 > 0, 𝑓 ∈ 𝐵(𝐸) + .

(8.37)

Moreover, if 𝜅 and 𝜇 are related by (8.37), we have 𝑆𝑡 (𝜅, 𝑓 ) = 𝜇(𝑈¯ 𝑡 (ℎ−1 𝑓 )),

𝑡 > 0, 𝑓 ∈ 𝐵(𝐸) + .

(8.38)

𝑇 ) and define the family of measures 𝜅 = (𝜅 ) Proof Let 𝜇 ∈ 𝑀 (𝐸 𝐷 𝑡 𝑡 >0 by (8.37). Observe that

𝜅𝑟 (𝑃𝑡 𝑓 ) = e𝑏0 𝑟 𝜇𝑇¯𝑟 (ℎ−1 𝑃𝑡 𝑓 ) = e𝑏0 (𝑟+𝑡) 𝜇𝑇¯𝑟 𝑇𝑡 (ℎ−1 𝑓 ) = 𝜅𝑟+𝑡 ( 𝑓 ) for all 𝑟, 𝑡 > 0 and 𝑓 ∈ 𝐵(𝐸). Moreover, by (8.33) and (8.37) it is easy to see that ∫

1

∫

1

𝜅 𝑠 (1)d𝑠 = lim 𝑟→0 𝑟 ∫

0

e𝑏0 𝑠 𝜇𝑇¯𝑠 (ℎ−1 )d𝑠 = lim

∫

1

𝜇𝑇¯𝑟 (ℎ−1 𝑃𝑠−𝑟 1)d𝑠

𝑟→0 𝑟 1

+ e ∥𝑏 ∥ (𝑠−𝑟) 𝜇𝑇¯𝑟 (ℎ−1 𝜋 𝑠−𝑟 1)d𝑠 𝑟→0 𝑟 + + lim e ∥𝑏 ∥ 𝜇𝑇¯𝑟 (1) = e ∥𝑏 ∥ 𝜇(1), 𝑟→0

≤ lim ≤

where we also used (8.10) for the first inequality. Then we have 𝜅 ∈ 𝒦(𝑃). Conversely, given 𝜅 ∈ 𝒦(𝑃), we first define an entrance law 𝜈 = (𝜈𝑡 )𝑡 >0 for the semigroup (𝑇𝑡 )𝑡 ≥0 by 𝜈𝑡 ( 𝑓 ) = e−𝑏0 𝑡 𝜅 𝑡 (ℎ 𝑓 ). Observe that 𝜈0+ (1) := ↑lim 𝜈𝑡 (1) = ↑lim e−𝑏0 𝑡 𝑡→0

𝑡→0

∫

1

∫

1

𝜅 𝑡 (𝜋 𝑠 1)d𝑠 = 0

𝜂 𝑠 (1)d𝑠 < ∞, 0

where 𝜂 ∈ 𝒦(𝜋) is defined by (8.8). For 𝑡 > 0 define 𝜈˜𝑡 ∈ 𝑀 (𝐸 𝜕 ) by 𝜈˜𝑡 | 𝐸 = 𝜈𝑡 and 𝜈˜𝑡 ({𝜕}) = 𝜈0+ (1) − 𝜈𝑡 (1). It is easy to see that ( 𝜈˜𝑡 )𝑡 >0 is a finite entrance law for the conservative Borel right semigroup (𝑇𝑡𝜕 )𝑡 ≥0 . By Theorem A.38 there exists a 𝜕,𝑇 ) such that 𝜈˜𝑡 = 𝜈˜0𝑇¯𝑡𝜕 for 𝑡 > 0. Then measure 𝜈˜0 ∈ 𝑀 (𝐸 𝐷 𝜅 𝑡 ( 𝑓 ) = e𝑏0 𝑡 𝜈𝑡 (ℎ−1 𝑓 ) = e𝑏0 𝑡 𝜇𝑇¯𝑡 (ℎ−1 𝑓 )

220

8 Entrance Laws and Kuznetsov Measures

𝑇 . Finally, assume that 𝜅 and 𝜇 are related by with 𝜇 being the restriction of 𝜈˜0 to 𝐸 𝐷 + (8.37). If 𝑓 ∈ 𝐵(𝐸) is bounded by const. · ℎ we can use Lemma 8.8 to see

𝑆𝑡 (𝜅, 𝑓 ) = lim 𝜅𝑟 (𝑉𝑡−𝑟 𝑓 ) = lim e𝑏0 𝑟 𝜇𝑇¯𝑟 (ℎ−1𝑉𝑡−𝑟 𝑓 ) 𝑟→0

𝑟→0

= lim 𝜇𝑇¯𝑟 (𝑈𝑡−𝑟 (ℎ−1 𝑓 )) = 𝜇(𝑈¯ 𝑡 (ℎ−1 𝑓 )). 𝑟→0

Then we obtain (8.38) for all 𝑓 ∈ 𝐵(𝐸) + by taking increasing limits.

□

Theorem 8.22 The entrance law 𝐾 ∈ 𝒦 1 (𝑄) is infinitely divisible if and only if it has the representation, for 𝑡 > 0 and 𝑓 ∈ 𝐵(𝐸) + ,   ∫ 𝐿 𝐾𝑡 ( 𝑓 ) = exp − 𝑆𝑡 (𝜅, 𝑓 ) − (8.39) (1 − e−𝑆𝑡 (𝜈, 𝑓 ) )𝐹 (d𝜈) , 𝒦 ( 𝑃) ◦

where 𝜅 ∈ 𝒦(𝑃) and 𝐹 (d𝜈) is a 𝜎-finite measure on the space 𝒦(𝑃) ◦ satisfying ∫

1

∫ 𝜈𝑠 (1)𝐹 (d𝜈) < ∞.

d𝑠 0

(8.40)

𝒦 ( 𝑃) ◦

Moreover, the entrance law 𝐾 ∈ 𝒦 1 (𝑄) defined by (8.39) has finite first-moments given by, for 𝑡 > 0 and 𝑓 ∈ 𝐵(𝐸), ∫ ∫ 𝑞 𝑡 (𝜈, 𝑓 )𝐹 (d𝜈), 𝜈( 𝑓 )𝐾𝑡 (d𝜈) = 𝑞 𝑡 (𝜅, 𝑓 ) + (8.41) 𝒦 ( 𝑃) ◦

𝑀 (𝐸)

where ∫

𝑡

𝑞 𝑡 (𝜅, 𝑓 ) = 𝜅 𝑡 ( 𝑓 ) +

𝜅 𝑡−𝑠 ((𝛾 − 𝑏)𝜋 𝑠 𝑓 )d𝑠.

(8.42)

0

Proof By Theorem 8.10 any entrance law 𝐾 ∈ 𝒦 1 (𝑄) corresponds to a probability measure 𝐽 on 𝒦(𝑃) such that ∫ ∫ exp{−𝑆𝑡 (𝜇, 𝑓 )}𝐽 (d𝜇) e−𝜈 ( 𝑓 ) 𝐾𝑡 (d𝜈) = 𝑀 (𝐸)

𝒦 ( 𝑃)

for every 𝑓 ∈ 𝐵(𝐸) + . Then by Lemma 8.21 there is a probability measure 𝐻 on 𝑇 ) such that 𝑀 (𝐸 𝐷 ∫ ∫ (8.43) e−𝜈 ( 𝑓 ) 𝐾𝑡 (d𝜈) = exp{−𝜇(𝑈¯ 𝑡 (ℎ−1 𝑓 ))}𝐻 (d𝜇). 𝑀 (𝐸)

𝑇) 𝑀 (𝐸𝐷

𝑇 ) + we can use the above equality to see For any 𝑓¯ ∈ 𝐶 (𝐸 𝐷 ∫ ∫ ¯ −𝜇 ( 𝑓¯) 𝐻 (d𝜇) = lim e e−𝜈 (ℎ 𝑓 ) 𝐾𝑡 (d𝜈). 𝑇) 𝑀 (𝐸𝐷

𝑡→0

𝑀 (𝐸)

(8.44)

8.4 Kuznetsov Measures and Excursion Laws

221

Using (8.43) and (8.44) one can see 𝐾 is an infinitely divisible probability entrance law if and only if 𝐻 is an infinitely divisible probability measure. In this case, we have the representation   ∫ ∫ −𝜇 ( 𝑓¯) 𝑇 ¯ −𝜈 ( 𝑓¯)
𝑇 e 𝐻 (d𝜇) = exp − 𝛾 𝐷 ( 𝑓 ) − 𝐺 𝐷 (d𝜈) , 1−e 𝑇 )◦ 𝑀 (𝐸𝐷

𝑇) 𝑀 (𝐸𝐷

𝑇 ∈ 𝑀 (𝐸 𝑇 ) and [1 ∧ 𝜈(1)]𝐺 𝑇 (d𝜈) is a finite measure on 𝑀 (𝐸 𝑇 ) ◦ . Then where 𝛾 𝐷 𝐷 𝐷 𝐷 (8.39) follows by (8.43) and another application of Lemma 8.21. By the calculations in the proof of Theorem 8.10, we can differentiate both sides of (8.39) and use (8.8) to obtain (8.41). By Corollary 8.7 one can show (8.40) is equivalent to (8.13). □

The theorem above gives a complete characterization of infinitely divisible entrance laws in 𝒦 1 (𝑄). This result also yields a representation for the entrance laws for the restricted semigroup (𝑄 ◦𝑡 )𝑡 ≥0 . Indeed, by Corollary 8.4 and Theorem 8.22, an entrance law 𝐻 ∈ 𝒦(𝑄 ◦ ) can always be represented as ∫ (1 − e−𝜈 ( 𝑓 ) )𝐻𝑡 (d𝜈) 𝑀 (𝐸) ◦ ∫ (1 − exp{−𝑆𝑡 (𝜈, 𝑓 )})𝐹 (d𝜈), = 𝑆𝑡 (𝜅, 𝑓 ) + (8.45) 𝒦 ( 𝑃) ◦

where 𝜅 ∈ 𝒦(𝑃) and 𝐹 (d𝜈) is a 𝜎-finite measure on 𝒦(𝑃) ◦ satisfying (8.40). By Corollaries 5.33 and 8.5 we have the following: Corollary 8.23 Suppose that Condition 5.31 holds with 𝜙∗′ (∞) = ∞. Then 𝐻 ∈ 𝒦(𝑄 ◦ ) if and only if it is given by (8.45) for 𝜅 ∈ 𝒦(𝑃) and for a 𝜎-finite measure 𝐹 (d𝜇) on 𝒦(𝑃) ◦ satisfying (8.40).

8.4 Kuznetsov Measures and Excursion Laws In this section, we prove some properties of the entrance rules and Kuznetsov measures associated with an MB-process, which bring useful insights into the structures of the process. Let 𝐸 be a Lusin topological space. Let 𝑊ˆ denote the space of paths 𝑤 : [0, ∞) → 𝑀 (𝐸) such that 𝑤 𝑡 takes values in 𝑀 (𝐸) ◦ and is right continuous in some interval (𝛼(𝑤), 𝜁 (𝑤)) or [𝛼(𝑤), 𝜁 (𝑤)) ⊂ [0, ∞) and takes the value 0 ∈ 𝑀 (𝐸) elsewhere. We include in 𝑊ˆ the path [0] that takes the value 0 ∈ 𝑀 (𝐸) constantly and understand 𝛼( [0]) = ∞ and 𝜁 ( [0]) = 0. We equip 𝑊ˆ with the natural 𝜎-algebras 𝒜 0 = 𝜎({𝑤(𝑠) : 𝑠 ≥ 0}) and 𝒜𝑡0 = 𝜎({𝑤(𝑠) : 0 ≤ 𝑠 ≤ 𝑡}) for 𝑡 ≥ 0. Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroups of a Borel right MB-process in 𝑀 (𝐸). Let (𝑉𝑡 )𝑡 ≥0 be the corresponding cumulant semigroup with canonical representation (2.5). By Theorem A.41, an entrance rule 𝐻 = (𝐻𝑡 )𝑡 >0 for (𝑄 ◦𝑡 )𝑡 ≥0 determines a ˆ 𝒜 0 ) such Kuznetsov measure Q(𝐻, ·), which is the unique 𝜎-finite measure on (𝑊, that Q(𝐻, {[0]}) = 0 and

222

8 Entrance Laws and Kuznetsov Measures

Q(𝐻, 𝑤 𝑡1 ∈ d𝜈1 , 𝑤 𝑡2 ∈ d𝜈2 , . . . , 𝑤 𝑡𝑛 ∈ d𝜈𝑛 ) = 𝐻𝑡1 (d𝜈1 )𝑄 ◦𝑡2 −𝑡1 (𝜈1 , d𝜈2 ) · · · 𝑄 ◦𝑡𝑛 −𝑡𝑛−1 (𝜈𝑛−1 , d𝜈𝑛 )

(8.46)

for every {𝑡 1 < · · · < 𝑡 𝑛 } ⊂ (0, ∞) and {𝜈1 , . . . , 𝜈𝑛 } ⊂ 𝑀 (𝐸) ◦ . Roughly speaking, the above formula means that {𝑤 𝑡 : 𝑡 > 0} under Q(𝐻, ·) is a Markov process in 𝑀 (𝐸) ◦ with transition semigroup (𝑄 ◦𝑡 )𝑡 ≥0 and one-dimensional distributions {𝐻𝑡 : 𝑡 > 0}. If 𝐻 = (𝐻𝑡 )𝑡 >0 is an entrance law, then Q(𝐻, ·) is carried by 𝑊ˆ 0 := {𝑤 ∈ 𝑊ˆ : 𝛼(𝑤) = 0}. A more convenient formulation of the Markov property (8.46) is given in the following:

Theorem 8.24 Let Q(𝐻, ·) be the Kuznetsov measure on 𝑊ˆ corresponding to the entrance rule 𝐻 = (𝐻𝑡 )𝑡 >0 for (𝑄 ◦𝑡 )𝑡 ≥0 determined by (8.2). Let 𝑡 ≥ 𝑟 > 0 and let 𝐹 ˆ Then for any 𝑓 ∈ 𝐵(𝐸) + we have be a positive 𝒜𝑟0 -measurable function on 𝑊.

Q 𝐻, 𝐹 (1 − e−𝑤𝑡 ( 𝑓 ) ) = Q 𝐻, 𝐹 (1 − e−𝑤𝑟 (𝑉𝑡−𝑟 𝑓 ) )   (8.47) + 𝐹 ( [0]) 𝜂𝑟 (𝑉𝑡−𝑟 𝑓 ) − 𝜂𝑡 ( 𝑓 ) . Proof Step 1. By Theorem A.40, there is a Radon measure 𝜌(d𝑠) on [0, ∞) and a countable set 𝑇 ⊂ (0, ∞) such that ∫ ∑︁ 𝐻𝑡𝑠 𝜌(d𝑠) + 𝐻𝑡 = 𝐺 𝑡𝑠 , 𝑡 > 0, [0,𝑡)

𝑠 ∈𝑇∩(0,𝑡 ]

where 𝐻 𝑠 = {𝐻𝑡𝑠 : 𝑡 > 𝑠} is an entrance laws at 𝑠 ≥ 0 and 𝐺 𝑠 = {𝐺 𝑡𝑠 : 𝑡 ≥ 𝑠} is a closed entrance law at 𝑠 ∈ 𝑇. Accordingly, we have the decomposition ∫ ∑︁ N𝑠 (d𝑤) 𝜌(d𝑠) + M𝑠 (d𝑤), Q(𝐻, d𝑤) = [0,𝑡)

𝑠 ∈𝑇∩(0,𝑡 ]

where N𝑠 is the Kuznetsov measure determined by the entrance law 𝐻 𝑠 and M𝑠 is the Kuznetsov measure determined by the closed entrance law 𝐺 𝑠 . Step 2. Let 0 ≤ 𝑟 1 ≤ 𝑟 2 ≤ · · · ≤ 𝑟 𝑛 ≤ 𝑟 ≤ 𝑠. For N𝑠 -a.e. 𝑤 ∈ 𝑊ˆ we have 𝑤 𝑟𝑖 = 0 = [0] 𝑟𝑖 , 𝑖 = 1, · · · , 𝑛. Then, for positive Borel functions 𝑔1 , . . . , 𝑔𝑛 on 𝑀 (𝐸),   N𝑠 𝑔1 (𝑤 𝑟1 ) · · · 𝑔𝑛 (𝑤 𝑟𝑛 ) = 𝑔1 ( [0] 𝑟1 ) · · · 𝑔𝑛 ( [0] 𝑟𝑛 ). ˆ by a monotone class argument For any positive 𝒜𝑟0 -measurable function 𝐹 on 𝑊, 𝑠 we infer N [𝐹 (𝑤)] = 𝐹 ( [0]). This implies further that N𝑠 [ℎ(𝐹 (𝑤))] = ℎ(𝐹 ( [0])) for any ℎ ∈ 𝐵[0, ∞) + , and so N𝑠 ({𝑤 ∈ 𝑊ˆ : 𝐹 (𝑤) ≠ 𝐹 ( [0])}) = 0. Similarly, one can show M𝑠 ({𝑤 ∈ 𝑊ˆ : 𝐹 (𝑤) ≠ 𝐹 ( [0])}) = 0 for 𝑠 > 𝑟.

8.4 Kuznetsov Measures and Excursion Laws

223

Step 3. Using the properties given in the two steps above, for any positive 𝒜𝑟0 -measurable function 𝐹 on 𝑊ˆ and any 𝑓 ∈ 𝐵(𝐸) + we have
Q 𝐻, 𝐹 (𝑤) (1 − e−𝑤𝑡 ( 𝑓 ) ) ∫ ∑︁     N𝑠 𝐹 (𝑤) (1 − e−𝑤𝑡 ( 𝑓 ) ) 𝜌(d𝑠) + = M𝑠 𝐹 (𝑤) (1 − e−𝑤𝑡 ( 𝑓 ) ) [0,𝑡)

∫ =

𝑠 ∈𝑇∩(0,𝑡 ]

N 𝐹 (𝑤) (1 − e−𝑤𝑟 (𝑉𝑡−𝑟 𝑓 ) ) 𝜌(d𝑠) + 𝐹 ( [0]) [0,𝑟) ∑︁   M𝑠 𝐹 (𝑤) (1 − e−𝑤𝑟 (𝑉𝑡−𝑟 𝑓 ) ) + 𝐹 ( [0]) +  𝑠



𝑠 ∈𝑇∩(0,𝑟 ]

∫

N𝑠 (1 − e−𝑤𝑡 ( 𝑓 ) ) 𝜌(d𝑠) [𝑟 ,𝑡)

∑︁

M𝑠 (1 − e−𝑤𝑡 ( 𝑓 ) )

𝑠 ∈𝑇∩(𝑟 ,𝑡 ]

∫

= Q 𝐻, 𝐹 (1 − e−𝑤𝑟 (𝑉𝑡−𝑟 𝑓 ) ) + 𝐹 ( [0]) N𝑠 (1 − e−𝑤𝑡 ( 𝑓 ) ) 𝜌(d𝑠) [0,𝑡) ∫ ∑︁ 𝑠 −𝑤𝑟 (𝑉𝑡−𝑟 𝑓 ) M𝑠 (1 − e−𝑤𝑡 ( 𝑓 ) ) − 𝐹 ( [0]) N (1 − e ) 𝜌(d𝑠) + 𝐹 ( [0])


[0,𝑟)

− 𝐹 ( [0])

∑︁

𝑠 ∈𝑇∩(0,𝑡 ] 𝑠

−𝑤𝑟 (𝑉𝑡−𝑟 𝑓 )

M (1 − e

)

𝑠 ∈𝑇∩(0,𝑟 ]

∫
= Q 𝐻, 𝐹 (1 − e−𝑤𝑟 (𝑉𝑡−𝑟 𝑓 ) ) + 𝐹 ( [0]) (1 − e−𝜈 ( 𝑓 ) )𝐻𝑡 (d𝜈) ◦ 𝑀 (𝐸) ∫ −𝜈 (𝑉𝑡−𝑟 𝑓 ) (1 − e )𝐻𝑟 (d𝜈) − 𝐹 ( [0]) 𝑀 (𝐸) ◦ ∫
= Q 𝐻, 𝐹 (1 − e−𝑤𝑟 (𝑉𝑡−𝑟 𝑓 ) ) + 𝐹 ( [0]) (1 − e−𝜈 ( 𝑓 ) )𝐻𝑡 (d𝜈) ◦ 𝑀 (𝐸) ∫ ◦ −𝜈 ( 𝑓 ) − 𝐹 ( [0]) (1 − e )𝐻𝑟 𝑄 𝑡−𝑟 (d𝜈) 𝑀 (𝐸) ◦ ∫
(1 − e−𝜈 ( 𝑓 ) )𝐻𝑡 (d𝜈) = Q 𝐻, 𝐹 (1 − e−𝑤𝑟 (𝑉𝑡−𝑟 𝑓 ) ) + 𝐹 ( [0]) 𝑀 (𝐸) ◦ ∫ − 𝐹 ( [0]) (1 − e−𝜈 (𝑉𝑡−𝑟 𝑓 ) )𝐻𝑟 (d𝜈) 𝑀 (𝐸) ◦ −𝑤𝑟 (𝑉𝑡−𝑟 𝑓 )

= Q 𝐻, 𝐹 (1 − e

 
) + 𝐹 ( [0]) 𝜂𝑟 (𝑉𝑡−𝑟 𝑓 ) − 𝜂𝑡 ( 𝑓 ) ,

where we have used Theorem 8.2 for the last equality.

□

Corollary 8.25 Let L(𝑥) = L(𝑥, ·) be the Kuznetsov measure corresponding to the canonical entrance rule 𝐿(𝑥) = {𝐿 𝑡 (𝑥, ·) : 𝑡 > 0} at 𝑥 ∈ 𝐸 defined by (2.5). Let ˆ Then for any 𝑡 ≥ 𝑟 > 0 and let 𝐹 be a positive 𝒜𝑟0 -measurable function on 𝑊. 𝑓 ∈ 𝐵(𝐸) + we have

L 𝑥, 𝐹 (1 − e−𝑤𝑡 ( 𝑓 ) ) = L 𝑥, 𝐹 (1 − e−𝑤𝑟 (𝑉𝑡−𝑟 𝑓 ) )   (8.48) + 𝐹 ( [0]) 𝜆𝑟 (𝑥, 𝑉𝑡−𝑟 𝑓 ) − 𝜆 𝑡 (𝑥, 𝑓 ) .

224

8 Entrance Laws and Kuznetsov Measures

Now let us consider the situation of a (𝜉, 𝜙)-superprocess, where 𝜉 is a Borel right process in 𝐸 and 𝜙 is a branching mechanism given by (2.29) or (2.30). Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroups of the (𝜉, 𝜙)-superprocess defined by (2.35) and (2.36). Let 𝛾(𝑥, d𝑦) be the kernel on 𝐸 defined by (2.31) and let (𝜋𝑡 )𝑡 ≥0 be the semigroup defined by (2.38). If the underlying semigroup (𝑃𝑡 )𝑡 ≥0 is conservative, the function (𝑠, 𝑥) ↦→ 𝜋 𝑠 1(𝑥) is bounded away from zero on [0, 𝑢] × 𝐸 for every 𝑢 > 0. In this case, we fix a constant 𝑢 > 0 and define the conservative inhomogeneous transition semigroup (𝑄 𝑟𝑢,𝑡 : 0 ≤ 𝑟 ≤ 𝑡 ≤ 𝑢) on 𝑀 (𝐸) ◦ by 𝑄 𝑟𝑢,𝑡 (𝜇, d𝜈) = 𝜇(𝜋𝑢−𝑟 1) −1 𝜈(𝜋𝑢−𝑡 1)𝑄 ◦𝑡−𝑟 (𝜇, d𝜈).

(8.49)

For any 𝑎 > 0 let 𝒦 𝑎 (𝑄 𝑢 ) denote the class of finite entrance laws 𝐻 := (𝐻𝑡 : 0 < 𝑡 ≤ 𝑢) for (𝑄 𝑟𝑢,𝑡 : 0 ≤ 𝑟 ≤ 𝑡 ≤ 𝑢) satisfying 𝐻𝑡 (𝑀 (𝐸) ◦ ) = 𝑎 for 0 < 𝑡 ≤ 𝑢. Given any 𝐾 ∈ 𝒦(𝑄 ◦ ) we can let ∫ 𝜈(1)𝐾𝑢 (d𝜈) 𝑎= 𝑀 (𝐸) ◦

and define 𝐻 𝑢 ∈ 𝒦 𝑎 (𝑄 𝑢 ) by 𝐻𝑡𝑢 (d𝜈) = 𝜈(𝜋𝑢−𝑡 1)𝐾𝑡 (d𝜈),

0 < 𝑡 ≤ 𝑢.

(8.50)

Theorem 8.26 Let 𝑥 ∈ 𝐸𝐶 and let 𝐿(𝑥) = {𝐿 𝑡 (𝑥, ·) : 𝑡 > 0} be the canonical entrance law defined by (2.11). Then the corresponding Kuznetsov measure L(𝑥) = L(𝑥, ·) is carried by 𝑊ˆ 0 and for L(𝑥)-a.e. 𝑤 ∈ 𝑊ˆ 0 we have 𝑤 𝑡 → 0 and 𝑤 𝑡 (1) −1 𝑤 𝑡 → 𝛿 𝑥 in 𝑀 (𝐸) as 𝑡 → 0. Proof By the general construction given by formula (A.28) we know that L(𝑥) is carried by 𝑊ˆ 0 . Recall that ∥𝜋𝑡 ∥ ≤ e𝑐0 𝑡 for 𝑡 ≥ 0 by Theorem A.53. We divide the following arguments into four steps. Step 1. We first assume the underlying semigroup (𝑃𝑡 )𝑡 ≥0 is conservative. For fixed 𝑢 > 0 define 𝐻 𝑢 (𝑥) ∈ 𝒦 𝑎 (𝑄 𝑢 ) by (8.50) with 𝐾 replaced by 𝐿 (𝑥), where 𝑎 = 𝜋𝑢 1(𝑥). Then Q𝑢 (𝑥, d𝑤) := 𝑎 −1 𝑤 𝑢 (1)L(𝑥, d𝑤) = 𝜋𝑢 1(𝑥) −1 𝑤 𝑢 (1)L(𝑥, d𝑤) defines a probability measure on 𝑊ˆ 0 . Under this measure, the coordinate process {𝑤 𝑡 : 0 < 𝑡 ≤ 𝑢} is Markovian with transition semigroup (𝑄 𝑟𝑢,𝑡 : 0 ≤ 𝑟 ≤ 𝑡 ≤ 𝑢) and one-dimensional distributions (𝑎 −1 𝐻𝑡𝑢 (𝑥) : 0 < 𝑡 ≤ 𝑢). By Corollary 8.19, 𝐿 (𝑥) ∈ 𝒦 ∞ (𝑄 ◦ ) is a minimal entrance law. Then 𝐻 𝑢 (𝑥) ∈ 𝒦 𝑎 (𝑄 𝑢 ) is a minimal 0 ; see Dynkin (1978, p. 724) or entrance law. It follows that Q𝑢 (𝑥) is trivial on 𝒜0+ + Sharpe (1988, p. 199). For any 𝑓 ∈ 𝐵(𝐸) we can use the martingale convergence theorem to get Q𝑢 (𝑥)-a.s. ∫
1 − e−𝜈 ( 𝑓 ) 𝜈(1) −1 𝐻𝑢𝑢 (𝑥, d𝜈) 𝑉𝑢 𝑓 (𝑥) = ◦ 𝑀 (𝐸)  
= 𝜋𝑢 1(𝑥)Q𝑢 𝑥, 1 − e−𝑤𝑢 ( 𝑓 ) 𝑤 𝑢 (1) −1

8.4 Kuznetsov Measures and Excursion Laws

225






= lim 𝜋𝑢 1(𝑥)Q𝑢 𝑥, 1 − e−𝑤𝑢 ( 𝑓 ) 𝑤 𝑢 (1) −1 𝒜𝑠0 𝑠→0 ∫
1 − e−𝜈 ( 𝑓 ) 𝜈(1) −1 𝑄 𝑢𝑠,𝑢 (𝑤 𝑠 , d𝜈) = lim 𝜋𝑢 1(𝑥) 𝑠→0 ◦ 𝑀 (𝐸) ∫
−1 1 − e−𝜈 ( 𝑓 ) 𝑄 ◦𝑢−𝑠 (𝑤 𝑠 , d𝜈) = lim 𝜋𝑢 1(𝑥)𝑤 𝑠 (𝜋𝑢−𝑠 1) 𝑠→0 ◦ 𝑀 (𝐸)
(8.51) = lim 𝜋𝑢 1(𝑥)𝑤 𝑠 (𝜋𝑢−𝑠 1) −1 1 − e−𝑤𝑠 (𝑉𝑢−𝑠 𝑓 ) , 𝑠→0

where we have used (8.49) for the second to last equality. Let 𝑊ˆ 𝑢 = {𝑤 ∈ 𝑊ˆ 0 : 𝑤 𝑢 (1) > 0}. Then Q𝑢 (𝑥, d𝑤) and L(𝑥, d𝑤) are absolutely continuous with respect to each other on 𝑊ˆ 𝑢 . Observe also that 𝑊ˆ 𝑣 ⊂ 𝑊ˆ 𝑢 for any 𝑣 ≥ 𝑢. Then (8.51) holds for L(𝑥)-a.e. 𝑤 ∈ 𝑊ˆ 𝑣 . Since (𝑃𝑡 )𝑡 ≥0 is conservative, (8.10) and (8.51) imply 𝑉𝑢 𝑓 (𝑥) ≤ lim inf e𝑐0 𝑢 𝑤 𝑠 (𝜋𝑢−𝑠 1) −1 ≤ lim inf e (𝑐0 + ∥𝑏 𝑠→0

+ ∥)𝑢

𝑤 𝑠 (1) −1 .

𝑠→0

Then letting 𝑢 → 0 and 𝑓 → ∞ we see 𝑤 𝑠 (1) → 0 as 𝑠 → 0 for L(𝑥)-a.e. 𝑤 ∈ 𝑊ˆ 𝑣 . Since 𝑊ˆ 0 = ∪𝑣>0𝑊ˆ 𝑣 , we have 𝑤 𝑠 (1) → 0 as 𝑠 → 0 for L(𝑥)-a.e. 𝑤 ∈ 𝑊ˆ 0 . Step 2. Let ℛ be a rational Ray cone for (𝑃𝑡 )𝑡 ≥0 that generates the Ray–Knight completion 𝐸¯ of 𝐸. Recall that each 𝑓 ∈ ℛ can be extended uniquely to a function ¯ + . By a similar reasoning as in the first step, for any 𝑣 ≥ 𝑢 we have 𝑓¯ ∈ 𝐶 ( 𝐸) ∫ 𝜈( 𝑓 )𝜈(1) −1 𝐻𝑢𝑢 (𝑥, d𝜈) 𝜋𝑢 𝑓 (𝑥) = 𝑀 (𝐸) ◦

= lim 𝜋𝑢 1(𝑥)𝑤 𝑠 (𝜋𝑢−𝑠 1) −1 𝑤 𝑠 (𝜋𝑢−𝑠 𝑓 )

(8.52)

𝑠→0

for L(𝑥)-a.e. 𝑤 ∈ 𝑊ˆ 𝑣 . Take any 𝑤 ∈ 𝑊ˆ 𝑣 along which the above relation holds for all 𝑓 ∈ ℛ and all rational 𝑢 ∈ (0, 𝑣]. Let 𝛼 = 𝛼( 𝑓 ) ≥ 0 be a constant such that 𝑓 ∈ ℛ is 𝛼-excessive for (𝑃𝑡 )𝑡 ≥0 . By the proof of Corollary 2.34, there is a constant 𝛽 ≥ 𝛼 such that 𝜋𝑡 𝑓 ≤ e2𝛽𝑡 𝑓 for all 𝑡 ≥ 0. Then by (8.10) and (8.52) we can see 𝜋𝑢 𝑓 (𝑥) ≤ lim inf e (𝑐0 − ∥𝑏

+ ∥+2𝛽)𝑢

𝑤 𝑠 (1) −1 𝑤 𝑠 ( 𝑓 )

(8.53)

𝑤 𝑠 (1) −1 𝑤 𝑠 (𝑃𝑢 𝑓 ),

(8.54)

𝑠→0

and e2𝛽𝑢 𝑓 (𝑥) ≥ lim sup e−(2 ∥𝑏

+ ∥+𝑐 )𝑢 0

𝑠→0

where we have also used the relation 𝑃𝑢−𝑠 𝑓 ≥ e−𝛼𝑠 𝑃𝑢 𝑓 for the second inequality. Step 3. Let 𝑠 𝑘 = 𝑠 𝑘 (𝑤) > 0 be any sequence such that 𝑠 𝑘 → 0 and 𝑤 𝑠𝑘 (1) −1 𝑤 𝑠𝑘 → ¯ By (8.53) we have ¯ as 𝑘 → ∞, where 𝑤ˆ 0 is a probability measure on 𝐸. 𝑤ˆ 0 in 𝑀 ( 𝐸) 𝜋𝑢 𝑓 (𝑥) ≤ e (𝑐0 − ∥𝑏

+ ∥+2𝛽)𝑢

𝑤ˆ 0 ( 𝑓¯).

Then letting 𝑢 → 0 gives 𝑓 (𝑥) ≤ 𝑤ˆ 0 ( 𝑓¯). Let (𝑈 𝛼 ) 𝛼>0 denote the resolvent of (𝑃𝑡 )𝑡 ≥0 and let (𝑈¯ 𝛼 ) 𝛼>0 denote its Ray extension. By (8.54) for any 𝜃 > 2𝛽 we have

226

8 Entrance Laws and Kuznetsov Measures

(𝜃 − 2𝛽) −1 𝑓 (𝑥) =

∫

∞

e−( 𝜃−2𝛽)𝑢 𝑓 (𝑥)d𝑢

∫0 ∞

+ lim sup e−( 𝜃+2 ∥𝑏 ∥+𝑐0 )𝑢 𝑤 𝑠 (1) −1 𝑤 𝑠 ( 𝑃¯𝑢 𝑓 )d𝑢 𝑠→0 0 ∫ ∞ + e−( 𝜃+2 ∥𝑏 ∥+𝑐0 )𝑢 𝑤 𝑠 (1) −1 𝑤 𝑠 ( 𝑃¯𝑢 𝑓 )d𝑢 ≥ lim sup

≥

0

𝑠→0

+ = lim sup 𝑤 𝑠 (1) −1 𝑤 𝑠 (𝑈¯ 𝜃+2 ∥𝑏 ∥+𝑐0 𝑓¯)

𝑠→0

≥ 𝑤ˆ 0 (𝑈¯ 𝜃+2 ∥𝑏

+ ∥+𝑐 0

𝑓¯).

Multiplying both sides of the above equality by 𝜃 and letting 𝜃 → ∞ we obtain 𝑓 (𝑥) ≥ 𝑤ˆ 0 ( 𝑓¯). It follows that 𝑓 (𝑥) = 𝑤ˆ 0 ( 𝑓¯) and hence 𝑤ˆ 0 = 𝛿 𝑥 because ¯ By standard arguments we infer { 𝑓¯1 − 𝑓¯2 : 𝑓1 , 𝑓2 ∈ ℛ} is uniformly dense in 𝐶 ( 𝐸). ¯ and then in 𝑀 (𝐸). Since 𝑣 > 0 𝑤 𝑠 (1) −1 𝑤 𝑠 → 𝛿 𝑥 for L(𝑥)-a.e. 𝑤 ∈ 𝑊ˆ 𝑣 first in 𝑀 ( 𝐸) was arbitrary, the theorem follows when (𝑃𝑡 )𝑡 ≥0 is conservative. Step 4. For a non-conservative underlying semigroup (𝑃𝑡 )𝑡 ≥0 , we extend it to a conservative semigroup ( 𝑃˜𝑡 )𝑡 ≥0 on the Lusin topological space 𝐸˜ := 𝐸 ∪ {𝜕} with 𝜕 being an isolated cemetery. Take a spatially constant local branching mechanism ˜ + let 𝜙(𝜕, ˜ 𝑓˜) = 𝜙∗ ( 𝑓˜(𝜕)) and let 𝜆 ↦→ 𝜙∗ (𝜆) satisfying 𝜙∗′ (∞) = ∞. For 𝑓˜ ∈ 𝐵( 𝐸) ˜ ˜ ˜ 𝜙(𝑥, 𝑓 ) = 𝜙(𝑥, 𝑓 | 𝐸 ) if 𝑥 ∈ 𝐸. From those extensions we can use an analogue of (2.36) to define the cumulant semigroup (𝑉˜𝑡 )𝑡 ≥0 , which can be represented by (2.5) with (𝜆 𝑡 , 𝐿 𝑡 ) replaced by (𝜆˜ 𝑡 , 𝐿˜ 𝑡 ). If we extend the definition of 𝑓 ∈ 𝐵(𝐸) + to 𝐸˜ by setting 𝑓 (𝜕) = 0, it is not hard to show 𝑉˜𝑡 𝑓 (𝑥) = 𝑉𝑡 𝑓 (𝑥) for 𝑡 ≥ 0 and 𝑥 ∈ 𝐸. Then 𝜆 𝑡 (𝑥, ·) = 𝜆˜ 𝑡 (𝑥, ·)| 𝐸 for 𝑡 ≥ 0 and 𝑥 ∈ 𝐸. In particular, we have 𝜆˜ 𝑡 (𝑥, 𝐸) = 0 for 𝑡 > 0 and 𝑥 ∈ 𝐸𝐶 . Let (𝑣 𝑡 )𝑡 ≥0 be the cumulant semigroup of the CB-process with branching mechanism 𝜆 ↦→ 𝜙∗ (𝜆), which admits the representation (3.15). Since 𝜕 is a cemetery for ( 𝑃˜𝑡 )𝑡 ≥0 , we have ∫ ∞
˜ ˜ ˜ +. ˜ 1 − e−𝑢 𝑓 (𝜕) 𝑙 𝑡 (d𝑢), 𝑡 > 0, 𝑓˜ ∈ 𝐵( 𝐸) 𝑉𝑡 𝑓 (𝜕) = 0

˜ = 0 for 𝑡 > 0. For any 𝑡 > 0 and 𝑥 ∈ 𝐸𝐶 one can choose sufficiently Thus 𝜆˜ 𝑡 (𝜕, 𝐸) small 𝑟 > 0 and use Corollary 2.7 to see ∫ ˜ = ˜ = 0. 𝜆˜ 𝑡 (𝑥, 𝐸) 𝜆˜ 𝑟 (𝑥, d𝑦) 𝜆˜ 𝑡−𝑟 (𝑦, 𝐸) 𝐸˜

It follows that 𝑉˜𝑡 𝑓˜(𝑥) =

∫

˜
1 − e−𝜈 ( 𝑓 ) 𝐿˜ 𝑡 (𝑥, d𝜈),

˜ +. 𝑓˜ ∈ 𝐵( 𝐸)

(8.55)

˜ ◦ 𝑀 ( 𝐸)

˜ having Let 𝑊˜ 0 be the space of right continuous paths 𝑡 ↦→ 𝑤˜ 𝑡 from (0, ∞) to 𝑀 ( 𝐸) ˜ ˜ ˜ zero as a trap and let L(𝑥) = L(𝑥, ·) be the Kuznetsov measure on 𝑊0 corresponding to the entrance law 𝐿˜ (𝑥) := { 𝐿˜ 𝑡 (𝑥, ·) : 𝑡 > 0} defined by (8.55). Then { 𝑤˜ 𝑡 | 𝐸 : 𝑡 > 0} ˜ ˜ is equivalent to {𝑤 𝑡 : 𝑡 > 0} under L(𝑥). By Steps 1 and 3, for L(𝑥)-a.e. under L(𝑥) ˜ = 0 and lim𝑡→0 𝑤˜ 𝑡 ( 𝐸) ˜ −1 𝑤˜ 𝑡 = 𝛿 𝑥 in 𝑀 ( 𝐸), ˜ and 𝑤˜ ∈ 𝑊˜ 0 we have lim𝑡→0 𝑤˜ 𝑡 ( 𝐸)

8.4 Kuznetsov Measures and Excursion Laws

227

hence lim𝑡→0 𝑤˜ 𝑡 (𝐸) = 0 and ˜ −1 𝑤˜ 𝑡 ( 𝑓 ) 𝑤˜ 𝑡 ( 𝐸) = 𝑓 (𝑥) ˜ −1 𝑤˜ 𝑡 (𝐸) 𝑡→0 𝑤 ˜ 𝑡 ( 𝐸)

lim 𝑤˜ 𝑡 (𝐸) −1 𝑤˜ 𝑡 ( 𝑓 ) = lim 𝑡→0

for any 𝑓 ∈ 𝐶 (𝐸). This proves the theorem.

□

By Theorem 8.26, for any 𝑥 ∈ 𝐸𝐶 the Kuznetsov measure L(𝑥) is actually carried by the paths 𝑤 ∈ 𝑊ˆ 0 satisfying 𝑤 𝑡 → 0 and 𝑤 𝑡 (1) −1 𝑤 𝑡 → 𝛿 𝑥 in 𝑀 (𝐸) as 𝑡 → 0. We call those paths excursions starting at 𝑥 ∈ 𝐸𝐶 and call L(𝑥) an excursion law for the (𝜉, 𝜙)-superprocess. The following theorem gives an alternate characterization of the excursion law. Theorem 8.27 Let 𝑡 > 0 and assume 𝜆 ∈ 𝑀 ( [0, 𝑡]) satisfies 𝜆({0}) = 0. For a bounded positive Borel function (𝑠, 𝑥) ↦→ 𝑓𝑠 (𝑥) on [0, 𝑡] × 𝐸 let (𝑟, 𝑥) ↦→ 𝑢𝑟 (𝑥) be the unique bounded positive solution of (5.26). Then for any 𝑥 ∈ 𝐸𝐶 we have  n ∫ 𝑡 o L 𝑥, 1 − exp − 𝑤 𝑠 ( 𝑓𝑠 )𝜆(d𝑠) = 𝑢 0 (𝑥). (8.56) 0

Proof Let 𝜉 = (Ω, ℱ, ℱ𝑟 ,𝑡 , 𝜉𝑡 , P𝑟 , 𝑥 ) and 𝑋 = (𝑊, 𝒢, 𝒢𝑟 ,𝑡 , 𝑋𝑡 , Q𝑟 , 𝜇 ) be right continuous realizations of the underlying spatial motion and the (𝜉, 𝜙)-superprocess, respectively, started from the arbitrary initial time 𝑟 ≥ 0. By the Markov property of L(𝑥) and Theorem 5.15 we have    ∫ 𝑡 L 𝑥, 1 − exp − 𝑤 𝑠 ( 𝑓𝑠 )𝜆(d𝑠) 0   ∫  = lim L 𝑥, 1 − exp − 𝑤 𝑠 ( 𝑓𝑠 )𝜆(d𝑠) 𝜀→0 [ 𝜀,𝑡   ] ∫   = lim L 𝑥, Q 𝜀,𝑤 𝜀 1 − exp − 𝑋𝑠 ( 𝑓𝑠 )𝜆(d𝑠) 𝜀→0 [ 𝜀,𝑡 ]    = lim L 𝑥, 1 − exp − 𝑤 𝜀 (𝑢 𝜀 ) = lim 𝑣 0 (𝑥, 𝑢 𝜀 ), (8.57) 𝜀→0

𝜀→0

where (𝑟, 𝑥) ↦→ 𝑢𝑟 (𝑥) is defined by (5.26) and (𝑟, 𝑥) ↦→ 𝑣 𝑟 (𝑥) = 𝑣 𝑟 (𝑥, 𝑢 𝜀 ) is defined by ∫ 𝜀 𝑣 𝑟 (𝑥) + P𝑟 , 𝑥 [𝜙(𝜉 𝑠 , 𝑣 𝑠 )]d𝑠 = P𝑟 , 𝑥 𝑢 𝜀 (𝜉 𝜀 ). (8.58) 𝑟

Combining (5.26) and (8.58) gives ∫ 𝜀 𝑣 0 (𝑥, 𝑢 𝜀 ) = P0, 𝑥 𝑢 𝜀 (𝜉 𝜀 ) − P0, 𝑥 [𝜙(𝜉 𝑠 , 𝑣 𝑠 (·, 𝑢 𝜀 ))]d𝑠 ∫ 0  ∫ 𝑡  = P0, 𝑥 P 𝜀, 𝜉 𝜀 𝑓𝑠 (𝜉 𝑠 )𝜆(d𝑠) − P0, 𝑥 P 𝜀, 𝜉𝜀 𝜙(𝜉 𝑠 , 𝑢 𝑠 )d𝑠 [ 𝜀,𝑡 ] 𝜀 ∫ 𝜀 − P0, 𝑥 [𝜙(𝜉 𝑠 , 𝑣 𝑠 (·, 𝑢 𝜀 ))]d𝑠 0

228

8 Entrance Laws and Kuznetsov Measures



∫ = P0, 𝑥 ∫ −

∫



𝑡

𝑓𝑠 (𝜉 𝑠 )𝜆(d𝑠) − P0, 𝑥

𝜙(𝜉 𝑠 , 𝑢 𝑠 )d𝑠

[ 𝜀,𝑡 ]

𝜀

𝜀

P0, 𝑥 [𝜙(𝜉 𝑠 , 𝑣 𝑠 (·, 𝑢 𝜀 ))]d𝑠,

0

which converges to 𝑢 0 (𝑥) as 𝜀 → 0. Then (8.56) follows from (8.57).

□

Example 8.1 Suppose that 𝜆 ↦→ 𝜙(𝜆) is a branching mechanism given by (3.1) satisfying 𝜙 ′ (∞) = ∞. By Theorem 3.14, the corresponding CB-process has cumulant semigroup admitting the representation (3.15). Let 𝐷 0 [0, ∞) + be the set of positive càdlàg paths {𝑤(𝑡) : 𝑡 ≥ 0} satisfying 𝑤(𝑡) = 𝑤(0) = 0 for 𝑡 ≥ inf{𝑠 > 0 : 𝑤(𝑠) = 0}. By Theorem 8.26, the entrance law (𝑙 𝑡 )𝑡 >0 defined by (3.15) corresponds to an excursion law Q(𝑙, d𝑤) that can be identified as a 𝜎-finite measure on 𝐷 0 [0, ∞) + . Example 8.2 Let 𝐸 be a complete separable metric space. Suppose that 𝜉 is a càdlàg Borel right process in 𝐸 satisfying Condition 4.7. We shall construct a variation of the 𝜉-Brownian snake. Let 𝑅 𝑎,𝑏 ((𝑢, 𝑦), d(𝑤, 𝑧)) be the kernel on 𝑆 defined as in Section 4.2. Let 𝐶 [0, ∞) + denote the space of positive continuous functions on [0, ∞) furnished with the topology of local uniform convergence. For 𝑔 ∈ 𝐶 [0, ∞) + 𝑔 and (𝑤 0 , 𝑧0 ) ∈ 𝑆 satisfying 𝑔(0) = 𝑧0 let Q (𝑤 ,𝑧 ) denote the law on 𝑆 [0,∞) of 0 0 the time-inhomogeneous Markov process started at (𝑤 0 , 𝑧0 ) whose transition kernel from time 𝑟 ≥ 0 to time 𝑡 ≥ 𝑟 is 𝑅𝑚(𝑟 ,𝑡),𝑔 (𝑡) ((𝑢, 𝑦), d(𝑤, 𝑧)),

𝑔(𝑟) = 𝑦, (𝑢, 𝑦) ∈ 𝑆, (𝑤, 𝑧) ∈ 𝑆,

where 𝑚(𝑟, 𝑡) = inf 𝑟 ≤𝑠 ≤𝑡 𝑔(𝑠). Let n(d𝑔) denote Itô’s excursion law, which is the Kuznetsov measure corresponding to the entrance law (A.31) for the absorbingbarrier Brownian motion. We think of n(d𝑔) as a 𝜎-finite measure carried by the set 𝐶0 [0, ∞) + of positive continuous paths {𝑔(𝑡) : 𝑡 ≥ 0} such that 𝑔(0) = 𝑔(𝑡) = 0 for every 𝑡 ≥ inf{𝑠 > 0 : 𝑔(𝑠) = 0}. For 𝑥 ∈ 𝐸 let N 𝑥 be the 𝜎-finite measure on 𝐶0 [0, ∞) + × 𝑆 [0,∞) defined by 𝑔

N 𝑥 (d𝑔, d(𝜂, 𝜁)) = n(d𝑔)Q (𝑤

0 ,0)

(d(𝜂, 𝜁)),

(8.59)

where 𝑤 0 (𝑡) = 𝑥 for all 𝑡 ≥ 0. Roughly speaking, under N 𝑥 the process {(𝜂 𝑠 , 𝜁 𝑠 ) : 𝑠 ≥ 0} behaves as the 𝜉-Brownian snake. The only difference is that {𝜁 𝑠 : 𝑠 ≥ 0} is distributed according to Itô’s excursion law. Let 𝜏0 (𝜁) = inf{𝑠 > 0 : 𝜁 𝑠 = 0} and let {2𝑙 𝑠 (𝑦, 𝜁) : 𝑠 ≥ 0, 𝑦 ≥ 0} be the local time of {𝜁 𝑠 : 𝑠 ≥ 0}. For 𝑡 ≥ 0 we define the measure 𝜈𝑡 (d𝑦) = 𝜈𝑡 (𝑔, 𝜂, 𝜁, d𝑦) on 𝐸 by ∫

𝜏0 (𝜁 )

𝑓 (𝜂 𝑠 (𝜁 𝑠 ))d𝑙 𝑠 (𝑡, 𝜁),

𝜈𝑡 ( 𝑓 ) =

𝑓 ∈ 𝐶 (𝐸) + .

(8.60)

0

Then {𝜈𝑡 : 𝑡 ≥ 0} is continuous in 𝑀 (𝐸). It was proved in Le Gall (1999, p. 63) that 1 𝜈𝑡 ( 𝑓 ) = lim 𝜀→0 𝜀

∫

𝜏0 (𝜁 )

1 [𝑡 ,𝑡+𝜀 ] (𝜁 𝑠 ) 𝑓 (𝜂 𝑠 (𝜁 𝑠 ))d𝑠 0

8.5 Cluster Representations of the Process

229

in N 𝑥 -measure. Consequently, for any ℎ ∈ 𝐶 (R+ ) + with bounded support, ∫

∞

ℎ(𝑡)𝜈𝑡 ( 𝑓 )d𝑡 = lim

∞ ∑︁

∫

𝑘→∞

0

∫

𝜏0 (𝜁 )

1 [𝑖/𝑘, (𝑖+1)/𝑘 ] (𝜁 𝑠 ) 𝑓 (𝜂 𝑠 (𝜁 𝑠 ))d𝑠

ℎ(𝑖/𝑘) 0

𝑖=0 𝜏0 (𝜁 )

ℎ(𝜁 𝑠 ) 𝑓 (𝜂 𝑠 (𝜁 𝑠 ))d𝑠.

= 0

From Proposition 3 of Le Gall (1999, p. 59) it follows that  n ∫ ∞ o N 𝑥 1 − exp − ℎ(𝑡)𝜈𝑡 ( 𝑓 )d𝑡 = 𝑢 0 (𝑥), 0

where (𝑟, 𝑥) ↦→ 𝑢𝑟 (𝑥) solves the integral equation ∫ ∞ ∫ ∞     2 P𝑟 , 𝑥 𝑢 𝑠 (𝜉 𝑠 ) d𝑠 = P𝑟 , 𝑥 ℎ(𝑠) 𝑓 (𝜉 𝑠 ) d𝑠. 𝑢𝑟 (𝑥) + 𝑟

𝑟

Thus {𝜈𝑡 : 𝑡 > 0} under N 𝑥 is distributed on 𝑊ˆ 0 according to the excursion law L(𝑥) characterized by (8.56) for this (𝜉, 𝜙)-superprocess with binary local branching mechanism 𝜙(𝑧) = 𝑧2 . This gives a representation of L(𝑥) using N 𝑥 . In view of (8.60), the result of Theorem 8.26 is obviously true for the binary local branching. It was pointed out in Le Gall (1999, p. 56) that N 𝑥 can be understood as an excursion law of the 𝜉-Brownian snake from the state (𝑤 0 , 0).

8.5 Cluster Representations of the Process In this section, we give some cluster representations for the MB-process in terms of Poisson random measures based on Kuznetsov measures. Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroups of a Borel right MB-process in 𝑀 (𝐸). Let (𝑉𝑡 )𝑡 ≥0 be the corresponding cumulant semigroup with canonical representation (2.5). Let 𝑊ˆ and 𝑊ˆ 0 be defined as in Section 8.4. For 𝜇 ∈ 𝑀 (𝐸) let L(𝜇, ·) be the Kuznetsov measure corresponding to the entrance rule 𝜇𝐿 := (𝜇𝐿 𝑡 )𝑡 >0 . Then we have ∫ 𝜇(d𝑥)L(𝑥, d𝑤), 𝑤 ∈ 𝑊ˆ . L(𝜇, d𝑤) = 𝐸

Let 𝑁 𝜇 (d𝑤) be a Poisson random measure on 𝑊ˆ with intensity L(𝜇, d𝑤). For 𝑡 ≥ 0 define ∫ 𝜇 𝑋𝑡 = 𝜇𝜆 𝑡 + (8.61) 𝑤 𝑡 𝑁 𝜇 (d𝑤) ˆ 𝑊

and 𝒢𝑡 = 𝜎({𝑁 𝜇 ( 𝐴 ∩ {𝛼 ≤ 𝑡}) : 𝐴 ∈ 𝒜𝑡0 }). 𝜇

230

8 Entrance Laws and Kuznetsov Measures 𝜇

𝜇

Theorem 8.28 The pair {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} defined above is an MB-process with 𝜇 transition semigroup (𝑄 𝑡 )𝑡 ≥0 and initial value 𝑋0 = 𝜇. 𝜇

Proof It is easy to check that 𝑋𝑡 has distribution 𝑄 𝑡 (𝜇, ·) on 𝑀 (𝐸). In fact, by the construction (8.61) for any 𝑓 ∈ 𝐵(𝐸) + we have   ∫ 𝜇 𝜇 𝑤 𝑡 ( 𝑓 )𝑁 (d𝑤) P exp{−𝑋𝑡 ( 𝑓 )} = P exp − 𝜇𝜆 𝑡 ( 𝑓 ) −   ∫ 𝑊ˆ −𝑤𝑡 ( 𝑓 ) )L(𝜇, d𝑤) = exp − 𝜇𝜆 𝑡 ( 𝑓 ) − (1 − e   ∫𝑊ˆ
= exp − 𝜇𝜆 𝑡 ( 𝑓 ) − 1 − e−𝜈 ( 𝑓 ) 𝜇𝐿 𝑡 (d𝜈) 𝑀 (𝐸) ◦  = exp − 𝜇(𝑉𝑡 𝑓 ) . Let 𝑡 ≥ 𝑟 ≥ 0 and let 𝑤 ↦→ ℎ(𝑤) be a bounded positive function on 𝑊ˆ measurable relative to 𝒜𝑟0 . Let ℎ𝑟 (𝑤) = ℎ(𝑤)1 { 𝛼(𝑤) ≤𝑟 } . It is easy to see that ℎ𝑟 ( [0]) = 0. Then we use (8.48) to see   ∫ 𝜇 𝜇 P exp − ℎ𝑟 (𝑤)𝑁 (d𝑤) − 𝑋𝑡 ( 𝑓 ) ˆ 𝑊   ∫ 𝜇 [ℎ𝑟 (𝑤) + 𝑤 𝑡 ( 𝑓 )]𝑁 (d𝑤) = P exp − 𝜇𝜆 𝑡 ( 𝑓 ) −   ∫ 𝑊ˆ = exp − 𝜇𝜆 𝑡 ( 𝑓 ) − (1 − e−ℎ𝑟 (𝑤)−𝑤𝑡 ( 𝑓 ) )L(𝜇, d𝑤)  ∫𝑊ˆ = exp − 𝜇𝜆 𝑡 ( 𝑓 ) − (1 − e−ℎ𝑟 (𝑤) )L(𝜇, d𝑤) ˆ 𝑊  ∫ −𝑤𝑡 ( 𝑓 ) −ℎ𝑟 (𝑤) )L(𝜇, d𝑤) (1 − e e − ˆ 𝑊  ∫ = exp − 𝜇𝜆 𝑡 ( 𝑓 ) − (1 − e−ℎ𝑟 (𝑤) )L(𝜇, d𝑤) ˆ 𝑊 ∫   e−ℎ𝑟 ( [0]) 𝜆𝑟 (𝑥, 𝑉𝑡−𝑟 𝑓 ) − 𝜆 𝑡 (𝑥, 𝑓 ) 𝜇(d𝑥) −  ∫𝐸 e−ℎ𝑟 (𝑤) (1 − e−𝑤𝑟 (𝑉𝑡−𝑟 𝑓 ) )L(𝜇, d𝑤) − ˆ 𝑊   ∫ = exp − 𝜇𝜆𝑟 (𝑉𝑡−𝑟 𝑓 ) − (1 − e−𝐺𝑟 ,𝑡 (𝑤) )L(𝜇, d𝑤) ˆ 𝑊   ∫ 𝜇 𝐺 𝑟 ,𝑡 (𝑤)𝑁 (d𝑤) = P exp − 𝜇𝜆𝑟 (𝑉𝑡−𝑟 𝑓 ) − ˆ 𝑊   ∫ 𝜇 𝜇 ℎ𝑟 (𝑤)𝑁 (d𝑤) − 𝑋𝑟 (𝑉𝑡−𝑟 𝑓 ) , = P exp − ˆ 𝑊 𝜇

𝜇

where 𝐺 𝑟 ,𝑡 (𝑤) = ℎ𝑟 (𝑤) + 𝑤 𝑟 (𝑉𝑡−𝑟 𝑓 ). Then {(𝑋𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a Markov process □ with transition semigroup (𝑄 𝑡 )𝑡 ≥0 .

8.5 Cluster Representations of the Process

231

The point of the cluster representation (8.61) is that it gives a construction of 𝜇 the whole path of the MB-process {𝑋𝑡 : 𝑡 ≥ 0}. From the construction we see that the trajectory of the MB-process is decomposed into two parts, the first part is deterministic and the second part consists of countably many pieces selected by the 𝜇 Poisson random measure. For any 𝑡 > 0 let 𝑁𝑡 (d𝜈) be the image of 𝑁 𝜇 (d𝑤) under 𝜇 the mapping 𝑤 ↦→ 𝑤 𝑡 . Then 𝑁𝑡 (d𝜈) is a Poisson random measure on 𝑀 (𝐸) ◦ with intensity 𝜇𝐿 𝑡 (d𝜈) and (8.61) implies ∫ 𝜇 𝜇 (8.62) 𝜈𝑁𝑡 (d𝜈). 𝑋𝑡 = 𝜇𝜆 𝑡 + 𝑀 (𝐸) ◦ 𝜇

This gives a cluster representation of the random measure 𝑋𝑡 for 𝑡 > 0. By slightly modifying the representation (8.61), we can construct an interesting family of correlated MB-processes on one probability space. To this end, let us fix a measure 𝜆 ∈ 𝑀 (𝐸). Let 𝑁 (d𝑥, d𝑤, d𝑢) be a Poisson random measure on 𝐸 × 𝑊ˆ × (0, ∞) with intensity 𝜆(d𝑥)L(𝑥, d𝑤)d𝑢. Suppose that 𝜇 ∈ 𝑀 (𝐸) is another measure absolutely continuous relative to 𝜆 with Radon–Nikodym derivative 𝜇¤ = d𝜇/d𝜆. For 𝑡 ≥ 0 let 𝜇 𝑋𝑡

𝜇¤ ( 𝑥)

∫ ∫ ∫ = 𝜇𝜆 𝑡 +

𝑤 𝑡 𝑁 (d𝑥, d𝑤, d𝑢) ˆ 𝑊

𝐸

(8.63)

0

and let 𝒢𝑡 be the 𝜎-algebra generated by {𝑁 (𝐵× ( 𝐴∩{𝛼 ≤ 𝑡}) ×𝐶) : 𝐵 ∈ ℬ(𝐸), 𝐴 ∈ 𝒜𝑡0 , 𝐶 ∈ ℬ(0, ∞)}. 𝜇

Theorem 8.29 The process 𝑋 𝜇 = {𝑋𝑡 : 𝑡 ≥ 0} defined above is an MB-process 𝜇 relative to {𝒢𝑡 : 𝑡 ≥ 0} with transition semigroup (𝑄 𝑡 )𝑡 ≥0 and initial value 𝑋0 = 𝜇. Moreover, a different choice of the Radon–Nikodym derivative 𝜇¤ gives a modification 𝜇 of {𝑋𝑡 : 𝑡 ≥ 0}. Proof The first assertion follows easily by a modification of the proof of Theorem 8.28. Suppose that 𝜇¤ 0 is another representative of the Radon–Nikodym derivative d𝜇/d𝜆. Let 𝜇¤ 1 = 𝜇¤ ∧ 𝜇¤ 0 and 𝜇¤ 2 = 𝜇¤ ∨ 𝜇¤ 0 . For 𝑖 = 0, 1, 2 let 𝑋𝑡(𝑖)

∫ ∫ ∫ = 𝜇𝜆 𝑡 +

𝜇¤ 𝑖 ( 𝑥)

𝑤 𝑡 𝑁 (d𝑥, d𝑤, d𝑢), 𝐸

ˆ 𝑊

𝑡 ≥ 0.

0

Then each {𝑋𝑡(𝑖) : 𝑡 ≥ 0} is an MB-processes with transition semigroup (𝑄 𝑡 )𝑡 ≥0 (𝑖) and initial value 𝑋0(𝑖) = 𝜇, and so P[e−𝑋𝑡 (1) ] = e−𝜇 (𝑉𝑡 1) . From the constructions 𝜇 of the processes it is clear that 𝑋𝑡(1) ≤ 𝑋𝑡 ≤ 𝑋𝑡(2) and 𝑋𝑡(1) ≤ 𝑋𝑡(0) ≤ 𝑋𝑡(2) . Then 𝜇 (𝑖) {𝑋𝑡 : 𝑡 ≥ 0} and {𝑋𝑡 : 𝑡 ≥ 0}, 𝑖 = 0, 1, 2, are all modifications of each other. □ Let 𝑀𝜆 (𝐸) denote the subset of 𝑀 (𝐸) consisting of measures absolutely continuous relative to the measure 𝜆 ∈ 𝑀 (𝐸). It is not hard to show that the family of MB-processes {𝑋 𝜇 : 𝜇 ∈ 𝑀𝜆 (𝐸)} constructed by (8.63) has the following properties:

232

8 Entrance Laws and Kuznetsov Measures

• For 𝜇 ≤ 𝜈 ∈ 𝑀𝜆 (𝐸), the difference 𝑋 𝜈 − 𝑋 𝜇 = {𝑋𝑡𝜈 − 𝑋𝑡 : 𝑡 ≥ 0} is an 𝜇 MB-process relative to {𝒢𝑡 : 𝑡 ≥ 0} with 𝑋0𝜈 − 𝑋0 = 𝜈 − 𝜇. • For 𝜇1 ≤ 𝜈1 ≤ 𝜇2 ≤ 𝜈2 ∈ 𝑀𝜆 (𝐸), the processes 𝑋 𝜈1 − 𝑋 𝜇1 and 𝑋 𝜈2 − 𝑋 𝜇2 are independent of each other. 𝜇

We now turn to a (𝜉, 𝜙)-superprocess, where 𝜉 is a Borel right process in 𝐸 and 𝜙 is a branching mechanism given by (2.29) or (2.30). We assume Condition 5.31 holds with 𝜙∗ satisfying Grey’s condition. By Corollary 5.33 the cumulant semigroup of 𝑔 the superprocess has representation (2.11). Let ( 𝑓 , 𝑔) ↦→ 𝑉𝑡 𝑓 be the operator from + + + 𝐵(𝐸) × 𝐵(𝐸) to 𝐵(𝐸) defined by ∫ 𝑔 𝑉𝑡 𝑓 (𝑥) = 𝜈(𝑔)e−𝜈 ( 𝑓 ) 𝐿 𝑡 (𝑥, d𝜈), 𝑥 ∈ 𝐸, 𝑓 , 𝑔 ∈ 𝐵(𝐸) + . (8.64) 𝑀 (𝐸) ◦

By Propositions 2.26 and 2.29 we have ∫ 𝑡 ∫ 𝑔 𝑔 d𝑠 𝑉𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑔(𝑥) − 𝜓(𝑦, 𝑉𝑠 𝑓 , 𝑉𝑠 𝑓 )𝑃𝑡−𝑠 (𝑥, d𝑦), 0

(8.65)

𝐸

where ( 𝑓 , 𝑔) ↦→ 𝜓(·, 𝑓 , 𝑔) is the operator from 𝐵(𝐸) + × 𝐵(𝐸) to 𝐵(𝐸) defined by (2.59). In this situation, for any 𝜇 ∈ 𝑀 (𝐸) the family {𝜇𝐿 𝑡 : 𝑡 > 0} is an entrance law for (𝑄 ◦𝑡 )𝑡 ≥0 . As an application of the cluster representation (8.62), we prove the following theorem on absolute continuities of the distributions associated with the (𝜉, 𝜙)-superprocess: Theorem 8.30 Suppose that Condition 5.31 holds with 𝜙∗ satisfying Grey’s condition. Then for any 𝜇1 , 𝜇2 ∈ 𝑀 (𝐸) the following properties are equivalent: (1) 𝜇1 𝜋𝑟 ≪ 𝜇2 𝜋𝑡 on 𝐸 for all 0 < 𝑟 ≤ 𝑡; (2) 𝜇1 𝐿 𝑟 ≪ 𝜇2 𝐿 𝑡 on 𝑀 (𝐸) ◦ for all 0 < 𝑟 ≤ 𝑡; (3) 𝑄 𝑟 (𝜇1 , ·) ≪ 𝑄 𝑡 (𝜇2 , ·) on 𝑀 (𝐸) for all 0 < 𝑟 ≤ 𝑡; (4) Q 𝜇1 (𝑋𝑟+· ∈ ·) ≪ Q 𝜇2 (𝑋𝑡+· ∈ ·) on 𝑊ˆ 0 for all 0 < 𝑟 ≤ 𝑡. Proof “(3) ⇔ (4)” The implication (4) ⇒ (3) is obvious. By the Markov property we have ∫ 𝑄 𝑟 (𝜇, d𝜈)Q𝜈 (𝑋· ∈ ·). Q 𝜇 (𝑋𝑟+· ∈ ·) = 𝑀 (𝐸)

This gives the implication (3) ⇒ (4). “(1) ⇒ (2)” Suppose that (1) holds. By Corollary 5.34, for 𝑡 > 0 we have 𝑓𝑡 := 𝐿 𝑡 (·, 1) ∈ 𝐵(𝐸) + and 𝜇𝜆 𝑡 = 0 in (8.62). Then, for any 𝐹 ∈ ℬ(𝑀 (𝐸) ◦ ),  𝜇 𝜇 (8.66) 𝑄 𝑡 (𝜇, 𝐹) ≥ P 𝑋𝑡 ∈ 𝐹, 𝑁𝑡 (1) = 1 = e−𝜇 ( 𝑓𝑡 ) 𝜇𝐿 𝑡 (𝐹). Take 0 < 𝑢 < 𝑟 ≤ 𝑡. From (8.66)) we have ∫ ∫ 𝜇2 𝐿 𝑡 = 𝑄 𝑢◦ (𝜇, ·)𝜇2 𝐿 𝑡−𝑢 (d𝜇) ≥ 𝑀 (𝐸) ◦

𝑀 (𝐸) ◦

e−𝜇 ( 𝑓𝑢 ) 𝜇𝐿 𝑢 (·)𝜇2 𝐿 𝑡−𝑢 (d𝜇).

8.5 Cluster Representations of the Process

233

Suppose that 𝜇2 𝐿 𝑡 (𝐹) = 0 for 𝐹 ∈ ℬ(𝑀 (𝐸) ◦ ). By the above relation we have ∫ ∫ ∫ 𝜇(d𝑥)𝐿 𝑢 (𝑥, 𝐹) = 𝜇2 𝐿 𝑡−𝑢 (d𝜇) 𝜇2 𝜋𝑡−𝑢 (d𝑥)𝐿 𝑢 (𝑥, 𝐹). 0= 𝑀 (𝐸) ◦

𝐸

𝐸

Then 𝐿 𝑢 (𝑥, 𝐹) = 0 for 𝜇2 𝜋𝑡−𝑢 -a.a. 𝑥 ∈ 𝐸 and property (1) implies 𝐿 𝑢 (𝑥, 𝐹) = 0 for 𝜇1 𝜋𝑟−𝑢 -a.a. 𝑥 ∈ 𝐸. A reversion of the above steps yields ∫ ∫ ∫ 𝜇1 𝐿 𝑟−𝑢 (d𝜇) 𝜇(d𝑥)𝐿 𝑢 (𝑥, 𝐹) 0= 𝜇1 𝜋𝑡−𝑢 (d𝑥)𝐿 𝑢 (𝑥, 𝐹) = 𝑀 (𝐸) ◦

𝐸

𝐸

and, consequently, ∫ 0= ∫𝑀 (𝐸) ◦ = ∫𝑀 (𝐸) ◦

e−𝜇 ( 𝑓𝑢 ) 𝜇𝐿 𝑢 (𝐹)𝜇1 𝐿 𝑟−𝑢 (d𝜇)
𝜇 𝜇 P 𝑋𝑢 ∈ 𝐹, 𝑁𝑢 (1) = 1 𝜇1 𝐿 𝑟−𝑢 (d𝜇) 



𝜇 𝜇 P 𝑋𝑢 ∈ 𝐹 − P 𝑁𝑢 (1) ≥ 2 𝜇1 𝐿 𝑟−𝑢 (d𝜇) 𝑀 (𝐸) ◦ ∫
𝜇 = 𝜇1 𝐿 𝑟 (𝐹) − P 𝑁𝑢 (1) ≥ 2 𝜇1 𝐿 𝑟−𝑢 (d𝜇),

≥

(8.67)

𝑀 (𝐸) ◦

𝜇

𝜇

where for the inequality we have used the fact that 𝑋𝑢 ∈ 𝐹 implies 𝑁𝑢 (1) ≥ 1. From (2.11), (2.36) and (8.65) we see that the last term on the right-hand side of (8.67) is equal to ∫
1 − e−𝜇 ( 𝑓𝑢 ) − 𝜇( 𝑓𝑢 )e−𝜇 ( 𝑓𝑢 ) 𝜇1 𝐿 𝑟−𝑢 (d𝜇) 𝑀 (𝐸) ◦ ∫ = 𝜇1 (𝑉𝑟−𝑢 𝑓𝑢 ) − 𝜇( 𝑓𝑢 )e−𝜇 ( 𝑓𝑢 ) 𝐿 𝑟−𝑢 (𝜇1 , d𝜇) ◦ 𝑀 (𝐸) ∫ 𝑟−𝑢 ∫ 𝑓 𝑑𝑠 = 𝜓(𝑦, 𝑉𝑠 𝑓𝑢 , 𝑉𝑠 𝑢 𝑓𝑢 )𝜇1 𝑃𝑡−𝑠 (d𝑦) 𝐸 ∫ 0 ∫ 𝑟−𝑢 𝜙(𝑦, 𝑉𝑠 𝑓𝑢 (𝑦))𝜇1 𝑃𝑡−𝑠 (d𝑦). 𝑑𝑠 − 0

𝐸

It is elementary to see that the right-hand side of the equation above tends to zero as 𝑢 → 𝑟. By (8.67) we conclude 𝐿 𝑟 (𝜇1 , 𝐹) = 0. “(2) ⇒ (3)” By the cluster decomposition of the superprocess we see that 𝑄 𝑟 (𝜇1 , ·) Í 𝜂1 1 Í 𝜂2 2 and 𝑄 𝑡 (𝜇2 , ·) are the laws of 𝑖=1 𝜈𝑖 , respectively, where 𝜂1 and 𝜂2 are 𝜈𝑖 and 𝑖=1 Poissonian random variables with means 𝐿 𝑟 (𝜇1 , 1) and 𝐿 𝑡 (𝜇2 , 1), respectively, and {𝜈𝑖1 : 𝑖 ≥ 1} and {𝜈𝑖2 : 𝑖 ≥ 1} are i.i.d. sequences with laws 𝜇1 ( 𝑓𝑟 ) −1 𝜇1 𝐿 𝑟 and 𝜇2 ( 𝑓𝑡 ) −1 𝜇2 𝐿 𝑡 , respectively. Clearly, (2) implies the 𝑛-fold product of 𝜇1 ( 𝑓𝑟 ) −1 𝜇1 𝐿 𝑟 is absolutely continuous with respect to that of 𝜇2 ( 𝑓𝑡 ) −1 𝜇2 𝐿 𝑡 . Therefore we can sum over the values of 𝜂1 and 𝜂2 to obtain 𝑄 𝑟 (𝜇1 , ·) ≪ 𝑄 𝑡 (𝜇2 , ·) as required. □ “(3) ⇒ (1)” This follows immediately from (2.53).

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8 Entrance Laws and Kuznetsov Measures

Corollary 8.31 For a local branching mechanism 𝜙 give by (2.49), the properties in Theorem 8.30 equivalent to: (5) 𝜇1 𝑃𝑟 ≪ 𝜇2 𝑃𝑡 on 𝐸 for all 0 < 𝑟 ≤ 𝑡. Proof For the local branching mechanism we have 𝜋𝑡 = 𝑃𝑡𝑏 for 𝑡 ≥ 0, which implies that 𝜋𝑡 (𝑥, d𝑦) and 𝑃𝑡 (𝑥, d𝑦) are absolutely continuous with respect to each other for every 𝑥 ∈ 𝐸. Then the result is obvious. □ Example 8.3 Let us continue the discussion in Example 8.2. Suppose that 𝜇 ∈ 𝑀 (𝐸) and 𝑁 (d𝑥, d𝑔, d(𝜂, 𝜁)) is a Poisson random measure on 𝐸 × 𝐶0 [0, ∞) + × 𝑆 [0,∞) with intensity 𝜇(d𝑥)N 𝑥 (d𝑔, d(𝜂, 𝜁)). Let 𝜈𝑡 = 𝜈𝑡 (𝑔, 𝜂, 𝜁) be given by (8.60) and define the measure-valued process {𝑋𝑡 : 𝑡 ≥ 0} by 𝑋0 = 𝜇 and ∫ ∫ ∫ 𝑋𝑡 = (8.68) 𝜈𝑡 𝑁 (d𝑥, d𝑔, d(𝜂, 𝜁)), 𝑡 > 0. 𝐸

𝐶0 [0,∞) +

𝑆 [0,∞)

Then {𝑋𝑡 : 𝑡 ≥ 0} is a (𝜉, 𝜙)-superprocess with local branching mechanism 𝜙(𝑧) = 𝑧2 ; see Le Gall (1999, pp. 61–62). This follows from Theorem 8.28 and the fact that {𝜈𝑡 : 𝑡 > 0} is distributed under N 𝑥 according to the excursion law L(𝑥) of the (𝜉, 𝜙)-superprocess. Let {(𝑥 𝑖 , 𝑔𝑖 , (𝜂𝑖 , 𝜁 𝑖 )) : 𝑖 = 1, 2, . . .} be the countable support of 𝑁 (d𝑥, d𝑔, d(𝜂, 𝜁)). Let 𝑇 = sup{𝑡 ≥ 0 : 𝑋𝑡 > 0} = inf{𝑡 ≥ 0 : 𝑋𝑡 = 0} denote the extinction time of {𝑋𝑡 : 𝑡 ≥ 0}. Then there is a unique label 𝑘 ≥ 1 such that 𝑇=

sup 0≤𝑠 ≤𝜏0

(𝜁 𝑘 )

𝜁 𝑘 (𝑠) = sup

sup

𝑖 ≥1 0≤𝑠 ≤𝜏0

𝜁 𝑖 (𝑠).

(𝜁 𝑖 )

Let 𝜎 be the unique value in [0, 𝜏0 (𝜁 𝑘 )] such that 𝜁 𝑘 (𝜎) = 𝑇. It is easy to see that lim 𝑋𝑡 (1) −1 𝑋𝑡 = 𝛿 𝜂 𝜎𝑘 (𝑇) . 𝑡 ↑𝑇

This gives a description of the behavior of the superprocess near its extinction time.

8.6 Super-Absorbing-Barrier Brownian Motions We first consider a bounded domain 𝐸 in the Euclidean space R𝑑 with twice continuously differentiable boundary 𝜕𝐸. Let 𝜉 be an absorbing-barrier Brownian motion in 𝐸. Let (𝑃𝑡 )𝑡 ≥0 denote the transition semigroup of 𝜉. It is well known that 𝑃𝑡 (𝑥, d𝑦) has a density 𝑝 𝑡 (𝑥, 𝑦) for 𝑡 > 0, which is the fundamental solution of the heat equation on 𝐸 with Dirichlet boundary condition. Moreover, 𝑝 𝑡 (𝑥, 𝑦) = 𝑝 𝑡 (𝑦, 𝑥) is continuously differentiable in 𝑥 and 𝑦 to the boundary 𝜕𝐸; see, e.g., Friedman (1964, p. 83). We shall use 𝜕𝑧 to denote the operator of inward normal differentiation at 𝑧 ∈ 𝜕𝐸. Clearly,

8.6 Super-Absorbing-Barrier Brownian Motions

∫

235

1

𝑃𝑠 1(𝑥)d𝑠,

ℎ(𝑥) =

𝑥∈𝐸

(8.69)

0

defines a bounded strictly positive excessive function for (𝑃𝑡 )𝑡 ≥0 and ℎ(𝑥) → 0 as 𝑥 → 𝑧 ∈ 𝜕𝐸. Let 𝑀ℎ (𝐸) denote the set of 𝜎-finite measures 𝜇 on 𝐸 such that 𝜇(ℎ) < ∞. Lemma 8.32 The function ℎ is continuously differentiable to the boundary and 𝑧 ↦→ 𝜕𝑧 ℎ is bounded above and bounded away from zero on 𝜕𝐸. Proof We only give the proof for 𝑑 ≥ 2 since the result for 𝑑 = 1 is well known. We shall use superscripts to indicate the dependence of the objects on the domain 𝐸. The arguments consist of three steps. Step 1. By Friedman (1964, p. 83) and the dominated convergence theorem it is easy to see that ℎ is continuously differentiable to the boundary and 1

∫ 𝜕𝑧 ℎ 𝐸 =

∫ 𝜕𝑧 𝑃𝑠 1d𝑠 =

0

1

∫ 𝜕𝑧 𝑝 𝑠𝐸 (·, 𝑥)d𝑥,

d𝑠 0

𝑧 ∈ 𝜕𝐸.

(8.70)

𝐸

Let (ℱ𝑡 , 𝜉 (𝑡), P 𝑥 ) be a standard Brownian motion on R𝑑 and let 𝜏𝐸 𝑐 denote its hitting time of 𝐸 𝑐 . For 𝑥 ∈ 𝐸 we have P 𝑥 -a.s. 𝜉 (𝜏𝐸 𝑐 ) ∈ 𝜕𝐸 and 𝜕𝑧 𝑝 𝑠𝐸 (·, 𝑥)d𝑠𝜎(d𝑧) = 𝜕𝑧 𝑝 𝑠𝐸 (𝑥, ·)d𝑠𝜎(d𝑧) = 2P 𝑥 {𝜏𝐸 𝑐 ∈ d𝑠, 𝜉 (𝜏𝐸 𝑐 ) ∈ d𝑧}, where 𝜎(d𝑧) is the volume element on 𝜕𝐸; see, e.g., Hsu (1986, p. 110). Since 𝐸 is bounded, integrating both sides of the equality above and using (8.70) we see that ∫ ∫ 𝐸 𝜕𝑧 ℎ 𝜎(d𝑧) = 2 0< (8.71) P 𝑥 {𝜏𝐸 𝑐 ≤ 1}d𝑥 < ∞. 𝜕𝐸

𝐸

Step 2. For 𝑤 ∈ R𝑑 and 𝑟 > 0 let 𝐵 𝑤 (𝑟) = {𝑥 ∈ R𝑑 : |𝑥 − 𝑤| < 𝑟}. By (8.70), (8.71) and the spatial homogeneity and symmetry of the Brownian motion, there is a constant 0 < 𝑐(𝑟) < ∞ such that 𝜕𝑧 ℎ 𝐵𝑤 (𝑟) = 𝑐(𝑟) for every 𝑧 ∈ 𝜕𝐵 𝑤 (𝑟). Consequently, the theorem holds for 𝐸 = 𝐵 𝑤 (𝑟). In the general case, the smoothness of the boundary 𝜕𝐸 implies the existence of a constant 𝑟 > 0 such that for each 𝑧 ∈ 𝜕𝐸 there exists a 𝑤 ∈ 𝐸 such that |𝑤 − 𝑧| = 𝑟 and 𝐵 𝑤 (𝑟) ⊂ 𝐸. Then using the absorbing-barrier Brownian motion we have 𝑝 𝑠𝐸 (𝑥, 𝑦)d𝑦 = P 𝑥 {𝜏𝐸 𝑐 > 𝑠, 𝜉 (𝑠) ∈ 𝑑𝑦} ≥ P 𝑥 {𝜏𝐵𝑤 (𝑟) 𝑐 > 𝑠, 𝜉 (𝑠) ∈ 𝑑𝑦} = 𝑝 𝑠𝐵𝑤 (𝑟) (𝑥, 𝑦)d𝑦 for every 𝑠 > 0 and 𝑥, 𝑦 ∈ 𝐵 𝑤 (𝑟). It follows that 𝜕𝑧 𝑝 𝑠𝐸 (·, 𝑦) ≥ 𝜕𝑧 𝑝 𝑠𝐵𝑤 (𝑟) (·, 𝑦) for every 𝑠 > 0 and 𝑦 ∈ 𝐵 𝑤 (𝑟). In view of (8.70) we have 𝜕𝑧 ℎ 𝐸 ≥ 𝜕𝑧 ℎ 𝐵𝑤 (𝑟) = 𝑐(𝑟). Then 𝜕𝑧 ℎ 𝐸 is bounded away from zero. Step 3. For 𝑤 ∈ R𝑑 and 0 < 𝑟 < 𝜌 < ∞ let 𝐵 𝑤 (𝑟, 𝜌) = {𝑥 ∈ R𝑑 : 𝑟 < |𝑥 − 𝑤| < 𝜌}. By the spatial homogeneity and symmetry of the Brownian motion, there are constants 0 < 𝛼(𝑟, 𝜌), 𝛽(𝑟, 𝜌) < ∞ such that

236

8 Entrance Laws and Kuznetsov Measures

𝜕𝑧 ℎ 𝐵𝑤 (𝑟 ,𝜌) =



𝛼(𝑟, 𝜌) 𝛽(𝑟, 𝜌)

if |𝑧 − 𝑤| = 𝑟, if |𝑧 − 𝑤| = 𝜌.

We can find constants 0 < 𝑟 < 𝜌 < ∞ such that for each 𝑧 ∈ 𝜕𝐸 there exists a 𝑤 ∈ 𝐸 𝑐 satisfying |𝑤 − 𝑧| = 𝑟 and 𝐵 𝑤 (𝑟, 𝜌) ⊃ 𝐸. By a comparison argument as in Step 2 we get 𝜕𝑧 ℎ 𝐸 ≤ 𝜕𝑧 ℎ 𝐵𝑤 (𝑟 ,𝜌) = 𝛼(𝑟, 𝜌). □ Theorem 8.33 For any 𝜇 ∈ 𝑀ℎ (𝐸) and 𝜂 ∈ 𝑀 (𝜕𝐸), we can define an entrance law 𝜅 ∈ 𝒦(𝑃) by ∫ 𝜅 𝑡 ( 𝑓 ) = 𝜇(𝑃𝑡 𝑓 ) + 𝜕𝑧 𝑃𝑡 𝑓 𝜂(d𝑧), 𝑡 > 0, 𝑓 ∈ 𝐵(𝐸). (8.72) 𝜕𝐸

Moreover, every 𝜅 ∈ 𝒦(𝑃) has the representation (8.72) for some 𝜇 ∈ 𝑀ℎ (𝐸) and 𝜂 ∈ 𝑀 (𝜕𝐸). Proof If 𝜅 = (𝜅 𝑡 )𝑡 >0 is given by (8.72), then clearly 𝜅 ∈ 𝒦(𝑃). For the converse, suppose that 𝜅 ∈ 𝒦(𝑃). Let (𝑇𝑡 )𝑡 ≥0 be the ℎ-transform of Doob defined from (𝑃𝑡 )𝑡 ≥0 and the function (8.69) and let 𝛾 = (𝛾𝑡 )𝑡 >0 be the bounded entrance law for (𝑇𝑡 )𝑡 ≥0 defined by 𝛾𝑡 ( 𝑓 ) = 𝜅 𝑡 (ℎ 𝑓 ) for 𝑡 > 0 and 𝑓 ∈ 𝐶 (𝐸). By the smoothness of the transition density 𝑝 𝑡 (𝑥, 𝑦), we can extend (𝑇𝑡 )𝑡 ≥0 to a transition semigroup (𝑇¯𝑡 )𝑡 ≥0 on 𝐸¯ := 𝐸 ∪ 𝜕𝐸 by letting 𝑇¯0 𝑓¯ ≡ 𝑓¯ and  if 𝑥 ∈ 𝐸, ℎ(𝑥) −1 𝑃𝑡 (ℎ 𝑓 ) (𝑥) 𝑇¯𝑡 𝑓¯(𝑥) = (8.73) (𝜕𝑥 ℎ) −1 𝜕𝑥 𝑃𝑡 (ℎ 𝑓 ) if 𝑥 ∈ 𝜕𝐸 ¯ to itself. Moreover, ¯ and 𝑓 = 𝑓¯| 𝐸 . Here 𝑇¯𝑡 maps 𝐶 ( 𝐸) for 𝑡 > 0, where 𝑓¯ ∈ 𝐶 ( 𝐸) ¯ ¯ ¯ since 𝐸 is compact, it is easy to show that 𝑡 ↦→ 𝑇𝑡 𝑓 is strongly continuous. By Theorem 1.14, the family (𝛾𝑡 )𝑡 >0 is relatively compact if we regard them as measures ¯ Choosing a sequence 𝑟 𝑛 → 0 such that 𝛾𝑟𝑛 converges weakly to some 𝛾0 ∈ on 𝐸. ¯ as 𝑛 → ∞ we get 𝑀 ( 𝐸) 𝛾𝑡 ( 𝑓 ) = lim 𝛾𝑟𝑛 (𝑇¯𝑡−𝑟𝑛 𝑓 ) = lim 𝛾𝑟𝑛 (𝑇¯𝑡 𝑓 ) = 𝛾0 (𝑇¯𝑡 𝑓 ), 𝑛→∞

𝑛→∞

and hence ∫ 𝜅𝑡 ( 𝑓 ) =

ℎ(𝑥) −1 𝑃𝑡 𝑓 (𝑥)𝛾0 (d𝑥) +

∫

(𝜕𝑥 ℎ) −1 𝜕𝑥 𝑃𝑡 𝑓 𝛾0 (d𝑥).

𝜕𝐸

𝐸

Then (8.72) follows with 𝜇(d𝑥) = ℎ(𝑥) −1 𝛾0 (d𝑥) and 𝜂(d𝑥) = (𝜕𝑥 ℎ) −1 𝛾0 (d𝑥). The □ extension to 𝑓 ∈ 𝐵(𝐸) is immediate. Let 𝜙 be a branching mechanism on 𝐸 given by (2.29) or (2.30) and let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup of the super-absorbing-barrier Brownian motion defined by (2.35) and (2.36). For the entrance law 𝜅 ∈ 𝒦(𝑃) given by (8.72) we have ∫ 𝜕𝑧 𝑉𝑡 𝑓 𝜂(d𝑧), 𝑡 > 0, 𝑓 ∈ 𝐵(𝐸) + . 𝑆𝑡 (𝜅, 𝑓 ) = 𝜇(𝑉𝑡 𝑓 ) + 𝜕𝐸

8.6 Super-Absorbing-Barrier Brownian Motions

237

Then Theorems 8.10 and 8.35 imply the following: Theorem 8.34 For any 𝜇 ∈ 𝑀ℎ (𝐸) and 𝜂 ∈ 𝑀 (𝜕𝐸), we can define an entrance law 𝐾 ∈ 𝒦𝑚1 (𝑄) by ∫ e−𝜈 ( 𝑓 ) 𝐾𝑡 (d𝜈) = exp{−𝜇(𝑉𝑡 𝑓 ) − 𝜂(𝜕· 𝑉𝑡 𝑓 )}. (8.74) 𝑀 (𝐸)

Moreover, every 𝐾 ∈ 𝒦𝑚1 (𝑄) has the representation (8.74) for 𝜇 ∈ 𝑀ℎ (𝐸) and 𝜂 ∈ 𝑀 (𝜕𝐸). The above theorem gives a complete characterization of minimal probability entrance laws for the super-absorbing-barrier Brownian motion. Using Theorems 8.22 and 8.35 we can also give a characterization of its infinitely divisible probability entrance laws. These results can be extended to some unbounded domains. For simplicity we only discuss briefly the extensions to the positive half line 𝐸 0 := (0, ∞). In the remainder of this section, let (𝑃𝑡 )𝑡 ≥0 be the transition semigroup of the absorbing-barrier Brownian motion in 𝐸 0 given by (A.29) and (A.30). In this case, the function 𝑥 ↦→ ℎ(𝑥) defined by (8.69) is a smooth function on 𝐸 0 with ℎ(0+) = 0. Theorem 8.35 For any 𝜇 ∈ 𝑀ℎ (𝐸 0 ) and any 𝛼 ≥ 0, we can define an entrance law 𝜅 ∈ 𝒦(𝑃) by 𝜅 𝑡 ( 𝑓 ) = 𝜇(𝑃𝑡 𝑓 ) + 𝛼𝜕0 𝑃𝑡 𝑓 ,

𝑡 > 0, 𝑓 ∈ 𝐵(𝐸 0 ).

(8.75)

Moreover, every 𝜅 ∈ 𝒦(𝑃) has the representation (8.75) with 𝜇 ∈ 𝑀ℎ (𝐸 0 ) and 𝛼 ≥ 0. Proof The proof is very similar to that of Theorem 8.33, so we only describe the difference. It suffices to prove each 𝜅 ∈ 𝒦(𝑃) has the representation (8.75). Let (𝑇𝑡 )𝑡 ≥0 and its bounded entrance law (𝛾𝑡 )𝑡 >0 be defined as in the proof of Theorem 8.33. By the smoothness of 𝑝 𝑡 (𝑥, 𝑦), we now extend (𝑇𝑡 )𝑡 ≥0 to a strongly continuous transition semigroup (𝑇¯𝑡 )𝑡 ≥0 on 𝐸¯ 0 := [0, ∞] by letting 𝑇¯0 𝑓¯ ≡ 𝑓¯ and  ℎ(𝑥) −1 𝑃𝑡 (ℎ 𝑓 ) (𝑥)    𝑇¯𝑡 𝑓¯(𝑥) = (𝜕0 ℎ) −1 𝜕0 𝑃𝑡 (ℎ 𝑓 )  ¯   𝑓 (∞)

if 0 < 𝑥 < ∞, if 𝑥 = 0, if 𝑥 = ∞

(8.76)

for 𝑡 > 0, where 𝑓¯ ∈ 𝐶 ( 𝐸¯ 0 ) and 𝑓 = 𝑓¯| 𝐸0 . Then 𝛾𝑡 ( 𝑓 ) = 𝛾0 (𝑇¯𝑡 𝑓¯) for some 𝛾0 ∈ 𝑀 ( 𝐸¯ 0 ). Since ∞ is a trap for (𝑇¯𝑡 )𝑡 ≥0 , we must have 𝛾0 ({∞}) = 0 and then □ (8.75) follows. Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup of the super-absorbing-barrier Brownian motion over 𝐸 0 with branching mechanism given by (2.29) or (2.30). By easy applications of Theorems 8.10 and 8.35 we get the following:

238

8 Entrance Laws and Kuznetsov Measures

Theorem 8.36 For any 𝜇 ∈ 𝑀ℎ (𝐸 0 ) and 𝛼 ≥ 0, we can define 𝐾 ∈ 𝒦𝑚1 (𝑄) by ∫ e−𝜈 ( 𝑓 ) 𝐾𝑡 (d𝜈) = exp{−𝜇(𝑉𝑡 𝑓 ) − 𝛼𝜕0𝑉𝑡 𝑓 }. (8.77) 𝑀 (𝐸0 )

Moreover, every 𝐾 ∈ 𝒦𝑚1 (𝑄) has the representation (8.77) with 𝜇 ∈ 𝑀ℎ (𝐸 0 ) and 𝛼 ≥ 0. Example 8.4 Let 𝑊ˆ 0 be the space of paths 𝑤 : [0, ∞) → 𝑀 (𝐸 0 ) such that 𝑤 𝑡 takes values in 𝑀 (𝐸 0 ) ◦ and is right continuous in some interval (0, 𝜁 (𝑤)) or [0, 𝜁 (𝑤)) ⊂ [0, ∞) and takes the value 0 ∈ 𝑀 (𝐸 0 ) elsewhere. Let (𝒜 0 , 𝒜𝑡0 ) be the natural 𝜎algebras on 𝑊ˆ 0+ generated by the coordinate process. By Theorem 8.36, for each 𝑢 > 0, ∫ e−𝜈 ( 𝑓 ) 𝐾𝑡𝑢 (d𝜈) = exp{−𝑢𝜕0𝑉𝑠 𝑓 }, 𝑡 > 0, 𝑓 ∈ 𝐵(𝐸 0 ) + , (8.78) 𝑀 (𝐸0 )

defines a probability entrance law (𝐾𝑡𝑢 )𝑡 >0 for the super-absorbing-barrier Brownian motion. Let Q𝑢 be the corresponding Kuznetsov measure, which is carried by 𝑊ˆ 0 . Recall that the branching mechanism 𝜙 is given by (2.29). Suppose that   ∫ h i 𝜈(ℎ) ∧ 𝜈(ℎ) 2 + 𝜈 𝑥 (ℎ) 𝐻 (𝑥, d𝜈) < ∞. (8.79) sup ℎ(𝑥) −1 𝜂(𝑥, ℎ) + 𝑀 (𝐸0 ) ◦

𝑥 ∈𝐸0

Then for any 𝜀 > 0 we have 𝑤 𝑡 ((0, 𝜀]) → ∞ and 𝑤 𝑡 ( [𝜀, ∞)) → 0 as 𝑡 → 0

(8.80)

for Q𝑢 -a.e. 𝑤 ∈ 𝑊ˆ 0 . To see this let (𝑇¯𝑡 )𝑡 ≥0 be defined as in the proof of Theorem 8.35 and use the same notation for its restriction to R+ . Under condition (8.79) we can ¯ 𝑓¯) = ℎ(𝑥) −1 𝜙(𝑥, ℎ 𝑓 ) for 𝑥 > 0 define a branching mechanism 𝜓¯ on R+ by 𝜓(𝑥, ¯ 𝑓¯) = 0, where 𝑓¯ ∈ 𝐵(R+ ) + and 𝑓 = 𝑓¯| 𝐸0 . Let (𝑈¯ 𝑡 )𝑡 ≥0 be the cumulant and 𝜓(0, ¯ Then for 𝑡 > 0 we have semigroup defined from (𝑇¯𝑡 )𝑡 ≥0 and 𝜓.  ℎ(𝑥) −1𝑉𝑡 (ℎ 𝑓 ) (𝑥) if 𝑥 > 0, (8.81) 𝑈¯ 𝑡 𝑓¯(𝑥) = (𝜕0 ℎ) −1 𝜕0𝑉𝑡 (ℎ 𝑓 ) if 𝑥 = 0. Theorem 5.13 implies that (𝑈¯ 𝑡 )𝑡 ≥0 determines a Borel right semigroup on 𝑀 (R+ ). We then define the measure-valued process { 𝑋¯ 𝑡 : 𝑡 > 0} by 𝑋¯ 𝑡 ({0}) = 0 and 𝑋¯ 𝑡 (d𝑥) = ℎ(𝑥)𝑤 𝑡 (d𝑥) for 𝑥 > 0. It is not hard to see that { 𝑋¯ 𝑡 : 𝑡 > 0} is a superprocess in 𝑀 (R+ ) with cumulant semigroup (𝑈¯ 𝑡 )𝑡 ≥0 . Using (8.76) one may check 𝜕0𝑉𝑠 (ℎ 𝑓 ) = 𝜕0 ℎ𝑈¯ 𝑡 𝑓¯(0). From (8.78) we have Q𝑢 exp{− 𝑋¯ 𝑡 ( 𝑓¯)} = exp{−𝑢𝜕0 ℎ𝑈¯ 𝑡 𝑓¯(0)},

𝑡 > 0, 𝑓¯ ∈ 𝐵(R+ ) + .

This implies Q𝑢 { 𝑋¯ 0+ = 𝑢𝜕0 ℎ𝛿0 } = 1. Since ℎ(0+) = 0 and ℎ(𝑥) > 0 for 𝑥 > 0, we have (8.80) for Q𝑢 -a.e. 𝑤 ∈ 𝑊ˆ 0 .

8.7 Notes and Comments

239

8.7 Notes and Comments The structures of entrance laws for Dawson–Watanabe superprocesses were investigated in Dynkin (1989b), Fitzsimmons (1988) and Li (1995/6, 1996, 1998b). Those for branching particle systems were studied in Li (1998a). Evans (1992) gave a characterization for the entrance laws of a conditioned superprocess. Theorem 8.2 first appeared in Li (1995/6). The proof of Theorem 8.10 follows Dynkin (1989b) and Li (1996, 1998b). Theorem 8.13 is a new result. Fitzsimmons (1988) proved Theorem 8.15 for local branching mechanisms. Theorem 8.22 was proved in Li (1996) for local branching mechanisms and in Li (1998b) for decomposable branching mechanisms. Theorem 8.24 was first given in Li (2019b) for CB-processes. The random field 𝜇 {𝑋𝑡 : 𝑡 ≥ 0, 𝜇 ∈ 𝑀𝜆 (𝐸)} constructed in Theorem 8.29 is a measure-valued counterpart of the stochastic flows studied in Bertoin and Le Gall (2006) and Dawson and Li (2012), where 𝐸 was assumed to be a singleton. Theorem 8.30 was first proved for binary local branching by Evans and Perkins (1991), who used it to prove propagation properties of the superprocess. Some of the results of Evans and Perkins (1991) were extended to general local branching mechanisms in Li and Zhou (2008). The existence of excursion laws of Dawson–Watanabe superprocesses was first observed by El Karoui and Roelly (1991). The presentation of Theorem 8.26 given here is in the spirit of Li (2001). A special form of this result was proved in Li and Shiga (1995) using a theorem of Perkins (1992), which asserts that a conditioned Dawson–Watanabe superprocess is a generalized Fleming–Viot superprocess. The proof of Lemma 8.32 was taken from Li (1998a). This results can also be obtained from the general estimates of heat kernels given in Zhang (2002). The proofs of Theorems 8.33 and 8.35 follow those of Li and Shiga (1995). It was shown in van der Hofstad (2006) that the canonical measure of the super-Brownian motion is a natural candidate for the scaling limits of various random systems; see also Hara and Slade (2000) and van der Hofstad and Slade (2003). The representation (8.68) of binary branching superprocesses was established in Le Gall (1991, 1999). This relies on the fact that the genealogical trees of Feller’s branching diffusion can be coded by the Brownian excursions. A similar coding for the genealogical structures of CB-processes with general branching mechanisms was given in Le Gall and Le Jan (1998a) using spectrally positive Lévy processes. Bertoin et al. (1997) provided a snake-like construction of superprocesses based on subordination. A direct construction of superprocesses with general local branching mechanisms was given by Le Gall and Le Jan (1998b) using the so-called Lévy snakes. The snake representation has become a powerful tool in the study of the superprocesses and the associated nonlinear partial differential equations; see Duquesne and Le Gall (2002), Le Gall (1999, 2005) and the references therein. This technique has also played important roles in the work of Le Gall (2007, 2010), Le Gall and Paulin (2008) and Miermont (2013) on the geometry of large planar maps; see also Le Gall (2014) and Le Gall and Miermont (2012). A Schilder type theorem on large deviations of the super-Brownian was established recently in Xiang (2010) using

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8 Entrance Laws and Kuznetsov Measures

the snake representation; see also Fleischmann et al. (1996) and Schied (1996). The behavior of superprocesses near extinction was studied by Liu and Mueller (1989) and Tribe (1992); see also Le Gall (1999, p. 72). In Abraham and Delmas (2008), Lévy snakes were used to construct fragmentation processes.

Chapter 9

Structures of Independent Immigration

In this chapter we study independent immigration structures associated with MBprocesses. We first give a formulation of the structures using skew convolution semigroups. Those semigroups are in one-to-one correspondence with infinitely divisible probability entrance laws for the MB-processes. We shall see that an immigration superprocess has a Borel right realization if the corresponding skew convolution semigroup is determined by a closable infinitely divisible probability entrance law. The trajectories of the immigration processes are constructed using stochastic integrals with respect to Poisson random measures determined by entrance rules. Ergodicities and exponential ergodicities are proved under suitable assumptions. Most of the properties of superprocess obtained before are extended to immigration superprocesses.

9.1 Skew Convolution Semigroups Suppose that 𝐸 is a Lusin topological space and (𝑄 𝑡 )𝑡 ≥0 is the transition semigroup of an MB-process given by (2.3) and (2.5). Let (𝑁𝑡 )𝑡 ≥0 be a family of probability measures on 𝑀 (𝐸). We call (𝑁𝑡 )𝑡 ≥0 a skew convolution semigroup (SC-semigroup) associated with (𝑄 𝑡 )𝑡 ≥0 provided 𝑁𝑟+𝑡 = (𝑁𝑟 𝑄 𝑡 ) ∗ 𝑁𝑡 ,

𝑟, 𝑡 ≥ 0.

(9.1)

The above equation is of interest because of the following: Theorem 9.1 The relation (9.1) is satisfied if and only if 𝑄 𝑡𝑁 (𝜇, ·) := 𝑄 𝑡 (𝜇, ·) ∗ 𝑁𝑡 ,

𝑡 ≥ 0, 𝜇 ∈ 𝑀 (𝐸)

(9.2)

defines a Markov semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 on 𝑀 (𝐸). Proof It is easy to see that (9.2) really defines a probability kernel on 𝑀 (𝐸). If (𝑁𝑡 )𝑡 ≥0 satisfies (9.1), then for any 𝑟, 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + we have © Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_9

241

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9 Structures of Independent Immigration

∫

𝑁 (𝜇, d𝜈) e−𝜈 ( 𝑓 ) 𝑄 𝑟+𝑡 𝑀 (𝐸) ∫ ∫ −𝜈1 ( 𝑓 ) e 𝑄 𝑟+𝑡 (𝜇, d𝜈1 ) = e−𝜈2 ( 𝑓 ) 𝑁𝑟+𝑡 (d𝜈2 ) (𝐸) (𝐸) 𝑀 𝑀 ∫ ∫ = e−𝜈1 ( 𝑓 ) 𝑄 𝑡 (𝛾, d𝜈1 ) 𝑄 𝑟 (𝜇, d𝛾) 𝑀 (𝐸) 𝑀 (𝐸) ∫ ∫ e−𝜈2 ( 𝑓 ) (𝑁𝑟 𝑄 𝑡 ) (d𝜈2 ) e−𝜈3 ( 𝑓 ) 𝑁𝑡 (d𝜈3 ) · (𝐸) (𝐸) 𝑀 𝑀 ∫ ∫ ∫ e−𝜈3 ( 𝑓 ) 𝑁𝑡 (d𝜈3 ) = 𝑄 𝑟𝑁 (𝜇, d𝛾) e−𝜈1 ( 𝑓 ) 𝑄 𝑡 (𝛾, d𝜈1 ) (𝐸) 𝑀 (𝐸) 𝑀 (𝐸) 𝑀 ∫ ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡𝑁 (𝛾, d𝜈), = 𝑄 𝑟𝑁 (𝜇, d𝛾) 𝑀 (𝐸)

𝑀 (𝐸)

where we used Theorem 2.1 for the third equality. Then (𝑄 𝑡𝑁 )𝑡 ≥0 satisfies the Chapman–Kolmogorov equation. For the converse, suppose the kernels (𝑄 𝑡𝑁 )𝑡 ≥0 constitute a transition semigroup. Since 𝑄 𝑡 (0, ·) = 𝛿0 , we have 𝑄 𝑡𝑁 (0, ·) = 𝑁𝑡 . Consequently, for any 𝑟, 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + , ∫ e−𝜈 ( 𝑓 ) 𝑁𝑟+𝑡 (d𝜈) 𝑀 (𝐸) ∫ ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡𝑁 (𝜇, d𝜈) = 𝑁𝑟 (d𝜇) 𝑀 𝑀 (𝐸) (𝐸) ∫ ∫ ∫ 𝑁𝑟 (d𝜇) e−𝜈2 ( 𝑓 ) 𝑁𝑡 (d𝜈2 ) e−𝜈1 ( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈1 ) = 𝑀 (𝐸) 𝑀 𝑀 (𝐸) (𝐸) ∫ ∫ e−𝜈2 ( 𝑓 ) 𝑁𝑡 (d𝜈2 ). = e−𝜈1 ( 𝑓 ) 𝑁𝑟 𝑄 𝑡 (d𝜈1 ) 𝑀 (𝐸)

Thus (𝑁𝑡 )𝑡 ≥0 satisfies (9.1).

𝑀 (𝐸)

□

Suppose that 𝑇 is an interval on the real line and (𝒢𝑡 )𝑡 ∈𝑇 is a filtration. If 𝑌 = {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ∈ 𝑇 } is a Markov process in 𝑀 (𝐸) with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 given by (9.2), we call it an immigration process associated with (𝑄 𝑡 )𝑡 ≥0 or the corresponding MB-process. In the special case where (𝑄 𝑡 )𝑡 ≥0 is the transition semigroup of a Dawson–Watanabe superprocess, we also call 𝑌 an immigration superprocess. The intuitive meaning of the model is clear in view of (9.1) and (9.2). From (9.2) we see that the population at any time 𝑡 ≥ 0 is made up of two parts; the native part generated by the mass 𝜇 ∈ 𝑀 (𝐸) has distribution 𝑄 𝑡 (𝜇, ·) and the immigration in the time interval (0, 𝑡] gives the distribution 𝑁𝑡 . In a similar way, the equation (9.1) decomposes the population immigrating to 𝐸 during the time interval (0, 𝑟 +𝑡] into two parts; the immigration in the interval (𝑟, 𝑟 +𝑡] gives the distribution 𝑁𝑡 while the immigration in the interval (0, 𝑟] generates the distribution 𝑁𝑟 at time 𝑟 and gives the distribution 𝑁𝑟 𝑄 𝑡 at time 𝑟 + 𝑡. It is not hard to see that (9.2) gives a general formulation of the immigration independent of the state of the population.

9.1 Skew Convolution Semigroups

243

Theorem 9.2 Suppose that {(𝑋𝑡 , ℱ𝑡 ) : 𝑡 ∈ 𝑇 } and {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ∈ 𝑇 } are two independent immigration processes associated with (𝑄 𝑡 )𝑡 ≥0 corresponding to the SC-semigroups (𝑀𝑡 )𝑡 ≥0 and (𝑁𝑡 )𝑡 ≥0 , respectively. Let 𝑍𝑡 = 𝑋𝑡 + 𝑌𝑡 and ℋ𝑡 = 𝜎(ℱ𝑡 ∪ 𝒢𝑡 ). Then {(𝑍𝑡 , ℋ𝑡 ) : 𝑡 ∈ 𝑇 } is an immigration process corresponding to the SC-semigroup (𝐿 𝑡 )𝑡 ≥0 defined by 𝐿 𝑡 = 𝑀𝑡 ∗ 𝑁𝑡 . Proof By Theorem 2.1 it is simple to show that (𝐿 𝑡 )𝑡 ≥0 is really an SC-semigroup associated with (𝑄 𝑡 )𝑡 ≥0 . Let (𝑄 𝑡𝐿 )𝑡 ≥0 be defined by (2.2) with (𝑁𝑡 )𝑡 ≥0 replaced by (𝐿 𝑡 )𝑡 ≥0 . By a modification of the proof of Theorem 2.3 one shows {(𝑍𝑡 , ℋ𝑡 ) : 𝑡 ∈ 𝑇 } □ is a Markov process with transition semigroup (𝑄 𝑡𝐿 )𝑡 ≥0 . Corollary 9.3 Let {(𝑋𝑡 , ℱ𝑡 ) : 𝑡 ∈ 𝑇 } be an MB-process with transition semigroup (𝑄 𝑡 )𝑡 ≥0 and let {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ∈ 𝑇 } be an immigration process with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 . Suppose the two processes are independent of each other. Let 𝑍𝑡 = 𝑋𝑡 + 𝑌𝑡 and ℋ𝑡 = 𝜎(ℱ𝑡 ∪ 𝒢𝑡 ). Then {(𝑍𝑡 , ℋ𝑡 ) : 𝑡 ∈ 𝑇 } is an immigration process with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 . Theorem 9.4 Suppose that 𝑡 ↦→ 𝑉𝑡 𝑓 (𝑥) is locally bounded and right continuous pointwise for every 𝑓 ∈ 𝐶 (𝐸) + . If (𝐾𝑠 ) 𝑠>0 is an infinitely divisible probability entrance law for (𝑄 𝑡 )𝑡 ≥0 satisfying ∫ 𝑡 (9.3) − 𝑡 ≥ 0, log 𝐿 𝐾𝑠 (1)d𝑠 < ∞, 0

then an SC-semigroup (𝑁𝑡 )𝑡 ≥0 associated with (𝑄 𝑡 )𝑡 ≥0 is defined by ∫ 𝑡   log 𝐿 𝑁𝑡 ( 𝑓 ) = log 𝐿 𝐾𝑠 ( 𝑓 ) d𝑠, 𝑡 ≥ 0, 𝑓 ∈ 𝐵(𝐸) + .

(9.4)

0

Conversely, for every SC-semigroup (𝑁𝑡 )𝑡 ≥0 associated with (𝑄 𝑡 )𝑡 ≥0 there is an infinitely divisible probability entrance law (𝐾𝑠 ) 𝑠>0 for (𝑄 𝑡 )𝑡 ≥0 satisfying (9.3) so that the Laplace functionals of (𝑁𝑡 )𝑡 ≥0 are given by (9.4). Proof If (𝐾𝑠 ) 𝑠>0 is an infinitely divisible probability entrance law for (𝑄 𝑡 )𝑡 ≥0 satisfying (9.3), then (9.4) clearly defines a family of infinitely divisible probability measures (𝑁𝑡 )𝑡 ≥0 on 𝑀 (𝐸). By the entrance law property of (𝐾𝑠 ) 𝑠>0 it is easy to show that (9.1) holds, so (𝑁𝑡 )𝑡 ≥0 is an SC-semigroup associated with (𝑄 𝑡 )𝑡 ≥0 . To prove the converse, let (𝑁𝑡 )𝑡 ≥0 be an SC-semigroup associated with (𝑄 𝑡 )𝑡 ≥0 and let ∫ e−𝜈 ( 𝑓 ) 𝑁𝑡 (d𝜈), 𝑡 ≥ 0, 𝑓 ∈ 𝐵(𝐸) + . (9.5) 𝐽𝑡 ( 𝑓 ) = − log 𝑀 (𝐸)

From (9.1) we have 𝐽𝑟+𝑡 ( 𝑓 ) = 𝐽𝑡 ( 𝑓 ) + 𝐽𝑟 (𝑉𝑡 𝑓 ),

𝑟, 𝑡 ≥ 0, 𝑓 ∈ 𝐵(𝐸) + .

(9.6)

Then 𝑡 ↦→ 𝐽𝑡 ( 𝑓 ) is increasing for every 𝑓 ∈ 𝐵(𝐸) + . Observe also 𝐽𝑡 ( 𝑓1 ) ≤ 𝐽𝑡 ( 𝑓2 ) for 𝑓1 ≤ 𝑓2 ∈ 𝐵(𝐸) + . For 𝑓 ∈ 𝐶 (𝐸) + we have 𝑉𝑡 𝑓 (𝑥) → 𝑓 (𝑥) pointwise as 𝑡 → 0. Then letting 𝑡 → 0 and 𝑟 → 0 in (9.6) and using (9.5) and dominated convergence

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9 Structures of Independent Immigration

we get lim 𝐽𝑡 ( 𝑓 ) = lim 𝐽𝑡 ( 𝑓 ) + lim 𝐽𝑟 ( 𝑓 ). 𝑡→0

𝑡→0

𝑟→0

Thus lim𝑡→0 𝐽𝑡 ( 𝑓 ) = 0 first for 𝑓 ∈ 𝐶 (𝐸) + and then for all 𝑓 ∈ 𝐵(𝐸) + . Let 𝑞 = 𝑞(𝑙, 𝑓 ) > 0 be such thatÍ∥𝑉𝑡 𝑓 ∥ ≤ 𝑞 for all 0 ≤ 𝑡 ≤ 𝑙. For 0 ≤ 𝑐 1 < 𝑑1 < · · · < 𝑛 𝑐 𝑛 < 𝑑 𝑛 < · · · ≤ 𝑙, set 𝜎𝑛 = 𝑖=1 (𝑑𝑖 − 𝑐 𝑖 ). We claim that 𝑛 ∑︁

[𝐽𝑑𝑖 ( 𝑓 ) − 𝐽𝑐𝑖 ( 𝑓 )] ≤ 𝐽 𝜎𝑛 (𝑞).

𝑖=1

For 𝑛 = 1 this follows from (9.6). If the above inequality is true when 𝑛 is replaced by 𝑛 − 1, using (9.6) we have 𝑛 ∑︁

𝑛−1 ∑︁

[𝐽𝑑𝑖 ( 𝑓 ) − 𝐽𝑐𝑖 ( 𝑓 )] =

𝑖=1

≤

[𝐽𝑑𝑖 ( 𝑓 ) − 𝐽𝑐𝑖 ( 𝑓 )] + 𝐽𝑑𝑛 −𝑐𝑛 (𝑉𝑐𝑛 𝑓 ) 𝑖=1 𝐽 𝜎𝑛−1 (𝑞) + 𝐽𝑑𝑛 −𝑐𝑛 (𝑉 𝜎𝑛−1 𝑞) = 𝐽 𝜎𝑛 (𝑞).

∫𝑡 Thus 𝑡 ↦→ 𝐽𝑡 ( 𝑓 ) is absolutely continuous. That is, we have 𝐽𝑡 ( 𝑓 ) = 0 𝐼𝑠 ( 𝑓 )d𝑠 for a positive Borel function 𝑠 ↦→ 𝐼𝑠 ( 𝑓 ) on [0, ∞). Let 𝑆2 (𝐸, 𝑟) be defined as in Section 1.2. Then there is a set 𝐴 ⊂ [0, ∞) with full Lebesgue measure such that for all 𝑠 ∈ 𝐴 and 𝑓 ∈ 𝑆2 (𝐸, 𝑟), i 1 1h 𝐽𝑠+𝑟 ( 𝑓 ) − 𝐽𝑠 ( 𝑓 ) = lim 𝐽𝑟 (𝑉𝑠 𝑓 ) 𝑟→0 𝑟 𝑟→0 𝑟  1 = lim 1 − exp{−𝐽𝑟 (𝑉𝑠 𝑓 )} 𝑟→0 𝑟 ∫ ∫
1 𝑁𝑟 (d𝜇) 1 − e−𝜈 ( 𝑓 ) 𝑄 𝑠 (𝜇, d𝜈). = lim 𝑟→0 𝑟 𝑀 (𝐸) 𝑀 (𝐸) ◦

𝐼𝑠 ( 𝑓 ) = lim

In view of (9.5) we have lim𝑛→∞ 𝐽𝑡 (1/𝑛) = 0 decreasingly. Thus, by taking a smaller set 𝐴 ⊂ [0, ∞) with full Lebesgue measure, we may assume lim𝑛→∞ 𝐼𝑠 (1/𝑛) = 0 for all 𝑠 ∈ 𝐴. By Proposition 1.32 there is a 𝑈𝑠 ∈ ℐ(𝐸) such that 𝐼𝑠 ( 𝑓 ) = 𝑈𝑠 ( 𝑓 ) for all 𝑠 ∈ 𝐴 and 𝑓 ∈ 𝑆2 (𝐸, 𝑟). Consequently, ∫ 𝑡 𝐽𝑡 ( 𝑓 ) = 𝑈𝑠 ( 𝑓 )d𝑠, 𝑡 ≥ 0, 0

first for 𝑓 ∈ 𝑆2 (𝐸, 𝑟) and then for all 𝑓 ∈ 𝐵(𝐸) + . Now the equation (9.6) yields ∫ 𝑟 ∫ 𝑟 𝑈𝑠+𝑡 ( 𝑓 )d𝑠 = 𝑈𝑠 (𝑉𝑡 𝑓 )d𝑠, 𝑟, 𝑡 ≥ 0, 𝑓 ∈ 𝐵(𝐸) + . 0

0

Then for any 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + there is a subset 𝐴𝑡 ( 𝑓 ) of [0, ∞) with full Lebesgue measure such that 𝑈𝑠+𝑡 ( 𝑓 ) = 𝑈𝑠 (𝑉𝑡 𝑓 ) for all 𝑠 ∈ 𝐴𝑡 ( 𝑓 ). By Fubini’s theorem, there are subsets 𝐵( 𝑓 ) and 𝐵𝑠 ( 𝑓 ) of [0, ∞) with full Lebesgue measures

9.1 Skew Convolution Semigroups

245

such that 𝑈𝑠+𝑡 ( 𝑓 ) = 𝑈𝑠 (𝑉𝑡 𝑓 ),

𝑠 ∈ 𝐵( 𝑓 ), 𝑡 ∈ 𝐵𝑠 ( 𝑓 ).

Since 𝑈𝑠+𝑡 and 𝑈𝑠 ◦ 𝑉𝑡 are determined by their restrictions to the countable set 𝑆2 (𝐸, 𝑟), for 𝐵 ⊂ [0, ∞) and 𝐵𝑠 ⊂ [𝑠, ∞) with full Lebesgue measures we have 𝑈𝑡 = 𝑈𝑠 ◦ 𝑉𝑡−𝑠 ,

𝑠 ∈ 𝐵, 𝑡 ∈ 𝐵𝑠 .

Choose a sequence {𝑠 𝑛 } ⊂ 𝐵 with 𝑠 𝑛 → 0. For any 𝑡 > 𝑠 𝑛 > 𝑠 𝑛+1 > 0, we may take 𝑠 ∈ 𝐵𝑠𝑛 ∩ 𝐵𝑠𝑛+1 ∩ [0, 𝑡] to see 𝑈𝑠𝑛 ◦ 𝑉𝑡−𝑠𝑛 = 𝑈𝑠𝑛 ◦ 𝑉𝑠−𝑠𝑛 ◦ 𝑉𝑡−𝑠 = 𝑈𝑠 ◦ 𝑉𝑡−𝑠 = 𝑈𝑠𝑛+1 ◦ 𝑉𝑠−𝑠𝑛+1 ◦ 𝑉𝑡−𝑠 = 𝑈𝑠𝑛+1 ◦ 𝑉𝑡−𝑠𝑛+1 . Then 𝑊𝑡 := 𝑈𝑠𝑛 ◦ 𝑉𝑡−𝑠𝑛 ∈ ℐ(𝐸) is independent of 𝑛 ≥ 1 and 𝑊𝑡 = 𝑈𝑡 for almost all 𝑡 > 0. Thus we have ∫ 𝑡 𝑊𝑠 ( 𝑓 )d𝑠, 𝐽𝑡 ( 𝑓 ) = 𝑡 ≥ 0, 𝑓 ∈ 𝐵(𝐸) + . 0

Moreover, it is easy to see that 𝑊𝑟+𝑡 = 𝑊𝑟 ◦𝑉𝑡 for all 𝑟, 𝑡 > 0. Let 𝐾𝑠 be the infinitely divisible probability measure on 𝑀 (𝐸) such that − log 𝐿 𝐾𝑠 = 𝑊𝑠 . Then (𝐾𝑠 ) 𝑠>0 is an entrance law for (𝑄 𝑡 )𝑡 ≥0 and (9.4) holds. □ It is clear that Theorem 9.4 establishes a one-to-one correspondence between the SC-semigroup (𝑁𝑡 )𝑡 ≥0 and the infinitely divisible probability entrance law (𝐾𝑠 ) 𝑠>0 satisfying (9.3). By Theorem 8.2 we may assume 𝐼𝑡 = − log 𝐿 𝐾𝑡 is given by (8.2). Then we can rewrite (9.4) as  ∫ 𝑡  ∫ −𝜈 ( 𝑓 ) 𝐼𝑠 ( 𝑓 )d𝑠 , 𝑓 ∈ 𝐵(𝐸) + . e 𝑁𝑡 (d𝜈) = exp − (9.7) 𝑀 (𝐸)

0

The corresponding transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 is characterized by   ∫ 𝑡 ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡𝑁 (𝜇, d𝜈) = exp − 𝜇(𝑉𝑡 𝑓 ) − 𝐼𝑠 ( 𝑓 )d𝑠 . 𝑀 (𝐸)

(9.8)

0

Recall that the cumulant semigroup (𝑉𝑡 )𝑡 ≥0 has representation (2.5). The following results are immediate: Proposition 9.5 The SC-semigroup (𝑁𝑡 )𝑡 ≥0 given by (9.4) has finite first-moments if and only if ∫ 𝑡 ∫ 𝜈(1)𝐾𝑠 (d𝜈) < ∞, d𝑠 𝑡 ≥ 0. (9.9) 0

𝑀 (𝐸)

246

9 Structures of Independent Immigration

In this case, we have ∫ ∫ 𝜈( 𝑓 )𝑁𝑡 (d𝜈) = 𝑀 (𝐸)

∫

𝑡

𝜈( 𝑓 )𝐾𝑠 (d𝜈),

d𝑠

𝑓 ∈ 𝐵(𝐸).

(9.10)

𝑀 (𝐸)

0

Proposition 9.6 Suppose that 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ for every 𝑡 ≥ 0. Let (𝑁𝑡 )𝑡 ≥0 be the SC-semigroup given by (9.4) such that (9.9) holds. Then the transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 defined by (9.8) has finite first-moments given by, for 𝜇 ∈ 𝑀 (𝐸) and 𝑓 ∈ 𝐵(𝐸), ∫ ∫ 𝑡 ∫ 𝑁 𝜈( 𝑓 )𝑄 𝑡 (𝜇, d𝜈) = 𝜇(𝜋𝑡 𝑓 ) + d𝑠 𝜈( 𝑓 )𝐾𝑠 (d𝜈), (9.11) 𝑀 (𝐸)

0

𝑀 (𝐸)

where (𝜋𝑡 )𝑡 ≥0 is the semigroup defined by (2.54). If (𝐾𝑡 )𝑡 >0 can be extended to a closed entrance law (𝐾𝑡 )𝑡 ≥0 , we say the SCsemigroup (𝑁𝑡 )𝑡 ≥0 is regular and we call 𝐼 := − log 𝐿 𝐾0 ∈ ℐ(𝐸) the immigration mechanism. Then an immigration mechanism has the representation ∫
(9.12) 𝐼 ( 𝑓 ) = 𝜂( 𝑓 ) + 1 − e−𝜈 ( 𝑓 ) 𝐻 (d𝜈), 𝑓 ∈ 𝐵(𝐸) + , 𝑀 (𝐸) ◦

where 𝜂 ∈ 𝑀 (𝐸) and [1 ∧ 𝜈(1)]𝐻 (d𝜈) is a finite measure on 𝑀 (𝐸) ◦ . In this case, we can rewrite (9.8) as   ∫ ∫ 𝑡 𝐼 (𝑉𝑠 𝑓 )d𝑠 . (9.13) e−𝜈 ( 𝑓 ) 𝑄 𝑡𝑁 (𝜇, d𝜈) = exp − 𝜇(𝑉𝑡 𝑓 ) − 𝑀 (𝐸)

0

By combining Theorem 9.4 and Proposition 9.5 with the results in Section 8.3, we get characterizations of SC-semigroups associated with the (𝜉, 𝜙)-superprocess under the first-moment assumption. In particular, from Theorem 8.22 we immediately derive the following: Theorem 9.7 Suppose that 𝜅 ∈ 𝒦(𝑃) and 𝐹 (d𝜈) is a 𝜎-finite measure on the space 𝒦(𝑃) ◦ satisfying (8.40). Then there is an SC-semigroup (𝑁𝑡 )𝑡 ≥0 associated with the (𝜉, 𝜙)-superprocess defined by  ∫ 𝑡  𝐼𝑟 (𝜅, 𝐹, 𝑓 )d𝑟 , 𝑡 ≥ 0, 𝑓 ∈ 𝐵(𝐸) + , (9.14) 𝐿 𝑁𝑡 ( 𝑓 ) = exp − 0

where ∫ 𝐼𝑟 (𝜅, 𝐹, 𝑓 ) = 𝑆𝑟 (𝜅, 𝑓 ) +

(1 − e−𝑆𝑟 (𝜈, 𝑓 ) )𝐹 (d𝜈).

(9.15)

𝒦 ( 𝑃) ◦

Moreover, the SC-semigroup (𝑁𝑡 )𝑡 ≥0 has finite first-moments given by, for 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸),

9.2 Properties of Transition Probabilities

∫

∫

𝑡

𝜈( 𝑓 )𝑁𝑡 (d𝜈) = 𝑀 (𝐸) ◦

247





∫ 𝑞 𝑟 (𝜅, 𝑓 ) +

𝑞 𝑟 (𝜈, 𝑓 )𝐹 (d𝜈) d𝑟,

(9.16)

𝒦 ( 𝑃) ◦

0

where 𝑞 𝑟 (𝜅, 𝑓 ) is defined as in (8.42). Conversely, if (𝑁𝑡 )𝑡 ≥0 is an SC-semigroup associated with the (𝜉, 𝜙)-superprocess with finite first-moments, then there exist 𝜅 ∈ 𝒦(𝑃) and a 𝜎-finite measure 𝐹 (d𝜈) on 𝒦(𝑃) ◦ satisfying (8.40) so that (𝑁𝑡 )𝑡 ≥0 is given by (9.14) and (9.15). Example 9.1 Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup of the 𝑑-dimensional CBprocess defined in Example 2.5. The branching mechanism of the process is given by (2.44). Since (𝑄 𝑡 )𝑡 ≥0 is a Feller semigroup, by Corollary A.39 the entrance law (𝐾𝑡 )𝑡 >0 in (9.4) has a closed extension (𝐾𝑡 )𝑡 ≥0 , where 𝐾0 is an infinitely divisible probability measure on R+𝑑 . Then the immigration mechanism 𝜓 := − log 𝐿 𝐾0 ∈ ℐ(R+𝑑 ) is a function on R+𝑑 with representation ∫
1 − e− ⟨𝑧,𝜆⟩ 𝑛(d𝑧), 𝜓(𝜆) = ⟨𝛽, 𝜆⟩ + (9.17) R+𝑑 \{0}

where 𝛽 ∈ R+𝑑 is a vector and 𝑛(d𝑢) = 𝑛(d𝑢 1 , · · · , d𝑢 𝑑 ) is a 𝜎-finite measure on R+𝑑 \ {0} such that ∫ (1 ∧ ⟨𝑧, 1⟩)𝑛(d𝑧) < ∞. R+𝑑 \{0}

The transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 of the corresponding 𝑑-dimensional CBI-process is given by   ∫ 𝑡 ∫ 𝜓(𝑣(𝑠, 𝜆))d𝑠 . (9.18) e− ⟨𝑦,𝜆⟩ 𝑄 𝑡𝑁 (𝑥, d𝑦) = exp − ⟨𝑥, 𝑣(𝑡, 𝜆)⟩ − R+𝑑

0

The CBI-process has been used widely in mathematical finance as models for interest rates and asset prices; see, e.g., Duffie et al. (2003).

9.2 Properties of Transition Probabilities Suppose that 𝐸 is a Lusin topological space and (𝑄 𝑡 )𝑡 ≥0 is the transition semigroup of an MB-process given by (2.3) and (2.5). Suppose that 𝑡 ↦→ 𝑉𝑡 𝑓 (𝑥) is locally bounded and right continuous pointwise for every 𝑓 ∈ 𝐶 (𝐸) + . Recall that (𝑄 ◦𝑡 )𝑡 ≥0 denotes the restriction of (𝑄 𝑡 )𝑡 ≥0 to 𝑀 (𝐸) ◦ . Theorem 9.8 If (𝑁𝑡 )𝑡 ≥0 is an SC-semigroup associated with (𝑄 𝑡 )𝑡 ≥0 , then each probability measure 𝑁𝑡 is infinitely divisible. Moreover, we have the canonical representation 𝑁𝑡 = 𝐼 (𝛾𝑡 , 𝐺 𝑡 ), where 𝑡 ↦→ 𝛾𝑡 is an increasing continuous path from [0, ∞) to 𝑀 (𝐸) and (𝐺 𝑡 )𝑡 ≥0 is a regular entrance rule for (𝑄 ◦𝑡 )𝑡 ≥0 .

248

9 Structures of Independent Immigration

Proof By Theorem 9.4 there is an infinitely divisible probability entrance law (𝐾𝑠 ) 𝑠>0 such that (9.4) holds. By Theorem 8.2 we may assume (𝐾𝑠 ) 𝑠>0 is given by (8.2) and (8.4). Then we have 𝑁𝑡 = 𝐼 (𝛾𝑡 , 𝐺 𝑡 ), where ∫ 𝑡 ∫ 𝑡 𝛾𝑡 = 𝐻𝑠 d𝑠. 𝜂 𝑠 𝑑𝑠 and 𝐺 𝑡 = (9.19) 0

0

In particular, the map 𝑡 ↦→ 𝛾𝑡 is increasing and continuous. For 𝑡 > 𝑟 ≥ 0 the relation (9.1) implies ∫ 𝛾𝑟 (d𝑥)𝜆 𝑡−𝑟 (𝑥, ·) (9.20) 𝛾𝑡 = 𝛾𝑡−𝑟 + 𝐸

and 𝐺 𝑡 = 𝐺 𝑡−𝑟 +

𝐺 𝑟 𝑄 ◦𝑡−𝑟

∫ 𝛾𝑟 (d𝑥)𝐿 𝑡−𝑟 (𝑥, ·).

+

(9.21)

𝐸

Thus 𝐺 𝑟 𝑄 ◦𝑡−𝑟 ≤ 𝐺 𝑡 . By the second equality in (9.19) we have ∫ 𝑡 ∫ 𝑟 𝐻𝑠 𝑄 ◦𝑡−𝑟 d𝑠. 𝐺 𝑟 𝑄 ◦𝑡−𝑟 = 𝐻𝑠 𝑄 ◦𝑡−𝑟 d𝑠 = 𝐺 𝑡 𝑄 ◦𝑡−𝑟 − 0

It follows that (𝑄 ◦𝑡 )𝑡 ≥0 .

𝐺 𝑟 𝑄 ◦𝑡−𝑟

𝑟

→ 𝐺 𝑡 as 𝑟 ↑ 𝑡. Then (𝐺 𝑡 )𝑡 ≥0 is a regular entrance rule for □

In the situation of the above theorem, we call (𝐺 𝑡 )𝑡 ≥0 the canonical entrance rule of the SC-semigroup (𝑁𝑡 )𝑡 ≥0 . The following theorem gives a general representation of the canonical entrance rule. Theorem 9.9 The canonical entrance rule (𝐺 𝑡 )𝑡 ≥0 of an SC-semigroup associated with (𝑄 𝑡 )𝑡 ≥0 has the representation ∫ 𝑡 𝑠 𝐺𝑡 = 𝜁 (d𝑠), 𝑡 ≥ 0, 𝐺 𝑡−𝑠 (9.22) 0

where 𝜁 (d𝑠) is a diffuse Radon measure on [0, ∞) and {(𝐺 𝑡𝑠 )𝑡 >0 : 𝑠 ≥ 0} is a family of entrance laws for (𝑄 ◦𝑡 )𝑡 ≥0 . Proof By Theorem A.40, there is a Radon measure 𝜁 (d𝑠) on [0, ∞) and a family of entrance laws {(𝐺 𝑡𝑠 )𝑡 >0 : 𝑠 ≥ 0} for (𝑄 ◦𝑡 )𝑡 ≥0 such that ∫ 𝑠 𝑡 ≥ 0. 𝐺 𝑡−𝑠 𝐺𝑡 = (9.23) 𝜁 (d𝑠), [0,𝑡)

For 𝑓 ∈ 𝐶 (𝐸) + we can use the second expression in (9.19) to see ∫ 𝑡 ∫ ∫ (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑡 (d𝜈) = 𝑡 ↦→ d𝑠 (1 − e−𝜈 ( 𝑓 ) )𝐻𝑠 (d𝜈) 𝑀 (𝐸) ◦

0

𝑀 (𝐸) ◦

9.2 Properties of Transition Probabilities

249

is continuous on [0, ∞). Then for any 𝑟 ≥ 0 we have ∫ ∫ (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑟 (d𝜈) (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑟+𝑡 (d𝜈) − 0 = lim ◦ 𝑡 ↓0 𝑀 (𝐸) ◦ 𝑀 (𝐸) ∫ ∫ 𝑠 (d𝜈) 𝜁 (d𝑠) = lim (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑟+𝑡−𝑠 ◦ 𝑡 ↓0 [0,𝑟+𝑡) 𝑀 (𝐸) ∫ ∫ 𝑠 − (d𝜈) (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑟−𝑠 𝜁 (d𝑠) ∫ 𝑀 (𝐸) ◦ ∫ [0,𝑟) 𝑠 𝜁 (d𝑠) (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑟+𝑡−𝑠 (d𝜈) = lim ◦ 𝑡 ↓0 [𝑟 ,𝑟+𝑡) 𝑀 (𝐸) ∫ ∫ 𝑠 𝜁 (d𝑠) + lim (d𝜇) (1 − e−𝜈 (𝑉𝑡 𝑓 ) )𝐺 𝑟−𝑠 ◦ 𝑡 ↓0 [0,𝑟) (𝐸) 𝑀 ∫ ∫ 𝑠 (d𝜈) 𝜁 (d𝑠) (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑟−𝑠 − ◦ [0,𝑟) 𝑀 (𝐸) ∫ ∫ 𝑠 (d𝜈) 𝜁 (d𝑠) (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑟+𝑡−𝑠 = lim ◦ 𝑡 ↓0 [𝑟 ,𝑟+𝑡) ∫ 𝑀 (𝐸) ≥ lim sup 𝜁 ({𝑟}) (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑟𝑡 (d𝜈). 𝑀 (𝐸) ◦

𝑡 ↓0

Now suppose that 𝜁 ({𝑟 }) > 0 for some 𝑟 ≥ 0. From the above it follows that ∫ lim sup (9.24) (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑟𝑡 (d𝜈) = 0. 𝑡 ↓0

𝑀 (𝐸) ◦

For 𝑡 ≥ 0 let 𝑞 𝑡 = sup0≤𝑠 ≤𝑡 ∥𝑉𝑠 𝑓 ∥. Then for 0 < 𝑠 < 𝑡 we have ∫ ∫ 𝑟 −𝜈 ( 𝑓 ) )𝐺 𝑡 (d𝜈) = (1 − e−𝜈 (𝑉𝑡−𝑠 𝑓 ) )𝐺 𝑟𝑠 (d𝜈) (1 − e ◦ 𝑀 (𝐸) ◦ (𝐸) 𝑀 ∫ (1 − e−𝜈 (𝑞𝑡 ) )𝐺 𝑟𝑠 (d𝜈). ≤ 𝑀 (𝐸) ◦

By (9.24) the right-hand side tends to zero as 𝑠 → 0, so (𝐺 𝑟𝑡 )𝑡 >0 is trivial. Therefore we can remove all the atoms of 𝜁 (d𝑠) and obtain (9.22) from (9.23). □ Corollary 9.10 For every SC-semigroup (𝑁𝑡 )𝑡 ≥0 associated with (𝑄 𝑡 )𝑡 ≥0 , we have the decomposition ∫ ∫ 𝑡 𝑠 (1 − e−𝜈 ( 𝑓 ) )𝐺 𝑡−𝑠 𝜁 (d𝑠) − log 𝐿 𝑁𝑡 ( 𝑓 ) = 𝛾𝑡 ( 𝑓 ) + (d𝜈), 0

𝑀 (𝐸) ◦

where 𝑡 ↦→ 𝛾𝑡 is an increasing continuous path from [0, ∞) to 𝑀 (𝐸), 𝜁 (d𝑠) is a diffuse Radon measure on [0, ∞) and {(𝐺 𝑡𝑠 )𝑡 >0 : 𝑠 ≥ 0} is a family of entrance laws for (𝑄 ◦𝑡 )𝑡 ≥0 .

250

9 Structures of Independent Immigration

We next give some estimates for the variations of transition probabilities of the MBI-process started from different initial states. Recall that the Wasserstein distance 𝑊1 (𝑄 1 , 𝑄 2 ) between two probability measures 𝑄 1 and 𝑄 2 on 𝑀 (𝐸) is defined by (2.71). For 𝜇, 𝜈 ∈ 𝑀 (𝐸) let |𝜇 − 𝜈| denote the total variation of 𝜇 − 𝜈. Then ∥𝜇 − 𝜈∥ := |𝜇 − 𝜈|(𝐸) is the total variation norm of 𝜇 − 𝜈.

Theorem 9.11 Suppose that (9.9) holds and 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ for every 𝑡 ≥ 0. Then for 𝜇, 𝜈 ∈ 𝑀 (𝐸) we have |(𝜇 − 𝜈) (𝜋𝑡 1)| ≤ 𝑊1 (𝑄 𝑡𝑁 (𝜇, ·), 𝑄 𝑡𝑁 (𝜈, ·)) ≤ |𝜇 − 𝜈|(𝜋𝑡 1),

(9.25)

where (𝜋𝑡 )𝑡 ≥0 is the semigroup of bounded kernels on 𝐸 defined by (2.54). Proof The first inequality in (9.25) follows by a first-moment calculation based on (9.11) as in the proof of Theorem 2.40. Let 𝑃𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) be the coupling of 𝑄 𝑡 (𝜇, d𝛾1 ) and 𝑄 𝑡 (𝜈, d𝛾2 ) defined in that proof. Let 𝑄 𝑡 (𝜇, 𝜈, d𝜂1 , d𝜂2 ) be the image of 𝑁𝑡 (d𝛾0 )𝑃𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) under the mapping (𝛾0 , 𝛾1 , 𝛾2 ) ↦→ (𝜂1 , 𝜂2 ) := (𝛾0 +𝛾1 , 𝛾+𝛾2 ). From (9.2) we see that 𝑄 𝑡 (𝜇, 𝜈, d𝜂1 , d𝜂2 ) is a coupling of 𝑄 𝑡𝑁 (𝜇, d𝜂1 ) and 𝑄 𝑡𝑁 (𝜈, d𝜂2 ). It follows that ∫ 𝑊1 (𝑄 𝑡𝑁 (𝜇, ·), 𝑄 𝑡𝑁 (𝜈, ·)) ≤ ∥𝜂1 − 𝜂2 ∥𝑄 𝑡 (𝜇, 𝜈, d𝜂1 , d𝜂2 ) ∫ ∫𝑀 (𝐸) 2 ∥𝛾1 − 𝛾2 ∥𝑃𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ) = 𝑁𝑡 (d𝛾0 ) 𝑀 (𝐸) 2 ∫𝑀 (𝐸) = ∥𝛾1 − 𝛾2 ∥𝑃𝑡 (𝜇, 𝜈, d𝛾1 , d𝛾2 ). 𝑀 (𝐸) 2

Then the second inequality in (9.25) follows by the calculations in the proof of □ Theorem 2.40. Corollary 9.12 Suppose that (9.9) holds and 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ for every 𝑡 ≥ 0. Then for any bounded Borel function 𝐹 on 𝑀 (𝐸) we have 𝐿 var (𝑄 𝑡𝑁 𝐹) ≤ ∥𝜋𝑡 1∥𝐿 var (𝐹), where 𝐿 var denotes the Lipschitz constant defined by (2.70). Theorem 9.13 Suppose that the function 𝑣¯ 𝑡 defined by (2.7) is bounded on 𝐸 for every 𝑡 > 0. Then for 𝜇, 𝜈 ∈ 𝑀 (𝐸) we have ∥𝑄 𝑡𝑁 (𝜇, ·) − 𝑄 𝑡𝑁 (𝜈, ·) ∥ ≤ 2(1 − e− | 𝜇−𝜈 | ( 𝑣¯𝑡 ) ) ≤ 2|𝜇 − 𝜈|( 𝑣¯ 𝑡 ).

9.3 Regular Immigration Superprocesses

251

Proof Let 𝐹 be a Borel function on 𝑀 (𝐸) satisfying |𝐹 | ≤ 1. In view of (9.2), we have ∫ ∫ 𝑁 𝑁 𝑄 𝑁 𝐹 (𝜇) − 𝑄 𝑁 𝐹 (𝜈) = 𝐹 (𝜂)𝑄 𝑡 (𝜈, d𝜂) 𝐹 (𝜂)𝑄 𝑡 (𝜇, d𝜂) − 𝑡 𝑡 𝑀 (𝐸) 𝑀 (𝐸) ∫ ∫ 𝐹 (𝜂 + 𝛾)𝑄 𝑡 (𝜇, d𝜂) = 𝑁𝑡 (d𝛾) 𝑀 (𝐸) ∫ 𝑀 (𝐸) ∫ − 𝑁𝑡 (d𝛾) 𝐹 (𝜂 + 𝛾)𝑄 𝑡 (𝜈, d𝜂) (𝐸) 𝑀 (𝐸) 𝑀 ∫ ∫ 𝐹 (𝜂 + 𝛾)𝑄 𝑡 (𝜇, d𝜂) ≤ 𝑀 (𝐸) 𝑀 (𝐸) ∫ − 𝐹 (𝜂 + 𝛾)𝑄 𝑡 (𝜈, d𝜂) 𝑁𝑡 (d𝛾) 𝑀 (𝐸)

≤ 𝑄 𝑡 (𝜇, ·) − 𝑄 𝑡 (𝜈, ·) . Then desired estimates follow by Theorem 2.45.

□

Corollary 9.14 Suppose that 𝑣¯ 𝑡 is bounded on 𝐸 for 𝑡 > 0. Then for any bounded Borel function 𝐹 on 𝑀 (𝐸) we have 𝐿 var (𝑄 𝑡𝑁 𝐹) ≤ 2∥ 𝑣¯ 𝑡 ∥ ∥𝐹 ∥, where 𝐿 var denotes the Lipschitz constant defined by (2.70). In the situation of Corollary 9.14, the operators (𝑄 𝑡𝑁 )𝑡 >0 map bounded Borel functions on 𝑀 (𝐸) into functions continuous in the total variation distance, that is, the semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 has the strong Feller property.

9.3 Regular Immigration Superprocesses In this section we present some basic properties of immigration superprocesses defined by regular SC-semigroups. Suppose that 𝜉 is a Borel right process in a Lusin topological space 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 . Let 𝜙 be a branching mechanism given by (2.29) or (2.30) and let (𝑉𝑡 )𝑡 ≥0 be the cumulant semigroup of the (𝜉, 𝜙)-superprocess defined by (2.36). Recall that 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)] and 𝑐+0 = 0 ∨ 𝑐 0 . Let 𝐼 ∈ ℐ(𝐸) be an immigration mechanism given by (9.12) with 𝜈(1)𝐻 (d𝜈) being a finite measure on 𝑀 (𝐸) ◦ . The associated regular immigration superprocess has transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 given by (9.13). In this case, it is easy to see that (𝑄 𝑡𝑁 )𝑡 ≥0 is actually a special case of the transition semigroup constructed in Theorem 6.10. Then the results given below for immigration processes are essentially consequences of those established in the preceding chapters.

252

9 Structures of Independent Immigration

Proposition 9.15 Let (𝑄 𝑡𝑁 )𝑡 ≥0 be defined by (9.13). Then for 𝑡 ≥ 0, 𝜇 ∈ 𝑀 (𝐸) and 𝑓 ∈ 𝐵(𝐸) we have ∫ ∫ 𝑡 𝑁 (9.26) 𝜈( 𝑓 )𝑄 𝑡 (𝜇, d𝜈) = 𝜇(𝜋𝑡 𝑓 ) + Γ(𝜋 𝑠 𝑓 )d𝑠, 𝑀 (𝐸)

0

where 𝑡 ↦→ 𝜋𝑡 𝑓 is defined by (2.38) and ∫ Γ( 𝑓 ) = 𝜂( 𝑓 ) +

𝜈( 𝑓 )𝐻 (d𝜈).

(9.27)

𝑀 (𝐸) ◦

Proposition 9.16 Let (𝑌𝑡 , 𝒢𝑡 , P) be an immigration superprocess with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 such that P[𝑌0 (1)] < ∞. Suppose that 𝛼 ∈ R and 𝑓 ∈ 𝐵(𝐸) + satisfy 𝜋𝑡 𝑓 (𝑥) ≤ e 𝛼𝑡 𝑓 (𝑥) for all 𝑡 ≥ 0 and 𝑥 ∈ 𝐸. Then the process ∫ 𝑡 −𝛼𝑡 𝑍𝑡 ( 𝑓 ) := e 𝑌𝑡 ( 𝑓 ) − Γ( 𝑓 ) 𝑡≥0 e−𝛼𝑠 d𝑠, 0

is a (𝒢𝑡 )-supermartingale. Proof By Proposition 9.15, for any 𝑡 ≥ 𝑟 ≥ 0 we have ∫ 𝑡   e−𝛼𝑠 d𝑠 P 𝑍𝑡 ( 𝑓 ) 𝒢𝑟 = e−𝛼𝑡 P[𝑌𝑡 ( 𝑓 )|𝒢𝑟 ] − Γ( 𝑓 )   ∫ 𝑡−𝑟 0 ∫ 𝑡 −𝛼𝑡 e−𝛼𝑠 d𝑠 Γ(𝜋 𝑠 𝑓 )d𝑠 − Γ( 𝑓 ) 𝑌𝑟 (𝜋𝑡−𝑟 𝑓 ) + =e ∫ 0𝑡 ∫0 𝑡−𝑟 −𝛼𝑟 𝛼(𝑠−𝑡) e−𝛼𝑠 d𝑠 e ≤ e 𝑌𝑟 ( 𝑓 ) + Γ( 𝑓 ) d𝑠 − Γ( 𝑓 ) 0 ∫0 𝑟 = e−𝛼𝑟 𝑌𝑟 ( 𝑓 ) − Γ( 𝑓 ) e−𝛼𝑠 d𝑠. 0

Then 𝑡 ↦→ 𝑍𝑡 ( 𝑓 ) is a (𝒢𝑡 )-supermartingale.

□

Corollary 9.17 Let (𝑌𝑡 , 𝒢𝑡 , P) be a right continuous immigration superprocess with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 such that P[𝑌0 (1)] < ∞. Then the process ∫ 𝑡 𝑧(𝑡) := e−𝑐0 𝑡 𝑌𝑡 (1) − Γ(1) e−𝑐0 𝑠 d𝑠, 𝑡 ≥ 0 0

is a (𝒢𝑡 )-supermartingale and for any 𝜆 ≥ 0 we have o

n

∫

𝜆P sup |𝑧(𝑡)| ≥ 𝜆 ≤ P[𝑌0 (1)] + 2Γ(1) 𝑡 ≥0

0

𝑡

e−𝑐0 𝑠 d𝑠.

9.3 Regular Immigration Superprocesses

253

Corollary 9.18 Let (𝑌𝑡 , 𝒢𝑡 , P) be an immigration superprocess with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 such that P[𝑌0 (1)] < ∞. Let 𝛼 ≥ 0 and let 𝑓 ∈ 𝐵(𝐸) + be an 𝛼-super-mean-valued function for (𝑃𝑡 )𝑡 ≥0 satisfying 𝜀 := inf 𝑥 ∈𝐸 𝑓 (𝑥) > 0. Then for any 𝛽 > 𝛼 + 𝑐+0 𝜀 −1 ∥ 𝑓 ∥ the process 𝑍𝑡 ( 𝑓 ) := e−2𝛽𝑡 𝑌𝑡 ( 𝑓 ) + (2𝛽) −1 e−2𝛽𝑡 Γ( 𝑓 ),

𝑡≥0

is a positive (𝒢𝑡 )-supermartingale. Let F be the set of functions 𝑓 ∈ 𝐵(𝐸) that are finely continuous relative to 𝜉. Fix 𝛽 > 0 and let ( 𝐴, 𝒟( 𝐴)) be the weak generator of (𝑃𝑡 )𝑡 ≥0 defined by 𝒟( 𝐴) = 𝑈 𝛽 F and 𝐴 𝑓 = 𝛽 𝑓 − 𝑔 for 𝑓 = 𝑈 𝛽 𝑔 ∈ 𝒟( 𝐴). Theorem 9.19 Suppose that (𝑌𝑡 , 𝒢𝑡 , P) is a progressive realization of the immigration superprocess with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 such that P[𝑌0 (1)] < ∞. Then for any 𝑓 ∈ 𝒟( 𝐴), the process ∫ 𝑡   𝑀𝑡 ( 𝑓 ) := 𝑌𝑡 ( 𝑓 ) − 𝑌0 ( 𝑓 ) − 𝑌𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 ) + Γ( 𝑓 ) d𝑠 0

is a (𝒢𝑡 )-martingale. Proposition 9.20 Suppose that the kernel 𝐻 (𝑥, d𝜈) in (2.29) and the measure 𝐻 (d𝜈) in (9.12) satisfy ∫ ∫ 2 (9.28) 𝜈( 𝑓 ) 2 𝐻 (d𝜈) < ∞. 𝜈( 𝑓 ) 𝐻 (𝑥, d𝜈) + sup 𝑥 ∈𝐸

𝑀 (𝐸) ◦

𝑀 (𝐸) ◦

Then for 𝑡 ≥ 0, 𝜇 ∈ 𝑀 (𝐸) and 𝑓 ∈ 𝐵(𝐸) we have 2  ∫ 𝑡 Γ(𝜋 𝑠 𝑓 )d𝑠 𝜈( 𝑓 ) 2 𝑄 𝑡𝑁 (𝜇, d𝜈) = 𝜇(𝜋𝑡 𝑓 ) + 𝑀 (𝐸) ∫ 𝑡 ∫ 0 d𝑠 + 𝑞(𝑥, 𝜋 𝑠 𝑓 )𝜇𝜋𝑡−𝑠 (d𝑥) ∫0 𝑡 ∫𝐸 𝑢 ∫ + d𝑢 d𝑠 𝑞(𝑥, 𝜋 𝑠 𝑓 )Γ𝜋𝑢−𝑠 (d𝑥) 𝐸 ∫0 𝑡 ∫ 0 𝜈(𝜋 𝑠 𝑓 ) 2 𝐻 (d𝜈), d𝑠 (9.29) +

∫

0

𝑀 (𝐸) ◦

where 𝑞(𝑥, 𝑓 ) is defined by (2.64). Theorem 9.21 The immigration superprocess with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 has a right realization in 𝑀 (𝐸). If 𝜉 is a Hunt process, then the immigration superprocess has a Hunt realization in 𝑀 (𝐸). When the spatial motion 𝜉 is conservative, we also have a counterpart of Theorem 5.11 for the immigration superprocess. In particular, it has a right process realization on the canonical space of paths that are right continuous in both 𝑀 (𝐸) ¯ and 𝑀 (𝐸 𝜌 ) and have left limits in 𝑀 ( 𝐸).

254

9 Structures of Independent Immigration

Let 𝜉 = (Ω, ℱ, ℱ𝑟 ,𝑡 , 𝜉𝑡 , P𝑟 , 𝑥 ) and 𝑌 = (𝑊, 𝒢, 𝒢𝑟 ,𝑡 , 𝑌𝑡 , Q𝑟𝑁, 𝜇 ) be respectively right continuous realizations of the underlying spatial motion and the immigration superprocess from an arbitrary initial time 𝑟 ≥ 0. The next theorem gives a characterization of the weighted occupation times of the immigration superprocess: Theorem 9.22 Suppose that 𝑡 ≥ 0 and 𝜆 ∈ 𝑀 ( [0, 𝑡]). Let (𝑠, 𝑥) ↦→ 𝑓𝑠 (𝑥) be a bounded positive Borel function on [0, 𝑡] × 𝐸. Then we have   ∫   ∫ 𝑡 𝑌𝑠 ( 𝑓𝑠 )𝜆(d𝑠) = exp − 𝜇(𝑢𝑟 ) − 𝐼 (𝑢 𝑠 )d𝑠 Q𝑟𝑁, 𝜇 exp − [𝑟 ,𝑡 ]

𝑟

for every 0 ≤ 𝑟 ≤ 𝑡, where (𝑟, 𝑥) ↦→ 𝑢𝑟 (𝑥) is the unique bounded positive solution on [0, 𝑡] × 𝐸 of (5.26). Corollary 9.23 Let 𝑌 = (𝑊, 𝒢, 𝒢𝑡 , 𝑌𝑡 , Q 𝜇𝑁 ) be a right continuous realization of the immigration superprocess started from time zero. Then for 𝑡 ≥ 0 and 𝑓 , 𝑔 ∈ 𝐵(𝐸) + we have     ∫ 𝑡 ∫ 𝑡 Q 𝜇𝑁 exp − 𝑌𝑡 ( 𝑓 ) − 𝐼 (𝑣 𝑠 )d𝑠 , 𝑌𝑠 (𝑔)d𝑠 = exp − 𝜇(𝑣 𝑡 ) − 0

0

where (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥) is the unique locally bounded positive solution of (5.32). We can also extend the immigration superprocesses to the state space of tempered measures. Let 𝛼 ≥ 0 and let ℎ ∈ pℬ(𝐸) be a strictly positive 𝛼-excessive function for 𝜉. Recall that 𝑀ℎ (𝐸) is the space of Borel measures 𝜇 on 𝐸 satisfying 𝜇(ℎ) < ∞ and 𝐵 ℎ (𝐸) is the set of Borel functions 𝑓 on 𝐸 satisfying | 𝑓 | ≤ const. · ℎ. Theorem 9.24 Let (𝑉𝑡 )𝑡 ≥0 be the cumulant semigroup defined in Theorem 6.3. Suppose that 𝜂 ∈ 𝑀ℎ (𝐸) and 𝜈(ℎ)𝐻 (d𝜈) is a finite measure on 𝑀ℎ (𝐸) ◦ := 𝑀ℎ (𝐸) \ {0} and write ∫
(9.30) 𝐼 ( 𝑓 ) = 𝜂( 𝑓 ) + 1 − e−𝜈 ( 𝑓 ) 𝐻 (d𝜈), 𝑓 ∈ 𝐵 ℎ (𝐸) + . 𝑀ℎ (𝐸) ◦

Then a Borel right transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 on 𝑀ℎ (𝐸) is defined by   ∫ ∫ 𝑡 𝐼 (𝑉𝑠 𝑓 )d𝑠 . e−𝜈 ( 𝑓 ) 𝑄 𝑡𝑁 (𝜇, d𝜈) = exp − 𝜇(𝑉𝑡 𝑓 ) − 𝑀ℎ (𝐸)

(9.31)

0

If, in addition, the semigroup ( 𝑃˜𝑡 )𝑡 ≥0 given by (6.10) has a Hunt realization, then (𝑄 𝑡𝑁 )𝑡 ≥0 has a Hunt realization. Suppose that 𝑇 ⊂ R is an interval and 𝐹 is a Lusin topological space. Let 𝐸˜ be a Borel subset of 𝑇 × 𝐹. Let (𝑃𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇) be an inhomogeneous Borel right transition semigroup with global state space 𝐸˜ and let 𝜉 = (Ω, ℱ, ℱ𝑟 ,𝑡 , 𝜉𝑡 , P𝑟 , 𝑥 ) be a right continuous realization of the semigroup. Let 𝜙 be given by (6.29). Suppose that 𝜂(𝑠, d𝑦) is a bounded kernel from 𝑇 to 𝐸˜ and 𝜈(1)𝐻 (𝑠, d𝜈) is a bounded kernel ˜ ◦ . For every 𝑠 ∈ 𝑇 we assume 𝜂(𝑠, d𝑦) is supported by {𝑠} × 𝐸 𝑠 and from 𝑇 to 𝑀 ( 𝐸)

9.3 Regular Immigration Superprocesses

255

𝐻 (𝑠, d𝜈) is supposed by 𝑀 ({𝑠} × 𝐸 𝑠 ) ◦ . Then we can regard 𝜂(𝑠, d𝑦) as a measure on 𝐸 𝑠 and regard 𝐻 (𝑠, d𝜈) as a measure on 𝑀 (𝐸 𝑠 ) ◦ . For 𝑠 ∈ 𝑇 and 𝑓 ∈ 𝐵(𝐸 𝑠 ) + define ∫
(9.32) 𝐼 (𝑠, 𝑓 ) = 𝜂(𝑠, 𝑓 ) + 1 − e−𝜈 ( 𝑓 ) 𝐻 (𝑠, d𝜈). 𝑀 (𝐸𝑠 ) ◦

Let 𝑀˜ = {(𝑡, 𝜇) : 𝑡 ∈ 𝑇, 𝜇 ∈ 𝑀 (𝐸 𝑡 )}. We identify this set with {(𝑡, 𝜇) ∈ 𝑇 × 𝑀 (𝐹) : 𝜇(𝐹 \ 𝐸 𝑡 ) = 0} furnished with the topology inherited from 𝑇 × 𝑀 (𝐹). Theorem 9.25 For 𝑡 ∈ 𝑇 and 𝑓 ∈ 𝐵(𝐸 𝑡 ) + let (𝑟, 𝑥) ↦→ 𝑉𝑟 ,𝑡 𝑓 (𝑥) be the unique locally bounded positive solution on 𝐸˜ ≤𝑡 of (6.30). Then an inhomogeneous Borel right transition semigroup (𝑄 𝑟𝑁,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇) with global state space 𝑀˜ is defined by   ∫ 𝑡 ∫ 𝐼 (𝑠, 𝑉𝑠,𝑡 𝑓 )d𝑠 . (9.33) e−𝜈 ( 𝑓 ) 𝑄 𝑟𝑁,𝑡 (𝜇, d𝜈) = exp − 𝜇(𝑉𝑟 ,𝑡 𝑓 ) − 𝑀 (𝐸𝑡 )

𝑟

If an inhomogeneous Markov process with global state space 𝑀˜ has transition semigroup (𝑄 𝑟𝑁,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇) defined by (9.33), we call it an inhomogeneous immigration superprocess with spatial motion 𝜉, branching mechanism 𝜙 and immigration mechanism 𝐼. Let (𝑄 𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇) be the transition semigroup defined by (6.30) and (6.31). Clearly, we have 𝑄 𝑟𝑁,𝑡 (𝜇, ·) = 𝑄 𝑟 ,𝑡 (𝜇, ·) ∗ 𝑁𝑟 ,𝑡 ,

𝑟 ≤ 𝑡 ∈ 𝑇,

(9.34)

𝑟 ≤ 𝑠 ≤ 𝑡 ∈ 𝑇.

(9.35)

where 𝑁𝑟 ,𝑡 := 𝑄 𝑟𝑁,𝑡 (0, ·) satisfies 𝑁𝑟 ,𝑡 = (𝑁𝑟 ,𝑠 𝑄 𝑠,𝑡 ) ∗ 𝑁 𝑠,𝑡 ,

In view of (9.35), it is natural to call (𝑁𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇) an inhomogeneous skewconvolution semigroup associated with (𝑄 𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇). Theorem 9.26 Let 𝑌 = (𝑊, 𝑌𝑡 , 𝒢, 𝒢𝑟 ,𝑡 , Q𝑟𝑁, 𝜇 ) be a right continuous inhomogeneous immigration superprocess with transition semigroup (𝑄 𝑟𝑁,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇) defined by (9.33). Suppose that 𝜆 is a Radon measure on 𝑇 and (𝑠, 𝑥) ↦→ 𝑓𝑠 (𝑥) is a locally ˜ Then for any 𝑡 ∈ 𝑇 we have bounded positive Borel function on 𝐸.   ∫ 𝑋𝑠 ( 𝑓𝑠 )𝜆(d𝑠) Q𝑟𝑁, 𝜇 exp −   [𝑟 ,𝑡 ] ∫ 𝑡 (9.36) 𝐼 (𝑠, 𝑢 𝑠 )d𝑠 , 𝑟 ∈ 𝑇≤𝑡 , = exp − 𝜇(𝑢𝑟 ) − 𝑟

where (𝑟, 𝑥) ↦→ 𝑢𝑟 (𝑥) is the unique locally bounded positive solution on 𝐸˜ ≤𝑡 of (6.35). Let us briefly discuss the special case of one-dimensional processes. Suppose that 𝛽 ∈ 𝐵(𝑇) + and 𝑧𝑛(𝑡, d𝑧) is a bounded kernel from 𝑇 to (0, ∞). For 𝑡 ∈ 𝑇 and 𝜆 ≥ 0 define

256

9 Structures of Independent Immigration

∫ 𝜓(𝑡, 𝜆) = 𝛽(𝑡)𝜆 +

∞


1 − e−𝜆𝑧 𝑛(𝑡, d𝑧).

(9.37)

0

Let 𝑟 ↦→ 𝑣 𝑟 ,𝑡 (𝜆) be the unique locally bounded positive solution of (6.38). By modifying the proof of Theorem 6.17 we obtain the following: Theorem 9.27 There is an inhomogeneous Borel right transition semigroup 𝛾 (𝑄 𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇) on [0, ∞) defined by, for 𝜆 ≥ 0, ∫ e

−𝜆𝑦

𝛾 𝑄 𝑟 ,𝑡 (𝑥, d𝑦)



∫

= exp − 𝑥𝑣 𝑟 ,𝑡 (𝜆) −

[0,∞)



𝑡

𝜓(𝑠, 𝑣 𝑠,𝑡 (𝜆))d𝑠 .

(9.38)

𝑟 𝛾

Moreover, the semigroup (𝑄 𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇) has a càdlàg realization. 𝛾

The transition semigroup (𝑄 𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇) defined by (9.38) is a natural generalization of the homogeneous semigroup given by (3.29). If an inhomogeneous 𝛾 Markov process in [0, ∞) has transition semigroup (𝑄 𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇), we call it an inhomogeneous CBI-process with branching mechanism 𝜙 and immigration mechanism 𝜓. As a special case of Theorem 9.26, we have: 𝛾

Theorem 9.28 Suppose that 𝑌 = (𝑊, 𝒢, 𝒢𝑟 ,𝑡 , 𝑦(𝑡), Q𝑟 , 𝑥 ) is a right continuous in𝛾 homogeneous CBI-process with transition semigroup (𝑄 𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝑇) defined by (9.38). Let 𝜆 ≥ 0 and let 𝑠 ↦→ 𝑓 (𝑠) be a locally bounded Borel function on 𝑇. Then for any 𝑡 ≥ 𝑟 ∈ 𝑇 we have   ∫ 𝑡 𝛾 Q𝑟 , 𝑥 exp − 𝜆𝑦(𝑡) − 𝑓 (𝑠)𝑦(𝑠)d𝑠 𝑟   ∫ 𝑡 𝜓(𝑠, 𝑢(𝑠, 𝜆, 𝑓 ))d𝑠 , = exp − 𝑥𝑢(𝑟, 𝜆, 𝑓 ) − (9.39) 𝑟

where 𝑟 ↦→ 𝑢(𝑟, 𝜆, 𝑓 ) is the unique locally bounded positive solution to (6.40).

9.4 Characterizations by Martingale Problems Suppose that 𝐸 is a locally compact separable metric space. Let 𝜉 be a Hunt process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 . We assume (𝑃𝑡 )𝑡 ≥0 preserves 𝐶0 (𝐸) and 𝑡 ↦→ 𝑃𝑡 𝑓 is continuous in the supremum norm for every 𝑓 ∈ 𝐶0 (𝐸), but the semigroup is not necessarily conservative. Let 𝜙 be a branching mechanism given by (2.29) or (2.30) satisfying Conditions 7.1 and 7.2. By Theorem 9.21 the immigration superprocess with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 given by (9.13) has a càdlàg realization. The characterizations of the immigration superprocess by martingale problems given below are essentially consequences of Theorem 6.10 and the results in Sections 7.2 and 7.4.

9.4 Characterizations by Martingale Problems

257

Proposition 9.29 If {𝑌𝑡 : 𝑡 ≥ 0} is a càdlàg Markov process in 𝑀 (𝐸) relative to a filtration (𝒢𝑡 )𝑡 ≥0 with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 , then {𝑌𝑡 : 𝑡 ≥ 0} is also a Markov process relative to the augmented right continuous filtration (𝒢¯ 𝑡+ )𝑡 ≥0 with the same transition semigroup. Let 𝐴 denote the strong generator of (𝑃𝑡 )𝑡 ≥0 with domain 𝐷 0 ( 𝐴) ⊂ 𝐶0 (𝐸). Let 𝒟0 be the class of functions on 𝑀 (𝐸) of the form (7.16). Let 𝐿 0 be the generator defined by (7.17). For 𝐹 ∈ 𝒟0 define ∫ 𝐹 ′ (𝜇; 𝑥)𝜂(d𝑥) 𝐿𝐹 (𝜇) = 𝐿 0 𝐹 (𝜇) + 𝐸 ∫ (9.40) + [𝐹 (𝜇 + 𝜈) − 𝐹 (𝜇)]𝐻 (d𝜈). 𝑀 (𝐸) ◦

Suppose that (𝑊, 𝒢, 𝒢𝑡 , P) is a filtered probability space satisfying the usual hypotheses and {𝑌𝑡 : 𝑡 ≥ 0} is a càdlàg process in 𝑀 (𝐸) that is adapted to (𝒢𝑡 )𝑡 ≥0 and satisfies P[𝑌0 (1)] < ∞. For this process we consider the following properties: (1) For every 𝑇 ≥ 0 and 𝑓 ∈ 𝐶0 (𝐸) + ,  ∫ exp − 𝑌𝑡 (𝑉𝑇−𝑡 𝑓 ) −

𝑇−𝑡

 𝐼 (𝑉𝑠 𝑓 )d𝑠 ,

0≤𝑡 ≤𝑇

0

is a martingale. (2) For every 𝑓 ∈ 𝐷 0 ( 𝐴) + ,   ∫ 𝑡 [𝑌𝑠 ( 𝐴 𝑓 − 𝜙( 𝑓 )) + 𝐼 ( 𝑓 )]d𝑠 , 𝐻𝑡 ( 𝑓 ) := exp − 𝑌𝑡 ( 𝑓 ) +

𝑡≥0

0

is a local martingale. (3) The process {𝑌𝑡 : 𝑡 ≥ 0} has no negative jumps. Moreover, we have: (a) Let 𝑁 (d𝑠, d𝜈) be the optional random measure on [0, ∞) × 𝑀 (𝐸) ◦ defined by ∑︁ 1 {Δ𝑌𝑠 ≠0} 𝛿 (𝑠,Δ𝑌𝑠 ) (d𝑠, d𝜈), (9.41) 𝑁 (d𝑠, d𝜈) = 𝑠>0

where Δ𝑌𝑠 = 𝑌𝑠 − 𝑌𝑠− . Then 𝑁 (d𝑠, d𝜈) has predictable compensator 𝑁ˆ (d𝑠, d𝜈) = d𝑠𝐾 (𝑌𝑠− , d𝜈) + d𝑠𝐻 (d𝜈), where ∫ 𝐾 (𝜇, d𝜈) =

𝜇(d𝑥)𝐻 (𝑥, d𝜈). 𝐸

258

9 Structures of Independent Immigration

(b) Let Γ be defined by (9.27) and let 𝑁˜ (d𝑠, d𝜈) = 𝑁 (d𝑠, d𝜈) − 𝑁ˆ (d𝑠, d𝜈). Then for any 𝑓 ∈ 𝐷 0 ( 𝐴), we have ∫ 𝑌𝑡 ( 𝑓 ) = 𝑌0 ( 𝑓 ) + 𝑀𝑡𝑐 ( 𝑓 ) + 𝑀𝑡𝑑 ( 𝑓 ) + Γ ( 𝑓 )𝑡 +

𝑡

𝑌𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠,

(9.42)

0

where 𝑡 ↦→ 𝑀𝑡𝑐 ( 𝑓 ) is a continuous local martingale with quadratic variation 2𝑌𝑡 (𝑐 𝑓 2 )d𝑡 and ∫ 𝑡∫ (9.43) 𝑡 ↦→ 𝑀𝑡𝑑 ( 𝑓 ) = 𝜈( 𝑓 ) 𝑁˜ (d𝑠, d𝜈) 0

𝑀 (𝐸) ◦

is a purely discontinuous local martingale. (4) For every 𝐹 ∈ 𝒟0 we have ∫ 𝑡 𝐹 (𝑌𝑡 ) = 𝐹 (𝑌0 ) + 𝐿𝐹 (𝑌𝑠 )d𝑠 + local mart. 0

(5) For every 𝐺 ∈ 𝐶 2 (R) and 𝑓 ∈ 𝐷 0 ( 𝐴), ∫ 𝑡 𝐺 ′ (𝑌𝑠 ( 𝑓 ))𝑌𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠 𝐺 (𝑌𝑡 ( 𝑓 )) = 𝐺 (𝑌0 ( 𝑓 )) + 0 ∫ 𝑡h i ′′ 𝐺 (𝑌𝑠 ( 𝑓 ))𝑌𝑠 (𝑐 𝑓 2 ) + 𝐺 ′ (𝑌𝑠 ( 𝑓 ))𝜂( 𝑓 ) d𝑠 + ∫ ∫0 𝑡 ∫ h d𝑠 + 𝑌𝑠 (d𝑥) 𝐺 (𝑌𝑠 ( 𝑓 ) + 𝜈( 𝑓 )) − 𝐺 (𝑌𝑠 ( 𝑓 )) 𝑀 (𝐸) ◦ 0 𝐸 ∫ 𝑡 ∫ h i − 𝜈( 𝑓 )𝐺 ′ (𝑌𝑠 ( 𝑓 )) 𝐻 (𝑥, d𝜈) + d𝑠 𝐺 (𝑌𝑠 ( 𝑓 ) + 𝜈( 𝑓 )) 0 𝑀 (𝐸) ◦ i − 𝐺 (𝑌𝑠 ( 𝑓 )) 𝐻 (d𝜈) + local mart. Theorem 9.30 The above properties (1), (2), (3), (4) and (5) are equivalent to each other. Those properties hold if and only if {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is an immigration superprocess with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 given by (9.13). In the following corollaries, we assume {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a càdlàg realization of the immigration superprocess satisfying P[𝑌0 (1)] < ∞. Corollary 9.31 The local martingales in the above properties (3), (4) and (5) are martingales. Corollary 9.32 For 𝑇 ≥ 0 and 𝑓 ∈ 𝐷 0 ( 𝐴) there is a constant 𝐶 (𝑇, 𝑓 ) ≥ 0 such that i h n P sup |𝑌𝑡 ( 𝑓 )| ≤ 𝐶 (𝑇, 𝑓 ) P[𝑌0 (1)] + Γ(1) 0≤𝑡 ≤𝑇 o √︁ √︁ + P[𝑌0 (1)] + Γ(1) .

9.4 Characterizations by Martingale Problems

259

Corollary 9.33 Suppose that (9.28) holds. Then for every 𝑓 ∈ 𝐷 0 ( 𝐴), ∫ 𝑡 𝑀𝑡 ( 𝑓 ) = 𝑌𝑡 ( 𝑓 ) − 𝑌0 ( 𝑓 ) − 𝑡Γ( 𝑓 ) − 𝑌𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠

(9.44)

0

is a square integrable (𝒢𝑡 )-martingale with quadratic variation process  ∫ 𝑡 ∫ ∫ 𝜈( 𝑓 ) 2 𝐻 (d𝜈) d𝑠, ⟨𝑀 ( 𝑓 )⟩𝑡 = 𝑞(𝑥, 𝑓 )𝑌𝑠 (d𝑥) + 0

𝑀 (𝐸) ◦

𝐸

where 𝑞(𝑥, 𝑓 ) is defined by (2.64). If 𝑃𝑡 1 ∈ 𝐶 (𝐸) for every 𝑡 ≥ 0 and there exists an 𝐴1 ∈ 𝐶 (𝐸) such that (7.36) holds, where the convergence is uniform, we can extend the operator 𝐴 to the linear span 𝐷 ( 𝐴) of 𝐷 0 ( 𝐴) and the constant functions. In this case, the results of Theorem 9.30 and its corollaries remain true with 𝐷 0 ( 𝐴) replaced by 𝐷 ( 𝐴). Theorem 9.34 The continuous martingale functional {𝑀𝑡𝑐 ( 𝑓 ) : 𝑡 ≥ 0; 𝑓 ∈ 𝐷 0 ( 𝐴)} defined by (9.42) induces a continuous orthogonal martingale measure with covariance measure 𝜂 𝑐 (d𝑠, d𝑥) = 2𝑐(𝑥)d𝑠𝑌𝑠 (d𝑥). Proposition 9.35 Let 𝑁 (d𝑠, d𝜈) be given by (9.41). Then for any 𝑎 > 0 we can define a càdlàg worthy martingale measure ∫ 𝑡∫ 𝑎 𝑍𝑡 (𝐵) = 𝜈(𝐵) 𝑁˜ (d𝑠, d𝜈), 𝑡 ≥ 0, 𝐵 ∈ ℬ(𝐸), 0

{𝜈 (1) ≤𝑎 }

which has covariance measure ∫ 𝑎

𝜂 (d𝑠, d𝑥, d𝑦) = d𝑠

∫ 𝜈(d𝑥)𝜈(d𝑦)𝐻 (𝑧, d𝜈)

𝑌𝑠 (d𝑧) {𝜈 (1) ≤𝑎 }

𝐸 ∫

𝜈(d𝑥)𝜈(d𝑦)𝐻 (d𝜈).

+ d𝑠 {𝜈 (1) ≤𝑎 }

Corollary 9.36 Suppose that (9.28) holds. Then we can define a càdlàg worthy martingale measure ∫ 𝑡∫ 𝑍𝑡 (𝐵) = 𝜈(𝐵) 𝑁˜ (d𝑠, d𝜈), 𝑡 ≥ 0, 𝐵 ∈ ℬ(𝐸), 0

𝑀 (𝐸) ◦

which has covariance measure ∫

∫ 𝑌𝑠 (d𝑧) 𝜈(d𝑥)𝜈(d𝑦)𝐻 (𝑧, d𝜈) 𝑀 (𝐸) ◦ 𝐸 ∫ + d𝑠 𝜈(d𝑥)𝜈(d𝑦)𝐻 (d𝜈).

𝜂(d𝑠, d𝑥, d𝑦) = d𝑠

𝑀 (𝐸) ◦

260

9 Structures of Independent Immigration

Theorem 9.37 Let (𝜋𝑡 )𝑡 ≥0 be the semigroup defined by (2.38). Then for any 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) we have a.s. ∫ 𝑡 ∫ 𝑡∫ 𝜋𝑡−𝑠 𝑓 (𝑥)𝑀 𝑐 (d𝑠, d𝑥) 𝑌𝑡 ( 𝑓 ) = 𝑌0 (𝜋𝑡 𝑓 ) + Γ(𝜋𝑡−𝑠 𝑓 )d𝑠 + 0 0 𝐸 ∫ 𝑡∫ + 𝜈(𝜋𝑡−𝑠 𝑓 ) 𝑁˜ (d𝑠, d𝜈). (9.45) 0

𝑀 (𝐸) ◦

The integral with respect to 𝑁˜ (d𝑠, d𝜈) in (9.45) is defined as in Section 7.4 by considering separately the sets {𝜈 ∈ 𝑀 (𝐸) ◦ : 𝜈(1) ≤ 1} and {𝜈 ∈ 𝑀 (𝐸) ◦ : 𝜈(1) > 1}. We can also give martingale problem formulations of the immigration superprocess on the tempered space. Suppose that ℎ ∈ 𝐷 0 ( 𝐴) is strictly positive and there is a constant 𝛼 > 0 such that 𝐴ℎ ≤ 𝛼ℎ. Recall that 𝐶ℎ (𝐸) is the set of continuous functions 𝑓 on 𝐸 satisfying | 𝑓 | ≤ const. · ℎ and 𝐷 ℎ ( 𝐴) = { 𝑓 ∈ 𝐷 0 ( 𝐴) ∩ 𝐶ℎ (𝐸) : 𝐴 𝑓 ∈ 𝐶ℎ (𝐸)}. Let 𝜙 be a branching mechanism given as in Section 6.1 with 𝜌 = ℎ and let (𝑄 𝑡𝑁 )𝑡 ≥0 be the transition semigroup on 𝑀ℎ (𝐸) defined by (9.30) and (9.31). Suppose that 𝑓 ↦→ ℎ−1 𝜙(·, ℎ 𝑓 ) −𝛼 𝑓 satisfies the conditions for the branching mechanism specified at the beginning of Section 7.1. Then we have: Theorem 9.38 Let (𝑊, 𝒢, 𝒢𝑡 , P) be a filtered probability space satisfying the usual hypotheses and let {𝑌𝑡 : 𝑡 ≥ 0} be a càdlàg process in 𝑀ℎ (𝐸) that is adapted to (𝒢𝑡 )𝑡 ≥0 and satisfies P[𝑌0 (ℎ)] < ∞. Then Theorem 9.30 still holds when 𝑀 (𝐸), 𝐶0 (𝐸) and 𝐷 0 ( 𝐴) are replaced by 𝑀ℎ (𝐸), 𝐶ℎ (𝐸) and 𝐷 ℎ ( 𝐴), respectively. Theorem 9.39 For every 𝜇 ∈ 𝑀ℎ (𝐸) the immigration superprocess with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 defined by (9.30) and (9.31) has a càdlàg realization {𝑌𝑡 : 𝑡 ≥ 0} in 𝑀ℎ (𝐸) with initial value 𝑌0 = 𝜇. We close this section with a characterization of the one-dimensional CBI-process in terms of a martingale problem. Let 𝜙 and 𝜓 be the branching and immigration mechanisms given by (3.1) and (3.26), respectively, with 𝑢𝑛(d𝑢) being a finite 𝛾 measure on (0, ∞). Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup defined by (3.3) and (3.29). As a consequence of Theorem 9.30, we have the following: Theorem 9.40 Suppose that {(𝑦(𝑡), 𝒢𝑡 ) : 𝑡 ≥ 0} is a positive càdlàg process such that P[𝑦(0)] < ∞. Then {(𝑦(𝑡), 𝒢𝑡 ) : 𝑡 ≥ 0} is a CBI-process with transition 𝛾 semigroup (𝑄 𝑡 )𝑡 ≥0 if and only if for every 𝑓 ∈ 𝐶 2 (R+ ) we have ∫ 𝑡 𝑓 (𝑦(𝑡)) = 𝑓 (𝑦(0)) + 𝐿 𝑓 (𝑦(𝑠))d𝑠 + local mart., (9.46) 0

where ′′

∫

∞

 𝑓 (𝑥 + 𝑧) − 𝑓 (𝑥) − 𝑧 𝑓 ′ (𝑥) 𝑚(d𝑧) 0 ∫ ∞   ′ + (𝛽 − 𝑏𝑥) 𝑓 (𝑥) + 𝑓 (𝑥 + 𝑧) − 𝑓 (𝑥) 𝑛(d𝑧).

𝐿 𝑓 (𝑥) = 𝑐𝑥 𝑓 (𝑥) + 𝑥



0

(9.47)

9.5 Constructions of the Trajectories

261

By Corollary 9.31, if {(𝑦(𝑡), 𝒢𝑡 ) : 𝑡 ≥ 0} is a CBI-process with transition 𝛾 semigroup (𝑄 𝑡 )𝑡 ≥0 , then the local martingale in (9.46) is actually a martingale.

9.5 Constructions of the Trajectories Suppose that 𝐸 is a Lusin topological space. Let (𝑉𝑡 )𝑡 ≥0 and (𝑄 𝑡 )𝑡 ≥0 denote respectively the cumulant and transition semigroups of a general Borel right MB-process in 𝑀 (𝐸). Recall that the cumulant semigroup (𝑉𝑡 )𝑡 ≥0 has canonical representation (2.5). Suppose that 𝑡 ↦→ 𝑉𝑡 𝑓 (𝑥) is locally bounded. Let 𝑊ˆ be the space of paths 𝑤 : [0, ∞) → 𝑀 (𝐸) such that 𝑤 𝑡 takes values in 𝑀 (𝐸) ◦ and is right continuous in some interval (𝛼(𝑤), 𝜁 (𝑤)) or [𝛼(𝑤), 𝜁 (𝑤)) ⊂ [0, ∞) and takes the value 0 ∈ 𝑀 (𝐸) ˆ We equip this space with elsewhere. Let 𝑤 𝑠 = {𝑤 𝑡∧𝑠 : 𝑡 ≥ 0} for 𝑠 ≥ 0 and 𝑤 ∈ 𝑊. 0 the natural 𝜎-algebras 𝒜 = 𝜎({𝑤(𝑠) : 𝑠 ≥ 0}) and 𝒜𝑡0 = 𝜎({𝑤(𝑠) : 0 ≤ 𝑠 ≤ 𝑡}) for 𝑡 ≥ 0. Let (𝑁𝑡 )𝑡 ≥0 be an SC-semigroup defined by (9.7) with 𝐼𝑡 = − log 𝐿 𝐾𝑡 for an infinitely divisible probability entrance law (𝐾𝑡 )𝑡 >0 , which is defined by (8.2) in terms of {(𝜂𝑡 , 𝐻𝑡 ) : 𝑡 > 0}. Let Q(𝐻, ·) be the Kuznetsov measure on 𝑊ˆ defined by (8.46) corresponding to the entrance rule 𝐻 = (𝐻𝑡 )𝑡 >0 . Suppose that 𝑁 (d𝑠, d𝑤) is a Poisson random measure on (0, ∞) × 𝑊ˆ with intensity d𝑠Q(𝐻, d𝑤). For 𝑡 ≥ 0 let ∫ 𝑡∫ ∫ 𝑡 𝑤 𝑡−𝑠 𝑁 (d𝑠, d𝑤) 𝜂𝑡−𝑠 d𝑠 + (9.48) 𝑌𝑡 = 0

0

ˆ 𝑊

and let 𝒢𝑡 be the 𝜎-algebra generated by random variables of the form ∫ ∞∫ ℎ𝑡 (𝑠, 𝑤)𝑁 (d𝑠, d𝑤), 0

(9.49)

ˆ 𝑊

where ℎ𝑡 (𝑠, 𝑤) = ℎ(𝑠, 𝑤 𝑡−𝑠 )1 {𝑠 ≤𝑡 , 𝛼(𝑤) ≤𝑡−𝑠 } for some ℎ ∈ p(ℬ(0, ∞) × 𝒜 0 ). Theorem 9.41 The pair {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} defined above is an immigration process with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 defined by (9.8). Proof For 𝑡 ≥ 𝑟 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + write 𝐹𝑟 ,𝑡 (𝑠, 𝑤) = ℎ𝑟 (𝑠, 𝑤) + 𝑤 𝑡−𝑠 ( 𝑓 ). Then we have  ∫ ∞∫  P exp − ℎ𝑟 (𝑠, 𝑤)𝑁 (d𝑠, d𝑤) − 𝑌𝑡 ( 𝑓 ) ˆ 0 𝑊  ∫ 𝑡  ∫ 𝑡∫ 𝐹𝑟 ,𝑡 (𝑠, 𝑤)𝑁 (d𝑠, d𝑤) = P exp − 𝜂𝑡−𝑠 ( 𝑓 )d𝑠 − ˆ 0 𝑊 0  ∫ 𝑡  ∫ 𝑡
= exp − Q 𝐻, 1 − e−𝐹𝑟 ,𝑡 (𝑠,𝑤) d𝑠 𝜂𝑡−𝑠 ( 𝑓 )d𝑠 − 0

0

262

9 Structures of Independent Immigration



∫

𝑡

∫

𝑟 −𝐹𝑟 ,𝑡 (𝑠,𝑤)




= exp − d𝑠 𝜂𝑡−𝑠 ( 𝑓 )d𝑠 − Q 𝐻, 1 − e 0 0   ∫ 𝑡
· exp − Q 𝐻, 1 − e−𝑤𝑡−𝑠 ( 𝑓 ) d𝑠 𝑟   ∫ 𝑡 ∫ 𝑟 −ℎ𝑟 (𝑠,𝑤)
= exp − 𝜂𝑡−𝑠 ( 𝑓 )d𝑠 − Q 𝐻, 1 − e d𝑠 0 0   ∫ 𝑟
Q 𝐻, e−ℎ𝑟 (𝑠,𝑤) [1 − e−𝑤𝑡−𝑠 ( 𝑓 ) ] d𝑠 · exp − 0  ∫ 𝑡  −𝑤𝑡−𝑠 ( 𝑓 )
Q 𝐻, 1 − e d𝑠 . · exp − 𝑟

For 0 < 𝑠 ≤ 𝑟 it is clear that ℎ𝑟 (𝑠, [0]) = 0 and 𝑤 ↦→ ℎ𝑟 (𝑠, 𝑤) is measurable relative 0 . By Theorem 8.24, to 𝒜𝑟−𝑠   ∫ ∞∫ P exp − ℎ𝑟 (𝑠, 𝑤)𝑁 (d𝑠, d𝑤) − 𝑌𝑡 ( 𝑓 ) 0  ∫ 𝑟 ∫𝑊ˆ 𝑡
= exp − 𝜂𝑡−𝑠 ( 𝑓 )d𝑠 − Q 𝐻, 1 − e−ℎ𝑟 (𝑠,𝑤) d𝑠 0   ∫0 𝑟
−ℎ𝑟 (𝑠,𝑤) [1 − e−𝑤𝑟−𝑠 (𝑉𝑡−𝑟 𝑓 ) ] d𝑠 Q 𝐻, e · exp −   ∫0 𝑟 · exp − [𝜂𝑟−𝑠 (𝑉𝑡−𝑟 𝑓 ) − 𝜂𝑡−𝑠 ( 𝑓 )]d𝑠  ∫0 𝑡  −𝑤𝑡−𝑠 ( 𝑓 )
d𝑠 · exp − Q 𝐻, 1 − e   ∫ 𝑟𝑟
= exp − Q 𝐻, 1 − exp{−𝐺 𝑟 ,𝑡 (𝑠, 𝑤)} d𝑠   ∫0 𝑟 ∫ 𝑡 𝜂𝑟−𝑠 (𝑉𝑡−𝑟 𝑓 )d𝑠 − · exp − 𝜂𝑡−𝑠 ( 𝑓 )d𝑠 𝑟   ∫0 𝑡 ∫ −𝜈 ( 𝑓 ) (1 − e )𝐻𝑡−𝑠 (d𝜈) · exp − d𝑠 ◦   ∫𝑟 𝑟 ∫ 𝑀 (𝐸) 𝐺 𝑟 ,𝑡 (𝑠, 𝑤)𝑁 (d𝑠, d𝑤) = P exp −  ∫ 0𝑟 𝑊ˆ  ∫ 𝑡 · exp − 𝜂𝑟−𝑠 (𝑉𝑡−𝑟 𝑓 )d𝑠 − 𝐼𝑡−𝑠 ( 𝑓 )d𝑠 𝑟   0∫ ∞ ∫  ℎ𝑟 (𝑠, 𝑤)𝑁 (d𝑠, d𝑤) = P exp − ˆ 𝑊   0 ∫ 𝑡 𝐼𝑡−𝑠 ( 𝑓 )d𝑠 , · exp − 𝑌𝑟 (𝑉𝑡−𝑟 𝑓 ) − 𝑟

where 𝐺 𝑟 ,𝑡 (𝑠, 𝑤) = ℎ𝑟 (𝑠, 𝑤) + 𝑤 𝑟−𝑠 (𝑉𝑡−𝑟 𝑓 ). Then {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is a Markov process with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 . □

9.5 Constructions of the Trajectories

263

Let 𝜂 ∈ 𝑀 (𝐸) and let L(𝜂, ·) be the Kuznetsov measure corresponding to the entrance rule 𝜂𝐿 := (𝜂𝐿 𝑡 )𝑡 >0 . Suppose that 𝑁 𝜂 (d𝑠, d𝑤) is a Poisson random measure on (0, ∞) × 𝑊ˆ with intensity d𝑠L(𝜂, d𝑤). For 𝑡 ≥ 0 let ∫ 𝑡∫ ∫ 𝑡 𝜂 𝜂𝜆 𝑡−𝑠 d𝑠 + 𝑤 𝑡−𝑠 𝑁 𝜂 (d𝑠, d𝑤) 𝑌𝑡 = (9.50) 0

0

ˆ 𝑊

𝜂

and let 𝒢𝑡 be the 𝜎-algebra generated by the random variables in (9.49) with 𝑁 (d𝑠, d𝑤) replaced by 𝑁 𝜂 (d𝑠, d𝑤). 𝜂

𝜂

Corollary 9.42 The pair {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is an immigration process with transi𝜂 tion semigroup (𝑄 𝑡 )𝑡 ≥0 defined by the special form of (9.13) with 𝐼 ( 𝑓 ) = 𝜂( 𝑓 ) for 𝑓 ∈ 𝐵(𝐸) + . Suppose that [1 ∧ 𝜈(1)]𝐻 (d𝜈) is a finite measure on 𝑀 (𝐸) ◦ . Let Q(𝐻, ·) be the Kuznetsov measure corresponding to the entrance law (𝐻𝑄 ◦𝑡 )𝑡 >0 . Suppose that 𝑁 𝐻 (d𝑠, d𝑤) is a Poisson random measure on (0, ∞) × 𝑊ˆ with intensity d𝑠Q(𝐻, d𝑤). For 𝑡 ≥ 0 let ∫ 𝑡∫ 𝑌𝑡𝐻 = 𝑤 𝑡−𝑠 𝑁 𝐻 (d𝑠, d𝑤) (9.51) 0

ˆ 𝑊

and let 𝒢𝑡𝐻 be the 𝜎-algebra generated by the random variables in (9.49) with 𝑁 (d𝑠, d𝑤) replaced by 𝑁 𝐻 (d𝑠, d𝑤). Corollary 9.43 The pair {(𝑌𝑡𝐻 , 𝒢𝑡𝐻 ) : 𝑡 ≥ 0} is an immigration process with transition semigroup (𝑄 𝑡𝐻 )𝑡 ≥0 defined by the special form of (9.13) with ∫
1 − e−𝜈 ( 𝑓 ) 𝐻 (d𝜈), 𝑓 ∈ 𝐵(𝐸) + . 𝐼( 𝑓 ) = (9.52) 𝑀 (𝐸) ◦

Corollary 9.44 Suppose that the two Poisson random measures in Corollaries 9.42 and 9.43 are defined on the same complete probability space and are independent 𝜂 𝜂 of each other. Let 𝑌𝑡 = 𝑌𝑡 + 𝑌𝑡𝐻 and 𝒢𝑡 = 𝜎(𝒢𝑡 ∪ 𝒢𝑡𝐻 ). Then {(𝑌𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} is an immigration process with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 defined by (9.12) and (9.13). From the constructions of the immigration processes given above it is clear that, except the deterministic parts, both the entry times and the evolutions of the immigrants are determined by the Poisson random measures based on the Kuznetsov measures. In the situation of the above corollaries, it is natural to expect a better behavior of the immigration process. We next discuss the constructions of immigration superprocesses. Suppose that 𝜉 is a Borel right process in 𝐸 and 𝜙 is a branching mechanism given by (2.29) or (2.30). Let 𝑋 = (𝑊, 𝒢, 𝒢𝑡 , 𝑋𝑡 , Q 𝜇 ) be a canonical realization of the (𝜉, 𝜙)-superprocess as a Borel right process, where 𝑊 is the space of right continuous paths from [0, ∞) into 𝑀 (𝐸). Suppose that [1 ∧ 𝜈(1)]𝐻 (d𝜈) is a finite measure on 𝑀 (𝐸) ◦ and let Q 𝐻 be the 𝜎-finite measure on 𝑊 defined by

264

9 Structures of Independent Immigration

∫ 𝐻 (d𝜈)Q𝜈 (d𝑤),

Q 𝐻 (d𝑤) =

𝑤 ∈ 𝑊.

(9.53)

𝑀 (𝐸) ◦

Since the (𝜉, 𝜙)-superprocess has the null measure as a trap, we may think of Q 𝐻 as a measure on 𝑊ˆ carried by 𝑊ˆ 0 = {𝑤 ∈ 𝑊ˆ : 𝛼(𝑤) = 0 and 𝑤 0 ∈ 𝑀 (𝐸) ◦ }. Then it is just the Kuznetsov measure corresponding to the closed entrance law (𝐻𝑄 ◦𝑡 )𝑡 ≥0 . Suppose that 𝑁 𝐻 (d𝑠, d𝑤) is a Poisson random measure on (0, ∞) × 𝑊ˆ with intensity d𝑠Q 𝐻 (d𝑤). For 𝑡 ≥ 0 let ∫ 𝑡∫ (9.54) 𝑤 𝑡−𝑠 𝑁 𝐻 (d𝑠, d𝑤) 𝑌𝐻 (𝑡) = 0

ˆ 𝑊

and let 𝒢𝐻 (𝑡) be the 𝜎-algebra generated by the random variables in (9.49) with 𝑁 (d𝑠, d𝑤) replaced by 𝑁 𝐻 (d𝑠, d𝑤). Let (𝒢¯ 𝐻 (𝑡))𝑡 ≥0 be the augmentation of the filtration (𝒢𝐻 (𝑡))𝑡 ≥0 . Theorem 9.45 The process {𝑌𝐻 (𝑡) : 𝑡 ≥ 0} defined above is an a.s. right continuous realization of the immigration superprocess in 𝑀 (𝐸) relative to (𝒢¯ 𝐻 (𝑡+))𝑡 ≥0 with transition semigroup (𝑄 𝑡𝐻 )𝑡 ≥0 defined by the special form of (9.13) with the immigration mechanism 𝐼 given by (9.52). Moreover, the process  ∫ 𝑠 ↦→ 1 − exp − ⟨𝑌𝐻 (𝑠), 𝑉𝑡−𝑠 𝑓 ⟩ −

∫

𝑡−𝑠

(1 − e−⟨𝜈,𝑉𝑟 𝑓 ⟩ ) 𝐻 (d𝜈)

d𝑟

 (9.55)

𝑀 (𝐸) ◦

0

is an a.s. right continuous (𝒢¯ 𝐻 (𝑠+))-martingale on [0, 𝑡]. ˆ Then {𝑤 + (𝑡) : 𝑡 ≥ 0} Proof Step 1. Let 𝑤 + (𝑡) = 𝑤 𝑡 1 {𝑡 >0} for 𝑡 ≥ 0 and 𝑤 ∈ 𝑊. under Q 𝐻 is distributed according to the Kuznetsov measure Q(𝐻, ·) corresponding to the entrance law (𝐻𝑄 ◦𝑡 )𝑡 >0 . Clearly, we have a.s. ∫ ∫ ∫ 𝑡∫ 𝑤 𝑡−𝑠 𝑁 𝐻 (d𝑠, d𝑤) = 𝑌𝐻 (𝑡) = 𝑤 + (𝑡 − 𝑠)𝑁 𝐻 (d𝑠, d𝑤). (0,𝑡)

ˆ 𝑊

0

ˆ 𝑊

Then {𝑌𝐻 (𝑡) : 𝑡 ≥ 0} is a modification of the process {𝑌𝑡𝐻 : 𝑡 ≥ 0} constructed by (9.51). From Corollary 9.43 we infer that {(𝑌𝐻 (𝑡), 𝒢¯ 𝐻 (𝑡)) : 𝑡 ≥ 0} is an immigration superprocess with transition semigroup (𝑄 𝑡𝐻 )𝑡 ≥0 defined by (9.13) with the immigration mechanism 𝐼 given by (9.52). Step 2. For 𝑘 ≥ 1 define {𝑌𝑘 (𝑡) : 𝑡 ≥ 0} by the right-hand side of (9.54) with 𝑁 𝐻 (d𝑠, d𝑤) replaced by 𝑁 𝑘 (d𝑠, d𝑤) := 1 {𝑤0 (𝐸) >1/𝑘 } 𝑁 𝐻 (d𝑠, d𝑤). Let (𝑄 𝑘 (𝑡))𝑡 ≥0 be the transition semigroup defined by the special form of (9.13) where 𝐼 is given by (9.52) with 𝑀 (𝐸) ◦ replaced by 𝑀𝑘 (𝐸) := {𝜈 ∈ 𝑀 (𝐸) ◦ : 𝜈(1) > 1/𝑘 }. Then the result of the first step implies that {(𝑌𝑘 (𝑡), 𝒢¯ 𝐻 (𝑡)) : 𝑡 ≥ 0} is an immigration superprocess with transition semigroup (𝑄 𝑘 (𝑡))𝑡 ≥0 . It is easy to see that 𝐻 (𝑀𝑘 (𝐸)) < ∞. ˆ < ∞ a.s. for every 𝑡 ≥ 0, and hence {𝑌𝑘 (𝑡) : 𝑡 ≥ 0} is a.s. Then 𝑁 𝑘 ((0, 𝑡] × 𝑊) right continuous in 𝑀 (𝐸). Since 𝑌𝑘 (𝑡) → 𝑌𝐻 (𝑡) increasingly, we conclude that 𝑡 ↦→ ⟨𝑌𝐻 (𝑡), 𝑓 ⟩ is a.s. right lower semi-continuous for every 𝑓 ∈ 𝐶 (𝐸) + .

9.5 Constructions of the Trajectories

265

Step 3. Consider the special case where 𝜈(1)𝐻 (d𝜈) is a finite measure on 𝑀 (𝐸) ◦ . Recall that 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)] and 𝑐+0 = 0 ∨ 𝑐 0 . By Corollary 9.18, for any 𝛽 > 𝑐+0 the a.s. right continuous positive process ∫ −1 −2𝛽𝑡 −2𝛽𝑡 ⟨𝜈, 1⟩𝐻 (d𝜈) 𝑍 𝑘 (𝑡) := e ⟨𝑌𝑘 (𝑡), 1⟩ + (2𝛽) e 𝑀𝑘 (𝐸)

is a (𝒢¯ 𝐻 (𝑡))-supermartingale. Then it is also a (𝒢¯ 𝐻 (𝑡+))-supermartingale by Dellacherie and Meyer (1982, p. 69). Note that 𝑍 𝑘 (𝑡) increases as 𝑘 → ∞ to ∫ −1 −2𝛽𝑡 −2𝛽𝑡 ⟨𝜈, 1⟩𝐻 (d𝜈). ⟨𝑌𝐻 (𝑡), 1⟩ + (2𝛽) e 𝑍 (𝑡) := e 𝑀 (𝐸) ◦

Thus {𝑍 (𝑡) : 𝑡 ≥ 0} is an a.s. right continuous positive (𝒢¯ 𝐻 (𝑡+))-supermartingale; see Dellacherie and Meyer (1982, p. 79). In particular, we conclude that 𝑡 ↦→ ⟨𝑌𝐻 (𝑡), 1⟩ is a.s. right continuous. For any 𝑓 ∈ 𝐶 (𝐸) + , choose a constant 𝑞 ≥ ∥ 𝑓 ∥. The arguments above imply that 𝑡 ↦→ ⟨𝑌𝐻 (𝑡), 𝑞⟩ = 𝑞⟨𝑌𝐻 (𝑡), 1⟩ is a.s. right continuous. As we saw in the second step, both 𝑡 ↦→ ⟨𝑌𝐻 (𝑡), 𝑓 ⟩ and 𝑡 ↦→ ⟨𝑌𝐻 (𝑡), 𝑞 − 𝑓 ⟩ are a.s. right lower semi-continuous. Those clearly yield the a.s. right continuity of 𝑡 ↦→ ⟨𝑌𝐻 (𝑡), 𝑓 ⟩. Then 𝑡 ↦→ 𝑌𝐻 (𝑡) is a.s. right continuous in 𝑀 (𝐸). Step 4. Consider the general case where [1 ∧ 𝜈(1)]𝐻 (d𝜈) is a finite measure on 𝑀 (𝐸) ◦ . Let 𝑁0 (d𝑠, d𝑤) = 1 𝑀0 (𝐸) (𝑤 0 )𝑁 𝐻 (d𝑠, d𝑤), where 𝑀0 (𝐸) = {𝜇 ∈ 𝑀 (𝐸) ◦ : 𝜇(1) ≤ 1}. Observe that 𝑌𝐻 (𝑡) = 𝑌0 (𝑡) + 𝑌1 (𝑡), where ∫ 𝑡∫ 𝑌0 (𝑡) = 𝑤 0𝑡−𝑠 𝑁0 (d𝑠, d𝑤). 0

ˆ 𝑊

Then 𝑡 ↦→ 𝑌0 (𝑡) is a.s. right continuous by the result proved in the third step. Since 𝑡 ↦→ 𝑌1 (𝑡) is a.s. right continuous by the second step, we see that 𝑡 ↦→ 𝑌𝐻 (𝑡) is a.s. right continuous in 𝑀 (𝐸). Step 5. Let 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + . Since the (𝜉, 𝜙)-superprocess is a Borel right process, by (9.53) and Theorem A.16 one can see that 𝑠 ↦→ exp{−⟨𝑤 𝑠 , 𝑉𝑡−𝑠 𝑓 ⟩} is ˆ Then right continuous on [0, 𝑡] for Q 𝐻 -a.e. 𝑤 ∈ 𝑊.  ∫ 𝑠 ↦→ 1 − exp − ⟨𝑌𝑘 (𝑠) , 𝑉𝑡−𝑠 𝑓 ⟩ − 0

∫

𝑡−𝑠

d𝑟

(1 − e−⟨𝜈,𝑉𝑟 𝑓 ⟩ ) 𝐻 (d𝜈)



𝑀𝑘 (𝐸)

is an a.s. right continuous (𝒢¯ 𝐻 (𝑠+))-martingale on [0, 𝑡]. By taking the increasing limit, we see as in the third step that (9.55) is an a.s. right continuous (𝒢¯ 𝐻 (𝑠+))martingale on [0, 𝑡]. This gives the desired Markov property of {(𝑌𝐻 (𝑡), 𝒢¯ 𝐻 (𝑡+)) : 𝑡 ≥ 0}. □ The next theorem improves the results of Theorem 9.21 by a weaker moment assumption on the immigration mechanism. Theorem 9.46 The immigration superprocess with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 defined by (9.12) and (9.13) has a right realization in 𝑀 (𝐸). If 𝜉 is a Hunt process, then the immigration superprocess has a Hunt realization in 𝑀 (𝐸).

266

9 Structures of Independent Immigration

Proof By Theorem 9.21, the results hold if 𝜈(1)𝐻 (d𝜈) is a finite measure on 𝑀 (𝐸) ◦ . In the general case, we first consider the immigration mechanism 𝐼0 ∈ ℐ(𝐸) given by, for 𝑓 ∈ 𝐵(𝐸) + , ∫
𝐼0 ( 𝑓 ) = 𝜂( 𝑓 ) + 1 − e−𝜈 ( 𝑓 ) 𝐻0 (d𝜈), 𝑀 (𝐸) ◦

where 𝐻0 (d𝜈) = 1 {𝜈 (1) ≤1} 𝐻 (d𝜈). By Theorem 9.21, the above immigration mechanism generates a Borel right immigration superprocess. Let 𝑌 𝜂,𝐻0 = 𝜂,𝐻 (𝑊, 𝒢, 𝒢𝑡 , 𝑌𝑡 , Q 𝜇 0 ) be the canonical right realization of the immigration superprocess, where 𝑊 is the space of right continuous paths from [0, ∞) into 𝑀 (𝐸) and (𝒢, 𝒢𝑡 ) are the augmentations of the natural 𝜎-algebras (𝒢0 , 𝒢𝑡0 ) by the set of 𝜂,𝐻 probability measures {Q𝐾 0 : 𝐾 is an initial law on 𝑀 (𝐸)}. By Theorem A.16, for any 𝑡 ≥ 0, 𝑓 ∈ 𝐵(𝐸) + and initial law 𝐾, the process   ∫ 𝑡−𝑠 𝑠 ↦→ 1 − exp − ⟨𝑌𝑠 , 𝑉𝑡−𝑠 𝑓 ⟩ − 𝐼0 (𝑉𝑟 𝑓 )d𝑟 0 𝜂,𝐻

is a Q𝐾 0 -a.s. right continuous (𝒢𝑠 )-martingale on [0, 𝑡]. In the sequel, we may assume 𝐻 ({𝜈 ∈ 𝑀 (𝐸) ◦ : 𝜈(1) > 1}) > 0, for otherwise the proof is over. Let 𝐻1 (d𝜈) = 1 {𝜈 (1) >1} 𝐻 (d𝜈) and let {𝑌1 (𝑡) : 𝑡 ≥ 0} be the a.s. right continuous immigration superprocess defined as in the proof of Theorem 9.45. Then  ∫ 𝑠 ↦→ 1 − exp − ⟨𝑌1 (𝑠) , 𝑉𝑡−𝑠 𝑓 ⟩ − 0

∫

𝑡−𝑠

d𝑟 𝑀 (𝐸) ◦

(1 − e−⟨𝜈,𝑉𝑟 𝑓 ⟩ ) 𝐻1 (d𝜈)



is an a.s. right continuous martingale on [0, 𝑡]. Let Q 𝐻1 denote the distribution of 𝜂,𝐻 𝑁 {𝑌1 (𝑡) : 𝑡 ≥ 0} on 𝑊 and let Q𝐾 = Q𝐾 0 ∗ Q 𝐻1 , where “∗” means convolution. Then 𝑌 0 = (𝑊, 𝒢0 , 𝒢𝑡0 , 𝑌𝑡 , Q 𝜇𝑁 ) is a right continuous immigration superprocess with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 by Theorem 9.2. Moreover, it is not hard to see that, for any 𝑡 ≥ 0, 𝑓 ∈ 𝐵(𝐸) + and initial law 𝐾, the process   ∫ 𝑡−𝑠 𝐼 (𝑉𝑟 𝑓 )d𝑟 𝑠 ↦→ 1 − exp − ⟨𝑌𝑠 , 𝑉𝑡−𝑠 𝑓 ⟩ − 0 𝑁 ¯ 𝒢¯ 𝑡 ) be the augmen-a.s. right continuous (𝒢𝑠0 )-martingale on [0, 𝑡]. Let (𝒢, is a Q𝐾 𝑁 0 0 tations of (𝒢 , 𝒢𝑡 ) by the set of probability measures {Q𝐾 : 𝐾 is an initial law on 𝑀 (𝐸)}. By a monotone class argument we see that, for any 𝐹 ∈ 𝐵(𝑀 (𝐸)), the pro𝑁 𝑁 𝐹 (𝑌𝑠 ) is a Q𝐾 -a.s. right continuous (𝒢¯ 𝑠 )-martingale on [0, 𝑡]. Then cess 𝑠 ↦→ 𝑄 𝑡−𝑠 𝑁 𝑁 ¯ ¯ 𝑌 = (𝑊, 𝒢, 𝒢𝑡 , 𝑌𝑡 , Q 𝜇 ) is a right process by Theorem A.16. From the construction given above, it is clear that 𝑌 𝑁 is a countable concatenation of the subprocess of 𝑌 𝜂,𝐻0 generated by the multiplicative functional 𝑡 ↦→ exp{−𝑡𝐻1 (𝑀 (𝐸) ◦ )} together with the transfer kernel 𝐽 (𝜇, d𝜈) on 𝑀 (𝐸) defined by, for 𝐹 ∈ 𝐵(𝑀 (𝐸)), ∫ ∫ 𝐹 (𝜇 + 𝜈)𝐻1 (d𝜈). 𝐹 (𝜈)𝐽 (𝜇, d𝜈) = 𝐻1 (𝑀 (𝐸) ◦ ) −1 𝑀 (𝐸)

𝑀 (𝐸) ◦

9.5 Constructions of the Trajectories

267

If 𝜉 is a Hunt process, then 𝑌 𝜂,𝐻0 has a Hunt realization by Theorem 9.45, and so □ 𝑌 𝑁 has a Hunt realization by Theorem A.44. In the following examples, we consider the situation where the underlying process 𝜉 is the absorbing-barrier Brownian motion in 𝐸 0 := (0, ∞) with transition semigroup defined by (A.29) and (A.30). Example 9.2 Let 𝜂 ∈ 𝑀 (𝐸 0 ) and let 𝛼 ≥ 0 be a constant. We can use the notation 𝜂, 𝛼 of Section 8.6 to define the transition semigroup (𝑄 𝑡 )𝑡 ≥0 of an immigration superprocess in 𝑀 (𝐸 0 ) by   ∫ ∫ 𝑡   𝜂, 𝛼 𝜂(𝑉𝑠 𝑓 ) + 𝛼𝜕0𝑉𝑠 𝑓 d𝑠 . e−𝜈 ( 𝑓 ) 𝑄 𝑡 (𝜇, d𝜈) = exp − 𝜇(𝑉𝑡 𝑓 ) − 𝑀 (𝐸0 )

0

Example 9.3 Suppose that 𝑢𝐹 (d𝑢) is a non-trivial finite measure on 𝐸 0 . We define another transition semigroup (𝑄 𝑡𝐹 )𝑡 ≥0 on 𝑀 (𝐸 0 ) by   ∫ 𝑡 ∫ e−𝜈 ( 𝑓 ) 𝑄 𝑡𝐹 (𝜇, d𝜈) = exp − 𝜇(𝑉𝑡 𝑓 ) − 𝐼𝑠 ( 𝑓 )d𝑠 , 𝑀 (𝐸0 )

0

where ∫ 𝐼𝑠 ( 𝑓 ) =

∞

(1 − e−𝑢𝜕0 𝑉𝑠 𝑓 )𝐹 (d𝑢).

0

The corresponding immigration superprocess can be constructed in the following way. From (8.78) one can see there is an entrance law 𝐻 ∈ 𝒦(𝑄 ◦ ) such that ∫ (1 − e−𝜈 ( 𝑓 ) )𝐻𝑠 (d𝜈) = 𝐼𝑠 ( 𝑓 ), 𝑠 > 0, 𝑓 ∈ 𝐵(𝐸 0 ) + . 𝑀 (𝐸0 ) ◦

Let 𝑊ˆ 0 be as in Example 8.4 and let Q𝐹 (d𝑤) be the Kuznetsov measure on 𝑊ˆ 0 corresponding to 𝐻 ∈ 𝒦(𝑄 ◦ ). Suppose that 𝑁 𝐹 (d𝑠, d𝑤) is a Poisson random measure on (0, ∞) × 𝑊ˆ 0 with intensity d𝑠Q𝐹 (d𝑤). By Theorem 9.41, we can define an immigration superprocess in 𝑀 (𝐸 0 ) with transition semigroup (𝑄 𝑡𝐹 )𝑡 ≥0 by ∫ 𝑡∫ 𝑡 ≥ 0. 𝑌𝑡 = (9.56) 𝑤 𝑡−𝑠 𝑁 𝐹 (d𝑠, d𝑤), 0

ˆ0 𝑊

Let {(𝑠𝑖 , 𝑤 𝑖 ) : 𝑖 = 1, 2, . . .} be an enumeration of the atoms of 𝑁 𝐹 (d𝑠, d𝑤). It is easy to see that ∫ ∞ Q𝑢 (d𝑤)𝐹 (d𝑢), 𝑤 ∈ 𝑊ˆ 0 , Q𝐹 (d𝑤) = 0

Q𝑢 (d𝑤)

where is as in Example 8.4. Let ℎ be defined by (8.69) and assume (8.79) holds. Then by (8.80) and (9.56), for every 𝜀 > 0 we have a.s. lim 𝑌𝑡 ((0, 𝜀]) ≥ lim 𝑤 𝑖,𝑡−𝑠𝑖 ((0, 𝜀]) = ∞. 𝑡→𝑠𝑖

𝑡→𝑠𝑖

(9.57)

268

9 Structures of Independent Immigration

If 𝐹 (d𝑢) is an infinite measure on (0, ∞), then {𝑠𝑖 : 𝑖 = 1, 2, . . . } ∩ (𝑟, 𝑡) is a.s. infinite for every 𝑡 > 𝑟 ≥ 0. Example 9.4 Let ℎ be defined by (8.69) and assume (8.79) holds. The immigration superprocess constructed in the last example is certainly not right continuous. Indeed, the transition semigroup (𝑄 𝑡𝐹 )𝑡 ≥0 has no right continuous realization. Otherwise, suppose that {𝑍𝑡 : 𝑡 ≥ 0} is such a realization with 𝑍0 = 0. Given 𝜇 ∈ 𝑀 (𝐸 0 ) we define 𝜇 ℎ ∈ 𝑀 (𝐸 0 ) by 𝜇 ℎ (d𝑥) = ℎ(𝑥)𝜇(d𝑥) for 𝑥 ∈ 𝐸 0 . In an obvious way, we also regard 𝜇 ℎ as a measure in 𝑀 (R+ ). Then {𝑍𝑡ℎ : 𝑡 ≥ 0} is an a.s. right continuous immigration superprocess in 𝑀 (R+ ) with semigroup ( 𝑄¯ 𝑡𝐹 )𝑡 ≥0 given by   ∫ ∫ 𝑡 ¯ 𝐼¯𝑠 ( 𝑓¯)d𝑠 , e−𝜈 ( 𝑓 ) 𝑄¯ 𝑡𝐹 (𝜇, d𝜈) = exp − 𝜇(𝑈¯ 𝑡 𝑓¯) − 𝑀 (R+ )

0

where (𝑈¯ 𝑡 )𝑡 ≥0 is defined by (8.81) and ∫ ∞ ¯ ¯ ¯𝐼𝑠 ( 𝑓¯) = (1 − e−𝑢𝜕0 ℎ𝑈𝑠 𝑓 (0) )𝐹 (d𝑢),

𝑓¯ ∈ 𝐵(R+ ) + .

0

It is easily seen that P{𝑍𝑡ℎ ({0}) = 0 for all 𝑡 ≥ 0} = 1. For any 𝑤 ∈ 𝑊ˆ 0 define  lim𝑡→0 𝑤 𝑡ℎ 𝑤 0ℎ = 0

(9.58)

if the limit exists in 𝑀 (R+ ), if the limit above does not exist.

By the discussions in Example 8.4 one can see {𝑤 𝑡ℎ : 𝑡 ≥ 0} under Q𝐹 is a superprocess in 𝑀 (R+ ) with cumulant semigroup (𝑈¯ 𝑡 )𝑡 ≥0 and 𝑤 0ℎ ({0}) > 0 for Q𝐹 -a.e. 𝑤 ∈ 𝑊ˆ 0 . Then by Theorem 9.45, ∫ 𝑡∫ ℎ ¯ 𝑡 ≥ 0, 𝑌𝑡 = 𝑤 𝑡−𝑠 𝑁 𝐹 (d𝑠, d𝑤), (9.59) 0

ˆ0 𝑊

defines another a.s. right continuous realization of (𝑄¯ 𝑡𝐹 )𝑡 ≥0 with 𝑌¯0 = 0. Let 𝑆0 = inf{𝑡 ≥ 0 : 𝑌¯𝑡 ({0}) > 0}. In view of (9.59) we have P{𝑆0 ≤ 𝑎} = 1 − e−𝑎𝐹 (𝐸0 ) ,

𝑢 > 0.

(9.60)

However, an application of Theorem 9.21 shows that (𝑄¯ 𝑡𝐹 )𝑡 ≥0 is a Borel right semigroup on 𝑀 (R+ ), so (9.58) and (9.60) are in contradiction.

9.6 Stationary Distributions and Ergodicities

269

9.6 Stationary Distributions and Ergodicities We first discuss some basic structures of the stationary distributions. Given two probability measures 𝐹1 and 𝐹2 on 𝑀 (𝐸), we write 𝐹1 ⪯ 𝐹2 if 𝐹1 ∗ 𝐺 = 𝐹2 for another probability measure 𝐺 on 𝑀 (𝐸). Clearly, the measure 𝐺 is unique if it exists. Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup of an MB-process defined by (2.3), where the cumulant semigroup (𝑉𝑡 )𝑡 ≥0 is given by (2.5). Let ℰ ∗ (𝑄) denote the set of probabilities 𝐹 on 𝑀 (𝐸) satisfying 𝐹𝑄 𝑡 ⪯ 𝐹 for all 𝑡 ≥ 0. Theorem 9.47 For each 𝐹 ∈ ℰ ∗ (𝑄) there is a unique SC-semigroup (𝑁𝑡 )𝑡 ≥0 associated with (𝑄 𝑡 )𝑡 ≥0 such that 𝐹𝑄 𝑡 ∗ 𝑁𝑡 = 𝐹 for all 𝑡 ≥ 0. Proof Since 𝐹 ∈ ℰ ∗ (𝑄), for each 𝑡 ≥ 0 there is a unique probability measure 𝑁𝑡 on 𝑀 (𝐸) satisfying 𝐹 = (𝐹𝑄 𝑡 ) ∗ 𝑁𝑡 . By Theorem 2.1, for 𝑟, 𝑡 ≥ 0 we have (𝐹𝑄 𝑟+𝑡 ) ∗ 𝑁𝑟+𝑡 = 𝐹 = (𝐹𝑄 𝑡 ) ∗ 𝑁𝑡 = {[(𝐹𝑄 𝑟 ) ∗ 𝑁𝑟 ]𝑄 𝑡 } ∗ 𝑁𝑡 = (𝐹𝑄 𝑟+𝑡 ) ∗ (𝑁𝑟 𝑄 𝑡 ) ∗ 𝑁𝑡 . Then (9.1) holds, that is, (𝑁𝑡 )𝑡 ≥0 is an SC-semigroup associated with (𝑄 𝑡 )𝑡 ≥0 .

□

The infinitely divisible probabilities in ℰ ∗ (𝑄) are closely related with excessive measures for (𝑄 ◦𝑡 )𝑡 ≥0 . Let ℰ(𝑄 ◦ ) denote the class of all excessive measures 𝐻 for (𝑄 ◦𝑡 )𝑡 ≥0 satisfying ∫ [1 ∧ 𝜈(1)]𝐻 (d𝜈) < ∞. (9.61) 𝑀 (𝐸) ◦

Proposition 9.48 Let 𝐹 = 𝐼 (𝜂, 𝐻) be an infinitely divisible probability measure on 𝑀 (𝐸). Then 𝐹 ∈ ℰ ∗ (𝑄) if and only if (𝜂, 𝐻) satisfy ∫ ∫ 𝜂(d𝑥)𝐿 𝑡 (𝑥, ·) + 𝐻𝑄 ◦𝑡 ≤ 𝐻. 𝜂(d𝑥)𝜆 𝑡 (𝑥, ·) ≤ 𝜂 and (9.62) 𝐸

𝐸

In particular, if 𝐻 ∈ ℰ(𝑄 ◦ ), then 𝐹 = 𝐼 (0, 𝐻) ∈ ℰ ∗ (𝑄). Proof By Proposition 2.6 we have 𝐹𝑄 𝑡 = 𝐼 (𝜂𝑡 , 𝐻𝑡 ), where ∫ ∫ 𝜂(d𝑥)𝜆 𝑡 (𝑥, ·) and 𝐻𝑡 = 𝜂𝑡 = 𝜂(d𝑥)𝐿 𝑡 (𝑥, ·) + 𝐻𝑄 ◦𝑡 . 𝐸

𝐸

Then 𝐹𝑄 𝑡 ⪯ 𝐹 holds if and only if (9.62) is satisfied. The second assertion is □ immediate. We write 𝐹 ∈ ℰ𝑖∗ (𝑄) if 𝐹 ∈ ℰ ∗ (𝑄) is a stationary distribution for (𝑄 𝑡 )𝑡 ≥0 , and write 𝐹 ∈ ℰ𝑝∗ (𝑄) if 𝐹 ∈ ℰ ∗ (𝑄) and lim𝑡→∞ 𝐹𝑄 𝑡 = 𝛿0 . Clearly, we have 𝛿0 ∈ ℰ𝑖∗ (𝑄), but there can be other non-trivial stationary distributions. Theorem 9.49 Let 𝐹 ∈ ℰ ∗ (𝑄) and let (𝑁𝑡 )𝑡 ≥0 be the SC-semigroup defined in Theorem 9.47. Then 𝐹 = 𝐹𝑖 ∗ 𝐹 𝑝 , where 𝐹𝑖 = lim𝑡→∞ 𝐹𝑄 𝑡 ∈ ℰ𝑖∗ (𝑄) and 𝐹 𝑝 = lim𝑡→∞ 𝑁𝑡 ∈ ℰ𝑝∗ (𝑄).

270

9 Structures of Independent Immigration

Proof By the definition of ℰ ∗ (𝑄) we have 𝐹𝑄 𝑟+𝑡 ⪯ 𝐹𝑄 𝑡 for 𝑟, 𝑡 ≥ 0. Thus for every 𝑓 ∈ 𝐵(𝐸) + the limits 𝐿 𝐹𝑖 ( 𝑓 ) = ↑lim 𝐿 𝐹𝑄𝑡 ( 𝑓 ) and 𝐿 𝐹𝑝 ( 𝑓 ) = ↓lim 𝐿 𝑁𝑡 ( 𝑓 ) 𝑡→∞

𝑡→∞

exist and they are the Laplace functionals of two probability measures 𝐹𝑖 and 𝐹 𝑝 on 𝑀 (𝐸). Clearly, 𝐹𝑖 ∈ ℰ𝑖∗ (𝑄) and 𝐹 = 𝐹𝑖 ∗ 𝐹 𝑝 . On the other hand, 𝐹𝑖 ∗ 𝐹 𝑝 = 𝐹 = (𝐹𝑄 𝑡 ) ∗ 𝑁𝑡 = 𝐹𝑖 ∗ (𝐹 𝑝 𝑄 𝑡 ) ∗ 𝑁𝑡 , so 𝐹 𝑝 = (𝐹 𝑝 𝑄 𝑡 ) ∗ 𝑁𝑡 . Therefore 𝐹 𝑝 ∈ ℰ ∗ (𝑄) and lim𝑡→∞ 𝐹 𝑝 𝑄 𝑡 = 𝛿0 .

□

It is easy to see that the measure 𝐹 𝑝 ∈ ℰ𝑝∗ (𝑄) in Theorem 9.49 is a stationary distribution of the transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 defined from (𝑄 𝑡 )𝑡 ≥0 and (𝑁𝑡 )𝑡 ≥0 by (9.2). Then the above theorem shows that any 𝐹 ∈ ℰ ∗ (𝑄) can be decomposed as the convolution of a stationary distribution of (𝑄 𝑡 )𝑡 ≥0 with a stationary distribution of an associated immigration process. Example 9.5 Let (𝑣 ∗𝑡 )𝑡 ≥0 be the cumulant semigroup of a one-dimensional CBprocess with branching mechanism 𝜙∗ . Let 𝜉 be the Markov process in [0, 1] defined by 𝜉𝑡 = (𝜉0 − 𝑡) ∨ ⌊𝜉0 ⌋ for 𝑡 ≥ 0, where “⌊·⌋” denotes the integer part. Let 𝜙 be the local branching mechanism on [0, 1] defined by 𝜙(𝑥, 𝜆) = 1 (0,1] (𝑥)𝜙∗ (𝜆) for 𝑥 ∈ [0, 1] and 𝜆 ≥ 0. Suppose that 𝑋 = (𝑊, 𝒢, 𝒢𝑡 , 𝑋𝑡 , Q 𝜇 ) is a right realization of the (𝜉, 𝜙)-superprocess. Then Q𝑎 𝛿 𝑥 {𝑋𝑡 = 𝑋𝑡 (1)𝛿 ( 𝑥−𝑡)∨ ⌊𝑥 ⌋ for 𝑡 ≥ 0} = 1 for 𝑥 ∈ [0, 1] and 𝑎 ≥ 0. The cumulant semigroup (𝑉𝑡 )𝑡 ≥0 of this (𝜉, 𝜙)-superprocess is given by 𝑉𝑡 𝑓 (𝑥) = 𝑣 ∗𝑡∧𝑥 ( 𝑓 ((𝑥 − 𝑡) ∨ 0)) for 𝑥 ∈ [0, 1) and 𝑉𝑡 𝑓 (𝑥) = 𝑣 ∗𝑡 ( 𝑓 (1)) for 𝑥 = 1. Clearly, for any 𝜇 ∈ 𝑀 ( [0, 1)) ⊂ 𝑀 ( [0, 1]), the limit 𝑄 ∞ (𝜇, ·) := lim𝑡→∞ 𝑄 𝑡 (𝜇, ·) exists and, for 𝑓 ∈ 𝐵( [0, 1]) + ,  ∫  ∫ ∗ −𝜈 ( 𝑓 ) e 𝑄 ∞ (𝜇, d𝜈) = exp − 𝑣 𝑧 ( 𝑓 (0))𝜇(d𝑧) . 𝑀 ( [0,1])

[0,1)

It is easy to see that 𝑄 ∞ (𝜇, ·) ∈ ℰ𝑖∗ (𝑄) is carried by 𝑀 ({0}) ⊂ 𝑀 ( [0, 1]). If 𝑏 ∗ := 𝜙∗′ (0) > 0, for each 𝛽 > 0 we can define 𝑁 𝛽 ∈ ℰ𝑝∗ (𝑄) by ∫ 𝑀 ( [0,1])

 ∫ e−𝜈 ( 𝑓 ) 𝑁 𝛽 (d𝜈) = exp − 𝛽

∞

 𝑣 ∗𝑠 ( 𝑓 (1))d𝑠 .

0

Then both ℰ𝑖∗ (𝑄) and ℰ𝑝∗ (𝑄) contain non-trivial elements. We next discuss the ergodicity of the MBI-process. Suppose that (𝑁𝑡 )𝑡 ≥0 is an SCsemigroup defined by (9.7) with 𝐼𝑡 = − log 𝐿 𝐾𝑡 for an infinitely divisible probability entrance law (𝐾𝑡 )𝑡 >0 given by (8.2). Let (𝑄 𝑡𝑁 )𝑡 ≥0 be the corresponding transition semigroup defined by (9.8).

9.6 Stationary Distributions and Ergodicities

271

Theorem 9.50 There is a probability measure 𝑁∞ on 𝑀 (𝐸) such that 𝑁𝑡 → 𝑁∞ weakly as 𝑡 → ∞ if and only if ∫ ∞ 𝐼𝑠 (1)d𝑠 < ∞. (9.63) 0

In this case, we have 

∫

𝐿 𝑁∞ ( 𝑓 ) = exp −

∞

 𝐼𝑠 ( 𝑓 )d𝑠 ,

𝑓 ∈ 𝐵(𝐸) +

(9.64)

0

and 𝑁∞ is a stationary distribution for the transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 . Proof Suppose that (9.63) holds. By Jensen’s inequality, for 𝑎 ≥ 1 and 𝑓 ∈ 𝐵 𝑎 (𝐸) + we have 𝐼𝑠 ( 𝑓 ) ≤ 𝐼𝑠 (𝑎) ≤ 𝑎𝐼𝑠 (1). Then 𝐿 𝑁𝑡 ( 𝑓 ) converges uniformly on 𝐵 𝑎 (𝐸) + to the right-hand side of (9.64). By Corollary 1.22 there is a probability measure 𝑁∞ given by (9.64) and lim𝑡→∞ 𝑁𝑡 = 𝑁∞ by weak convergence. Conversely, suppose that 𝑁𝑡 converges weakly to a probability measure 𝑁∞ on 𝑀 (𝐸) as 𝑡 → ∞. From (9.11) we see (9.64) holds for 𝑓 ∈ 𝐶 (𝐸) + , so it holds all 𝑓 ∈ 𝐵(𝐸) + . □ Corollary 9.51 The MBI-process with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 is ergodic with 𝑁∞ as its unique stationary distribution if and only if (9.63) holds and lim𝑡→∞ 𝜇(𝑉𝑡 1) = 0 for every 𝜇 ∈ 𝑀 (𝐸). Proof This follows immediately from (9.8) and Theorems 1.21 and 9.50.

□

Theorem 9.52 Let (𝑁𝑡 )𝑡 ≥0 be a regular SC-semigroup given by (9.7) with 𝐼𝑠 ( 𝑓 ) = 𝐼 (𝑉𝑠 𝑓 ) for 𝑓 ∈ 𝐵(𝐸) + , where 𝐼 ∈ ℐ(𝐸) is represented by (9.12). Suppose there is a constant 𝑐 ∗ > 0 such that 𝑉𝑡 1(𝑥) ≤ e−𝑐∗ 𝑡 for every 𝑡 ≥ 0 and 𝑥 ∈ 𝐸. Then 𝑁𝑡 converges weakly to the stationary distribution 𝑁∞ given by (9.64) as 𝑡 → ∞ if ∫ (9.65) log[1 + 𝜈(1)]𝐻 (d𝜈) < ∞. 𝑀 (𝐸) ◦

Proof By Theorem 9.50 it suffices to show (9.65) implies (9.63). For any 𝑐 > 0 we can use a change of the variable to get  ∫ ∞ ∫ −𝑐𝑠 (1 − e−e 𝜈 (1) )𝐻 (d𝜈) d𝑠 e−𝑐𝑡 𝜂(1) + 𝐹 (𝜂, 𝐻, 𝑐) := ◦ 0 ∫ 𝜈 (1) ∫ 𝑀 (𝐸) −1 −1 (9.66) 𝑓 (𝑧)d𝑧, = 𝑐 𝜂(1) + 𝑐 𝐻 (d𝜈) 𝑀 (𝐸) ◦

0

where 𝑓 (𝑧) = 𝑧−1 (1 − e−𝑧 ). Observe that 𝑓 (𝑧) = 1 + 𝑜(1) as 𝑧 → 0 and 𝑓 (𝑧) = 𝑧−1 + 𝑜(𝑧−1 ) as 𝑧 → ∞. Then 𝐹 (𝜂, 𝐻, 𝑐) < ∞ if and only if (9.65) holds. Clearly, we have ∫ ∞ 𝐼𝑠 (1)d𝑠 ≤ 𝐹 (𝜂, 𝐻, 𝑐 ∗ ). 0

272

9 Structures of Independent Immigration

Then (9.65) implies (9.63).

□

Corollary 9.53 Let (𝑁𝑡 )𝑡 ≥0 be given as in Theorem 9.52. Suppose there are constants ∗ 𝑐∗ ≥ 𝑐 ∗ > 0 such that e−𝑐 𝑡 ≤ 𝑉𝑡 1(𝑥) ≤ e−𝑐∗ 𝑡 for every 𝑡 ≥ 0 and 𝑥 ∈ 𝐸. Then 𝑁𝑡 converges weakly to the stationary distribution 𝑁∞ given by (9.64) as 𝑡 → ∞ if and only if (9.65) holds. Proof Let 𝐹 (𝜂, 𝐻, 𝑐) be defined by (9.66). Under the conditions of the corollary, we have ∫ ∞ 𝐼𝑠 (1)d𝑠 ≥ 𝐹 (𝜂, 𝐻, 𝑐∗ ). 0

Then the result follows by Theorem (9.52).

□

Theorem 9.54 Suppose that (9.63) holds and the function 𝑣¯ 𝑡 defined by (2.7) is bounded on 𝐸 for every 𝑡 > 0. Then we have     2𝐿 𝑁∞ (1) 1 − 𝐿 𝑁∞ (𝑉𝑡 1) ≤ ∥𝑁𝑡 − 𝑁∞ ∥ ≤ 2 1 − 𝐿 𝑁∞ ( 𝑣¯ 𝑡 ) , (9.67) where 𝐿 𝑁∞ ( 𝑣¯ 𝑡 ) → 1 as 𝑡 → ∞. Proof By (9.14) and (9.64) it is easy to show that 𝑁𝑡 ∗ (𝑁∞ 𝑄 𝑡 ) = 𝑁∞ . Let 𝑀𝑡 (d𝜂1 , d𝜂2 ) be the image of the product measure 𝑁𝑡 (d𝜈1 ) (𝑁∞ 𝑄 𝑡 ) (d𝜈2 ) under the mapping (𝜈1 , 𝜈2 ) ↦→ (𝜂1 , 𝜂2 ) := (𝜈1 , 𝜈1 + 𝜈2 ). Then 𝑀𝑡 (d𝜂1 , d𝜂2 ) is a coupling of 𝑁𝑡 (d𝜂1 ) and 𝑁∞ (d𝜂2 ). By Theorem 5.7 in Chen (2004, p. 179) we have ∫ ∥𝑁𝑡 − 𝑁∞ ∥ ≤ 2 1 { 𝜂1 ≠𝜂2 } 𝑀𝑡 (d𝜂1 , d𝜂2 ) ∫ ∫𝑀 (𝐸) 2 =2 𝑁𝑡 (d𝜈1 ) 1 {𝜈2 ≠0} 𝑁∞ 𝑄 𝑡 (d𝜈2 ) ∫ 𝑀 (𝐸) ∫𝑀 (𝐸) 1 {𝜈2 ≠0} 𝑄 𝑡 (𝜈, d𝜈2 ) =2 𝑁∞ (d𝜈) 𝑀 (𝐸) ∫𝑀 (𝐸) (1 − e−𝜈 ( 𝑣¯𝑡 ) )𝑁∞ (d𝜈), =2 𝑀 (𝐸)

where the last equality follows from (2.6). Then we get the upper bound in (9.67). By applying Theorems 5.7 and 5.10 in Chen (2004, pp. 179–181) to the discrete metric on 𝑀 (𝐸) we have ∫ (1 − e−𝜈 (1) ) (𝑁∞ − 𝑁𝑡 ) (d𝜈) ∥𝑁𝑡 − 𝑁∞ ∥ ≥ 2 𝑀 (𝐸)   ∫ 𝑡   ∫ ∞ = 2 exp − 𝐼𝑠 (1)d𝑠 − exp − 𝐼𝑠 (1)d𝑠   ∫0 𝑡   0∫ ∞ = 2 exp − 𝐼𝑠 (1)d𝑠 1 − exp − 𝐼𝑠 (1)d𝑠  ∫0 ∞    ∫𝑡 ∞ 𝐼𝑠 (1)d𝑠 1 − exp − ≥ 2 exp − 𝐼𝑠 (𝑉𝑡 1)d𝑠 . 0

0

9.6 Stationary Distributions and Ergodicities

273

This gives the lower bound in (9.67). For any 𝑡 ≥ 𝑟 > 0 we have ∫ ∞ ∫ ∞ 𝐼𝑠 (𝑉𝑡−𝑟 𝑣¯ 𝑟 )d𝑠 𝐼𝑠 ( 𝑣¯ 𝑡 )d𝑠 = − log 𝐿 𝑁∞ ( 𝑣¯ 𝑡 ) = 0 ∫ ∫0 ∞ ∞ 𝐼𝑠 ( 𝑣¯ 𝑟 )d𝑠, 𝐼𝑠+𝑡−𝑟 ( 𝑣¯ 𝑟 )d𝑠 = = 0

𝑡−𝑟

which vanishes as 𝑡 → ∞. Then 𝐿 𝑁∞ ( 𝑣¯ 𝑡 ) → 1 as 𝑡 → ∞.

□

Corollary 9.55 Suppose that (9.63) holds and the function 𝑣¯ 𝑡 is bounded on 𝐸 for every 𝑡 > 0. Then we have   ∥𝑄 𝑡𝑁 (𝜇, ·) − 𝑁∞ ∥ ≤ 2(1 − e−𝜇 ( 𝑣¯𝑡 ) ) + 2 1 − 𝐿 𝑁∞ ( 𝑣¯ 𝑡 ) , 𝜇 ∈ 𝑀 (𝐸). Proof By Theorem 9.13 and the relation 𝑁𝑡 = 𝑄 𝑡𝑁 (0, ·), we have ∥𝑄 𝑡𝑁 (𝜇, ·) − 𝑁𝑡 ∥ ≤ 2(1 − e−𝜇 ( 𝑣¯𝑡 ) ). Then the estimate follows by Theorem 9.54 and the triangle inequality.

□

It is not hard to show that the probability measure 𝑁∞ given by (9.64) has finite first-moment if and only if ∫ ∞ ∫ 𝜈(1)𝐾𝑠 (d𝜈) < ∞, (9.68) 𝑡 ≥ 0. d𝑠 𝑀 (𝐸)

0

In this case, we have ∫ ∫ 𝜈( 𝑓 )𝑁∞ (d𝜈) = 𝑀 (𝐸)

∞

0

∫ 𝜈( 𝑓 )𝐾𝑠 (d𝜈),

d𝑠

𝑓 ∈ 𝐵(𝐸).

(9.69)

𝑀 (𝐸)

The next theorem gives an accurate evaluations of the Wasserstein distance between 𝑁𝑡 and the limit distribution 𝑁∞ . Theorem 9.56 Suppose that (9.68) holds and 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ for every 𝑡 ≥ 0. Then ∫ ∫ ∞ ∫ 𝜈(1)𝐾𝑠 (d𝜈), 𝜇(𝜋𝑡 1)𝑁∞ (d𝜇) = 𝑊1 (𝑁𝑡 , 𝑁∞ ) = d𝑠 𝑀 (𝐸)

𝑡

𝑀 (𝐸)

which vanishes as 𝑡 → ∞. Proof Let 𝑀𝑡 (d𝜂1 , d𝜂2 ) be the coupling of 𝑁𝑡 (d𝜂1 ) and 𝑁∞ (d𝜂2 ) defined in the proof of Theorem 9.54. Then we have ∫ 𝑊1 (𝑁𝑡 , 𝑁∞ ) ≤ ∥𝜂1 − 𝜂2 ∥𝑀𝑡 (d𝜂1 , d𝜂2 ) ∫𝑀 (𝐸) 2 ∫ 𝑁𝑡 (d𝜈1 ) = ∥𝜈2 ∥𝑁∞ 𝑄 𝑡 (d𝜈2 ) 𝑀 (𝐸)

𝑀 (𝐸)

274

9 Structures of Independent Immigration

∫

∫ 𝜈2 (1)𝑄 𝑡 (𝜇, d𝜈2 )

𝑁∞ (d𝜇)

=

𝑀 (𝐸)

∫𝑀 (𝐸)

𝜇(𝜋𝑡 1)𝑁∞ (d𝜇).

= 𝑀 (𝐸)

On the other hand, for any coupling 𝑃𝑡 (d𝜂1 , d𝜂2 ) of 𝑁𝑡 (d𝜂1 ) and 𝑁∞ (d𝜈2 ) we have ∫ ∥𝜂1 − 𝜂2 ∥𝑃𝑡 (d𝜈1 , d𝜂2 ) 𝑀 (𝐸) 2 ∫ ≥ [𝜂2 (1) − 𝜂1 (1)]𝑃𝑡 (d𝜂1 , d𝜂2 ) ∫𝑀 (𝐸) 2 𝜈(1) (𝑁∞ − 𝑁𝑡 ) (d𝜈) ≥ ∫ 𝑡 ∫ ∫ 𝑀∞(𝐸) ∫ 𝜈(1)𝐾𝑠 (d𝜈) − = d𝑠 𝜈(1)𝐾𝑠 (d𝜈) d𝑠 𝑀 (𝐸) 0 ∫0 ∞ ∫𝑀 (𝐸) 𝜈(1)𝐾𝑠+𝑡 (d𝜈) = d𝑠 ∫0 ∞ ∫𝑀 (𝐸) ∫ d𝑠 𝐾𝑠 (d𝜇) = 𝜈(1)𝑄 𝑡 (𝜇, d𝜈) 𝑀 (𝐸) 𝑀 (𝐸) ∫0 𝜇(𝜋𝑡 1)𝑁∞ (d𝜇). = 𝑀 (𝐸)

Then we get the first equality. The second equality follows immediately by the above □ calculations. Corollary 9.57 Suppose that (9.68) holds and 𝜈(1)𝐿 𝑡 (𝑥, d𝜈) is a bounded kernel from 𝐸 to 𝑀 (𝐸) ◦ for every 𝑡 ≥ 0. Then ∫ 𝑁 𝜈(𝜋𝑡 1)𝑁∞ (d𝜈), 𝜇 ∈ 𝑀 (𝐸). 𝑊1 (𝑄 𝑡 (𝜇, ·), 𝑁∞ ) ≤ 𝜇(𝜋𝑡 1) + 𝑀 (𝐸)

Proof Since 𝑁𝑡 = 𝑄 𝑡𝑁 (0, ·), this follows by Theorems 9.11 and 9.56 and the triangle □ inequality. Now let us briefly discuss the immigration structures associated with a (𝜉, 𝜙)superprocess, where 𝜉 is a Borel right process in 𝐸 and 𝜙 is a branching mechanism given by (2.29) or (2.30). For the SC-semigroup (𝑁𝑡 )𝑡 ≥0 defined by (9.14) and (9.15), the transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 is given by   ∫ ∫ 𝑡 −𝜈 ( 𝑓 ) 𝑁 𝑄 𝑡 (𝜇, d𝜈) = exp − 𝜇(𝑉𝑡 𝑓 ) − e 𝐼𝑟 (𝜅, 𝐹, 𝑓 )d𝑟 , (9.70) 𝑀 (𝐸)

0

where 𝐼𝑟 (𝜅, 𝐹, 𝑓 ) is defined by (9.15). Let 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)]. By Theorem A.53 and Corollary 5.34 we have ∥𝜋𝑡 ∥ ≤ e𝑐0 𝑡 for 𝑡 ≥ 0 and ∥ 𝑣¯ 𝑡 ∥ ≤ e𝑐0 (𝑡−𝑟) ∥ 𝑣¯ 𝑟 ∥ for 𝑡 ≥ 𝑟 > 0. Recall that the functional 𝑞 𝑡 (𝜅, 𝑓 ) is defined by (8.42). The following results are immediate consequences of the general results established above.

9.6 Stationary Distributions and Ergodicities

275

Theorem 9.58 There is a probability measure 𝑁∞ on 𝑀 (𝐸) with finite first-moment such that 𝑁𝑡 → 𝑁∞ weakly as 𝑡 → ∞ if and only if  ∫ ∞ ∫ (9.71) 𝑞 𝑠 (𝜅, 1) + 𝑞 𝑠 (𝜈, 1)𝐹 (d𝜈) d𝑠 < ∞. 𝒦 ( 𝑃) ◦

0

In this case, we have, for 𝑓 ∈ 𝐵(𝐸) + ,  ∫ 𝐿 𝑁∞ ( 𝑓 ) = exp −

∞

 𝐼𝑠 (𝜅, 𝐹, 𝑓 )d𝑠

(9.72)

0

and, for 𝑓 ∈ 𝐵(𝐸), ∫ ∫ 𝜈( 𝑓 )𝑁∞ (d𝜈) = 𝑀 (𝐸) ◦

∞



 𝑞 𝑠 (𝜈, 𝑓 )𝐹 (d𝜈) d𝑠. (9.73)

∫ 𝑞 𝑠 (𝜅, 𝑓 ) + 𝒦 ( 𝑃) ◦

0

Theorem 9.59 Suppose that the condition (9.71) is satisfied. Then we have   ∫ 𝜈(1)𝑁∞ (d𝜈) . 𝑊1 (𝑄 𝑡𝑁 (𝜇, ·), 𝑁∞ ) = e𝑐0 𝑡 𝜇(1) + 𝑀 (𝐸)

Theorem 9.60 Suppose that (9.71) and Condition 5.31 hold with 𝜙∗ satisfying Grey’s condition. Then, for 𝑡 ≥ 𝑟 > 0 and 𝜇 ∈ 𝑀 (𝐸),   ∫ 𝜈(1)𝑁∞ (d𝜈) . ∥𝑄 𝑡𝑁 (𝜇, ·) − 𝑁∞ ∥ ≤ 2e𝑐0 (𝑡−𝑟) ∥ 𝑣¯ 𝑟 ∥ 𝜇(1) + 𝑀 (𝐸)

Clearly, for 𝑐 0 < 0 the estimates in Theorems 9.59 and 9.60 yield the exponential ergodicities of the immigration superprocess in the Wasserstein distance 𝑊1 and the total variation distance ∥ · ∥, respectively. Example 9.6 Let (𝑄 𝑡𝑁 )𝑡 ≥0 be the transition semigroup of the 𝑑-dimensional CBIprocess defined as in Example 9.1. The branching and immigration mechanisms of the process are given by (2.44) and (9.17), respectively. We assume that ⟨𝑢, 1⟩𝑛(d𝑢) is a finite measure on R+𝑑 \ {0}. Let ∫ 𝑖, 𝑗 = 1, · · · , 𝑑. 1 {𝑖≠ 𝑗 } 𝑢 𝑗 𝐻𝑖 (d𝑢), 𝛾𝑖 𝑗 = 𝜂 𝑖 𝑗 + R+𝑑 \{0}

Suppose that there is a spatially constant local branching mechanism 𝜙∗ satisfying Grey’s condition so that, for 𝜆 ≥ 0 and 𝑖 = 1, · · · , 𝑑, ∫
e−𝜆𝑧𝑖 − 1 + 𝜆𝑧𝑖 𝐻𝑖 (d𝑧) ≥ 𝜙∗ (𝜆). [𝑏 𝑖 − ⟨𝛾𝑖 , 1)⟩]𝜆 + 𝑐 𝑖 𝜆2 + R+𝑑 \{0}

By Corollaries 5.34 and 9.14, the semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 has the strong Feller property. If 𝑐 0 := max1≤𝑖 ≤𝑑 [⟨𝛾𝑖 , 1⟩ − 𝑏 𝑖 ] < 0 in addition, the CBI-process is exponentially ergodic in the total variation distance by Theorem 9.60.

276

9 Structures of Independent Immigration

9.7 Notes and Comments The concept of SC-semigroups was introduced in Li (1995/6), where Theorem 9.4 was proved. The structures of those semigroups associated with Dawson–Watanabe superprocesses were studied in Li (1996, 1998b). Theorem 9.8 was proved in Li (2001). Theorems 9.11 and 9.13 are from Li (2021). Theorems 9.21 and 9.45 were given in Li (1996). Example 9.3 can also be found in Li (1996). Most of other results in Sections 9.5 and 9.6 first appeared in Li (2001, 2021). The results on strong Feller properties and ergodicities generalize those for CBI-processes established in Li and Ma (2015) by a coupling constructed by strong solutions of a stochastic equation. Some results for special branching mechanisms were obtained earlier in Stannat (2003a, 2003b). Extensions of the results to the space of tempered measures were discussed in Friesen (2022+). The equilibrium behavior of a population with immigration on a hierarchical group was studied in Dawson et al. (2004a). A general criterion for transience or recurrence of CBI-processes was given by Duhalde et al. (2014). The exponential ergodicities in two suitably chosen Wasserstein distances for a finite-dimensional affine process were established in Friesen et al. (2020). The strong Feller property and ergodicities in the total variation distance of a two-factor affine process were proved in Chen and Li (2022+) by extensions of the coupling of Li and Ma (2015); see also Barczy et al. (2014), Handa (2012) and Jin et al. (2017). The methods of couplings and distances for Markov processes have been developed systematically in Chen (2004, 2005). Let (𝐹𝑡 )𝑡 ≥0 be the composition semigroup of probability generating functions of a continuous-time branching process. A probability generating function 𝑔 is called self-decomposable relative to (𝐹𝑡 )𝑡 ≥0 by van Harn et al. (1982) if for each 𝑡 ≥ 0 there is another probability generating function 𝑔𝑡 such that 𝑔(𝑧) = (𝑔 ◦ 𝐹𝑡 ) (𝑧)𝑔𝑡 (𝑧),

|𝑧| ≤ 1.

(9.74)

This generalizes the classical concept of self-decomposability; see, e.g., Loève (1977) and Sato (1999). A general representation of self-decomposable probability generating functions for a critical or subcritical branching process was given in van Harn et al. (1982); see also the earlier work of Steutel and van Harn (1979). In view of (9.1) and (9.74), we may regard (9.64) as a counterpart in the setting of measure-valued processes of the representation (6.1b) in van Harn et al. (1982). The structure of immigration was studied in Li and Shiga (1995) in the setting of 𝜂, 𝛼 measure-valued diffusions. In particular, the transition semigroup (𝑄 𝑡 )𝑡 ≥0 given in Example 9.2 was considered in Li and Shiga (1995) for binary local branching with constant branching rate. There it was proved the corresponding immigration superprocess {𝑌𝑡 : 𝑡 ≥ 0} has a continuous density field {𝑌 (𝑡, 𝑥) : 𝑡 > 0, 𝑥 > 0} satisfying the following stochastic partial differential equation: √︁ 1 𝜕 𝑌 (𝑡, 𝑥) = 𝑌 (𝑡, 𝑥)𝑊¤ (𝑡, 𝑥) + Δ𝑌 (𝑡, 𝑥) + 𝜂(𝑥) ¤ − 𝛼 𝛿¤0 , 𝜕𝑡 2

9.7 Notes and Comments

277

where {𝑊 (𝑡, 𝑥) : 𝑡 ≥ 0, 𝑥 > 0} is a time–space Gaussian white noise based on the Lebesgue measure and the “dot” denotes the derivative in the distribution sense. The random measures {𝑌𝑡 : 𝑡 ≥ 0} have bounded supports if and only if 𝑌0 and 𝜂 have bounded supports. In this case, let 𝑅𝑡 = ∪0≤𝑠 ≤𝑡 supp(𝑌𝑡 ) and 𝑅ˆ𝑡 = sup{𝑥 > 0 : 𝑥 ∈ 𝑅𝑡 }. It was proved in Li and Shiga (1995) that the distribution of 𝑡 −1/3 𝑅ˆ𝑡 converges −3 as 𝑡 → ∞ to the Fréchet distribution 𝐹 (𝑧) = e−𝛾𝑧 with ∫ ∞  1  Γ(1/3)Γ(1/6)  3  𝛾= 𝛼+ 𝑥𝜂(d𝑥) . 18 Γ(1/2) 0 Motivated by applications to statistical physics models, the asymptotics of the maximum processes of branching models has been studied by many researchers; see, e.g., Bovier (2017), Shi (2015) and the references therein. A central limit theorem for the super-Brownian motion with immigration was also given in Li and Shiga (1995). The corresponding large and moderate deviations were studied in Zhang (2004a, 2004b). A number of functional central limit theorems for related processes were proved in Zhang (2005a, 2008), which gave rise to distribution-valued Gaussian processes. Li (1998a) studied immigration structures associated with branching particle systems. A large-deviation principle for a Brownian branching immigration particle system was established in Zhang (2005b). Some central limit theorems for supercritical branching superprocesses were established by Ren et al. (2017a, 2017b, 2019) and Wang (2018).

Chapter 10

One-Dimensional Stochastic Equations

In this chapter we establish several stochastic equations for one-dimensional CBIprocesses. By considering a particular stochastic equation, we establish the classical Lamperti transformations of CB-processes and spectrally positive Lévy processes. We also derive the distributions of some important random variables including the number of jumps with specified jump sizes and the size of the maximal jump in a time interval. For the CB-process, some properties of its global maximal jump are proved. By a slight modification of one of the stochastic equations, we construct a kind of generalized CBI-processes with stochastic immigration rates.

10.1 Existence and Uniqueness of Solutions Let 𝜙 and 𝜓 be the branching and immigration mechanisms given by (3.1) and (3.26), 𝛾 respectively. Let (𝑄 𝑡 )𝑡 ≥0 be the transition semigroup defined by (3.3) and (3.29). Suppose that (Ω, 𝒢, 𝒢𝑡 , P) is a filtered probability space satisfying the usual hypotheses. Let {𝐵(𝑡)} be a standard (𝒢𝑡 )-Brownian motion and let {𝑀 (d𝑠, d𝑧, d𝑢)} be a time–space (𝒢𝑡 )-Poisson random measure on (0, ∞) 3 with intensity d𝑠𝑚(d𝑧)d𝑢. We denote by 𝑀˜ (d𝑠, d𝑧, d𝑢) = 𝑀 (d𝑠, d𝑧, d𝑢) − d𝑠𝑚(d𝑧)d𝑢 the compensated random measure. Let {𝜂(𝑡)} be a positive increasing (𝒢𝑡 )-Lévy process with Laplace transform given by P[e−𝜆𝜂 (𝑡) ] = e−𝑡 𝜓 (𝜆) ,

𝑡 ≥ 0, 𝜆 ≥ 0.

By the Lévy–Itô decomposition, there is a time–space (𝒢𝑡 )-Poisson random measure {𝑁 (d𝑠, d𝑧)} on (0, ∞) 2 with intensity d𝑠𝑛(d𝑧) such that ∫ 𝑡∫ ∞ 𝑧𝑁 (d𝑠, d𝑧). 𝜂(𝑡) = 𝛽𝑡 + (10.1) 0

0

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_10

279

280

10 One-Dimensional Stochastic Equations

We assume that {𝐵(𝑡)}, {𝑀 (d𝑠, d𝑧, d𝑢)} and {𝜂(𝑡)} are independent of each other. Given a 𝒢0 -measurable random variable 𝑦(0) ≥ 0, we consider the stochastic integral equation ∫ 𝑡 √︁ ∫ 𝑡 𝑦(𝑡) = 𝑦(0) + 2𝑐𝑦(𝑠−)d𝐵(𝑠) − 𝑏 𝑦(𝑠−)d𝑠 0 ∫ 𝑡 ∫ 0∞ ∫ 𝑦 (𝑠−) + 𝑧 𝑀˜ (d𝑠, d𝑧, d𝑢) + 𝜂(𝑡). (10.2) 0

0

0

We understand the fourth term on the right-hand side of (10.2) as an integral over the random set {(𝑠, 𝑧, 𝑢) : 0 < 𝑠 ≤ 𝑡, 0 < 𝑧 < ∞, 0 < 𝑢 ≤ 𝑦(𝑠−)} and give similar interpretations for other stochastic integrals with respect to time– space noises. By a (positive) solution to the stochastic equation (10.2), we mean a positive càdlàg (𝒢𝑡 )-adapted process {𝑦(𝑡) : 𝑡 ≥ 0} that satisfies the stochastic equation almost surely for every 𝑡 ≥ 0. A solution {𝑦(𝑡)} is called a strong solution if it is adapted to the filtration (ℱ𝑡 ), where ℱ𝑡 is the 𝜎-algebra generated by the set of random variables {𝑦(0), 𝐵(𝑠), 𝜂(𝑠), 𝑀 ((0, 𝑠] × 𝐴) : 0 < 𝑠 ≤ 𝑡, 𝐴 ∈ ℬ((0, ∞) 2 )}. We say the pathwise uniqueness of solutions holds for the equation if two solutions {𝑦 1 (𝑡)} and {𝑦 2 (𝑡)} are indistinguishable whenever they have the same initial value 𝑦 1 (0) = 𝑦 2 (0). We refer to Ikeda and Watanabe (1989) and Situ (2005) for the basic theory of stochastic equations. See also Barczy et al. (2015a) and Kurtz (2014) for updated treatments of weak and strong solutions. Theorem 10.1 A solution {𝑦(𝑡) : 𝑡 ≥ 0} to (10.2) is a CBI-process relative to the 𝛾 filtration (𝒢𝑡 )𝑡 ≥0 with transition semigroup (𝑄 𝑡 )𝑡 ≥0 . Proof Let (𝑟, 𝑦) ↦→ 𝑓 (𝑟, 𝑦) be a function on [0, ∞) 2 with bounded continuous derivatives up to the first order relative to 𝑟 ≥ 0 and up to the second order relative to 𝑦 ≥ 0. By (10.2) and Itô’s formula, ∫ 𝑟   𝑓 (𝑟, 𝑦(𝑟)) = 𝑓 (0, 𝑦(0)) + 𝑦(𝑠) 𝑐 𝑓 𝑦′′𝑦 (𝑠, 𝑦(𝑠)) − 𝑏 𝑓 𝑦′ (𝑠, 𝑦(𝑠)) d𝑠 ∫ 𝑟 ∫0 ∞  + 𝑦(𝑠)d𝑠 𝑓 (𝑠, 𝑦(𝑠) + 𝑧) − 𝑓 (𝑠, 𝑦(𝑠)) 0 0 ∫ 𝑟  ′ − 𝑧 𝑓 𝑦 (𝑠, 𝑦(𝑠)) 𝑚(d𝑧) + 𝛽 𝑓 𝑦′ (𝑠, 𝑦(𝑠))d𝑠 0 ∫ 𝑟 ∫ ∞   + d𝑠 𝑓 (𝑠, 𝑦(𝑠) + 𝑧) − 𝑓 (𝑠, 𝑦(𝑠)) 𝑛(d𝑧) 0 ∫0 𝑟 ′ + 𝑓𝑠 (𝑠, 𝑦(𝑠))d𝑠 + local mart. (10.3) 0

10.1 Existence and Uniqueness of Solutions

281

Given 𝑡 ≥ 0 and 𝜆 ≥ 0, we define 𝑓𝜆 (𝑟, 𝑦) = e−𝑦𝑣𝑡−𝑟 (𝜆) ,

0 ≤ 𝑟 ≤ 𝑡, 𝑦 ≥ 0.

It is elementary to see that d d2 𝑓𝜆 (𝑠, 𝑦) = − 𝑓𝜆 (𝑠, 𝑦)𝑣 𝑡−𝑠 (𝜆), 𝑓𝜆 (𝑠, 𝑦) = 𝑓𝜆 (𝑠, 𝑦)𝑣 𝑡−𝑠 (𝜆) 2 d𝑦 d𝑦 2 and, by (3.5), d 𝑓𝜆 (𝑠, 𝑦) = −𝑦 𝑓𝜆 (𝑠, 𝑦)𝜙(𝑣 𝑡−𝑠 (𝜆)). d𝑠 By applying (10.3) to any smooth extension of (𝑟, 𝑦) ↦→ 𝑓𝜆 (𝑟, 𝑦) on [0, ∞) 2 we have ∫ 𝑟∧𝑡 −𝑦 (𝑟) 𝑣𝑡−𝑟∧𝑡 (𝜆) −𝑦 (0) 𝑣𝑡 (𝜆) e =e − e−𝑦 (𝑠) 𝑣𝑡−𝑠 (𝜆) 𝜓(𝑣 𝑡−𝑠 (𝜆))d𝑠 + local mart. 0

Then an application of integration by parts shows that   ∫ 𝑡 𝑟 ↦→ exp − 𝑦(𝑟)𝑣 𝑡−𝑟∧𝑡 (𝜆) − 𝜓(𝑣 𝑡−𝑠 (𝜆))d𝑠

(10.4)

𝑟∧𝑡

is a bounded (𝒢𝑡 )-martingale. It follows that, for 𝑡 ≥ 𝑟 ≥ 0,   ∫ 𝑡   P e−𝜆𝑦 (𝑡) |𝒢𝑟 = exp − 𝑦(𝑟)𝑣 𝑡−𝑟 (𝜆) − 𝜓(𝑣 𝑡−𝑠 (𝜆))d𝑠 . 𝑟

This gives the desired result.

□

Theorem 10.2 For any initial value 𝑦(0) ≥ 0, there is a pathwise unique positive strong solution to (10.2). Proof Step 1. Let us consider the special case where 𝑧𝑛(d𝑧) is a finite measure on (0, ∞). By considering the conditional law given 𝒢0 , we may assume 𝑦(0) ≥ 0 is a deterministic constant. Suppose that {𝑦(𝑡)} is a càdlàg realization of the CBI𝛾 process with transition semigroup (𝑄 𝑡 )𝑡 ≥0 . Let Δ𝑦(𝑠) = 𝑦(𝑠) − 𝑦(𝑠−) for 𝑠 > 0. By Theorem 9.30, the process {𝑦(𝑡)} has no negative jumps and the random measure ∑︁ 𝑁0 (d𝑠, d𝑧) := 1 {Δ𝑦 (𝑠)≠0} 𝛿 (𝑠,Δ𝑦 (𝑠)) (d𝑠, d𝑧) 𝑠>0

has predictable compensator 𝑁ˆ 0 (d𝑠, d𝑧) = 𝑦(𝑠−)d𝑠𝑚(d𝑧) + d𝑠𝑛(d𝑧).

282

10 One-Dimensional Stochastic Equations

Moreover, we have  ∫ 𝑦(𝑡) = 𝑦(0) + 𝑡 𝛽 +

∞

 ∫ 𝑡 𝑏𝑦(𝑠)d𝑠 𝑧𝑛(d𝑧) − 0∫ ∫ 0 𝑡 ∞ 𝑧 𝑁˜ 0 (d𝑠, d𝑧), + 𝑀 𝑐 (𝑡) + 0

0

where 𝑁˜ 0 (d𝑠, d𝑧) = 𝑁0 (d𝑠, d𝑧) − 𝑁ˆ 0 (d𝑠, d𝑧) and 𝑡 ↦→ 𝑀 𝑐 (𝑡) is a continuous local martingale with quadratic variation 2𝑐𝑦(𝑡−)d𝑡. By Theorem III.7.1′ in Ikeda and Watanabe (1989, p. 90), on an extension of the original probability space there is a standard Brownian motion {𝐵(𝑡)} such that ∫ 𝑡 √︁ 𝑐 𝑀 (𝑡) = 2𝑐𝑦(𝑠−)d𝐵(𝑠). 0

By Theorem III.7.4 in Ikeda and Watanabe (1989, p. 93), on a further extension of the probability space we can define independent Poisson random measures {𝑀 (d𝑠, d𝑧, d𝑢)} and {𝑁 (d𝑠, d𝑧)} with intensities d𝑠𝑚(d𝑧)d𝑢 and d𝑠𝑛(d𝑧), respectively, so that ∫ 0

𝑡

∫

∞

𝑧 𝑁˜ 0 (d𝑠, d𝑧) =

∫

0

𝑡

0∫

∫ 𝑡

∞ ∫ 𝑦 (𝑠−)

0∫

+ 0

∞

𝑧 𝑀˜ (d𝑠, d𝑧, d𝑢)

0

𝑧 𝑁˜ (d𝑠, d𝑧).

0

Moreover, an application of Theorem III.6.3 in Ikeda and Watanabe (1989, p. 77) shows that {𝐵(𝑡)} is independent of {𝑀 (d𝑠, d𝑧, d𝑢)} and {𝑁 (d𝑠, d𝑧)}. Then {𝑦(𝑡)} is a solution to (10.2).

Step 2. Consider again the special case where 𝑧𝑛(d𝑧) is a finite measure on (0, ∞). We shall prove the pathwise uniqueness of the solution to (10.2). Suppose that {𝑥(𝑡)} and {𝑦(𝑡)} are two positive solutions to the stochastic equation on the same probability space. By Theorem 10.1, both of them are CBI-processes. We may assume 𝑥(0) and 𝑦(0) are deterministic upon taking a conditional probability. Then the processes have locally bounded first-moments. For 𝑘 ≥ 0 set 𝑎 𝑘 = exp{−𝑘 (𝑘 + 1)/2}. Observe that 𝑎 𝑘 → 0 decreasingly as 𝑘 → ∞ and ∫ 𝑎𝑘−1 𝑎  𝑘−1 = 𝑘, 𝑧−1 d𝑧 = log 𝑘 ≥ 1. 𝑎 𝑘 𝑎𝑘 Then for each 𝑘 ≥ 1 there is a positive continuous function 𝑥 ↦→ 𝑔 𝑘 (𝑥) supported by (𝑎 𝑘 , 𝑎 𝑘−1 ) such that 𝑔 𝑘 (𝑥) ≤ 2(𝑘𝑥) −1 and ∫ 𝑎𝑘−1 𝑔 𝑘 (𝑥)d𝑥 = 1. 𝑎𝑘

10.1 Existence and Uniqueness of Solutions

283

For 𝑘 ≥ 1 and 𝑧 ∈ R let ∫

|𝑧 |

𝑓 𝑘 (𝑧) =

∫

0

𝑦

𝑔 𝑘 (𝑥)d𝑥.

d𝑦 0

Then 𝑓 𝑘 (𝑧) → |𝑧| increasingly as 𝑘 → ∞. Moreover, we have | 𝑓 𝑘′ (𝑧)| ≤ 1 and 0 ≤ |𝑧| 𝑓 𝑘′′ (𝑧) = |𝑧|𝑔 𝑘 (|𝑧|) ≤ 2/𝑘.

(10.5)

𝐷 𝑧 𝑓 𝑘 (𝜁) = 𝑓 𝑘 (𝜁 + 𝑧) − 𝑓 𝑘 (𝜁) − 𝑧 𝑓 𝑘′ (𝜁).

(10.6)

For 𝑧, 𝜁 ∈ R write

It is easy to see that |𝐷 𝑧 𝑓 𝑘 (𝜁)| ≤ | 𝑓 𝑘 (𝜁 + 𝑧) − 𝑓 𝑘 (𝜁)| + |𝑧 𝑓 𝑘′ (𝜁)| ≤ 2|𝑧|. By Taylor’s expansion, when 𝑧𝜁 ≥ 0, there is an 𝜂 between 𝜁 and 𝜁 + 𝑧 such that |𝜁 ||𝐷 𝑧 𝑓 𝑘 (𝜁)| ≤ |𝜁 || 𝑓 𝑘′′ (𝜂)|𝑧2 /2 ≤ |𝜂|| 𝑓 𝑘′′ (𝜂)|𝑧2 /2 ≤ 𝑧2 /𝑘, where we have used (10.5) for the last inequality. It follows that, when 𝑧𝜁 ≥ 0, |𝜁 ||𝐷 𝑧 𝑓 𝑘 (𝜁)| ≤ (2|𝑧𝜁 |) ∧ (𝑧2 /𝑘) ≤ (1 + 2|𝜁 |) [|𝑧| ∧ (𝑧2 /𝑘)].

(10.7)

Let 𝜁 (𝑡) = 𝑥(𝑡) − 𝑦(𝑡) for 𝑡 ≥ 0. From (10.2) we have ∫ 𝑡 √︁ √ ∫ 𝑡 √︁  𝜁 (𝑠−)d𝑠 + 2𝑐 𝜁 (𝑡) = 𝜁 (0) − 𝑏 𝑥(𝑠−) − 𝑦(𝑠−) d𝐵(𝑠) 0 ∫ 𝑡 0∫ ∞ ∫ 𝑥 (𝑠−) + 𝑧1 {𝜁 (𝑠−) >0} 𝑀˜ (d𝑠, d𝑧, d𝑢) 0 0 𝑦 (𝑠−) ∫ 𝑡 ∫ ∞ ∫ 𝑦 (𝑠−) − 𝑧1 {𝜁 (𝑠−) 0} d𝑠 𝐷 𝑧 𝑓 𝑘 (𝜁 (𝑠))𝑚(d𝑧) ∫0 𝑡 ∫0 ∞ − 𝜁 (𝑠)1 {𝜁 (𝑠) 0} 𝑀˜ (d𝑠, d𝑧, d𝑢) 0 0 𝑦 (𝑠−) ∫ 𝑡 ∫ ∞ ∫ 𝑦 (𝑠−) − 𝑓 𝑘′ (𝜁 (𝑠−))𝑧1 {𝜁 (𝑠−) 0} 𝑀˜ (d𝑠, d𝑧, d𝑢) 0 0 𝑦 (𝑠−) ∫ 𝑡 ∫ ∞ ∫ 𝑦 (𝑠−) + 𝐷 −𝑧 𝑓 𝑘 (𝜁 (𝑠))1 {𝜁 (𝑠−) 0} 𝑊 (d𝑠, d𝑢) 𝑦2 (𝑠−) ∫ 𝑡 ∫ 0𝑦2 (𝑠−) ∫ 𝑡 − 𝜁 (𝑠−)d𝑠 1 {𝜁 (𝑠−) 0} 𝑧 𝑀˜ (d𝑠, d𝑧, d𝑢) + 0 𝑦2 (𝑠−) 0 ∫ 𝑡 ∫ ∞ ∫ 𝑦2 (𝑠−) − 1 {𝜁 (𝑠−) 0} d𝑠 + 𝐷 𝑧 𝑓 𝑘 (𝜁 (𝑠))𝑚(d𝑧) ∫0 ∞ ∫0 𝑡 𝜁 (𝑠)1 {𝜁 (𝑠) 0,

10.1 Existence and Uniqueness of Solutions

287

where the coefficient has been chosen so that P[e−𝜆𝑧 (𝑡) ] = e𝑡𝜆

𝛼 /𝛼

,

𝑡, 𝜆 ≥ 0.

We assume that {𝐵(𝑡)}, {𝑧(𝑡)} and {𝜂(𝑡)} are independent of each other. Consider the stochastic differential equation √︁ √︁ d𝑦(𝑡) = 2𝑐𝑦(𝑡−)d𝐵(𝑡) + 𝛼 𝛼𝜎𝑦(𝑡−)d𝑧(𝑡) − 𝑏𝑦(𝑡−)d𝑡 + d𝜂(𝑡). (10.10) Theorem 10.7 A solution {𝑦(𝑡) : 𝑡 ≥ 0} to (10.10) is a CBI-process relative to the filtration (𝒢𝑡 )𝑡 ≥0 with branching mechanism 𝜙(𝜆) = 𝑏𝜆 + 𝑐𝜆2 + 𝜎𝜆 𝛼 and with immigration mechanism 𝜓 given by (3.26). Proof For any 𝜆 > 0 let 𝑡 ↦→ 𝑣 𝑡 (𝜆) be the unique positive solution to (3.5) with 𝜙 specified as in the theorem. As in the proof of Theorem 10.1, one can use Itô’s formula to see that (10.4) is a bounded (𝒢𝑡 )-martingale, which implies that {𝑦(𝑡)} is a CBI-process relative to (𝒢𝑡 ) with branching mechanism 𝜙 and immigration mechanism 𝜓. □ Theorem 10.8 For any initial value 𝑦(0) ≥ 0, there is a pathwise unique positive strong solution to (10.10). Proof By Theorem 10.2 there is a solution {𝑦(𝑡)} to (10.2) with {𝑀 (d𝑠, d𝑧, d𝑢)} being a Poisson random measure with intensity d𝑠𝑚(d𝑧)d𝑢 = 𝛼𝜎d𝑠𝛾(d𝑧)d𝑢. We may assume 𝜎 > 0, for otherwise the proof is simpler. Define the random measure {𝑀0 (d𝑠, d𝑧)} on (0, ∞) 2 by ∫

𝑡

∞ ∫ 𝑦 (𝑠−)

∫

𝑀0 ((0, 𝑡] × 𝐵) = 0

0

∫

𝑡

0

∫

 1 {𝑦 (𝑠−) >0} 1 𝐵 √︁ 𝛼

𝑧



𝑀 (d𝑠, d𝑧, d𝑢)

𝛼𝜎𝑦(𝑠−)

∞ ∫ 1/𝛼 𝜎

+

1 {𝑦 (𝑠−)=0} 1 𝐵 (𝑧)𝑀 (d𝑠, d𝑧, d𝑢), 0

0

0

where 𝑡 ≥ 0 and 𝐵 ∈ ℬ(0, ∞). It is easy to compute that {𝑀0 (d𝑠, d𝑧)} has predictable compensator { 𝑀ˆ 0 (d𝑠, d𝑧)} defined by   ∫ 𝑡∫ ∞ 𝑧 𝛼𝜎𝑦(𝑠−) (𝛼 − 1)d𝑠d𝑧 ˆ 𝑀0 ((0, 𝑡] × 𝐵) = 1 {𝑦 (𝑠−) >0} 1 𝐵 √︁ 𝛼 Γ(2 − 𝛼)𝑧 1+𝛼 0 0 𝛼𝜎𝑦(𝑠−) ∫ 𝑡∫ ∞ (𝛼 − 1)d𝑠d𝑧 + 1 {𝑦 (𝑠−)=0} 1 𝐵 (𝑧) Γ(2 − 𝛼)𝑧1+𝛼 0 ∫ 𝑡0 ∫ ∞ (𝛼 − 1)d𝑠d𝑧 = 1 𝐵 (𝑧) . Γ(2 − 𝛼)𝑧 1+𝛼 0 0 Thus {𝑀0 (d𝑠, d𝑧)} is a Poisson random measure with intensity d𝑠𝛾(d𝑧); see, e.g., Theorem III.6.2 in Ikeda and Watanabe (1989, p. 75). Now define the Lévy process ∫ 𝑡∫ ∞ 𝑧(𝑡) = 𝑧 𝑀˜ 0 (d𝑠, d𝑧), (10.11) 0

0

288

10 One-Dimensional Stochastic Equations

where 𝑀˜ 0 (d𝑠, d𝑧) = 𝑀0 (d𝑠, d𝑧) − 𝑀ˆ 0 (d𝑠, d𝑧). It is easy to see that ∫ 𝑡 √︁ ∫ 𝑡 ∫ ∞ √︁ 𝛼 𝛼 𝛼𝜎𝑦(𝑠−)d𝑧(𝑠) = 𝛼𝜎𝑦(𝑠−) 𝑧 𝑀˜ 0 (d𝑠, d𝑧) 0 0 0 ∫ 𝑡 ∫ ∞ ∫ 𝑦 (𝑠−) = 𝑧 𝑀˜ (d𝑠, d𝑧, d𝑢). 0

0

0

Then {𝑦(𝑡)} solves (10.10). This gives the existence of the solution. We next prove the pathwise uniqueness for (10.10). By the Lévy–Itô decomposition, the one-sided 𝛼-stable process {𝑧(𝑡)} has representation (10.11) with { 𝑀˜ 0 (d𝑠, d𝑧)} being a compensated Poisson random measure on (0, ∞) 2 with intensity d𝑠𝛾(d𝑧). For 𝑡 ≥ 0 let ∫ 𝑡∫ 1 ∫ 𝑡∫ ∞ 𝑧 1 (𝑡) = 𝑧 𝑀˜ 0 (d𝑠, d𝑧), 𝑧2 (𝑡) = 𝑧𝑀0 (d𝑠, d𝑧). 0

0

0

1

Since 𝑡 ↦→ 𝑧2 (𝑡) has at most finitely many jumps in each bounded interval, we only need to prove the pathwise uniqueness for √︁ √︁ d𝑦(𝑡) = 2𝑐𝑦(𝑡−)d𝐵(𝑡) + 𝛼 𝛼𝜎𝑦(𝑡−)d𝑧 1 (𝑡) − 𝑏𝑦(𝑡−)d𝑡 √︁ − 𝛼−1 (𝛼 − 1)Γ(2 − 𝛼) −1 𝛼 𝛼𝜎𝑦(𝑡−)d𝑡 + d𝜂(𝑡). (10.12) Suppose that {𝑥(𝑡)} and {𝑦(𝑡)} are two positive solutions of (10.12) defined on the same√︁probability √︁ space with deterministic initial value 𝑥(0) = 𝑦(0) ≥ 0. Let 𝜁 𝜃 (𝑡) = 𝜃 𝑥(𝑡) − 𝜃 𝑦(𝑡) for 0 < 𝜃 ≤ 2 and 𝑡 ≥ 0. Then we have d𝜁1 (𝑡) =

√ √ 2𝑐𝜁2 (𝑡−)d𝐵(𝑡) + 𝛼 𝛼𝜎𝜁 𝛼 (𝑡−)d𝑧1 (𝑡) − 𝑏𝜁1 (𝑡−)d𝑡 √ − 𝛼−1 (𝛼 − 1)Γ(2 − 𝛼) −1 𝛼 𝛼𝜎𝜁 𝛼 (𝑡−)d𝑡.

For 𝑘 ≥ 1 let 𝑓 𝑘 be the function defined as in the proof of Theorem 10.2. By Itô’s formula, it is not hard to see that ∫ 𝑡 ∫ 𝑡 𝑓 𝑘 (𝜁1 (𝑡)) = 𝑐 𝑓 𝑘′′ (𝜁1 (𝑠−))𝜁2 (𝑠−) 2 d𝑠 − 𝑏 𝑓 𝑘′ (𝜁1 (𝑠−))𝜁1 (𝑠−)d𝑠 0 ∫ 𝑡0 𝛼 −1 −1 √ 𝛼𝜎 − 𝛼 (𝛼 − 1)Γ(2 − 𝛼) 𝑓 𝑘′ (𝜁1 (𝑠−))𝜁 𝛼 (𝑠−)d𝑠 0 ∫ 𝑡 ∫ 1h √ + d𝑠 𝑓 𝑘 (𝜁1 (𝑠−) + 𝛼 𝛼𝜎𝜁 𝛼 (𝑠−)𝑧) − 𝑓 𝑘 (𝜁1 (𝑠−)) 0 0 i √ − 𝛼 𝛼𝜎𝜁 𝛼 (𝑠−)𝑧 𝑓 𝑘′ (𝜁1 (𝑠−)) 𝛾(d𝑧)+𝑀𝑘 (𝑡), (10.13) where 𝑡 ↦→ 𝑀𝑘 (𝑡) is a local martingale with localization sequence {𝜏𝑖 } defined by 𝜏𝑖 = inf{𝑠 ≥ 0 : 𝑥(𝑠) ≥ 𝑖 or 𝑦(𝑠) ≥ 𝑖}. Clearly, for any 0 < 𝑠 ≤ 𝜏𝑖 we have |𝜁1 (𝑠−)| ≤ 𝑖. By Taylor’s expansion, there exists 0 < 𝜉 < 𝑧 such that

10.1 Existence and Uniqueness of Solutions

289

√ 𝛼

√ 𝛼𝜎𝜁 𝛼 (𝑠−)𝑧) − 𝑓 𝑘 (𝜁1 (𝑠−)) − 𝛼 𝛼𝜎𝜁 𝛼 (𝑠−)𝑧 𝑓 𝑘′ (𝜁1 (𝑠−)) √ = 2−1 (𝛼𝜎) 2/𝛼 𝑓 𝑘′′ (𝜁1 (𝑠−) + 𝛼 𝛼𝜎𝜁 𝛼 (𝑠−)𝜉)𝜁 𝛼 (𝑠−) 2 𝑧2 √ ≤ 2−1 (𝛼𝜎) 2/𝛼 𝑓 𝑘′′ (𝜁1 (𝑠−) + 𝛼 𝛼𝜎𝜁 𝛼 (𝑠−)𝜉)|𝜁1 (𝑠−)| 2/𝛼 𝑧2 √ ≤ 2−1 (𝛼𝜎) 2/𝛼 𝑖 2/𝛼−1 𝑓 𝑘′′ (𝜁1 (𝑠−) + 𝛼 𝛼𝜎𝜁 𝛼 (𝑠−)𝜉)|𝜁1 (𝑠−)|𝑧2

𝑓 𝑘 (𝜁1 (𝑠−) +

≤ (𝛼𝜎) 2/𝛼 𝑖 2/𝛼−1 𝑘 −1 𝑧2 , where we have used (10.5) and the fact 𝜁1 (𝑠−)𝜁 𝛼 (𝑠−) ≥ 0. Taking the expectation in both sides of (10.13) at time 𝑡 ∧ 𝜏𝑖 gives  ∫ 𝑡∧𝜏𝑖  P[ 𝑓 𝑘 (𝜁1 (𝑡 ∧ 𝜏𝑖 ))] ≤ 2𝑐𝑘 −1 𝑡 + |𝑏|P |𝜁1 (𝑠−)|d𝑠 0 ∫ 𝑡 ∫ 1 2/𝛼 2/𝛼−1 −1 + (𝛼𝜎) 𝑖 𝑘 d𝑠 𝑧2 𝛾(d𝑧) 0 0 ∫ 𝑡 ≤ 2𝑐𝑘 −1 𝑡 + |𝑏| P[|𝜁1 (𝑠 ∧ 𝜏𝑖 )|]d𝑠 0 ∫ 𝑡 ∫ 1 + (𝛼𝜎) 2/𝛼 𝑖 2/𝛼−1 𝑘 −1 d𝑠 𝑧2 𝛾(d𝑧). 0

0

Now we can let 𝑘 → ∞ in the inequality above to get ∫ 𝑡 P[|𝜁1 (𝑡 ∧ 𝜏𝑖 )|] ≤ |𝑏| P[|𝜁1 (𝑠 ∧ 𝜏𝑖 )|]d𝑠. 0

Then P[|𝑥(𝑡 ∧𝜏𝑖 ) − 𝑦(𝑡 ∧𝜏𝑖 )|] = P[|𝜁1 (𝑡 ∧𝜏𝑖 )|] = 0 for 𝑡 ≥ 0 by Gronwall’s inequality. This implies the pathwise uniqueness for (10.10). □ The stochastic integral equations (10.2) and (10.9) give convenient constructions of the trajectories of the CBI-process. In particular, the immigration structure is represented by the increasing Lévy process {𝜂(𝑡)}, which is decomposed by (10.1) into the continuous part determined by the drift coefficient 𝛽 and the discontinuous part given by the Poisson random measure {𝑁 (d𝑠, d𝑧)}. Let {𝐿(d𝑠, d𝑢)} be the spectrally positive time–space (𝒢𝑡 )-Lévy white noise on (0, ∞) 2 defined by ∫ 𝐿(d𝑠, d𝑢) = 𝑊 (d𝑠, d𝑢) − 𝑏d𝑠d𝑢 + 𝑧 𝑀˜ (d𝑠, d𝑧, d𝑢). {0 0 : 𝑌𝑠 = 0 or 𝑌𝑠− = 0} and let 𝑍𝑡 = 𝑌𝑡∧𝜏 for 𝑡 ≥ 0. Recall that 𝐷 ( [0, ∞), R+ ) is the space of càdlàg paths from [0, ∞) to R+ furnished with the Skorokhod topology.

10.2 The Lamperti Transformations

291

Theorem 10.9 For any 𝑡 ≥ 0 let 𝑧(𝑡) = 𝑥(𝜅(𝑡)), where ∫ 𝑣 ∫ 𝑣 o n 𝜅(𝑡) = inf 𝑣 ≥ 0 : 𝑥(𝑠−)d𝑠 = 𝑥(𝑠)d𝑠 > 𝑡 .

(10.16)

0

0

Then {𝑧(𝑡) : 𝑡 ≥ 0} is distributed identically on 𝐷 ( [0, ∞), R+ ) with {𝑍𝑡 : 𝑡 ≥ 0}. Proof By Theorems 10.1 and 10.2, we may construct a realization of {𝑥(𝑡)} by the pathwise unique positive solution to the stochastic equation ∫ 𝑡 √︁ ∫ 𝑡 𝑥(𝑡) = 𝑥 + 2𝑐𝑥(𝑠−)d𝐵(𝑠) − 𝑏𝑥(𝑠−)d𝑠 0 ∫ 𝑡0∫ ∞ ∫ 𝑥 (𝑠−) + 𝑧 𝑀˜ (d𝑠, d𝑧, d𝑢), (10.17) 0

0

0

which is a special form of (10.2). It follows that ∫ 𝜅 (𝑡) √︁ 2𝑐𝑥(𝑠−)d𝐵(𝑠) − 𝑧(𝑡) = 𝑥 + 𝑏𝑥(𝑠−)d𝑠 0 ∫ 𝜅0(𝑡) ∫ ∞ ∫ 𝑥 (𝑠−) + 𝑧 𝑀˜ (d𝑠, d𝑧, d𝑢) 0 0 0 ∫ 𝑡 √ = 𝑥 + 2𝑐𝑊 (𝑡) − 𝑏 𝑧(𝑠−)d𝜅(𝑠) ∫ 𝑡 ∫ ∞ ∫ 𝑧 (𝑠−) 0 + 𝑧 𝑀˜ (d𝜅(𝑠), d𝑧, d𝑢), ∫

𝜅 (𝑡)

0

0

𝜅 (𝑡)

∫ √︁ 𝑥(𝑠−)d𝐵(𝑠) =

(10.18)

0

where ∫ 𝑊 (𝑡) =

𝑡

√︁

𝑧(𝑠−)d𝐵(𝜅(𝑠))

0

0

is a continuous martingale. From (10.16) we have d𝜅(𝑠) = 𝑥(𝜅(𝑠)−) −1 1 {𝑥 (𝜅 (𝑠)−) >0} d𝑠 = 1 {𝑧 (𝑠−) >0} 𝑧(𝑠−) −1 d𝑠. Let 𝜏0 = inf{𝑡 ≥ 0 : 𝑧(𝑡) = 0} = inf{𝑡 ≥ 0 : 𝑧(𝑡−) = 0}. Since zero is a trap for {𝑧(𝑡)}, it follows that ∫ 𝑡 ∫ 𝑡 ∫ 𝑡 𝑧(𝑠−)d𝜅(𝑠) = 1 {𝑧 (𝑠−) >0} d𝑠 = 1 {𝑠0} 𝑀 (d𝜅(𝑠), d𝑧, d𝑢), 0

𝑎

0

292

10 One-Dimensional Stochastic Equations

where 𝑡 ≥ 0 and 𝑏 ≥ 𝑎 > 0. It is easy to compute that {𝑁0 (d𝑠, d𝑧)} has predictable compensator { 𝑁ˆ 0 (d𝑠, d𝑧)} defined by ∫ 𝑡 ˆ 𝑚(𝑎, 𝑏]𝑧(𝑠−)d𝜅(𝑠) = 𝑚(𝑎, 𝑏] (𝑡 ∧ 𝜏0 ). 𝑁0 ((0, 𝑡] × (𝑎, 𝑏]) = 0

Then we can extend {𝑁0 (d𝑠, d𝑧)} to a Poisson random measure on (0, ∞) 2 with intensity d𝑠𝑚(d𝑧); see, e.g., Ikeda and Watanabe (1989, p. 93). From (10.18) we get ∫ 𝑡∧𝜏0 ∫ ∞ √ 𝑧(𝑡) = 𝑥 + 2𝑐𝑊 (𝑡 ∧ 𝜏0 ) − 𝑏(𝑡 ∧ 𝜏0 ) + 𝑧 𝑁˜ 0 (d𝑠, d𝑧). 0

0

Therefore {𝑧(𝑡)} is distributed on 𝐷 ( [0, ∞), R+ ) identically with {𝑍𝑡 : 𝑡 ≥ 0}. Theorem 10.10 For any 𝑡 ≥ 0 let 𝑋𝑡 = 𝑍 𝜃 (𝑡) , where ∫ 𝑣 ∫ 𝑣 n o −1 𝜃 (𝑡) = inf 𝑣 ≥ 0 : 𝑍 𝑠− d𝑠 = 𝑍 𝑠−1 d𝑠 > 𝑡 . 0

□

(10.19)

0

Then {𝑋𝑡 : 𝑡 ≥ 0} is distributed identically on 𝐷 ( [0, ∞), R+ ) with {𝑥(𝑡) : 𝑡 ≥ 0}. Proof By the Lévy–Itô decomposition, up to an extension of the original probability space we may assume {𝑌𝑡 } is given by ∫ √ 𝑌𝑡 = 𝑥 + 2𝑐𝑊 (𝑡) − 𝑏𝑡 +

∞∫ 1

∫

𝑡

0

0

𝑧 𝑀˜ 0 (d𝑠, d𝑧, d𝑢),

0

where {𝑊 (𝑡)} is a standard Brownian motion and { 𝑀˜ 0 (d𝑠, d𝑧, d𝑢)} is a compensated Poisson random measure on (0, ∞) 3 with intensity d𝑠𝑚(d𝑧)d𝑢. It follows that ∫ √ 𝑋𝑡 = 𝑥 + 2𝑐𝑊 (𝜃 (𝑡)) − 𝑏𝜃 (𝑡) + 0

𝑡

∫

∞∫ 1

0

𝑧 𝑀˜ 0 (d𝜃 (𝑠), d𝑧, d𝑢).

(10.20)

0

From (10.19) we have ∫

∫

𝑡

𝜃 (𝑡) =

𝑍 𝜃 (𝑠) d𝑠 = 0

𝑡

𝑋𝑠 d𝑠. 0

Then the continuous martingale {𝑊 (𝜃 (𝑡))} has the representation ∫ 𝑡 √︁ 𝑋𝑠 d𝐵(𝑠), 𝑊 (𝜃 (𝑡)) = 𝑡 ≥ 0, 0

where {𝐵(𝑡)} is another standard Brownian motion. Next we take an independent Poisson random measure {𝑀1 (d𝑠, d𝑧, d𝑢)} on (0, ∞) 3 with intensity d𝑠𝑚(d𝑧)d𝑢 and define the random measure −1 𝑀 (d𝑠, d𝑧, d𝑢) = 1 {𝑢 ≤𝑋𝑠− } 𝑀0 (d𝜃 (𝑠), d𝑧, 𝑋𝑠− d𝑢) + 1 {𝑢>𝑋𝑠− } 𝑀1 (d𝑠, d𝑧, d𝑢).

10.3 Distributional Properties of Jumps

293

It is easy to see that {𝑀 (d𝑠, d𝑧, d𝑢)} has deterministic compensator d𝑠𝑚(d𝑧)d𝑢, so it is a Poisson random measure. From (10.20) we see that {𝑋𝑡 } is a solution of (10.17). □ This gives the desired result. The random time changes presented in the two theorems above are called Lamperti transformations.

10.3 Distributional Properties of Jumps In this section, we prove some distributional properties of jumps of the CBI-process. Suppose that 𝜙 and 𝜓 are branching and immigration mechanisms given by (3.1) and (3.26), respectively. Let {𝑦(𝑡) : 𝑡 ≥ 0} be the CBI-process defined by the pathwise unique solution to (10.2) with deterministic initial value 𝑦(0) = 𝑥 ≥ 0. Then {𝑦(𝑡) : 𝑡 ≥ 0} reduces to a CB-process when 𝜓 ≡ 0. Suppose that 𝐴 ∈ ℬ(0, ∞) satisfies 𝑚( 𝐴) + 𝑛( 𝐴) < ∞. For 𝑡 ≥ 0 let 𝑦 𝐴 (𝑡) denote the number of jumps of the process by time 𝑡 with jump sizes belonging to the set 𝐴. In view of (10.2), we have ∫

𝑡

∫ ∫

𝑦 (𝑠−)

∫

𝑡

∫

𝑀 (d𝑠, d𝑧, d𝑢) +

𝑦 𝐴 (𝑡) = 0

𝐴

0

𝑁 (d𝑠, d𝑧). 0

(10.21)

𝐴

For 𝜆1 , 𝜆2 ≥ 0, set Φ 𝐴 (𝜆1 , 𝜆2 ) = 𝑏𝜆 1 + 𝑐𝜆21 +

∞

∫


e−𝑧 (𝜆1 +𝜆2 1 𝐴 (𝑧)) − 1 + 𝑧𝜆1 𝑚(d𝑧)

0

and ∫ Ψ 𝐴 (𝜆1 , 𝜆2 ) = 𝛽𝜆1 +

∞


1 − e−𝑧 (𝜆1 +𝜆2 1 𝐴 (𝑧)) 𝑛(d𝑧).

0

We can define the transition semigroup (𝑄 𝑡𝐴)𝑡 ≥0 of a two-dimensional CBI-process by ∫ e−𝜆1 𝑦1 −𝜆2 𝑦2 𝑄 𝑡𝐴 (𝑥1 , 𝑥2 , d𝑦 1 , d𝑦 2 ) R+2   ∫ 𝑡 Ψ 𝐴 (𝑣 1 (𝑠), 𝜆2 )d𝑠 , (10.22) = exp − 𝑥1 𝑣 1 (𝑡) − 𝑥2 𝜆2 − 0

where 𝑡 ↦→ 𝑣 1 (𝑡) = 𝑣 1 (𝑡, 𝜆1 , 𝜆2 ) is the unique positive solution to d 𝑣 1 (𝑡) = −Φ 𝐴 (𝑣 1 (𝑡), 𝜆2 ), d𝑡

𝑣 1 (0) = 𝜆1 .

(10.23)

This is clearly a very special case of the transition semigroup given by (9.18).

294

10 One-Dimensional Stochastic Equations

Theorem 10.11 The process {(𝑦(𝑡), 𝑦 𝐴 (𝑡)) : 𝑡 ≥ 0} defined by (10.2) and (10.21) is a two-dimensional CBI-process relative to the filtration (𝒢𝑡 )𝑡 ≥0 with transition semigroup (𝑄 𝑡𝐴)𝑡 ≥0 defined by (10.22) and (10.23). Proof Following the proof of Theorem 10.1, one can show by Itô’s formula and integration by parts that   ∫ 𝑡 Ψ 𝐴 (𝑣 1 (𝑡 − 𝑠), 𝜆2 )d𝑠 𝑟 ↦→ exp − 𝑦(𝑟)𝑣 1 (𝑡 − 𝑟 ∧ 𝑡) − 𝑦 𝐴 (𝑠)𝜆2 − 𝑟∧𝑡

is a bounded (𝒢𝑡 )-martingale, which implies the desired result.

□

Corollary 10.12 For any 𝑡 ≥ 0 the joint distribution of (𝑦(𝑡), 𝑦 𝐴 (𝑡)) is determined by   ∫ 𝑡 −𝜆1 𝑦 (𝑡)−𝜆2 𝑦 𝐴 (𝑡) = exp − 𝑥𝑣 1 (𝑡) − Pe (10.24) Ψ 𝐴 (𝑣 1 (𝑠), 𝜆2 )d𝑠 , 0

where 𝑡 ↦→ 𝑣 1 (𝑡) = 𝑣 1 (𝑡, 𝜆1 , 𝜆2 ) is the unique positive solution to (10.23). For the set 𝐴 ∈ ℬ(0, ∞) with 𝑚( 𝐴) + 𝑛( 𝐴) < ∞ given as above, we can define the branching mechanism 𝜙 𝐴 and the immigration mechanism 𝜓 𝐴 by ∫ 𝜙 𝐴 (𝜆) = 𝜙(𝜆) + (1 − e−𝑧𝜆 )𝑚(d𝑧) (10.25) 𝐴

and ∫ 𝜓 𝐴 (𝜆) = 𝜓(𝜆) −

(1 − e−𝑧𝜆 )𝑛(d𝑧).

(10.26)

(e−𝑧𝜆 − 1 + 𝑧𝜆)𝑚(d𝑧),

(10.27)

𝐴

It is easy to see that 𝜙 𝐴 (𝜆) = 𝑏 𝐴𝜆 + 𝑐𝜆2 +

∫ 𝐴𝑐

where 𝐴𝑐 = (0, ∞) \ 𝐴 and ∫ 𝑏𝐴 = 𝑏 +

𝑧𝑚(d𝑧). 𝐴

Theorem 10.13 Let 𝜏𝐴 = min{𝑠 > 0 : Δ𝑦(𝑠) = 𝑦(𝑠) − 𝑦(𝑠−) ∈ 𝐴}. Then for any 𝑡 ≥ 0 we have   ∫ 𝑡 𝜓 𝐴 (𝑣 𝐴 (𝑠))d𝑠 , (10.28) P{𝜏𝐴 > 𝑡} = exp − 𝑥𝑣 𝐴 (𝑡) − 𝑡𝑛( 𝐴) − 0

where 𝑡 ↦→ 𝑣 𝐴 (𝑡) is the unique positive solution of d 𝑣 𝐴 (𝑡) = 𝑚( 𝐴) − 𝜙 𝐴 (𝑣 𝐴 (𝑡)), d𝑡

𝑣 𝐴 (0) = 0.

(10.29)

10.3 Distributional Properties of Jumps

295

Proof From (10.22) we see that 𝑣 1 (𝑡, 𝜆1 , 𝜆2 ) is increasing in both 𝜆1 ≥ 0 and 𝜆2 ≥ 0. The differential equation (10.23) is equivalent to ∫ 𝑡 𝑣 1 (𝑡, 𝜆1 , 𝜆2 ) = 𝜆1 − Φ 𝐴 (𝑣 1 (𝑠, 𝜆1 , 𝜆2 ), 𝜆2 )d𝑠. 0

Observe also that Φ 𝐴 (𝜆1 , ∞) = 𝜙 𝐴 (𝜆1 ) − 𝑚( 𝐴), Ψ 𝐴 (𝜆1 , ∞) = 𝜓 𝐴 (𝜆1 ) + 𝑛( 𝐴). Then 𝑡 ↦→ 𝑣 𝐴 (𝑡) = 𝑣 1 (𝑡, 0, ∞) is the unique locally bounded positive solution to ∫ 𝑡 𝜙 𝐴 (𝑣 𝐴 (𝑠))d𝑠, 𝑣 𝐴 (𝑡) = 𝑚( 𝐴)𝑡 − 0

which is just the integral form of (10.29). By letting 𝜆1 → 0 and 𝜆2 → ∞ in (10.24) we obtain (10.28). □ Corollary 10.14 (1) If 𝑚( 𝐴)+𝑛( 𝐴) = 0, then P{𝜏𝐴 = ∞} = 1. (2) If 𝑚( 𝐴)+𝑛( 𝐴) > 0 and 𝜓(𝜆) > 0 for all 𝜆 > 0, then P{𝜏𝐴 < ∞} = 1. Proof (1) Since 𝑚( 𝐴) = 0, by (10.29) and Corollary 5.19 we have 𝑣 𝐴 (𝑡) = 0 for all 𝑡 ≥ 0. By Theorem 10.13 it follows that P{𝜏𝐴 > 𝑡} = 1 for all 𝑡 ≥ 0. Then P{𝜏𝐴 = ∞} = lim𝑡→∞ P{𝜏𝐴 > 𝑡} = 1. (2) By choosing a smaller set if it is necessary, we may assume 0 < 𝑚( 𝐴) +𝑛( 𝐴) < ∞. In the case of 𝑛( 𝐴) > 0, we have 𝑡𝑛( 𝐴) → ∞ as 𝑡 → ∞. In the case of 𝑛( 𝐴) = 0, we must have 𝑚( 𝐴) > 0. By (10.29) and Corollary 5.19 we see that 𝑠 ↦→ 𝑣 𝐴 (𝑠) is increasing and 𝑣 𝐴 (𝑠) > 0 for 𝑠 > 0. Since 𝜓 𝐴 (𝜆) = 𝜓(𝜆) > 0 for all 𝜆 > 0, we have ∫ 𝑡 𝜓 𝐴 (𝑣 𝐴 (𝑠))d𝑠 = ∞. lim 𝑡→∞

0

In view of (10.28), we get P{𝜏𝐴 = ∞} = lim𝑡→∞ P{𝜏𝐴 > 𝑡} = 0 in both cases.

□

Corollary 10.15 Suppose that 𝜓 ≡ 0 and 𝑚( 𝐴) < ∞. Then for any 𝑡 ≥ 0 we have P{𝜏𝐴 > 𝑡} = exp{−𝑥𝑣 𝐴 (𝑡)}, where 𝑡 ↦→ 𝑣 𝐴 (𝑡) is the unique positive solution of (10.29). Corollary 10.16 Suppose that 𝐴 ⊂ 𝐵 ∈ ℬ(0, ∞) and 𝑚(𝐵) < ∞. Then 𝑣 𝐴 (𝑡) ≤ 𝑣 𝐵 (𝑡) for 𝑡 ≥ 0. Proof Since 𝜏𝐴 ≥ 𝜏𝐵 , we have exp{−𝑥𝑣 𝐴 (𝑡)} ≥ exp{−𝑥𝑣 𝐵 (𝑡)} by Corollary 10.15, □ and so 𝑣 𝐴 (𝑡) ≤ 𝑣 𝐵 (𝑡) for 𝑡 ≥ 0. Corollary 10.17 Suppose that 𝜓 ≡ 0 and 0 < 𝑚( 𝐴) < ∞. Then we have P{𝜏𝐴 = ∞} = exp{−𝑥𝜙−1 𝐴 (𝑚( 𝐴))} with 0 · ∞ = 0 by convention.

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10 One-Dimensional Stochastic Equations

Proof Since 0 < 𝑚( 𝐴) < ∞, we can apply Theorem 10.13 to see that P{𝜏𝐴 = ∞} = lim P{𝜏𝐴 > 𝑡} = lim exp{−𝑥𝑣 𝐴 (𝑡)}. 𝑡→∞

𝑡→∞

Then the result follows by Proposition 5.20 or Corollary 5.21.

□

10.4 Local and Global Maximal Jumps Suppose that 𝜙 and 𝜓 are branching and immigration mechanisms given by (3.1) and (3.26), respectively. Let {𝑦(𝑡) : 𝑡 ≥ 0} be the CBI-process defined by the pathwise unique solution to (10.2) with deterministic initial value 𝑦(0) = 𝑥 ≥ 0. We are interested in the distributional properties of the local and global maximal jumps of the process. For any measure 𝜇 on (0, ∞) let sup(𝜇) = sup{𝑥 > 0 : 𝜇(𝑥, ∞) > 0}. Theorem 10.18 Suppose that 𝑟 ≥ 0 satisfies 𝑚(𝑟, ∞)+𝑛(𝑟, ∞) < ∞. Let 𝜙𝑟 = 𝜙 (𝑟 ,∞) and 𝜓𝑟 = 𝜓 (𝑟 ,∞) be given by (10.25) and (10.26), respectively, with 𝐴 = (𝑟, ∞). Then, for any 𝑡 ≥ 0,   ∫ 𝑡 o n 𝜓𝑟 (𝑣 𝑟 (𝑠))d𝑠 , P max Δ𝑦(𝑡) ≤ 𝑟 = exp − 𝑥𝑣 𝑟 (𝑡) − 𝑡𝑛(𝑟, ∞) − 𝑠 ∈ (0,𝑡 ]

0

where 𝑡 ↦→ 𝑣 𝑟 (𝑡) is the unique positive solution of d 𝑣 𝑟 (𝑡) = 𝑚(𝑟, ∞) − 𝜙𝑟 (𝑣 𝑟 (𝑡)), d𝑡

𝑣 𝑟 (0) = 0.

(10.30)

Proof Since P{max𝑠 ∈ (0,𝑡 ] Δ𝑦(𝑡) ≤ 𝑟 } = P{𝜏(𝑟 ,∞) > 𝑡}, the result follows by Theorem 10.13. □ Corollary 10.19 Suppose that 𝜓(𝜆) > 0 for all 𝜆 > 0. Then we have n o P sup Δ𝑦(𝑠) = sup(𝑚 + 𝑛) = 1. 𝑠>0

Proof Since 𝑚(sup(𝑚 + 𝑛), ∞) = 𝑛(sup(𝑚 + 𝑛), ∞) = 0, by Theorem 10.18 for any 𝑡 > 0 we have n o P sup Δ𝑦(𝑠) ≤ sup(𝑚 + 𝑛) = 1, 𝑠 ∈ (0,𝑡 ]

and hence n o P sup Δ𝑦(𝑠) ≤ sup(𝑚 + 𝑛) = 1. 𝑠>0

10.4 Local and Global Maximal Jumps

297

On the other hand, for any 𝑟 < sup(𝑚 + 𝑛) we have (𝑚 + 𝑛) (𝑟, sup(𝑚 + 𝑛)] > 0. By Corollary 10.14 (2), n o  P sup Δ𝑦(𝑠) ∈ (𝑟, sup(𝑚 + 𝑛)] = P 𝜏(𝑟 ,sup(𝑚+𝑛) ] < ∞ = 1, 𝑠>0

which implies the desired result.

□

Corollary 10.20 Suppose that 𝜓 ≡ 0 and 𝑟 ≥ 0 satisfies 0 < 𝑚(𝑟, ∞) < ∞. Then n o  P sup Δ𝑦(𝑠) ≤ 𝑟 = exp − 𝑥𝜙𝑟−1 (𝑚(𝑟, ∞)) . 𝑠>0

Corollary 10.21 Suppose that 𝜓 ≡ 0 and 𝑚(0, ∞) > 0. Then n o  P sup Δ𝑦(𝑠) = sup(𝑚) = 1 − exp − 𝑥𝜙−1 {sup(𝑚) } (𝑚({sup(𝑚)})) 𝑠>0

with 𝜙 {∞} = 𝜙 and 𝑚({∞}) = 0 by convention. Proof Since 𝑚(sup(𝑚), ∞) = 𝑛(sup(𝑚), ∞) = 0, as in the proof of Corollary 10.19 one can see that P{sup𝑠>0 Δ𝑦(𝑠) ≤ sup(𝑚)} = 1. For 0 < 𝑟 < sup(𝑚) we have 𝑚(𝑟, sup(𝑚)] > 0. From Corollary 10.20 it follows that o n n o P sup Δ𝑦(𝑠) ∈ (𝑟, sup(𝑚)] = P sup Δ𝑦(𝑠) > 𝑟 𝑠>0 𝑠>0  = 1 − exp − 𝑥𝜙𝑟−1 (𝑚(𝑟, ∞))  = 1 − exp − 𝑥𝜙−1 (𝑟 ,sup(𝑚) ] (𝑚(𝑟, sup(𝑚)]) . Then we get the desired result by letting 𝑟 → sup(𝑚).

□

Corollary 10.22 Suppose that 𝜓 ≡ 0, 𝑏 > 0 and sup(𝑚) = ∞. If 𝑦(0) = 𝑥 > 0, then as 𝑟 → ∞ we have o n 𝑥 P sup Δ𝑦(𝑠) > 𝑟 = 1 − exp{−𝑥𝜙𝑟−1 (𝑚(𝑟, ∞))} ∼ 𝑚(𝑟, ∞). 𝑏 𝑠>0 Proof By (10.27) we see by dominated convergence that (𝜕/𝜕𝑧)𝜙𝑟 (0) = 𝑏 (𝑟 ,∞) . It follows that (𝜕/𝜕𝑧)𝜙𝑟−1 (0) = 1/𝑏 (𝑟 ,∞) . Then, as 𝑟 → ∞, 𝜙𝑟−1 (𝑚(𝑟, ∞)) ∼ 𝑚(𝑟, ∞)/𝑏 (𝑟 ,∞) ∼ 𝑚(𝑟, ∞)/𝑏, and the desired result follows from Corollary 10.20.

□

The next theorem establishes the equivalence of the distribution of the local maximal jump of the CBI-process and the total Lévy measure 𝑚 + 𝑛. In view of Theorem 10.18, it may be true that P{max𝑠 ∈ (0,𝑡 ] Δ𝑦(𝑠) = 0} > 0, so we only discuss the absolute continuity on the set (0, ∞).

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10 One-Dimensional Stochastic Equations

Theorem 10.23 Suppose that 𝛽 > 0 or 𝑦(0) = 𝑥 > 0. Then for any 𝑡 > 0 the restriction of the distribution P{max𝑠 ∈ (0,𝑡 ] Δ𝑦(𝑠) ∈ ·} to (0, ∞) is equivalent to 𝑚 + 𝑛. Proof We first take a set 𝐴 ∈ ℬ(0, ∞) satisfying 𝑚( 𝐴) + 𝑛( 𝐴) = 0. By Corollary 10.14 (1) we have n o P max Δ𝑦(𝑠) ∈ 𝐴 ≤ P{𝜏𝐴 ≤ 𝑡} = 0. 𝑠 ∈ (0,𝑡 ]

Then P{max𝑠 ∈ (0,𝑡 ] Δ𝑦(𝑠) ∈ ·}| (0,∞) is absolutely continuous with respect to 𝑚+𝑛. To prove the absolute continuity of 𝑚 + 𝑛 with respect to P{max𝑠 ∈ (0,𝑡 ] Δ𝑦(𝑠) ∈ ·}| (0,∞) , consider a set 𝐴 ∈ ℬ(0, ∞) such that 𝑚( 𝐴) + 𝑛( 𝐴) > 0. Take a sufficiently small 𝑟 > 0 such that 0 < 𝑚( 𝐴𝑟 )+𝑛( 𝐴𝑟 ) < ∞, where 𝐴𝑟 = 𝐴∩(𝑟, ∞). Let 𝐵𝑟 = (𝑟, ∞)\𝐴𝑟 . Then 𝑣 𝑟 (𝑠) ≥ 𝑣 𝐵𝑟 (𝑠) for 𝑠 ≥ 0 by Corollary 10.16. By (10.28) we have   ∫ 𝑡 𝜓𝑟 (𝑣 𝑟 (𝑠))s. P{𝜏(𝑟 ,∞) > 𝑡} = exp − 𝑥𝑣 𝑟 (𝑡) − 𝑡𝑛(𝑟, ∞) −  ∫ 𝑡0 𝑟 𝜓 𝐵𝑟 (𝑣 𝑟 (𝑠))d𝑠 = exp − 𝑥𝑣 𝑟 (𝑡) − 𝑡𝑛(𝐵 ) − 0  ∫ 𝑡 ∫ −𝑧𝑣𝑟 (𝑠) − 𝑛(d𝑧) . d𝑠 e (10.31) 0

𝐴𝑟

In the case of 𝑛( 𝐴𝑟 ) > 0, it follows that   ∫ 𝑡 𝜓 𝐵𝑟 (𝑣 𝑟 (𝑠))d𝑠 P{𝜏(𝑟 ,∞) > 𝑡} < exp − 𝑥𝑣 𝑟 (𝑡) − 𝑡𝑛(𝐵𝑟 ) − 0   ∫ 𝑡 𝑟 ≤ exp − 𝑥𝑣 𝐵𝑟 (𝑡) − 𝑡𝑛(𝐵 ) − 𝜓 𝐵𝑟 (𝑣 𝐵𝑟 (𝑠))d𝑠 . 0

Using (10.28) again we see that P{𝜏(𝑟 ,∞) > 𝑡} < P{𝜏𝐵𝑟 > 𝑡}.

(10.32)

In the case of 𝑛( 𝐴𝑟 ) = 0, we must have 𝑚( 𝐴𝑟 ) > 0 and so 𝑚(𝐵𝑟 ) < 𝑚(𝑟, ∞). By Corollary 5.19 and the right continuity the CB-process, we see that, for all 𝑠 > 0, 𝑣 𝑟 (𝑠) = 𝑣(𝑠, 𝑚(𝑟, ∞)) > 𝑣(𝑠, 𝑚(𝐵𝑟 )) = 𝑣 𝐵𝑟 (𝑠). Since 𝛽 > 0 or 𝑥 > 0, from (10.31) we get   ∫ 𝑡 𝜓 𝐵𝑟 (𝑣 𝑟 (𝑠))d𝑠 P{𝜏(𝑟 ,∞) > 𝑡} = exp − 𝑥𝑣 𝑟 (𝑡) − 𝑡𝑛(𝐵𝑟 ) − 0   ∫ 𝑡 𝑟 𝜓 𝐵𝑟 (𝑣 𝐵𝑟 (𝑠))d𝑠 , < exp − 𝑥𝑣 𝐵𝑟 (𝑡) − 𝑡𝑛(𝐵 ) − 0

10.4 Local and Global Maximal Jumps

299

which also gives (10.32). Then P{𝜏(𝑟 ,∞) ≤ 𝑡} > P{𝜏𝐵𝑟 ≤ 𝑡} in both cases. By Theorem 10.11 we have P{𝑦 𝐴𝑟 (𝑡) < ∞} = 1. It follows that o n n o P max Δ𝑦(𝑠) ∈ 𝐴 ≥ P max Δ𝑦(𝑠) ∈ 𝐴𝑟 𝑠 ∈ (0,𝑡 ]  𝑠 ∈ (0,𝑡 ] ≥ P 𝜏𝐴𝑟 ≤ 𝑡, 𝑦 𝐴𝑟 (𝑡) < ∞, 𝜏𝐵𝑟 > 𝑡  = P 𝜏𝐴𝑟 ≤ 𝑡, 𝜏𝐵𝑟 > 𝑡  = P 𝜏(𝑟 ,∞) ≤ 𝑡, 𝜏𝐵𝑟 > 𝑡   = P 𝜏(𝑟 ,∞) ≤ 𝑡 − P 𝜏𝐵𝑟 ≤ 𝑡 > 0. This shows the absolute continuity of the total Lévy measure 𝑚 + 𝑛 with respect to P{max𝑠 ∈ (0,𝑡 ] Δ𝑦(𝑠) ∈ ·}| (0,∞) . □ For critical and subcritical branching CB-processes without immigration, we may also discuss the absolute continuity of the distribution of its global maximal jump. Such a result is presented in the following:

Theorem 10.24 Suppose that 𝜓 ≡ 0 and 𝑏 ≥ 0. If 𝑦(0) = 𝑥 > 0, then the restriction of P{sup𝑠>0 Δ𝑦(𝑠) ∈ ·} to (0, ∞) is equivalent to 𝑚. Proof Since 𝜓 ≡ 0, the process {𝑦(𝑡) : 𝑡 ≥ 0} is actually a CB-process. The result is trivially true if 𝜙 ≡ 0. Then we may assume 𝜙(𝑧0 ) ≠ 0 for some 𝑧0 > 0 in the proof. By Corollary 5.24 we have P{lim𝑡→∞ 𝑦(𝑡) = 0} = 1 and hence P{sup𝑠>0 Δ𝑦(𝑠) = max𝑠>0 Δ𝑦(𝑠)} = 1. For 𝐴 ∈ ℬ(0, ∞) with 𝑚( 𝐴) = 0, we can use Corollary 10.14 (1) to see o n P max Δ𝑦(𝑠) ∈ 𝐴 ≤ P{𝜏𝐴 < ∞} = 0. 𝑠>0

Then the restriction of P{sup𝑠>0 Δ𝑦(𝑠) ∈ ·} = P{max𝑠>0 Δ𝑦(𝑠) ∈ ·} to (0, ∞) is absolutely continuous with respect to 𝑚. Now suppose that 𝐴 ∈ ℬ(0, ∞) satisfies 𝑚( 𝐴) > 0. For any 𝑟 > 0, one can see as in the proof of Theorem 10.23 that n o P max Δ𝑦(𝑠) ∈ 𝐴 ≥ P{𝜏(𝑟 ,∞) < ∞} − P{𝜏(𝑟 ,∞)\𝐴 < ∞}. 𝑠>0

If P(max𝑠>0 Δ𝑦(𝑠) ∈ 𝐴) = 0, we have P{𝜏(𝑟 ,∞) < ∞} = P{𝜏(𝑟 ,∞)\𝐴 < ∞}, and so Corollary 10.17 implies 𝜙𝑟−1 (𝑚(𝑟, ∞)) = 𝜙−1 (𝑟 ,∞)\𝐴 (𝑚((𝑟, ∞) \ 𝐴)) =: 𝑎(𝑟).

300

10 One-Dimensional Stochastic Equations

It follows that 𝜙𝑟 (𝑎(𝑟)) = 𝑚(𝑟, ∞) = 𝑚((𝑟, ∞) \ 𝐴) + 𝑚( 𝐴 ∩ (𝑟, ∞)) = 𝜙 (𝑟 ,∞)\𝐴 ◦ 𝜙−1 (𝑟 ,∞)\𝐴 (𝑚((𝑟, ∞) \ 𝐴)) + 𝑚( 𝐴 ∩ (𝑟, ∞)) = 𝜙 (𝑟 ,∞)\𝐴 (𝑎(𝑟)) + 𝑚( 𝐴 ∩ (𝑟, ∞)). Then, as in the proof of Theorem 10.23, we must have 𝑚( 𝐴 ∩ (𝑟, ∞)) = 0. Since 𝑟 > 0 was arbitrary, this contradicts 𝑚( 𝐴) > 0. It then follows that P{sup𝑠>0 Δ𝑦(𝑠) ∈ 𝐴} > 0. □

10.5 A Generalized CBI-process Suppose that (Ω, 𝒢, 𝒢𝑡 , P) is a filtered probability space satisfying the usual hypotheses. Let ℒ 1 denote the set of (𝒢𝑡 )-progressive processes 𝜌 = {𝜌(𝑡) : 𝑡 ≥ 0} that are locally integrable in the sense that ∫ 𝑡 P[|𝜌(𝑠)|]d𝑠 < ∞, 𝑡 ≥ 0. 0

Let 𝒫 = 𝒫(𝒢𝑡 ) denote the 𝜎-algebra on Ω × [0, ∞) generated by all real-valued left continuous processes adapted to (𝒢𝑡 ). A process {𝜌(𝑡) : 𝑡 ≥ 0} is predictable if the mapping (𝜔, 𝑡) ↦→ 𝜌(𝜔, 𝑡) is 𝒫-measurable. For each 𝜌 ∈ ℒ 1 there is an (𝒢𝑡 )-predictable process 𝜌0 ∈ ℒ 1 such that ∫ 𝑡 P[|𝜌(𝑠) − 𝜌0 (𝑠)|]d𝑠 = 0, 𝑡 ≥ 0. (10.33) 0

ˆ For instance, we can let 𝜌0 (𝑡) = 1 { | 𝜌(𝑡) where ˆ | 0 and 𝑐 𝑖 ≥ 0, 𝑖 = 0, 1, 2, the process was studied earlier by Li (2019a). The study of the nonlinear branching model is much more difficult than the classical linear one because of the lack of explicit characterization of the transition probabilities by Laplace transforms. The reader may also refer to Foucart et al. (2020, 2021), Li and Wang (2020), Li and Zhou (2021) and the references therein for the properties of the model such as extinction, explosion, ergodicity, exponential ergodicity and entrance from zero or infinity. Discrete-state branching processes in random environments were introduced by Smith (1968) and Smith and Wilkinson (1969) as an extension of the classical population model. Those extensions possess many interesting new properties; see, e.g., Afanasyev et al. (2005, 2012), Guivarc’h and Liu (2001) and Vatutin (2003). It was observed in Bansaye et al. (2013) that a kind of random environment in the continuous-state setting can be modeled by Lévy processes. Let 𝐿 = {𝐿 (𝑡) : 𝑡 ≥ 0}

10.6 Notes and Comments

307

be a Lévy process with no jump smaller than −1. By a result of He et al. (2018), for any 𝑡 ≥ 0 and 𝜆 ≥ 0, there is a pathwise unique positive solution 𝑟 ↦→ 𝑣 𝑟𝐿,𝑡 (𝜆) on [0, 𝑡] to the stochastic integral equation ∫ 𝑡 ∫ 𝑣 𝑡−𝑠,𝑡 (𝜆)d𝐿 𝑡 (𝑠), 𝑣 𝑟 ,𝑡 (𝜆) = 𝜆 − 𝜙(𝑣 𝑠,𝑡 (𝜆))d𝑠 + [0,𝑡−𝑟)

𝑟

where 𝐿 𝑡 (𝑠) = 𝐿(𝑡−) − 𝐿 ((𝑡 − 𝑠)−). He et al. (2018) showed that a stochastic transition semigroup (𝑄 𝑟𝐿,𝑡 : 𝑡 ≥ 𝑟 ≥ 0) can be defined by (6.37) from the stochastic cumulant semigroup (𝑣 𝑟𝐿,𝑡 : 𝑡 ≥ 𝑟 ≥ 0). A Markov process is called a CB-process in Lévy environment if it has conditional transition semigroup (𝑄 𝑟𝐿,𝑡 : 𝑡 ≥ 𝑟 ≥ 0) given the Lévy process 𝐿. Suppose that {𝐵(𝑡)} and {𝑀 (d𝑠, d𝑢)} are as in (10.2) and they are independent of the Lévy process 𝐿. It was proved in He et al. (2018) that a realization of the CB-process in Lévy environment with stochastic cumulant semigroup (𝑣 𝑟𝐿,𝑡 : 𝑡 ≥ 𝑟 ≥ 0) can be given as the pathwise unique positive solution to the stochastic equation ∫ 𝑡 √︁ ∫ 𝑡 𝑋 (𝑡) = 𝑋 (0) + 2𝑐𝑋 (𝑠−)d𝐵(𝑠) − 𝑏 𝑋 (𝑠−)d𝑠 0∫ ∫ 𝑡 ∫ ∞0∫ 𝑋 (𝑠−) 𝑡 + 𝑧 𝑀˜ (d𝑠, d𝑧, d𝑢) + 𝑋 (𝑠−)d𝐿 (𝑠). 0

0

0

0

This process is a natural generalization of the branching model with catastrophe introduced by Bansaye et al. (2013). See also the work of Palau and Pardo (2018), who constructed CB-processes with competition in Lévy environments. The asymptotic behavior of the survival probabilities of the CB-process in Lévy environment was studied in Bansaye et al. (2013), Li and Xu (2018) and Palau and Pardo (2017). It is still an open problem to identify the class of all possible continuous-state scaling limits of the discrete-state branching processes in random environments introduced by Smith (1968) and Smith and Wilkinson (1969).

Chapter 11

Path-Valued Processes and Stochastic Flows

In this chapter, we study a class of stochastic flows by means of stochastic equations and measure-valued processes. They are reformulations of the tree-valued processes of Abraham and Delmas (2012) and Aldous and Pitman (1998) and generalize naturally the flows studied by Bertoin and Le Gall (2000, 2006). We define such a flow by a path-valued growing process, which is actually a family of correlated CBI-processes parameterized by an interval. In a special case, the total population of the process evolves along the parameter according to an increasing inhomogeneous CBI-process. The path-valued growing process is reconstructed by strong solutions of a system of stochastic equations. A representation of the stochastic flow is given in terms of a homogeneous non-local branching immigration superprocess. Under natural conditions, we give rigorous computations of some interesting conditional laws.

11.1 Path-Valued Growing Processes Let 𝐸 ⊂ R be a nonempty interval and let 𝐹 (𝐸) denote the set of positive increasing càdlàg functions on 𝐸. Given 𝜇 ∈ 𝐹 (𝐸), we can define a Radon measure 𝜇 on 𝐸 such that 𝜇( 𝑝, 𝑞] = 𝜇(𝑞) − 𝜇( 𝑝) for 𝑞 ≥ 𝑝 ∈ 𝐸. Let {𝜙𝑞 : 𝑞 ∈ 𝐸 } be a family of branching mechanisms, where 𝜙𝑞 is given by (3.1) with (𝑏, 𝑚) = (𝑏 𝑞 , 𝑚 𝑞 ) depending on 𝑞 ∈ 𝐸 and with 𝑐 ≥ 0 remaining constant. We assume that {𝜙𝑞 : 𝑞 ∈ 𝐸 } is an admissible family in the sense that for each 𝜆 ≥ 0, the function 𝑞 ↦→ 𝜙𝑞 (𝜆) is decreasing and continuously differentiable with the derivative 𝜁𝑞 (𝜆) := −(d/d𝑞)𝜙𝑞 (𝜆) of the form ∫ ∞ 𝑞 ∈ 𝐸, 𝜆 ≥ 0, (11.1) 𝜁𝑞 (𝜆) = 𝛽𝑞 𝜆 + (1 − e−𝑧𝜆 )𝑛𝑞 (d𝑧), 0

where 𝛽𝑞 ≥ 0 and 𝑛𝑞 (d𝑧) is a 𝜎-finite kernel from 𝐸 to (0, ∞) satisfying © Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_11

309

310

11 Path-Valued Processes and Stochastic Flows

sup

h

∫

∞

𝛽𝑦 +

𝑝 ≤𝑦 ≤𝑞

i 𝑧𝑛 𝑦 (d𝑧) < ∞,

𝑝 ≤ 𝑞 ∈ 𝐸.

(11.2)

0

For such an admissible family {𝜙𝑞 : 𝑞 ∈ 𝐸 }, we clearly have ∫ 𝑞 𝜙 𝑝,𝑞 (𝜆) := 𝜙 𝑝 (𝜆) − 𝜙𝑞 (𝜆) = 𝜁 𝑦 (𝜆)d𝑦.

(11.3)

𝑝

It follows that ∫

∫

𝑞

𝑏𝑞 = 𝑏 𝑝 −

∫

𝑞

𝛽 𝑦 d𝑦 −

(11.4)

0

𝑝

𝑝

∞

𝑧𝑛 𝑦 (d𝑧)

d𝑦

and ∫

𝑞

𝑚 𝑞 (d𝑧) = 𝑚 𝑝 (d𝑧) +

𝑛 𝑦 (d𝑧)d𝑦.

(11.5)

𝑝

We say 𝑦 ∈ 𝐸 is a critical point of the admissible family {𝜙𝑞 : 𝑞 ∈ 𝐸 } if 𝑏 𝑦 = 0, which means 𝜙 𝑦 is a critical branching mechanism. By (11.4) one can see 𝑞 ↦→ 𝑏 𝑞 is a continuous decreasing function, so the set consisting of all critical points can only be a subinterval of 𝐸. Our purpose is to construct a natural family of correlated CBI-processes with branching mechanisms {𝜙𝑞 : 𝑞 ∈ 𝐸 }. From (11.3) we see that for any 𝑞 ≥ 𝑝 ∈ 𝐸 the branching mechanism 𝜙𝑞 is more productive than 𝜙 𝑝 . We shall give a modeling of the evolution process of the population along the parameter 𝑞 ∈ 𝐸 by an increasing pathvalued Markov process. The model involves some natural branching and immigration structures. Let 𝐷 [0, ∞) + = 𝐷 ( [0, ∞), R+ ) be the space of positive càdlàg paths on [0, ∞) endowed with the Skorokhod topology. Let 𝜂 ∈ 𝐹 (𝐸) and 𝜌 ∈ 𝐷 [0, ∞) + be fixed. By Theorem 9.27, for any 𝑞 ≥ 𝑝 ∈ 𝐸 there is an inhomogeneous Borel right transition 𝑝,𝑞, 𝜂,𝜌 semigroup (𝑄 𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ≥ 0) on [0, ∞) defined by ∫ 𝑝,𝑞, 𝜂,𝜌 e−𝜆𝑦 𝑄 𝑟 ,𝑡 (𝑥, d𝑦) [0,∞)  ∫ 𝑡 𝑣 𝑞 (𝑡 − 𝑠, 𝜆))d𝑠 = exp − 𝑥𝑣 𝑞 (𝑡 − 𝑟, 𝜆) − 𝜂( 𝑝, 𝑞] 𝑟  ∫ 𝑡 − 𝜙 𝑝,𝑞 (𝑣 𝑞 (𝑡 − 𝑠, 𝜆)) 𝜌(𝑠)d𝑠 , (11.6) 𝑟

where 𝑡 ↦→ 𝑣 𝑞 (𝑡, 𝜆) is the unique positive solution of (3.4) with 𝜙 = 𝜙𝑞 . The semigroup generates an inhomogeneous càdlàg CBI-process {𝑦(𝑡) : 𝑡 ≥ 0}. Let 𝑝,𝑞, 𝜂,𝜌 Q𝑥 (d𝑤) denote the distribution on 𝐷 [0, ∞) + of such a process with initial state 𝑦(0) = 𝑥 ≥ 0. By Theorem 9.28, for any 𝑓 ∈ 𝐵[0, ∞) + with compact support we have

11.1 Path-Valued Growing Processes

311

n o ∞ 𝑝,𝑞, 𝜂,𝜌 𝑓 (𝑠)𝑤(𝑠)d𝑠 Q 𝑥 (d𝑤) exp − + 𝐷 [0,∞)  0 ∫ ∞ = exp − 𝑥𝑢 𝑞 (0, 𝑓 ) − 𝜂( 𝑝, 𝑞] 𝑢 𝑞 (𝑠, 𝑓 )d𝑠 0  ∫ ∞ 𝜙 𝑝,𝑞 (𝑢 𝑞 (𝑠, 𝑓 )) 𝜌(𝑠)d𝑠 , − ∫

∫

(11.7)

0

where 𝑠 ↦→ 𝑢 𝑞 (𝑠, 𝑓 ) is the unique positive solution to ∫ ∫ ∞ 𝜙𝑞 (𝑢 𝑞 (𝑡, 𝑓 ))d𝑡 = 𝑢 𝑞 (𝑠, 𝑓 ) + 𝑠

∞

𝑓 (𝑡)d𝑡.

(11.8)

𝑠

It is easy to see that 𝑢 𝑞 (𝑠, 𝑓 ) = 𝑢 𝑝,𝑞 (𝑠, 𝑓 ) = 0 for 𝑠 > sup{𝑡 ≥ 0 : 𝑓 (𝑡) > 0}. Now suppose that we are given another function 𝜇 ∈ 𝐹 (𝐸). We define the 𝜇, 𝜂 probability measure P 𝑝,𝑞 (𝜌, ·) on 𝐷 + [0, ∞) by ∫ 𝑝,𝑞, 𝜂,𝜌 𝜇, 𝜂 1 𝐵 (𝜌 + 𝑤)Q 𝜇 ( 𝑝,𝑞] (d𝑤), 𝐵 ∈ ℬ(𝐷 + [0, ∞)). (11.9) P 𝑝,𝑞 (𝜌, 𝐵) = 𝐷 + [0,∞)

For 𝑓 ∈ 𝐵[0, ∞) + with compact support write ∫ ∞ 𝑓 (𝑠)𝑤(𝑠)d𝑠, ⟨𝑤, 𝑓 ⟩ =

𝑤 ∈ 𝐷 [0, ∞) + .

0

From (11.7) and (11.9) it follows that ∫  𝜇, 𝜂 e− ⟨𝑤, 𝑓 ⟩ P 𝑝,𝑞 (𝜌, d𝑤) = exp − ⟨𝜌, 𝑢 𝑝,𝑞 (·, 𝑓 )⟩ − 𝜇( 𝑝, 𝑞]𝑢 𝑞 (0, 𝑓 ) + 𝐷 [0,∞) (11.10) − 𝜂( 𝑝, 𝑞] ⟨1, 𝑢 𝑞 (·, 𝑓 )⟩ , where 𝑢 𝑝,𝑞 (𝑠, 𝑓 ) = 𝑓 (𝑠) + 𝜙 𝑝,𝑞 (𝑢 𝑞 (𝑠, 𝑓 )),

𝑠 ≥ 0.

(11.11)

Clearly, the right-hand side of (11.10) is continuous in 𝜌 ∈ 𝐷 [0, ∞) + . Therefore 𝜇, 𝜂 P 𝑝,𝑞 (𝜌, d𝑤) is a Borel kernel on 𝐷 [0, ∞) + . Proposition 11.1 For any 𝑓 ∈ 𝐵[0, ∞) + with compact support, we have 𝑢 𝑝 (𝑠, 𝑢 𝑝,𝑞 (·, 𝑓 )) = 𝑢 𝑞 (𝑠, 𝑓 ),

𝑠 ≥ 0, 𝑝 ≤ 𝑞 ∈ 𝐸

(11.12)

and 𝑢 𝑝,𝑦 (𝑠, 𝑢 𝑦,𝑞 (·, 𝑓 )) = 𝑢 𝑝,𝑞 (𝑠, 𝑓 ),

𝑠 ≥ 0, 𝑝 ≤ 𝑦 ≤ 𝑞 ∈ 𝐸 .

(11.13)

Proof From (11.3) and (11.8) we can see that 𝑠 ↦→ 𝑢 𝑞 (𝑠, 𝑓 ) is the unique positive solution of

312

11 Path-Valued Processes and Stochastic Flows

∫ 𝑢 𝑞 (𝑠, 𝑓 ) =

∞

∫ [ 𝑓 (𝑡) + 𝜙 𝑝,𝑞 (𝑢 𝑞 (𝑡, 𝑓 ))]d𝑡 −

𝑠

∞

𝜙 𝑝 (𝑢 𝑞 (𝑡, 𝑓 ))d𝑡. (11.14) 𝑠

On the other hand, by (11.8) and (11.11) we have ∫ ∞ ∫ ∞ 𝑢 𝑝 (𝑠, 𝑢 𝑝,𝑞 (·, 𝑓 )) = 𝜙 𝑝 (𝑢 𝑝 (𝑡, 𝑢 𝑝,𝑞 (·, 𝑓 )))d𝑡 𝑢 𝑝,𝑞 (𝑡, 𝑓 )d𝑡 − 𝑠 ∫𝑠 ∞ = [ 𝑓 (𝑡) + 𝜙 𝑝,𝑞 (𝑢 𝑞 (𝑡, 𝑓 ))]d𝑡 𝑠 ∫ ∞ 𝜙 𝑝 (𝑢 𝑝 (𝑡, 𝑢 𝑝,𝑞 (·, 𝑓 )))d𝑡. − 𝑠

Then 𝑠 ↦→ 𝑢 𝑝 (𝑠, 𝑢 𝑝,𝑞 (·, 𝑓 )) is also a positive solution to (11.14). By the uniqueness of the solution we get (11.12). It follows that 𝑢 𝑝,𝑦 (𝑠, 𝑢 𝑦,𝑞 (·, 𝑓 )) = 𝑢 𝑦,𝑞 (𝑠, 𝑓 ) + 𝜙 𝑝,𝑦 (𝑢 𝑦 (𝑠, 𝑢 𝑦,𝑞 (·, 𝑓 ))) = 𝑓 (𝑠) + 𝜙 𝑦,𝑞 (𝑢 𝑞 (𝑠, 𝑓 )) + 𝜙 𝑝,𝑦 (𝑢 𝑞 (𝑠, 𝑓 )) = 𝑓 (𝑠) + 𝜙 𝑝,𝑞 (𝑢 𝑞 (𝑠, 𝑓 )). Then we have (11.13) by (11.11).

□

Corollary 11.2 Let 𝑞 ∈ 𝐸 and let 𝑓 ∈ 𝐵[0, ∞) + with compact support. Then ( 𝑝, 𝑠) ↦→ 𝑢 𝑝,𝑞 (𝑠, 𝑓 ) is the unique locally bounded positive solution on 𝐸 ≤𝑞 × [0, ∞) of ∫ 𝑞 (11.15) 𝑢 𝑝,𝑞 (𝑠, 𝑓 ) = 𝑓 (𝑠) + 𝜁 𝑦 ◦ 𝑢 𝑦 (𝑠, 𝑢 𝑦,𝑞 (·, 𝑓 ))d𝑦. 𝑝

Proof By (11.3) and (11.11) one can see 𝑝 ↦→ 𝑢 𝑝,𝑞 (𝑠, 𝑓 ) is a decreasing function. By the relations established in Proposition 11.1, for 𝑞 > 𝑦 > 𝑝 ∈ 𝐸 we have 𝑢 𝑝,𝑞 (𝑠, 𝑓 ) = 𝑢 𝑝,𝑦 (𝑠, 𝑢 𝑦,𝑞 (·, 𝑓 )) = 𝑢 𝑦,𝑞 (𝑠, 𝑓 ) + 𝜙 𝑝,𝑦 (𝑢 𝑦 (𝑠, 𝑢 𝑦,𝑞 (·, 𝑓 ))). Then we can differentiate both sides to see d d 𝑢 𝑝,𝑞 (𝑠, 𝑓 ) 𝜙 𝑝,𝑦 (𝑢 𝑦 (𝑠, 𝑢 𝑦,𝑞 (·, 𝑓 ))) = 𝑝=𝑦 𝑝=𝑦 d𝑝 d𝑝 = −𝜁 𝑦 (𝑢 𝑦 (𝑠, 𝑢 𝑦,𝑞 (·, 𝑓 ))). This shows ( 𝑝, 𝑠) ↦→ 𝑢 𝑝,𝑞 (𝑠, 𝑓 ) is a locally bounded positive solution of (11.15). The uniqueness of the solution is a standard application of Gronwall’s inequality. □ 𝜇, 𝜂

Theorem 11.3 The family of kernels (P 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸) defined by (11.9) or (11.10) constitute an inhomogeneous transition semigroup on 𝐷 [0, ∞) + . 𝜇, 𝜂

Proof It suffices to check the Chapman–Kolmogorov equation for (P 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸), which is immediate by (11.10) and the relations given in Proposition 11.1. □

11.1 Path-Valued Growing Processes

313 𝜇

Corollary 11.4 There is an inhomogeneous transition semigroup (P 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸) on 𝐷 [0, ∞) + defined by, for 𝑓 ∈ 𝐵[0, ∞) + with compact support, ∫ 𝜇 e− ⟨𝑤, 𝑓 ⟩ P 𝑝,𝑞 (𝜌, d𝑤) 𝐷 [0,∞) +  = exp − ⟨𝜌, 𝑢 𝑝,𝑞 (·, 𝑓 )⟩ − 𝜇( 𝑝, 𝑞]𝑢 𝑞 (0, 𝑓 ) . (11.16) Corollary 11.5 Let 𝜙 be a branching mechanism given by (3.1). Then there is an 𝜇 inhomogeneous transition semigroup (P 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸) on 𝐷 [0, ∞) + defined by, + for 𝑓 ∈ 𝐵[0, ∞) with compact support, ∫  𝜇 e− ⟨𝑤, 𝑓 ⟩ P 𝑝,𝑞 (𝜌, d𝑤) = exp − ⟨𝜌, 𝑓 ⟩ − 𝜇( 𝑝, 𝑞]𝑢(0, 𝑓 ) , (11.17) 𝐷 [0,∞) +

where 𝑠 ↦→ 𝑢(𝑠, 𝑓 ) is the unique positive solution to ∫ ∫ ∞ 𝜙(𝑢(𝑡, 𝑓 ))d𝑡 = 𝑢(𝑠, 𝑓 ) + 𝑠

∞

𝑓 (𝑡)d𝑡.

(11.18)

𝑠

For 𝑞 ∈ 𝐸 let K𝑞 denote the distribution on 𝐷 [0, ∞) + of a CBI-process {𝑋𝑡 (𝑞) : 𝑡 ≥ 0} with initial value 𝑋0 (𝑞) = 𝜇(𝑞) and with transition semigroup 𝑞, 𝜂 (𝑄 𝑡 )𝑡 ≥0 given by   ∫ 𝑡 ∫ 𝑞, 𝜂 e−𝜆𝑦 𝑄 𝑡 (𝑥, d𝑦) = exp − 𝑥𝑣 𝑞 (𝑡, 𝜆) − 𝜂(𝑞) 𝑣 𝑞 (𝑠, 𝜆))d𝑠 , (11.19) [0,∞)

0

where 𝑡 ↦→ 𝑣 𝑞 (𝑡, 𝜆) is the unique positive solution of (3.4) with 𝜙 = 𝜙𝑞 . Then K𝑞 is the distribution of a CBI-process with initial value 𝜇(𝑞), branching mechanism 𝜙𝑞 and immigration rate 𝜂(𝑞). By Theorem 9.28, we have, for 𝑓 ∈ 𝐵[0, ∞) + with compact support, ∫  e− ⟨𝑤, 𝑓 ⟩ K𝑞 (d𝑤) = exp − 𝜇(𝑞)𝑢 𝑞 (0, 𝑓 ) − 𝜂(𝑞)⟨1, 𝑢 𝑞 (·, 𝑓 )⟩ , (11.20) 𝐷 [0,∞) +

where 𝑠 ↦→ 𝑢 𝑞 (𝑠, 𝑓 ) is the unique positive solution to (11.8). Theorem 11.6 The family (K𝑞 : 𝑞 ∈ 𝐸) is a probability entrance law for the 𝜇, 𝜂 inhomogeneous semigroup (P 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸) defined by (11.9) or (11.10). Proof For any 𝑝 ≤ 𝑞 ∈ 𝐸 and any 𝑓 ∈ 𝐵[0, ∞) + with compact support, by (11.10) and Theorem 9.28 we have ∫ ∫ 𝜇, 𝜂 e− ⟨𝑤, 𝑓 ⟩ P 𝑝,𝑞 (𝜌, d𝑤) K 𝑝 (d𝜌) 𝐷 [0,∞) + 𝐷 [0,∞) + ∫  exp − ⟨𝜌, 𝑢 𝑝,𝑞 (·, 𝑓 )⟩ K 𝑝 (d𝜌) = + 𝐷 [0,∞)  · exp − 𝜇( 𝑝, 𝑞]𝑢 𝑞 (0, 𝑓 ) − 𝜂( 𝑝, 𝑞] ⟨1, 𝑢 𝑞 (·, 𝑓 )⟩

314

11 Path-Valued Processes and Stochastic Flows

 = exp − 𝜇( 𝑝)𝑢 𝑝 ◦ 𝑢 𝑝,𝑞 (0, 𝑓 ) − 𝜂( 𝑝)⟨1, 𝑢 𝑞 (·, 𝑓 )⟩  · exp − 𝜇( 𝑝, 𝑞]𝑢 𝑞 (0, 𝑓 ) − 𝜂( 𝑝, 𝑞] ⟨1, 𝑢 𝑞 (·, 𝑓 )⟩  = exp − 𝜇(𝑞)𝑢 𝑞 (0, 𝑓 ) − 𝜂(𝑞)⟨1, 𝑢 𝑞 (·, 𝑓 )⟩ ∫ e− ⟨𝑤, 𝑓 ⟩ K𝑞 (d𝑤). = 𝐷 [0,∞) + 𝜇, 𝜂

Then (K𝑞 : 𝑞 ∈ 𝐸) is a probability entrance law for (P 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸).

□

If a Markov process {𝑋 (𝑞) : 𝑞 ∈ 𝐸 } in 𝐷 [0, ∞) + has transition semigroup : 𝑞 ≥ 𝑝 ∈ 𝐸) defined by (11.9) or (11.10), we call it a path-valued growing process. From (11.9) we see that {𝑋 (𝑞) : 𝑞 ∈ 𝐸 } has positive increments. Let 𝑢˜ 𝑞 ( 𝑓 ) = ⟨1, 𝑢 𝑞 (·, 𝑓 )⟩. In view of (11.12), we can rewrite (11.10) as ∫ 𝜇, 𝜂 e− ⟨𝑤, 𝑓 ⟩ P 𝑝,𝑞 (𝜌, d𝑤) 𝐷 [0,∞) +  ∫ 𝑞 𝑢 𝑦 (0, 𝑢 𝑦,𝑞 (·, 𝑓 ))𝜇(d𝑦) = exp − ⟨𝜌, 𝑢 𝑝,𝑞 (·, 𝑓 )⟩ − 𝑝  ∫ 𝑞 − (11.21) 𝑢˜ 𝑦 (𝑢 𝑦,𝑞 (·, 𝑓 ))𝜂(d𝑦) . 𝜇, 𝜂 (P 𝑝,𝑞

𝑝

The above formula reveals some branching and immigration structures similar to those of the inhomogeneous transition semigroup defined by (9.33). Example 11.1 There is an inhomogeneous transition semigroup (P 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸) on 𝐷 [0, ∞) + defined by, for 𝑓 ∈ 𝐵[0, ∞) + with compact support, ∫  e− ⟨𝑤, 𝑓 ⟩ P 𝑝,𝑞 (𝜌, d𝑤) = exp − ⟨𝜌, 𝑢 𝑝,𝑞 (·, 𝑓 )⟩ . 𝐷 [0,∞) +

This is obtained by letting 𝜇 ≡ 0 in (11.16) or letting 𝜇 = 𝜂 ≡ 0 in (11.10). Clearly, the semigroup defined above satisfies the branching property: P 𝑝,𝑞 (𝜌1 + 𝜌2 , ·) = P 𝑝,𝑞 (𝜌1 , ·) ∗ P 𝑝,𝑞 (𝜌2 , ·),

𝜌1 , 𝜌2 ∈ 𝐷 [0, ∞) + .

Example 11.2 Let 𝜙 be a branching mechanism given by (3.1) and let 𝜙𝑞 (𝜆) = 𝜙(𝜆) − 𝑞𝜆 for 𝜆 ≥ 0. Then {𝜙𝑞 : 𝑞 ∈ R} is an admissible family of branching mechanisms. Example 11.3 Let 𝜙 be a branching mechanism given by (3.1) and let 𝐸 = 𝐸 (𝜙) be the set of 𝑞 ∈ R such that ∫ ∞ 𝑧e𝑞𝑧 𝑚(d𝑧) < ∞. 1

11.2 The Total Population Process

315

Using analytical extensions, we can define an admissible family of branching mechanisms {𝜙𝑞 : 𝑞 ∈ 𝐸 } by 𝜙𝑞 (𝜆) = 𝜙(𝜆 − 𝑞) − 𝜙(−𝑞),

𝜆 ≥ 0.

(11.22)

In this case, we have 𝐸 = (−∞, sup 𝐸] or (−∞, sup 𝐸).

11.2 The Total Population Process Let {𝜙𝑞 : 𝑞 ∈ [0, 𝑎]} be an admissible family of branching mechanisms. Throughout this section, we assume 𝜙𝑞 (∞) := lim𝜆→∞ 𝜙𝑞 (𝜆) = ∞ for every 𝑞 ∈ 𝐸. For 𝜃 ≥ 0 let 𝜙−1 𝑞 (𝜃) be defined by (5.33) with 𝜙 = 𝜙 𝑞 . Suppose that {(𝑋 (𝑞), ℱ𝑞 ) : 𝑞 ∈ 𝐸 } is a path-valued growing process with 𝜇 transition semigroup (P 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸) given by (11.16), where the filtration {ℱ𝑞 : 𝑞 ∈ 𝐸 } has been augmented. For the random path 𝑋 (𝑞) = {𝑋𝑡 (𝑞) : 𝑡 ≥ 0} we define ∫ 𝑡 𝑋𝑠 (𝑞)d𝑠, 0 ≤ 𝑡 ≤ ∞. 𝑥 𝑡 (𝑞) = 0

Then 𝑥∞ (𝑞) is the total population of 𝑋 (𝑞). In view of (11.9), for any 𝑝 ≤ 𝑞 ∈ 𝐸 we have P{𝑥 𝑡 ( 𝑝) ≤ 𝑥 𝑡 (𝑞) : 0 ≤ 𝑡 ≤ ∞} = 1. Theorem 11.7 Let 𝑝 ≤ 𝑞 ∈ 𝐸 and 𝐹 ∈ bℱ𝑝 . Then for any 𝜃 > 0 we have    −1 P[𝐹e−𝜃 𝑥∞ (𝑞) ] = P 𝐹 exp − 𝑥∞ ( 𝑝)𝜙 𝑝 (𝜙−1 𝑞 (𝜃)) − 𝜇( 𝑝, 𝑞]𝜙 𝑞 (𝜃) .

(11.23)

Proof Let 𝑡 ↦→ 𝑣 𝑞 (𝑡, 𝜃) be the unique positive solution to (5.37) with 𝜙 = 𝜙𝑞 . Then 𝑢 𝑞 (𝑠, 𝜃1 [0,𝑡 ] ) = 𝑣 𝑞 (𝑡 − 𝑠, 𝜃) for 0 ≤ 𝑠 ≤ 𝑡. By Proposition 5.20 we have lim 𝑢 𝑞 (𝑠, 𝜃1 [0,𝑡 ] ) = lim 𝑣 𝑞 (𝑡 − 𝑠, 𝜃) = 𝜙−1 𝑞 (𝜃).

(11.24)

𝑡→∞

𝑡→∞

Then (11.11) implies lim 𝑢 𝑝,𝑞 (𝑠, 𝜃1 [0,𝑡 ] ) = 𝜃 + 𝜙 𝑝,𝑞 (𝜙−1 𝑞 (𝜃)). 𝑡→∞

Since 𝜙 𝑝,𝑞 (𝑧) = 𝜙 𝑝 (𝑧) − 𝜙𝑞 (𝑧), we obtain lim 𝑢 𝑝,𝑞 (𝑠, 𝜃1 [0,𝑡 ] ) = 𝜙 𝑝 (𝜙−1 𝑞 (𝜃)).

(11.25)

𝑡→∞

From (11.16) and the Markov property of {(𝑋 (𝑞), ℱ𝑞 ) : 𝑞 ∈ 𝐸 } it follows that   P[𝐹e−𝜃 𝑥𝑡 (𝑞) ] = P 𝐹 exp{−⟨𝑋 ( 𝑝), 𝑢 𝑝,𝑞 (·, 𝜃1 [0,𝑡 ] )⟩ − 𝜇( 𝑝, 𝑞]𝑢 𝑞 (0, 𝜃1 [0,𝑡 ] )} . Then letting 𝑡 → ∞ gives (11.23).

□

316

11 Path-Valued Processes and Stochastic Flows

Corollary 11.8 The pair {(𝑥 ∞ (𝑞), ℱ𝑞 ) : 𝑞 ∈ 𝐸 } is a Markov process in [0, ∞] with 𝜇 ∞ as a cemetery and with transition semigroup (𝑃 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸) such that, for 𝜃 ≥ 0, ∫  𝜇 −1 e−𝜃 𝑦 𝑃 𝑝,𝑞 (𝑥, d𝑦) = exp − 𝑥𝜙 𝑝 ◦ 𝜙−1 𝑞 (𝜃) − 𝜇( 𝑝, 𝑞]𝜙 𝑞 (𝜃) . (11.26) [0,∞)

𝜇

From (11.26) we see that 𝑃 𝑝,𝑞 (𝑥, ·) → 𝛿 𝑥 in the weak convergence as 𝑞 ↓ 𝑝. Then the process {𝑥∞ (𝑞) : 𝑞 ∈ 𝐸 } has an increasing càdlàg modification. Moreover, for 𝑥 ∈ [0, ∞) we have  𝜇 −1 (11.27) 𝑃 𝑝,𝑞 (𝑥, [0, ∞)) = exp − 𝑥𝜙 𝑝 ◦ 𝜙−1 𝑞 (0) − 𝜇( 𝑝, 𝑞]𝜙 𝑞 (0) .

Now let us consider the path-valued growing process {(𝑋 (𝑞), ℱ𝑞 ) : 𝑞 ∈ 𝐸 } with one-dimensional distributions (K𝑞 : 𝑞 ∈ 𝐸) defined by, for 𝑓 ∈ 𝐵[0, ∞) + with compact support, ∫  (11.28) e− ⟨𝑤, 𝑓 ⟩ K𝑞 (d𝑤) = exp − 𝜇(𝑞)𝑢 𝑞 (0, 𝑓 ) , 𝐷 [0,∞) +

where 𝑠 ↦→ 𝑢 𝑞 (𝑠, 𝑓 ) is the unique positive solution to (11.8). This is obtained by letting 𝜂 ≡ 0 in (11.20). Let {𝑥(𝑞) : 𝑞 ∈ 𝐸 } be the increasing càdlàg modification its total population process. Let 𝐴 = inf{𝑞 ∈ 𝐸 : 𝑥(𝑞) = ∞} be the explosion time. By Corollary 5.21, for any 𝜃 ≥ 0 we have     −1 P e−𝜃 𝑥 (𝑞) 1 { 𝐴>𝑞 } = P e−𝜃 𝑥 (𝑞) 1 {𝑥 (𝑞) 0 and 𝜙 𝑞 (𝜙 𝑞 (0)) > 0. In the case of 𝜇(𝑞) > 𝜇(𝑞−), we have P( 𝐴 = 𝑞) > 0 and



P e

−𝜃 𝑥 (𝑞−)



|𝐴 = 𝑞 =

−1 ( 𝜃)

− e−𝜇 (𝑞) 𝜙𝑞

−1 (0)

− e−𝜇 (𝑞) 𝜙𝑞

e−𝜇 (𝑞−) 𝜙𝑞

e−𝜇 (𝑞−) 𝜙𝑞

−1 ( 𝜃)

−1 (0)

,

(11.32)

which characterizes the conditional distribution of the total population size just before its explosion time. In the case of 𝜇(𝑞) = 𝜇(𝑞−) > 0, by differentiating both sides of (11.30) in 𝜃 ≥ 0 we obtain

11.2 The Total Population Process

317 −1

  𝜇(𝑞)e−𝜇 (𝑞) 𝜙𝑞 ( 𝜃) P 𝑥(𝑞−)e−𝜃 𝑥 (𝑞−) 1 { 𝐴≥𝑞 } = , 𝜙𝑞′ (𝜙−1 𝑞 (𝜃))

(11.33)

and hence −1



0 < P 𝑥(𝑞−)1 { 𝐴≥𝑞 }



𝜇(𝑞)e−𝜇 (𝑞) 𝜙𝑞 (0) = < ∞. 𝜙𝑞′ (𝜙−1 𝑞 (0))

(11.34)

Under natural conditions, the next theorem gives a rigorous characterization of the restriction to ℱ𝑞− of the conditional law P(·| 𝐴 = 𝑞) when P( 𝐴 = 𝑞) = 0. Theorem 11.9 Suppose that 𝜇 ∈ 𝐹 (𝐸) is differentiable at some point 𝑞 ∈ 𝐸 \{inf 𝐸 } and 𝜇(𝑞) > 0, 𝑏 𝑞 = 𝜙𝑞′ (0) < 0,

d −1 𝜙 (0) 𝑟=𝑞 + 𝜇 ′ (𝑞) > 0. d𝑟 𝑟

(11.35)

Then for any 𝐺 ∈ bℱ𝑞− we have oi h n d 𝜙𝑟−1 (0) 𝑟=𝑞 + 𝜇′ (𝑞) 𝜙𝑞−1 (0) P 𝐺1{ 𝐴≥𝑞} 𝑥 (𝑞−) 𝜙𝑞′ ( 𝜙𝑞−1 (0)) d𝑟 h n oi . P 𝐺|𝐴 = 𝑞 = d P 1{ 𝐴≥𝑞} 𝑥 (𝑞−) 𝜙𝑞′ ( 𝜙𝑞−1 (0)) d𝑟 𝜙𝑟−1 (0) 𝑟=𝑞 + 𝜇′ (𝑞) 𝜙𝑞−1 (0)


Proof By (11.31), we have P( 𝐴 = 𝑞) = 0 for 𝑞 ∈ 𝐸 \ {inf 𝐸 }. We first consider 𝐺 ∈ bℱ𝑝 for some 𝑞 > 𝑝 ∈ 𝐸. By (11.27) and the Markov property,  
P 𝐺1 { 𝐴>𝑟 } = P 𝐺1 { 𝐴>𝑞 } exp{−𝑥(𝑞)𝜙𝑞 ◦ 𝜙𝑟−1 (0) − 𝜇(𝑞, 𝑟]𝜙−1 𝑞 (0)} . Since 𝜇(𝑞) = 𝜇(𝑞−), the above relation implies  
P 𝐺1 { 𝐴≥𝑟 } = P 𝐺1 { 𝐴≥𝑞 } exp{−𝑥(𝑞−)𝜙𝑞 ◦ 𝜙𝑟−1 (0) − 𝜇[𝑞, 𝑟]𝜙−1 𝑞 (0)} . By taking the derivatives we obtain −

h n o i
d d P 𝐺1{ 𝐴≥𝑟 } 𝑟=𝑞 = P 𝐺1{ 𝐴≥𝑞} 𝑥 (𝑞−) 𝜙𝑞 ◦ 𝜙𝑟−1 (0) + 𝜇′ (𝑟) 𝜙𝑞−1 (0) 𝑟=𝑞 d𝑟 d𝑟

and −

h n o i d d P( 𝐴 ≥ 𝑟) | 𝑟=𝑞 = P 1{ 𝐴≥𝑞} 𝑥 (𝑞−) 𝜙𝑞 ◦ 𝜙𝑟−1 (0) + 𝜇′ (𝑟) 𝜙𝑞−1 (0) , 𝑟=𝑞 d𝑟 d𝑟

where d d −1 𝜙𝑞 ◦ 𝜙𝑟−1 (0) = 𝜙𝑞′ (𝜙−1 (0)) 𝜙 (0) . 𝑞 𝑟=𝑞 𝑟=𝑞 d𝑟 d𝑟 𝑟

(11.36)

′ −1 Under condition (11.35), we have 𝜙−1 𝑞 (0) > 0 and 𝜙 𝑞 (𝜙 𝑞 (0)) > 0. From (11.34) and (11.36) we see that − d𝑟d P( 𝐴 ≥ 𝑟)| 𝑟=𝑞 > 0. Then the result holds for 𝐺 ∈ bℱ𝑝 . By a monotone class argument we get the desired result for 𝐺 ∈ bℱ𝑞− . □

318

11 Path-Valued Processes and Stochastic Flows

Corollary 11.10 In the setup of Theorem 11.9, if 𝜇 ′ (𝑞) = 0, then for any 𝜃 ≥ 0 we have −1 ( 𝜃)



P e

−𝜃 𝑥 (𝑞−)



|𝐴 = 𝑞 =

−𝜇 (𝑞) 𝜙𝑞 𝜙𝑞′ (𝜙−1 𝑞 (0))e

−1 (0)

(11.37)

.

−𝜇 (𝑞) 𝜙𝑞 𝜙𝑞′ (𝜙−1 𝑞 (𝜃))e

Proof Since 𝜇 ′ (𝑞) = 0, by the result of Theorem 11.9 we have P(e−𝜃 𝑥 (𝑞−) | 𝐴 = 𝑞) =

P[e−𝜃 𝑥 (𝑞−) 𝑥(𝑞−)1 { 𝐴≥𝑞 } ] . P[𝑥(𝑞−)1 { 𝐴≥𝑞 } ]

Then (11.37) follows from (11.33) and (11.34).

□

Example 11.4 Let {𝜙𝑞 : 𝑞 ∈ 𝐸 } be the admissible family defined by (11.22). For −1 𝑞 ≥ 𝑝 ∈ 𝐸 it is elementary to see that 𝜙−1 𝑞 (𝜃) = 𝑞 + 𝜙 (𝜃 + 𝜙(−𝑞)) and −1 𝜙 𝑝 (𝜙−1 𝑞 (𝜃)) = 𝜙(𝑞 − 𝑝 + 𝜙 (𝜃 + 𝜙(−𝑞))) − 𝜙(−𝑝),

𝜃 ≥ 0.

Suppose that 𝜙 ′ (−𝑞) < 0 and 𝜇 ′ (𝑞) = 0 for some 𝑞 ∈ 𝐸. By Corollary 11.10 we have P[e−𝜃 𝑥 (𝑞−) | 𝐴 = 𝑞] =

𝜙 ′ (𝜙−1 (𝜙(−𝑞))) exp{−𝜇(𝑞) [𝑞 + 𝜙−1 (𝜃 + 𝜙(−𝑞))]} . 𝜙 ′ (𝜙−1 (𝜃 + 𝜙(−𝑞))) exp{−𝜇(𝑞) [𝑞 + 𝜙−1 (𝜙(−𝑞))]} 𝜇

In a typical situation, the transition semigroup (𝑃 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸) given by (11.26) can be characterized in terms of an inhomogeneous CBI-process. To see this, we set 𝑢 𝑝,𝑞 (𝜃) = 𝜙 𝑝 ◦ 𝜙−1 𝑞 (𝜃) and write 𝜇( 𝑝, 𝑞]𝜙−1 𝑞 (𝜃)

∫

∫

𝑞

𝜙−1 𝑞 (𝜃)𝜇(d𝑠)

=

𝑞

𝜙−1 𝑠 (𝑢 𝑠,𝑞 (𝜃))𝜇(d𝑠).

=

𝑝

𝑝

By (11.26) we have   ∫ ∫ 𝑞 −𝜃 𝑦 𝜇 e 𝑃 𝑝,𝑞 (𝑥, d𝑦) = exp −𝑥𝑢 𝑝,𝑞 (𝜃) − 𝐼 (𝑠, 𝑢 𝑠,𝑞 (𝜃))𝜇(d𝑠) [0,∞) 𝑝  ∫ 𝑞  · exp − 𝜙−1 (0)𝜇(d𝑠) , (11.38) 𝑠 𝑝 −1 where 𝐼 (𝑠, ·) = 𝜙−1 𝑠 (·) − 𝜙 𝑠 (0) ∈ ℐ by Corollary 5.23.

Proposition 11.11 For any 𝑞 ∈ 𝐸 and 𝜃 ≥ 0, the function 𝑝 ↦→ 𝑢 𝑝,𝑞 (𝜃) = 𝜙 𝑝 ◦ 𝜙−1 𝑞 (𝜃) is the unique locally bounded positive solution on 𝐸 ≤𝑞 of ∫

𝑞

𝜁 𝑠 (𝜙−1 𝑠 (0))d𝑠 +

𝑢 𝑝,𝑞 (𝜃) = 𝜃 + 𝑝

−1 where 𝜓(𝑠, ·) = 𝜁 𝑠 ◦ 𝜙−1 𝑠 − 𝜁 𝑠 ◦ 𝜙 𝑠 (0) ∈ ℐ.

∫

𝑞

𝜓(𝑠, 𝑢 𝑠,𝑞 (𝜃))d𝑠, 𝑝

(11.39)

11.2 The Total Population Process

319

−1 Proof Since 𝜁 𝑠 ∈ ℐ, we have 𝐽𝑠 := 𝜁 𝑠 (· + 𝜙−1 𝑠 (0)) − 𝜁 𝑠 (𝜙 𝑠 (0)) ∈ ℐ, and so 𝜓(𝑠, ·) = 𝐽𝑠 ◦𝐼 (𝑠, ·) ∈ ℐ by Theorem 1.38. Using the relation (d/d𝑠)𝜙 𝑠 (𝜃) = −𝜁 𝑠 (𝜃), we get

d d −1 −1 𝑢 𝑠,𝑞 (𝜃) = 𝜙 𝑠 (𝜙−1 𝑞 (𝜃)) = −𝜁 𝑠 ◦ 𝜙 𝑞 (𝜃) = −𝜁 𝑠 ◦ 𝜙 𝑠 (𝑢 𝑠,𝑞 (𝜃)). d𝑠 d𝑠 This proves (11.39).

□

Let 𝑈 ⊂ 𝐸 be an interval that does not contain critical points. By elementary calculations, for 𝑠 ∈ 𝑈 we have 1 d d −1 𝐼 (𝑠, 𝜃) 𝜙 (𝜃) = ′ −1 = 0.

This process can be obtained from two homogeneous CB-processes by simple transformations. To see this, let √ 𝑣 −𝑡 (𝜃) = e−2𝑡 𝜃 + 2e−𝑡 (1 − e−𝑡 ) ( 1 + 𝜃 − 1), 𝑡 ≥ 0, 𝜃 ≥ 0. It is easy to check that −2𝑡 𝑣 −𝑡−𝑟 (𝜃) = e2𝑟 𝜙−e−𝑟 ◦ 𝜙−1 𝜃), −e−𝑡 (e

𝜃 ≥ 0, 𝑡 ≥ 𝑟 ∈ R.

From (11.26) one can see that {e−2𝑡 𝑥(−e−𝑡 ) : 𝑡 ∈ R} has homogeneous transition semigroup (𝑃𝑡− )𝑡 ≥0 defined by ∫ − e−𝜃 𝑦 𝑃𝑡− (𝑥, d𝑦) = e−𝑥𝑣𝑡 ( 𝜃) , 𝜃 ≥ 0. [0,∞)

Moreover, we have d − 𝑣 (𝜃) = −𝜙− (𝑣 −𝑡 (𝜃)), d𝑡 𝑡

𝜃 ≥ 0, 𝑡 ≥ 0,

11.3 Construction by Stochastic Equations

321

where √ 𝜙− (𝜆) = 2𝜆 − 2( 1 + 𝜆 − 1),

𝜆 ≥ 0.

Then {e−2𝑡 𝑥(−e−𝑡 ) : 𝑡 ∈ R} is actually a homogeneous CB-process in [0, ∞) with branching mechanism 𝜙− . Similarly, one can see {e2𝑡 𝑥(𝑒 𝑡 ) : 𝑡 ∈ R} is a homogeneous sub-Markov process with transition semigroup (𝑃𝑡+ )𝑡 ≥0 defined by ∫ + e−𝜃 𝑦 𝑃𝑡+ (𝑥, d𝑦) = e−𝑥𝑣𝑡 ( 𝜃) , 𝜃 ≥ 0, [0,∞)

where √ 𝑣 +𝑡 (𝜃) = e2𝑡 𝜃 + 2e𝑡 (e𝑡 − 1) ( 1 + 𝜃 + 1). One can also see that d + 𝑣 (𝜃) = −𝜙+ (𝑣 +𝑡 (𝜃)) + 4, d𝑡 𝑡 where √ 𝜙+ (𝜃) = −2𝜃 − 2( 1 + 𝜃 + 1) + 4. Then {e2𝑡 𝑥(𝑒 𝑡 ) : 𝑡 ∈ R} is the subprocess of a homogeneous CB-process {𝑦(𝑡) : 𝑡 ∈ R} with branching mechanism 𝜙+ generated by the multiplicative functional   ∫ 𝑡 𝑀 (𝑟, 𝑡) ↦→ exp − 4 𝑦(𝑠)d𝑠 , 𝑡 ≥ 𝑟 ∈ R. 𝑟

11.3 Construction by Stochastic Equations Let 𝐸 = [0, 𝑎] for some 𝑎 > 0. Suppose that 𝜇, 𝜂 ∈ 𝐹 [0, 𝑎] and {𝜙𝑞 : 𝑞 ∈ [0, 𝑎]} is an admissible family of branching mechanisms, where 𝜙𝑞 is given by (3.1) with the parameters (𝑏, 𝑚) = (𝑏 𝑞 , 𝑚 𝑞 ) depending on 𝑞 ∈ [0, 𝑎]. We shall give a reconstruction of the path-valued growing process with transition semigroup 𝜇, 𝜂 (P 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ [0, 𝑎]) and one-dimensional distributions {K𝑞 : 𝑞 ∈ [0, 𝑎]} by strong solutions to a system of stochastic equations. Suppose that (Ω, 𝒢, 𝒢𝑡 , P) is a filtered probability space satisfying the usual hypotheses. Let {𝑊 (d𝑠, d𝑢)} be a time–space (𝒢𝑡 )-Gaussian white noise on (0, ∞) 2 with intensity 2𝑐d𝑠d𝑢. Let { 𝑀˜ (d𝑠, d𝑦, d𝑧, d𝑢)} be a compensated time–space (𝒢𝑡 )Poisson random measure on (0, ∞) × [0, 𝑎] × (0, ∞) 2 with intensity d𝑠𝑚(d𝑦, d𝑧)d𝑢, where 𝑚(d𝑦, d𝑧) is the unique 𝜎-finite measure on [0, 𝑎] × (0, ∞) such that 𝑚( [0, 𝑞] × 𝐵) = 𝑚 𝑞 (𝐵),

𝑞 ∈ [0, 𝑎], 𝐵 ∈ ℬ(0, ∞).

322

11 Path-Valued Processes and Stochastic Flows

Proposition 11.13 For any 𝑝 ≤ 𝑞 ∈ [0, 𝑎] and 𝜌 ∈ 𝐷 [0, ∞) + , there is a pathwise unique positive solution {𝜉 𝑝,𝑞 (𝑡) : 𝑡 ≥ 0} to ∫ 𝑡 ∫ 𝜉 (𝑠−) 𝑊 (d𝑠, 𝜌(𝑠−) + d𝑢) + 𝜂( 𝑝, 𝑞]𝑡 𝜉 (𝑡) = 𝜇( 𝑝, 𝑞] + ∫ 𝑡 0 0 ∫ 𝑞 ∫ 𝑡 − 𝑏𝑞 𝛽 𝑦 d𝑦 𝜌(𝑠−)d𝑠 𝜉 (𝑠−)d𝑠 + 0 0 𝑝 ∫ 𝑡∫ ∫ ∞ ∫ 𝜉 (𝑠−) 𝑧 𝑀˜ (d𝑠, d𝑦, d𝑧, 𝜌(𝑠−) + d𝑢) + 0 [0,𝑞] 0 0 ∫ 𝑡 ∫ 𝑞 ∫ ∞ ∫ 𝜌(𝑠−) + 𝑧𝑀 (d𝑠, d𝑦, d𝑧, d𝑢). 0

𝑝

0

(11.43)

0

Moreover, the solution {𝜉 𝑝,𝑞 (𝑡) : 𝑡 ≥ 0} is an inhomogeneous CBI-process with 𝑝,𝑞, 𝜂,𝜌 transition semigroup (𝑄 𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ≥ 0) defined by (11.6). Proof It is known that 𝑊1 (d𝑠, d𝑢) = 𝑊 (d𝑠, 𝜌(𝑠−) + d𝑢) defines a time–space (𝒢𝑡 )-Gaussian white noise on (0, ∞) 2 with intensity 2𝑐d𝑠d𝑢. Observe also that ∫ 𝑀1 (d𝑠, d𝑧, d𝑢) = 𝑀 (d𝑠, d𝑦, d𝑧, 𝜌(𝑠−) + d𝑢) {0≤𝑦 ≤𝑞 }

defines a time–space (𝒢𝑡 )-Poisson random measure on (0, ∞) 3 with intensity d𝑠𝑚 𝑞 (d𝑧)d𝑢, and ∫ ∫ 𝑀 (d𝑠, d𝑦, d𝑧, d𝑢) 𝑁1 (d𝑠, d𝑧) = { 𝑝 0, define 𝑍𝑡 ∈ 𝐹 [0, 𝑎] by 𝑍𝑡 (𝑎) = 𝑋𝑡− (𝑎) and 0 ≤ 𝑞 < 𝑎.

𝑍𝑡 (𝑞) = inf{𝑋𝑡− (𝑣) : 𝑣 ∈ 𝑄 𝐸 ∩ (𝑞, 𝑎]}, By Proposition 11.16, for each 𝑞 ∈ 𝐸 we have

P{𝑌𝑡 (𝑞) = 𝑋𝑡 (𝑞) and 𝑍𝑡 (𝑞) = 𝑋𝑡− (𝑞) for all 𝑡 ≥ 0} = 1.

(11.48)

Consequently, for every 𝑞 ∈ [0, 𝑎] the process {𝑌𝑡 (𝑞) : 𝑡 ≥ 0} is a.s. càdlàg and solves (11.47), so it is a CBI-process with distribution K𝑞 on 𝐷 [0, ∞) + given by (11.20). In view of (11.4) and (11.5), for each 𝑞 ∈ [0, 𝑎] we have ∫ 𝑡 ∫ 𝑞 ∫ 𝑡 ∫ 𝑍𝑠 (𝑞) 𝑍 𝑠 (𝑞)d𝑠 𝑊 (d𝑠, d𝑢) + 𝑌𝑡 (𝑞) = 𝜇(𝑞) + 𝛽 𝑦 d𝑦 0 0 ∫ 𝑡0 0 ∫ 𝑡∫ ∫ ∞∫ 𝑍𝑠 (𝑞) − 𝑏0 𝑍 𝑠 (𝑞)d𝑠 + 𝑧 𝑀˜ (d𝑠, d𝑦, d𝑧, d𝑢) {0} 0 0 0 0 ∫ 𝑡 ∫ 𝑞 ∫ ∞ ∫ 𝑍𝑠 (𝑞) + 𝜂(𝑞)𝑡 + 𝑧𝑀 (d𝑠, d𝑦, d𝑧, d𝑢). (11.49) 0

0

0

0

Let 𝑌𝑡 (d𝑥) and 𝑍𝑡 (d𝑥) denote the random measures on [0, 𝑎] induced by 𝑌𝑡 and 𝑍𝑡 ∈ 𝐹 [0, 𝑎], respectively. For any 𝑓 ∈ 𝐶 1 [0, 𝑎] one can use Fubini’s theorem to see ∫ 𝑎 (11.50) ⟨𝑌𝑡 , 𝑓 ⟩ = 𝑓 (𝑎)𝑌𝑡 (𝑎) − 𝑓 ′ (𝑞)𝑌𝑡 (𝑞)d𝑞. 0

There is a similar relation for 𝑍𝑡 . Fix an integer 𝑛 ≥ 1 and let 𝑞 𝑖 = 𝑖𝑎/2𝑛 for 𝑖 = 0, 1, · · · , 2𝑛 . From (11.49) it follows that 2𝑛 ∑︁ 𝑖=1

′

𝑓 (𝑞 𝑖 )𝑌𝑡 (𝑞 𝑖 ) =

2𝑛 ∑︁

2𝑛 ∑︁

′

𝑓 (𝑞 𝑖 )𝜇(𝑞 𝑖 ) +

𝑖=1

+

2𝑛 ∑︁

𝑓 ′ (𝑞 𝑖 )

+

𝑓 ′ (𝑞 𝑖 )

+

′

+

𝑖=1

∫

𝑡

{0}

0

𝑞𝑖

∫

∫

𝐸

∞ ∫ 𝑍𝑠 (𝑞𝑖 )

∫

∫

𝑧 𝑀˜ (d𝑠, d𝑦, d𝑧, d𝑢)

0 ∞ ∫ 𝑍𝑠 (𝑞𝑖 )

𝑧𝑀 (d𝑠, d𝑦, d𝑧, d𝑢)

𝑓 (𝑞 𝑖 ) 0 ′

1 {𝑥 ≤𝑞𝑖 } 𝑍 𝑠 (d𝑥)

d𝑠 0

𝑡

0

∫

𝑡

𝛽 𝑦 d𝑦

0

𝑖=1 2𝑛 ∑︁

∫

𝑞𝑖

∫

𝑍𝑠 (𝑞𝑖 )

∫

𝑊 (d𝑠, d𝑢) 0

0

𝑖=1 2𝑛 ∑︁

∫

𝑡

𝑓 (𝑞 𝑖 )

𝑖=1

𝑖=1 2𝑛 ∑︁

∫

′

∫

0

0

0

𝑡

[𝜂(𝑞 𝑖 ) − 𝑏 0 𝑍 𝑠 (𝑞 𝑖 )]d𝑠

𝑓 (𝑞 𝑖 ) 0

326

11 Path-Valued Processes and Stochastic Flows

=

2𝑛 ∑︁

∫

′

𝑡

𝑍𝑠 (𝑎)

∫

𝑓 (𝑞 𝑖 )𝜇(𝑞 𝑖 ) +

𝐹𝑛 (𝑠, 0, 𝑢)𝑊 (d𝑠, d𝑢) 0

0

𝑖=1

∫ +

∫

∫

𝑡

0

∫

𝑎

𝐹𝑛 (𝑠, 𝑥 ∨ 𝑦, 0) 𝛽 𝑦 d𝑦

𝑍 𝑠 (d𝑥)

d𝑠 𝐸 𝑡∫

∞∫

∫

0 𝑍𝑠 (𝑎)

𝑧𝐹𝑛 (𝑠, 0, 𝑢) 𝑀˜ (d𝑠, d𝑦, d𝑧, d𝑢)

+ 0

∫

0 {0} 0 𝑡 ∫ 𝑎 ∫ ∞ ∫ 𝑍𝑠 (𝑎)

𝑧𝐹𝑛 (𝑠, 𝑦, 𝑢) 𝑀 (d𝑠, d𝑦, d𝑧, d𝑢)

+ 0

∫ +

0 2𝑛 𝑡 ∑︁

0

0

0

  𝑓 ′ (𝑞 𝑖 ) 𝜂(𝑞 𝑖 ) − 𝑏 0 𝑍 𝑠 (𝑞 𝑖 ) d𝑠,

(11.51)

𝑖=1

where 2𝑛 ∑︁

𝐹𝑛 (𝑠, 𝑦, 𝑢) =

𝑓 ′ (𝑞 𝑖 )1 {𝑦 ≤𝑞𝑖 } 1 {𝑢≤𝑍𝑠 (𝑞𝑖 ) } .

𝑖=1

By the right continuity of 𝑞 ↦→ 𝑍 𝑠 (𝑞) it is not hard to see that, as 𝑛 → ∞, ∫ 𝑎 2−𝑛 𝐹𝑛 (𝑠, 𝑦, 𝑢) → 𝐹 (𝑠, 𝑦, 𝑢) := 1 {𝑢 ≤𝑍𝑠 (𝑞) } 𝑓 ′ (𝑞)d𝑞. 𝑦

Then we can multiply (11.51) by ∫

∫

𝑎 ′

2−𝑛

and let 𝑛 → ∞ to get ∫

𝑎 ′

𝑡

𝑓 (𝑞)𝜇(𝑞)d𝑞 + 0 0∫ ∫ ∫ ∞ ∫ 𝑍𝑠 (𝑎) 𝑡

𝑓 (𝑞)𝑌𝑡 (𝑞)d𝑞 = 0

+

∫

𝑍𝑠 (𝑎)

𝐹 (𝑠, 0, 𝑢)𝑊 (d𝑠, d𝑢) 0

𝑧𝐹 (𝑠, 0, 𝑢) 𝑀˜ (d𝑠, d𝑦, d𝑧, d𝑢)

∫0 𝑡 ∫{0}𝑎 ∫0 ∞ ∫0 𝑍𝑠 (𝑎) +

𝑧𝐹 (𝑠, 𝑦, 𝑢)𝑀 (d𝑠, d𝑦, d𝑧, d𝑢) ∫0 𝑡

+

0∫ 0

0

∫

𝑎

𝑍 𝑠 (d𝑥) 𝐹 (𝑠, 𝑥 ∨ 𝑦, 0) 𝛽 𝑦 d𝑦 0 ∫𝐸 𝑎 d𝑠 𝑓 ′ (𝑞) [𝜂(𝑞) − 𝑏 0 𝑍 𝑠 (𝑞)]d𝑞.

d𝑠 ∫0 𝑡

+ 0

(11.52)

0

From (11.49), (11.50) and (11.52) it follows that ∫ 𝑡 ∫ 𝑍𝑠 (𝑎) ⟨𝑌𝑡 , 𝑓 ⟩ = ⟨𝜇, 𝑓 ⟩ + [ 𝑓 (𝑎) − 𝐹 (𝑠, 0, 𝑢)]𝑊 (d𝑠, d𝑢) ∫ 𝑡 ∫ 0∫ ∞0 ∫ 𝑍𝑠 (𝑎) 𝑧[ 𝑓 (𝑎) − 𝐹 (𝑠, 0, 𝑢)] 𝑀˜ (d𝑠, d𝑦, d𝑧, d𝑢) + 0 0 {0} 0 ∫ 𝑡 ∫ 𝑎 ∫ ∞ ∫ 𝑍𝑠 (𝑎) 𝑧[ 𝑓 (𝑎) − 𝐹 (𝑠, 𝑦, 𝑢)] 𝑀 (d𝑠, d𝑦, d𝑧, d𝑢) + 0

0

0

0

11.4 A Stochastic Flow of Measures

∫

∫

𝑡

+

𝐸

𝑎

𝑓 (𝑥 ∨ 𝑦) 𝛽 𝑦 d𝑦

𝑍 𝑠 (d𝑥)

d𝑠 ∫0 𝑡

327

∫ 0

[⟨𝜂, 𝑓 ⟩ − 𝑏 0 ⟨𝑍 𝑠 , 𝑓 ⟩]d𝑠.

+

(11.53)

0

Proposition 11.18 The measure-valued process {𝑌𝑡 : 𝑡 ≥ 0} has a càdlàg modification. Proof By (11.53) one can see {⟨𝑌𝑡 , 𝑓 ⟩ : 𝑡 ≥ 0} has a càdlàg modification for every 𝑓 ∈ 𝐶 1 (𝐸). Let 𝒰 be the countable set of polynomials having rational coefficients. Then 𝒰 is uniformly dense in both 𝐶 1 (𝐸) and 𝐶 (𝐸). For 𝑓 ∈ 𝒰 let {𝑌𝑡∗ ( 𝑓 ) : 𝑡 ≥ 0} be a càdlàg modification of {⟨𝑌𝑡 , 𝑓 ⟩ : 𝑡 ≥ 0}. By removing a null set from Ω if necessary, we obtain a càdlàg process {𝑌𝑡∗ : 𝑡 ≥ 0} of rational linear functionals on 𝒰, which can immediately be extended to a càdlàg process of real linear functionals on 𝐶 (𝐸). By Riesz’s representation, the latter determines a càdlàg measure-valued □ process, which is clearly a modification of {𝑌𝑡 : 𝑡 ≥ 0}. Proposition 11.19 The càdlàg modification of {𝑌𝑡 : 𝑡 ≥ 0} solves the following martingale problem: For every 𝐺 ∈ 𝐶 2 (R) and 𝑓 ∈ 𝐶 (𝐸), ∫ 𝑡 ∫ ∫ 𝑌𝑠 (d𝑥) 𝐺 (⟨𝑌𝑡 , 𝑓 ⟩) = 𝐺 (⟨𝜇, 𝑓 ⟩) + 𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩)d𝑠 𝑓 (𝑥 ∨ 𝑦) 𝛽 𝑦 d𝑦 0 𝐸∫ 𝐸 ∫ 𝑡 𝑡 𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩)⟨𝑌𝑠 , 𝑓 ⟩d𝑠 + 𝑐 − 𝑏0 𝐺 ′′ (⟨𝑌𝑠 , 𝑓 ⟩)⟨𝑌𝑠 , 𝑓 2 ⟩d𝑠 0 ∫ ∞ ∫ 𝑡 0 ∫ 
+ 𝐺 ⟨𝑌𝑠 , 𝑓 ⟩ + 𝑧 𝑓 (𝑥) − 𝐺 (⟨𝑌𝑠 , 𝑓 ⟩) 𝑌𝑠 (d𝑥) d𝑠 𝐸 0 0 − 𝑧 𝑓 (𝑥)𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩) 𝑚 0 (d𝑧) + local mart. ∫ ∫ 𝑡 ∫ ∫ ∞ 
d𝑠 𝑌𝑠 (d𝑥) 𝐺 ⟨𝑌𝑠 , 𝑓 ⟩ + 𝑧 𝑓 (𝑥 ∨ 𝑦) d𝑦 + 𝐸 0 𝐸 ∫ 0 𝑡  𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩)⟨𝜂, 𝑓 ⟩d𝑠. − 𝐺 (⟨𝑌𝑠 , 𝑓 ⟩) 𝑛 𝑦 (d𝑧) + (11.54) 0

Proof We first assume 𝑓 ∈ 𝐶 1 (𝐸). Since the càdlàg process {𝑌𝑡 : 𝑡 ≥ 0} has at most countably many discontinuity points, by (11.53) and Itô’s formula, we get ∫ 𝑡   𝐺 (⟨𝑌𝑡 , 𝑓 ⟩) = 𝐺 (⟨𝜇, 𝑓 ⟩) + 𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩) ⟨𝜂, 𝑓 ⟩ − 𝑏 0 ⟨𝑍 𝑠 , 𝑓 ⟩ d𝑠 ∫ 𝑡 ∫ 𝑍0𝑠 (𝑎)  2 𝐺 ′′ (⟨𝑌𝑠 , 𝑓 ⟩) 𝑓 (𝑎) − 𝐹 (𝑠, 0, 𝑢) d𝑢 d𝑠 +𝑐 0 ∫ ∫ ∫ 𝑡0 ′ 𝑍 𝑠 (d𝑥) 𝐺 (⟨𝑌𝑠 , 𝑓 ⟩)d𝑠 + 𝑓 (𝑥 ∨ 𝑦) 𝛽 𝑦 d𝑦 𝐸 ∫0 𝑡 ∫ 𝑍𝑠 (𝑎) ∫ 𝐸∞ 
+ d𝑢 d𝑠 𝐺 ⟨𝑌𝑠 , 𝑓 ⟩ + 𝑧[ 𝑓 (𝑎) − 𝐹 (𝑠, 0, 𝑢)] 0 0 0  − 𝐺 (⟨𝑌𝑠 , 𝑓 ⟩) − 𝑧[ 𝑓 (𝑎) − 𝐹 (𝑠, 0, 𝑢)]𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩) 𝑚 0 (d𝑧)

11 Path-Valued Processes and Stochastic Flows

328

∫ +

𝑍𝑠 (𝑎)

∫

𝑡

∫

∫ d𝑢

d𝑠

d𝑦

∞

𝐺 ⟨𝑌𝑠 , 𝑓 ⟩ + 𝑧[ 𝑓 (𝑎) − 𝐹 (𝑠, 𝑦, 𝑢)]




0 0 𝐸 0  − 𝐺 (⟨𝑌𝑠 , 𝑓 ⟩) 𝑛 𝑦 (d𝑧) + local mart.

For 𝑠, 𝑢 > 0 let 𝑍 𝑠−1 (𝑢) = inf{𝑞 ≥ 0 : 𝑍 𝑠 (𝑞) > 𝑢}. It is easy to see that {𝑞 ≥ 0 : 𝑢 ≤ 𝑍 𝑠 (𝑞)} = [𝑍 𝑠−1 (𝑢), ∞), except for at most countably many 𝑢 > 0. Then the above equality remains true when ∫ 𝑎 1 {𝑢 ≤𝑍𝑠 (𝑞) } 𝑓 ′ (𝑞)d𝑞 𝑓 (𝑎) − 𝐹 (𝑠, 𝑦, 𝑢) = 𝑓 (𝑎) − 𝑦

is replaced by ∫ 𝑓 (𝑎) −

𝑎

1 {𝑍𝑠−1 (𝑢) ≤𝑞 } 𝑓 ′ (𝑞)d𝑞 = 𝑓 (𝑍 𝑠−1 (𝑢) ∨ 𝑦).

𝑦

It follows that ∫ 𝑡   𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩) ⟨𝜂, 𝑓 ⟩ − 𝑏 0 ⟨𝑍 𝑠 , 𝑓 ⟩ d𝑠 𝐺 (⟨𝑌𝑡 , 𝑓 ⟩) = 𝐺 (⟨𝜇, 𝑓 ⟩) + ∫ 𝑡 ∫ 𝑍0𝑠 (𝑎) +𝑐 𝐺 ′′ (⟨𝑌𝑠 , 𝑓 ⟩) 𝑓 (𝑍 𝑠−1 (𝑢)) 2 d𝑢 d𝑠 0 ∫ ∫ 𝑡0 ∫ ′ 𝑓 (𝑥 ∨ 𝑦) 𝛽 𝑦 d𝑦 + 𝐺 (⟨𝑌𝑠 , 𝑓 ⟩)d𝑠 𝑍 𝑠 (d𝑥) 𝐸 ∫0 𝑡 ∫ 𝑍𝑠 (𝑎) ∫ 𝐸∞ 
d𝑢 𝐺 ⟨𝑌𝑠 , 𝑓 ⟩ + 𝑧 𝑓 (𝑍 𝑠−1 (𝑢)) d𝑠 + 0 0 0  −1 − 𝐺 (⟨𝑌𝑠 , 𝑓 ⟩) − 𝑧 𝑓 (𝑍 𝑠 (𝑢))𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩) 𝑚 0 (d𝑧) ∫ 𝑡 ∫ 𝑍𝑠 (𝑎) ∫ ∫ ∞ 
d𝑢 + d𝑠 d𝑦 𝐺 ⟨𝑌𝑠 , 𝑓 ⟩ + 𝑧 𝑓 (𝑍 𝑠−1 (𝑢) ∨ 𝑦) 0 0 𝐸 0  − 𝐺 (⟨𝑌𝑠 , 𝑓 ⟩) 𝑛 𝑦 (d𝑧) + local mart. ∫ 𝑡   𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩) ⟨𝜂, 𝑓 ⟩ − 𝑏 0 ⟨𝑍 𝑠 , 𝑓 ⟩ d𝑠 = 𝐺 (⟨𝜇, 𝑓 ⟩) + ∫ 𝑡 ∫ 0 d𝑠 +𝑐 𝐺 ′′ (⟨𝑌𝑠 , 𝑓 ⟩) 𝑓 (𝑥) 2 𝑍 𝑠 (d𝑥) 0 𝐸 ∫ ∫ 𝑡 ∫ 𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩)d𝑠 𝑓 (𝑥 ∨ 𝑦) 𝛽 𝑦 d𝑦 + 𝑍 𝑠 (d𝑥) 𝐸 ∫ 𝐸∞ ∫0 𝑡 ∫ 
d𝑠 + 𝑍 𝑠 (d𝑥) 𝐺 ⟨𝑌𝑠 , 𝑓 ⟩ + 𝑧 𝑓 (𝑥) 0 𝐸 0  − 𝐺 (⟨𝑌𝑠 , 𝑓 ⟩) − 𝑧 𝑓 (𝑥)𝐺 ′ (⟨𝑌𝑠 , 𝑓 ⟩) 𝑚 0 (d𝑧) ∫ ∞ ∫ ∫ 𝑡 ∫ 
+ d𝑦 𝐺 ⟨𝑌𝑠 , 𝑓 ⟩ + 𝑧 𝑓 (𝑥 ∨ 𝑦) d𝑠 𝑍 𝑠 (d𝑥) 0 𝐸 0 𝐸  − 𝐺 (⟨𝑌𝑠 , 𝑓 ⟩) 𝑛 𝑦 (d𝑧) + local mart.

11.4 A Stochastic Flow of Measures

329

For each 𝑞 ∈ [0, 𝑎] the càdlàg process {𝑋𝑡 (𝑞) : 𝑡 ≥ 0} has at most countably many discontinuity points 𝐴𝑞 := {𝑡 > 0 : 𝑋𝑡− (𝑞) ≠ 𝑋𝑡 (𝑞)}. Then 𝑍𝑡 (𝑞) = 𝑌𝑡 (𝑞) for all 𝑞 ∈ [0, 𝑎] and all 𝑡 ∈ [0, ∞) \ 𝐴, where 𝐴 := 𝐴 𝑎 ∪ (∪𝑣 ∈𝑄𝐸 𝐴𝑣 ) is a countable subset of [0, ∞). It follows that (11.54) holds for 𝑓 ∈ 𝐶 1 (𝐸). For an arbitrary 𝑓 ∈ 𝐶 (𝐸), we get (11.54) by an approximation argument. □ The martingale problem (11.54) is clearly a special case of one of those discussed in Theorem 7.16. Let us define the branching mechanism Φ on 𝐸 by 𝑥 ∈ 𝐸, 𝑓 ∈ 𝐵(𝐸) + ,

Φ(𝑥, 𝑓 ) = 𝜙0 ( 𝑓 (𝑥)) − Ψ(𝑥, 𝑓 ),

(11.55)

where ∫

∫

∫ 𝑓 (𝑥 ∨ 𝑦) 𝛽 𝑦 d𝑦 +

Ψ(𝑥, 𝑓 ) =

d𝑦 𝐸

𝐸

∞


1 − e−𝑧 𝑓 ( 𝑥∨𝑦) 𝑛 𝑦 (d𝑧).

0

By Theorem 7.16 we have the following: Theorem 11.20 The measure-valued process {𝑌𝑡 : 𝑡 ≥ 0} is a non-local branching 𝜂 immigration superprocess in 𝑀 (𝐸) with transition semigroup (𝑄 𝑡 )𝑡 ≥0 defined by, + for 𝑓 ∈ 𝐵(𝐸) ,   ∫ ∫ 𝑡 − ⟨𝜈, 𝑓 ⟩ 𝜂 𝑄 𝑡 (𝜇, d𝜈) = exp − ⟨𝜇, 𝑉𝑡 𝑓 ⟩ − ⟨𝜂, 𝑉𝑠 𝑓 ⟩d𝑠 , (11.56) e 𝑀 (𝐸)

0

where (𝑡, 𝑥) ↦→ 𝑉𝑡 𝑓 (𝑥) is the unique locally bounded positive solution of ∫ 𝑡 Φ(𝑥, 𝑉𝑠 𝑓 )d𝑠, 𝑡 ≥ 0, 𝑥 ∈ 𝐸 . 𝑉𝑡 𝑓 (𝑥) = 𝑓 (𝑥) −

(11.57)

0

The above theorem gives a representation of the stochastic flow in terms of a nonlocal branching immigration superprocess. The branching mechanism Φ defined by (11.55) has local part (𝑥, 𝑓 ) ↦→ 𝜙0 ( 𝑓 (𝑥)) and non-local part (𝑥, 𝑓 ) ↦→ Ψ(𝑥, 𝑓 ); see Example 2.8. The spatial motion of {𝑌𝑡 : 𝑡 ≥ 0} is trivial. Heuristically, when an infinitesimal particle dies at site 𝑥 ∈ 𝐸, some offspring are born at this site according to the local branching mechanism and some are born in the interval [𝑥, 𝑎] according to the non-local branching mechanism. The measure 𝜂 governs the immigration. Therefore the branching of an infinitesimal particle located at 𝑥 ∈ 𝐸 does not make any influence on the population in the interval [0, 𝑥). This explains the Markov property of the path-valued growing process   {𝑋 (𝑞) : 𝑞 ∈ 𝐸 } = (𝑋𝑡 (𝑞))𝑡 ≥0 : 𝑞 ∈ 𝐸 = (𝑌𝑡 [0, 𝑞] : 𝑡 ≥ 0) : 𝑞 ∈ 𝐸 . For any 𝑞 ∈ [0, 𝑎] and 𝜇 ∈ 𝑀 [0, 𝑎] we define the restriction 𝜇| 𝑞 ∈ 𝑀 [0, 𝑞] by 𝜇| 𝑞 (d𝑥) = 1 [0,𝑞] (𝑥)𝜇(d𝑥),

𝑥 ∈ [0, 𝑎].

(11.58)

330

11 Path-Valued Processes and Stochastic Flows 𝜂

Theorem 11.21 Let 𝑋 = (Ω, 𝒢, 𝒢𝑡 , 𝑌𝑡 , Q 𝜇 ) be a right continuous realization of the 𝜂 immigration superprocess with transition semigroup (𝑄 𝑡 )𝑡 ≥0 given by (11.56) and (11.57). Let 𝑞 ≥ 𝑝 ∈ [0, 𝑎]. Then for any 𝑓 ∈ 𝐵[0, ∞) + and 𝑔 ∈ 𝐵( [0, ∞) × 𝐸) + with compact supports, we have   ∫ ∞
𝜂 Q 𝜇 exp − ⟨𝑌𝑠 | 𝑝 , 𝑔(𝑠, ·)⟩ + 𝑓 (𝑠)𝑌𝑠 [0, 𝑞] d𝑠 0   ∫ ∞
𝜂 ⟨𝑌𝑠 | 𝑝 , 𝑔(𝑠, ·)⟩ + 𝑢 𝑝,𝑞 (𝑠, 𝑓 )𝑌𝑠 [0, 𝑝] d𝑠 = Q 𝜇 exp − 0   ∫ ∞ · exp − 𝜇( 𝑝, 𝑞]𝑢 𝑞 (0, 𝑓 ) − 𝜂( 𝑝, 𝑞] 𝑢 𝑞 (𝑠, 𝑓 )d𝑠 , 0

where 𝑠 ↦→ 𝑢 𝑞 (𝑠, 𝑓 ) and 𝑠 ↦→ 𝑢 𝑝,𝑞 (𝑠, 𝑓 ) are defined by (11.8) and (11.11), respectively. Proof This follows from (11.10) and Theorem 11.17 together with the construction □ of the immigration superprocess. The above theorem gives a convenient formulation of the Markov property of the path-valued growing process. In view of (11.55), we have Φ(·, 1 [0, 𝑝] 𝑓 ) = 1 [0, 𝑝] Φ(·, 𝑓 ) for 𝑝 ∈ [0, 𝑎] and 𝑓 ∈ 𝐵(𝐸) + . By this observation and Corollary 9.23 it is not hard to see that     ∫ ∞ ∫ ∞ 𝜂 ⟨𝜂| 𝑝 , 𝑢 𝑠 ⟩d𝑠 , ⟨𝑌𝑠 | 𝑝 , 𝑔(𝑠, ·)⟩d𝑠 = exp − ⟨𝜇| 𝑝 , 𝑢 0 ⟩ − Q 𝜇 exp − 0

0

where (𝑟, 𝑥) ↦→ 𝑢𝑟 (𝑥) is the unique bounded positive solution to ∫ ∞ ∫ ∞ 𝑢𝑟 (𝑥) + Φ(𝑥, 𝑢 𝑠 )d𝑠 = 𝑔𝑠 (𝑥)d𝑠, 𝑟 ≥ 0, 𝑥 ∈ 𝐸 . 𝑟

𝑟

Given an admissible family of branching mechanisms {𝜙𝑞 : 𝑞 ∈ 𝐸 } indexed by 𝐸 = [0, 𝑎) for 0 < 𝑎 ≤ ∞, we can take an increasing sequence {𝑎 𝑘 } ⊂ [0, 𝑎) such that 𝑎 𝑘 → 𝑎 as 𝑘 → ∞. By Theorem 11.20, for each 𝑘 ≥ 1 we construct an immigration superprocess {𝑌𝑡(𝑘) : 𝑡 ≥ 0} in 𝑀 [0, 𝑎 𝑘 ]. Those processes determine a Markov process {𝑌𝑡 : 𝑡 ≥ 0} in ℳ(𝐸), the space of Radon measures on 𝐸 furnished with the topology of vague convergence. The results established above can be extended to {𝑌𝑡 : 𝑡 ≥ 0} with suitable modifications.

11.5 The Excursion Law Let 𝐸 = [0, 𝑎] for some 𝑎 > 0. Let {𝜙𝑞 : 𝑞 ∈ 𝐸 } be an admissible family of branching mechanisms such that 𝜙0′ (∞) = ∞. By Theorem 11.20, we can define the transition semigroup (𝑄 𝑡 )𝑡 ≥0 of a non-local branching superprocess by, for 𝑓 ∈ 𝐵(𝐸) + ,

11.5 The Excursion Law

331

∫

 e− ⟨𝜈, 𝑓 ⟩ 𝑄 𝑡 (𝜇, d𝜈) = exp − ⟨𝜇, 𝑉𝑡 𝑓 ⟩ ,

𝑀 (𝐸)

where (𝑡, 𝑥) ↦→ 𝑉𝑡 𝑓 (𝑥) is the unique locally bounded positive solution of (11.57). Let (𝑄 ◦𝑡 )𝑡 ≥0 denote the restriction of the semigroup to 𝑀 (𝐸) ◦ . Theorem 11.22 The cumulant semigroup (𝑉𝑡 )𝑡 ≥0 defined by (11.57) admits the representation, for 𝑓 ∈ 𝐵(𝐸) + , ∫ (11.59) 𝑉𝑡 𝑓 (𝑥) = (1 − e− ⟨𝜈, 𝑓 ⟩ )𝐿 𝑡 (𝑥, d𝜈), 𝑡 > 0, 𝑥 ∈ 𝐸, 𝑀 (𝐸) ◦

where (𝐿 𝑡 (𝑥, ·))𝑡 >0 is a 𝜎-finite entrance law for (𝑄 ◦𝑡 )𝑡 ≥0 . Proof It is easy to see that the branching mechanism Φ defined by (11.55) has local projection  ∫  ∫ ∞ 𝑧𝑛 𝑦 (d𝑧) d𝑦 𝛽𝑦 + Φ1 (𝑥, 𝜆) = 𝜙0 (𝜆) − 𝜆 0 𝐸 ∫ 𝑥 ∫ ∞
−𝜆𝑧 + d𝑦 e − 1 + 𝜆𝑧 𝑛 𝑦 (d𝑧), 0

0

which is bounded below by the constant local branching mechanism  ∫  ∫ ∞ 𝜙∗ (𝜆) := 𝜙0 (𝜆) − 𝜆 𝑧𝑛 𝑦 (d𝑧) d𝑦. 𝛽𝑦 + 𝐸

0

Since 𝜙0′ (∞) = ∞ implies 𝜙∗′ (∞) = ∞, the result follows by Corollary 5.33.

□

In the situation of Theorem 11.22, the entrance law (𝐿 𝑡 (0, ·))𝑡 >0 determines an excursion law N0 on 𝐷 ( [0, ∞), 𝑀 (𝐸)). Recall that for 𝑞 ∈ [0, 𝑎] and 𝜇 ∈ 𝑀 [0, 𝑎] the restriction 𝜇| 𝑞 ∈ 𝑀 [0, 𝑞] is defined by (11.58). Let ℱ𝑞 be the 𝜎-algebra on 𝐷 ( [0, ∞), 𝑀 (𝐸)) generated by the restricted coordinate process {𝑤 𝑡 | 𝑞 : 𝑡 ≥ 0}. For 𝑤 ∈ 𝐷 ( [0, ∞), 𝑀 (𝐸)) let ∫ ∞ 𝑤 𝑠 [0, 𝑞]d𝑠. (11.60) 𝜎𝑞 (𝑤) = 0

Given 𝑝 ∈ 𝐸 and 𝑔 ∈ 𝐵( [0, ∞) × 𝐸) + with compact support, we write ∫ ∞ 𝐻 𝑝 (𝑤) = ⟨𝑤 𝑠 | 𝑝 , 𝑔(𝑠, ·)⟩d𝑠.

(11.61)

0

Let 𝑋 = (Ω, 𝒢, 𝒢𝑡 , 𝑋𝑡 , Q 𝜇 ) be a right continuous realization of the superprocess with transition semigroup (𝑄 𝑡 )𝑡 ≥0 . By Theorems 5.15 and 8.27, we have   ∫ ∞   ⟨𝑋𝑠 | 𝑝 , 𝑔(𝑠, ·)⟩d𝑠 . (11.62) N0 1 − e−𝐻 𝑝 (𝑤) = − log Q 𝛿0 exp − 0

332

11 Path-Valued Processes and Stochastic Flows

Theorem 11.23 Let 𝑞 ≥ 𝑝 ∈ 𝐸 and 𝐺 ∈ pℱ𝑝 . Then for any 𝑓 ∈ 𝐵[0, ∞) + with compact support, we have   ∫ ∞   𝑤 𝑠 [0, 𝑞] 𝑓 (𝑠)d𝑠 N0 𝐺 1 − exp − 0     ∫ ∞ = N0 𝐺 1 − exp − 𝑤 𝑠 [0, 𝑝]𝑢 𝑝,𝑞 (𝑠, 𝑓 )d𝑠 . (11.63) 0

Proof Let 𝐻 𝑝 (𝑤) be given by (11.61) for 𝑔 ∈ 𝐵( [0, ∞)×𝐸) + with compact supports. By Theorem 11.21 and the relation (11.62), for any 𝑓 ∈ 𝐵[0, ∞) + we have    ∫ ∞ N0 1 − exp − 𝐻 𝑝 (𝑤) − 𝑤 𝑠 [0, 𝑞] 𝑓 (𝑠)d𝑠   ∫ 0∞
= − log Q 𝛿0 exp − ⟨𝑋𝑠 | 𝑝 , 𝑔(𝑠, ·)⟩ + 𝑋𝑠 [0, 𝑞] 𝑓 (𝑠) d𝑠  ∫0 ∞ 
⟨𝑋𝑠 | 𝑝 , 𝑔(𝑠, ·)⟩ + 𝑢 𝑝,𝑞 (𝑠, 𝑓 ) 𝑋𝑠 [0, 𝑝] d𝑠 = − log Q 𝛿0 exp − 0    ∫ ∞ 𝑢 𝑝,𝑞 (𝑠, 𝑓 )𝑤 𝑠 [0, 𝑝]d𝑠 . = N0 1 − exp − 𝐻 𝑝 (𝑤) − 0

It follows that   ∫ ∞    𝑤 𝑠 [0, 𝑞] 𝑓 (𝑠)d𝑠 N0 exp − 𝐻 𝑝 (𝑤) 1 − exp −    ∫ 0∞ 𝑤 𝑠 [0, 𝑞] 𝑓 (𝑠)d𝑠 = N0 1 − exp − 𝐻 𝑝 (𝑤) − 0 h i  − N0 1 − exp − 𝐻 𝑝 (𝑤)    ∫ ∞ 𝑤 𝑠 [0, 𝑝]𝑢 𝑝,𝑞 (𝑠, 𝑓 )d𝑠 = N0 1 − exp − 𝐻 𝑝 (𝑤) − 0 h i  − N0 1 − exp − 𝐻 𝑝 (𝑤)  ∫ ∞     = N0 exp − 𝐻 𝑝 (𝑤) 1 − exp − 𝑤 𝑠 [0, 𝑝]𝑢 𝑝,𝑞 (𝑠, 𝑓 )d𝑠 . 0

Then we get (11.63) by a monotone class argument.

□

Corollary 11.24 In the setup of Theorem 11.23, for any 𝜃 > 0 we have h      i (𝑤) (𝜃)𝜎 N0 𝐺 (1 − e−𝜃 𝜎𝑞 (𝑤) ) = N0 𝐺 1 − exp − 𝜙 𝑝 ◦ 𝜙−1 . 𝑝 𝑞 Proof We first apply Theorem 11.23 for 𝑓 = 1 [0,𝑡 ] . Then, by letting 𝑡 → ∞ and using (11.25), we get the result. □ According to Theorem 11.17, for any 𝑥 ≥ 0 and 𝑞 ∈ [0, 𝑎] the process {𝑋𝑡 [0, 𝑞] : 𝑡 ≥ 0} under Q 𝑥 𝛿0 is a CB-process with branching mechanism 𝜙𝑞 and initial value 𝑋0 [0, 𝑞] = 𝑥. For 𝑤 ∈ 𝐷 ( [0, ∞), 𝑀 (𝐸)) let 𝐴(𝑤) = inf{𝑞 ∈ 𝐸 : 𝜎𝑞 (𝑤) = ∞},

11.5 The Excursion Law

333

where 𝜎𝑞 (𝑤) is defined by (11.60). By Corollary 5.21 and the relation (11.62), we have   (11.64) N0 1 − e−𝜃 𝜎𝑞 (𝑤) = 𝜙−1 𝜃 > 0, 𝑞 (𝜃), which implies  N0 {𝐴(𝑤) ≤ 𝑞} = N0 𝜎𝑞 (𝑤) = ∞ = 𝜙−1 𝑞 (0).

(11.65)

From (11.64) it follows that, for 𝑞 ∈ [0, 𝑎] and 𝜃 > 0,   N0 𝜎𝑞 (𝑤)e−𝜃 𝜎𝑞 (𝑤) 1 { 𝐴(𝑤) >𝑞 } =

1 𝜙𝑞′ (𝜙−1 𝑞 (𝜃))

.

By the continuity of 𝜆 ↦→ 𝜙𝑞′ (𝜆) and 𝜃 ↦→ 𝜙−1 𝑞 (𝜃), for 𝑞 ∈ (0, 𝑎] and 𝜃 > 0,   N0 𝜎𝑞− (𝑤)e−𝜃 𝜎𝑞− (𝑤) 1 { 𝐴(𝑤) ≥𝑞 } =

1 𝜙𝑞′ (𝜙−1 𝑞 (𝜃))

.

In particular, if 𝑏 𝑞 = 𝜙𝑞′ (0) < 0, we have 𝜙𝑞′ (𝜙−1 𝑞 (0)) > 0 and   0 < N0 𝜎𝑞− (𝑤)1 { 𝐴(𝑤) ≥𝑞 } =

1

< ∞.

𝜙𝑞′ (𝜙−1 𝑞 (0))

(11.66)

By a modification of the proof of Theorem 11.9 we have the following:

Theorem 11.25 Suppose that for some 𝑞 ∈ (0, 𝑎] we have 𝑏 𝑞 = 𝜙𝑞′ (0) < 0,

d −1 𝜙 (0) 𝑟=𝑞 > 0. d𝑟 𝑟

Then, for any 𝐺 ∈ bℱ𝑞− ,   N0 [𝐺 | 𝐴(𝑤) = 𝑞] = 𝜙𝑞′ (𝜙−1 𝑞 (0))N0 𝐺𝜎𝑞− (𝑤)1 { 𝐴(𝑤) ≥𝑞 } . Corollary 11.26 In the setup of Theorem 11.25, we have N0 [e−𝜃 𝜎𝑞− (𝑤) | 𝐴(𝑤) = 𝑞] =

𝜙𝑞′ (𝜙−1 𝑞 (0)) 𝜙𝑞′ (𝜙−1 𝑞 (𝜃))

,

𝜃 ≥ 0.

Example 11.6 Let {𝜙𝑞 : 𝑞 ∈ [0, 𝑎]} be the admissible family of branching mechanisms defined by (11.22). For 𝑞 ∈ [0, 𝑎] and 𝜃 > 0 we have   N0 1 − e−𝜃 𝜎𝑞 (𝑤) = 𝑞 + 𝜙−1 (𝜃 + 𝜙(−𝑞))

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11 Path-Valued Processes and Stochastic Flows

and   N0 𝜎𝑞− (𝑤)e−𝜃 𝜎𝑞− (𝑤) 1 { 𝐴(𝑤) ≥𝑞 } =

𝜙 ′ (𝜙−1 (𝜃

1 . + 𝜙(−𝑞)))

In particular, if 𝑏 𝑞 = 𝜙 ′ (−𝑞) < 0, we have 𝜙 ′ (𝜙−1 (𝜙(−𝑞))) > 0 and   N0 e−𝜃 𝜎𝑞− (𝑤) | 𝐴(𝑤) = 𝑞 =

𝜙 ′ (𝜙−1 (𝜙(−𝑞))) , + 𝜙(−𝑞)))

𝜙 ′ (𝜙−1 (𝜃

𝜃 ≥ 0.

11.6 Notes and Comments The main results of this chapter unify those of Dawson and Li (2012) and Li (2014). Under the conditions of Corollary 11.5, the stochastic flow was constructed by Bertoin and Le Gall (2000) using Bochner’s subordination. They considered a more general branching mechanism to include Neveu’s CB-process and established connections between the model and the coalescent process of Bolthausen and Sznitman (1998). These and related structures were investigated intensively in the series of papers by Bertoin and Le Gall (2003, 2005, 2006). Suppose that 𝑐, 𝑏 ≥ 0 are constants, 𝑞 ↦→ 𝛾(𝑞) is a continuous increasing map from [0, 1] into itself and 𝑧2 𝜈(d𝑧) is a finite measure on (0, 1]. Let {𝑊 (d𝑠, d𝑢)} be a time–space Gaussian white noise on (0, ∞) × (0, 1] with intensity 2𝑐d𝑠d𝑢 and let {𝑀 (d𝑠, d𝑧, d𝑢)} be a Poisson random measure on (0, ∞) × (0, 1] 2 with intensity d𝑠𝜈(d𝑧)d𝑢. It was proved in Dawson and Li (2012) that for any 𝑞 ∈ [0, 1] there is a pathwise unique positive solution 𝑋 (𝑞) = {𝑋𝑡 (𝑞) : 𝑡 ≥ 0} to the stochastic equation ∫

𝑡

∫

𝑋𝑡 (𝑞) = 𝑞 + ∫0 𝑡 +𝑏 ∫ + 0

1



 1 {𝑢≤𝑋𝑠− (𝑞) } − 𝑋𝑠− (𝑞) 𝑊 (d𝑠, d𝑢)

0



 𝛾(𝑞) − 𝑋𝑠− (𝑞) d𝑠 0 𝑡∫ 1∫ 1   𝑧 1 {𝑢≤𝑋𝑠− (𝑞) } − 𝑋𝑠− (𝑞) 𝑀 (d𝑠, d𝑧, d𝑢). 0

0

This gives an explicit construction of the generalized Fleming–Viot flows introduced by Bertoin and Le Gall (2003, 2005). By the results of Bertoin and Le Gall (2006) and Dawson and Li (2012), the flow of CBI-processes arises naturally in a hydrodynamic limit theorem of the generalized Fleming–Viot flows. The study of the flows originated from the investigation of the generalised random energy models introduced by Derrida (1985) and Ruelle (1987) in the theory of spin glasses; see also Bovier (2017) for the related backgrounds. In the situation of Example 11.1, the path-valued growing process is a counterpart of the time reversal of the tree-valued decreasing process studied by Abraham and Delmas (2012), who considered the admissible family (11.22) obtained from a critical branching mechanism. The explosion time was defined in Abraham and

11.6 Notes and Comments

335

Delmas (2012) as the smallest negative time when the tree or the total population of the corresponding CB-process becomes finite. They gave some characterizations of the evolution of the tree after this time under an excursion law. In particular, they also derived the formulas in Example 11.6. Their results extended those of Aldous and Pitman (1998), who studied similar models in the setting of Galton–Watson trees. Let ℳ[0, ∞) be the space of Radon measures on [0, ∞) endowed with the topology of vague convergence. We can embed 𝐷 [0, ∞) + continuously into ℳ[0, ∞) by identifying the path 𝑤 ∈ 𝐷 [0, ∞) + and the measure 𝜈 ∈ ℳ[0, ∞) such that 𝜈(d𝑠) = 𝑤(𝑠)d𝑠 for 𝑠 ≥ 0. By an approximation argument, we can extend the 𝜇, 𝜂 transition semigroup (P 𝑝,𝑞 : 𝑞 ≥ 𝑝 ∈ 𝐸) defined by (11.10) or (11.21) to an inhomogeneous transition semigroup on ℳ[0, ∞). A Markov process {𝑍 𝑞 : 𝑞 ∈ 𝐸 } in ℳ[0, ∞) with this transition semigroup can be regarded as an extended immigration superprocess; see Li (2014) for the details. The long-term behavior of flow of CB-processes was discussed in Foucart and Ma (2019). In Foucart et al. (2019), an inverse of the flow was identified by its Laplace transform and shown to be a Markov process. A nice derivation of the stochastic equation for continuous CBI-processes was given in Aïdékon et al. (2020+) as a reformulation of Tanaka’s formula with an explicit construction of the time–space Gaussian white noise. There has also been some related progresses in constructing Dawson–Watanabe superprocesses by strong solutions of stochastic equations. Let {𝑋𝑡 : 𝑡 ≥ 0} be a super Brownian motion in 𝑀 (R) with density field {𝑢 𝑡 (𝑥) : 𝑡 > 0, 𝑥 ∈ R} solving weakly the stochastic partial differential equation: √︁ 𝜕 ¤ 𝑥) + 1 Δ𝑢 𝑡 (𝑥), 𝑢 𝑡 (𝑥) = 𝑢 𝑡 (𝑥) 𝐵(𝑡, 𝜕𝑡 2

(11.67)

which is a special form of (7.70). Let ∫

𝑥

𝑋𝑡 (𝑥) = 𝑋𝑡 (−∞, 𝑥] =

𝑢 𝑡 (𝑦)d𝑦,

𝑡 ≥ 0, 𝑥 ∈ R.

−∞

It is not hard to show that on an extension of the probability space the following stochastic equation is satisfied: ∫

𝑡

∫

𝑋𝑡 (𝑥) = 𝑋0 (𝑥) + 0

0

𝑋𝑠 ( 𝑥)

1 𝑊 (d𝑠, d𝑢) + 2

∫

𝑡

Δ𝑋𝑠 (𝑥)d𝑠,

(11.68)

0

where {𝑊 (d𝑠, d𝑢) : 𝑡 > 0, 𝑢 > 0} is another time–space Gaussian white noise based on the Lebesgue measure. The pathwise uniqueness for (11.68) was proved in Xiong (2013) by a clever idea using a backward doubly stochastic equation. This gives a construction of the super Brownian motion by a strong solution. In fact, Xiong (2013) studied more general stochastic equations including that of a Fleming–Viot superprocess with Brownian mutation. The results of Xiong (2013) were generalized in He et al. (2014) to the Lévy spatial motion and general branching mechanism. Wang et al. (2017) proved a comparison theorem for the equation

336

11 Path-Valued Processes and Stochastic Flows

and used the theorem to obtain well-posedness of martingale problems for some interacting measure-valued processes. Unfortunately, it still not known whether the pathwise uniqueness holds for (11.67). Branching processes with logistic growth in both discrete and continuous state spaces were introduced by Lambert (2005). The processes add quadratic death rates to classical branching processes and model the evolution of populations with competition. The continuous-state process can be constructed as a time-changed Ornstein–Uhlenbeck type process in the Lamperti fashion. A systematic treatment of general discrete- and continuous-state branching models with competition was given in the monograph by Pardoux (2016). The continuous-state model was constructed in Pardoux (2016) by a generalization of (10.9) with an additional nonlinear drift term. A very interesting property of the model with competition is that the extinction time may remain finite when the ancestral population tends to infinity as long as the competition is strong enough. The exploration of continuous-state models in Pardoux (2016) is mainly focused on processes with continuous sample paths. A part of the monograph is devoted to representations of the genealogical forest of the model, which result in several versions of the Ray–Knight theorem. For discontinuous competition models with continuous-state, a representation theorem of Ray–Knight type was established by Berestycki et al. (2018) in terms of local times of suitably pruned forests. Their proof of the result is based on iteration and fixed point arguments. A different construction of the genealogical forest was given in Li et al. (2022) by solving a stochastic integral equation. Those extend the results of Duquesne and Le Gall (2002) and Le Gall and Le Jan (1998a) for CB-processes without competition. Structured populations were studied in the monograph by Bansaye and Méléard (2015). The first part of the monograph concerns one-dimensional models like birth-and-death processes and variations of CB-processes. In the second part of the monograph, the authors used measure-valued processes to model populations with individuals carrying types such as heritable traits subject to selection and mutation.

Chapter 12

State-Dependent Immigration Structures

In this chapter we give the construction of a class of Dawson–Watanabe superprocesses with state-dependent immigration. The basic idea is to construct such a process by adding up measure-valued paths in a suitable manner according to Poisson random measures defined by entrance rules. This is realized by introducing a stochastic equation driven by the Poisson random measures and proving the existence and uniqueness of the solution. The approach provides a way of changing the branching mechanism of a superprocess. We shall deal with processes with càdlàg paths.

12.1 Inhomogeneous Immigration Rates Let 𝐸 be a locally compact separable metric space. We shall use ∥·∥ to denote both the supremum norm of functions and the total variation norm of signed measures on 𝐸. Let 𝜉 be a Hunt process in 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 . We assume (𝑃𝑡 )𝑡 ≥0 preserves 𝐶0 (𝐸) and 𝑡 ↦→ 𝑃𝑡 𝑓 is continuous in the supremum norm for every 𝑓 ∈ 𝐶0 (𝐸). Let 𝐴 denote the strong generator of (𝑃𝑡 )𝑡 ≥0 with domain 𝐷 0 ( 𝐴) ⊂ 𝐶0 (𝐸). Let 𝜙 be a branching mechanism given by (2.29) or (2.30) satisfying Conditions 7.1 and 7.2. Let (𝑄 𝑡 )𝑡 ≥0 and (𝑉𝑡 )𝑡 ≥0 denote the transition semigroup and the cumulant semigroup of the (𝜉, 𝜙)-superprocess, respectively. By Theorem 5.13 one can see that the superprocess has a realization as a Hunt process. Suppose that 𝜂0 is a 𝜎-finite measure on 𝐸 and 𝐻0 is a 𝜎-finite measure on 𝑀 (𝐸) ◦ . Let ∥ · ∥ 𝜂0 denote the norm of the Banach space 𝐿 1 (𝜂0 ) of 𝜂0 -integrable functions on 𝐸. Let ∥ · ∥ 𝐻1 denote the norm of the Banach space 𝐿 1 (𝐻1 ) of 𝐻1 -integrable functions on 𝑀 (𝐸) ◦ , where 𝐻1 (d𝜈) = 𝜈(1)𝐻0 (d𝜈). Suppose that (𝑠, 𝑦) ↦→ 𝑞 𝑠 (𝑦) is a positive Borel function on [0, ∞) × 𝐸 and (𝑠, 𝜈) ↦→ 𝑔𝑠 (𝜈) is a positive Borel function on [0, ∞) × 𝑀 (𝐸) ◦ such that 𝑠 ↦→ ∥𝑞 𝑠 ∥ 𝜂0 + ∥𝑔𝑠 ∥ 𝐻1 is locally bounded on [0, ∞). For 𝑠 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) + write © Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_12

337

338

12 State-Dependent Immigration Structures

∫

𝑞,𝑔

𝐼𝑠 ( 𝑓 ) = 𝜂0 (𝑞 𝑠 𝑓 ) +

𝑀 (𝐸) ◦

(1 − e−𝜈 ( 𝑓 ) )𝑔𝑠 (𝜈)𝐻0 (d𝜈).

(12.1)

𝑞,𝑔

Using (𝑉𝑡 )𝑡 ≥0 and the set of functionals {𝐼𝑠 : 𝑠 ≥ 0} given in (12.1), we can define 𝑞,𝑔 the transition semigroup (𝑄 𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ≥ 0) of an inhomogeneous immigration superprocess by   ∫ ∫ 𝑡 𝑞,𝑔 −𝜈 ( 𝑓 ) 𝑞,𝑔 e 𝑄 𝑟 ,𝑡 (𝜇, d𝜈) = exp − 𝜇(𝑉𝑡−𝑟 𝑓 ) − 𝐼𝑠 (𝑉𝑡−𝑠 𝑓 )d𝑠 , (12.2) 𝑀 (𝐸)

𝑟

where 𝜇 ∈ 𝑀 (𝐸) and 𝑓 ∈ 𝐵(𝐸) + . This is essentially a special case of the transition semigroup defined by (9.33). Let 𝑊ˆ be the space of paths 𝑤 : [0, ∞) → 𝑀 (𝐸) such that 𝑤 𝑡 takes values in 𝑀 (𝐸) ◦ and is càdlàg in some interval (𝛼(𝑤), 𝜁 (𝑤)) or [𝛼(𝑤), 𝜁 (𝑤)) ⊂ [0, ∞) and takes the value 0 ∈ 𝑀 (𝐸) elsewhere. The constant path [0] taking value 0 ∈ 𝑀 (𝐸) is included in 𝑊ˆ with 𝛼( [0]) = ∞ and 𝜁 ( [0]) = 0. Let 𝑤 𝑠 = {𝑤 𝑡∧𝑠 : 𝑡 ≥ 0} for ˆ We equip 𝑊ˆ with its natural 𝜎-algebras 𝒜 0 = 𝜎({𝑤 𝑠 : 𝑠 ≥ 0}) 𝑠 ≥ 0 and 𝑤 ∈ 𝑊. 0 and 𝒜𝑡 = 𝜎({𝑤 𝑠 : 0 ≤ 𝑠 ≤ 𝑡}) for 𝑡 ≥ 0. For 𝑥 ∈ 𝐸 let {𝜆 𝑡 (𝑥, ·) : 𝑡 ≥ 0} and {𝐿 𝑡 (𝑥, ·) : 𝑡 ≥ 0} be determined by the canonical representation (2.5) of the cumulant semigroup. Let L(𝑥, ·) be the Kuznetsov measure corresponding to the canonical entrance rule {𝐿 𝑡 (𝑥, ·) : 𝑡 > 0}. In the current ˆ Suppose that {𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤)} is a Poissituation, this measure is carried by 𝑊. son random measure on (0, ∞) × 𝐸 × (0, ∞) × 𝑊ˆ with intensity d𝑠𝜂0 (d𝑦)d𝑢L(𝑦, d𝑤). For 𝑡 ≥ 0 let 𝑌𝑡𝑞 =

∫

∫

𝑡

0

∫

𝑡

∫ ∫

𝑞𝑠 ( 𝑦)𝜆𝑡−𝑠 ( 𝑦, ·) 𝜂0 (d𝑦) +

d𝑠

0

𝐸

𝐸

0

𝑞𝑠 ( 𝑦)

∫ ˆ 𝑊

𝑤𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤).

We understand the second term on the right-hand side as an integral over the set {(𝑠, 𝑦, 𝑢, 𝑤) : 0 < 𝑠 ≤ 𝑡, 𝑦 ∈ 𝐸, 0 < 𝑢 ≤ 𝑞 𝑠 (𝑦), 𝑤 ∈ 𝑊ˆ } and give similar interpretations for other integrals with respect to Poisson random 𝜂 measures in this chapter. Let 𝒢𝑡 0 be the 𝜎-algebra generated by random variables of the form ∫ ∞∫ ∫ ∞∫ ℎ𝑡 (𝑠, 𝑦, 𝑢, 𝑤)𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤), 0

𝐸

0

ˆ 𝑊

where ℎ𝑡 (𝑠, 𝑦, 𝑢, 𝑤) = ℎ(𝑠, 𝑦, 𝑢, 𝑤 𝑡−𝑠 )1 {𝑠 ≤𝑡 , 𝛼(𝑤) ≤𝑡−𝑠 }

(12.3)

for some ℎ ∈ p(ℬ((0, ∞) × 𝐸 × (0, ∞)) × 𝒜 0 ). 𝜂

Proposition 12.1 The pair {(𝑌𝑡𝑞 , 𝒢𝑡 0 ) : 𝑡 ≥ 0} defined above is an inhomogeneous immigration superprocess with transition semigroup (𝑄 𝑡𝑞,0 )𝑡 ≥0 given by (12.1) and (12.2) with 𝑔𝑠 ≡ 0.

12.1 Inhomogeneous Immigration Rates

339

Proof The arguments are based on similar ideas as the proof of Theorem 9.41, but more careful calculations are needed here. Let 𝑡 ≥ 𝑟 ≥ 0 and let 𝐹 be a 𝜂 𝒢𝑟 0 -measurable random variable of the form  ∫ ∞∫ ∫ ∞∫  ℎ𝑟 (𝑠, 𝑦, 𝑢, 𝑤)𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) , 𝐹 = exp − 0

𝐸

ˆ 𝑊

0

where ℎ𝑟 (𝑠, 𝑦, 𝑢, 𝑤) = ℎ(𝑠, 𝑦, 𝑢, 𝑤 𝑟−𝑠 )1 {𝑠 ≤𝑟 , 𝛼(𝑤) ≤𝑟−𝑠 } for some ℎ ∈ p(ℬ((0, ∞) × 𝐸 × (0, ∞)) × 𝒜 0 ). For 𝑓 ∈ 𝐵(𝐸) + we shall prove    ∫ 𝑡   𝑞 (12.4) 𝜂0 (𝑞 𝑠 𝑉𝑡−𝑠 𝑓 )d𝑠 , P 𝐹e−𝑌𝑡 ( 𝑓 ) = P 𝐹 exp − 𝑌𝑟𝑞 (𝑉𝑡−𝑟 𝑓 ) − 𝑟 𝜂

which implies the desired Markov property of {(𝑌𝑡𝑞 , 𝒢𝑡 0 )}. Let 𝑓𝑡 (𝑠, 𝑦, 𝑢, 𝑤) = 𝑤 𝑡−𝑠 ( 𝑓 )1 {00

where Δ𝑌𝑠 = 𝑌𝑠 − 𝑌𝑠− . Following the proof of Theorem 7.16 one can see there is a càdlàg process {𝑈𝑡 ( 𝑓 )} with locally bounded variations such that 𝑌𝑡 ( 𝑓 ) = 𝑌0 ( 𝑓 ) + 𝑈𝑡 ( 𝑓 ) + 𝑀𝑡𝑐 ( 𝑓 ) + 𝑀𝑡𝑑 ( 𝑓 ), where {𝑀𝑡𝑐 ( 𝑓 )} is a continuous local martingale and

344

12 State-Dependent Immigration Structures

∫

𝑡∫

𝜈( 𝑓 ) 𝑁˜ (d𝑠, d𝜈),

𝑀𝑡𝑑 ( 𝑓 ) = 0

𝑡 ≥ 0,

𝑆 (𝐸) ◦

is a purely discontinuous local martingale. By Itô’s formula, for any 𝐺 ∈ 𝐶 2 (R) we have ∫ 𝑡 𝐺 (𝑌𝑡 ( 𝑓 )) = 𝐺 (𝑌0 ( 𝑓 )) + 𝐺 ′ (𝑌𝑠− ( 𝑓 ))d𝑈𝑠 ( 𝑓 ) 0 ∫ 𝑡 1 + 𝐺 ′′ (𝑌𝑠− ( 𝑓 ))d⟨𝑀 𝑐 ( 𝑓 )⟩𝑠 + local mart. 2∫ 0 ∫ h 𝑡 + d𝑠 𝐺 (𝑌𝑠− ( 𝑓 ) + 𝜈( 𝑓 )) 0 𝑆 (𝐸) ◦ i − 𝐺 (𝑌𝑠− ( 𝑓 )) − 𝜈( 𝑓 )𝐺 ′ (𝑌𝑠− ( 𝑓 )) 𝑁ˆ (d𝑠, d𝜈). By comparing this with (12.12) we see that d𝑈𝑠 ( 𝑓 ) = 𝑌𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠,

d⟨𝑀 𝑐 ( 𝑓 )⟩𝑠 = 2𝑌𝑠 (𝑐 𝑓 2 )d𝑠

and 𝑁ˆ (d𝑠, d𝜈) is carried by [0, ∞) × 𝑀 (𝐸) ◦ with the representation (12.9). Then property (1) follows. □

It is easy to see that ∥ · ∥ 𝜂0 ,𝑡 is a seminorm on ℒ𝜂10 (𝐸) and ∥ · ∥ 𝐻1 ,𝑡 is a seminorm on ℒ𝐻1 1 (𝑀 (𝐸) ◦ ). We identify 𝑞 1 , 𝑞 2 ∈ ℒ𝜂10 (𝐸) if ∥𝑞 1 − 𝑞 2 ∥ 𝜂0 ,𝑡 = 0 for every 𝑡 ≥ 0 and define the metric 𝑑1 on ℒ𝜂10 (𝐸) by 𝑑1 (𝑞 1 , 𝑞 2 ) =

∞ ∑︁
1 1 ∧ ∥𝑞 1 − 𝑞 2 ∥ 𝜂0 ,𝑖 . 𝑖 2 𝑖=1

(12.13)

Similarly, we identify 𝑔1 , 𝑔2 ∈ ℒ𝐻1 1 (𝑀 (𝐸) ◦ ) if ∥𝑔1 − 𝑔2 ∥ 𝐻1 ,𝑡 = 0 for every 𝑡 ≥ 0 and define the metric 𝐷 1 on ℒ𝐻1 1 (𝑀 (𝐸) ◦ ) by 𝐷 1 (𝑔1 , 𝑔2 ) =

∞ ∑︁
1 1 ∧ ∥𝑔1 − 𝑔2 ∥ 𝐻1 ,𝑖 . 𝑖 2 𝑖=1

(12.14)

Let ℒ𝜂00 (𝐸) denote the set of step processes in ℒ𝜂10 (𝐸) and let ℒ𝐻0 1 (𝑀 (𝐸) ◦ ) denote the set of step processes in ℒ𝐻1 1 (𝑀 (𝐸) ◦ ). It is not hard to show as in the proof of Proposition 7.26 that both (ℒ𝜂10 (𝐸), 𝑑1 ) and (ℒ𝐻1 1 (𝑀 (𝐸) ◦ ), 𝐷 1 ) are complete metric spaces. One can also see that ℒ𝜂00 (𝐸) and ℒ𝐻0 1 (𝑀 (𝐸) ◦ ) are dense subsets of ℒ𝜂10 (𝐸) and ℒ𝐻1 1 (𝑀 (𝐸) ◦ ), respectively. In the sequel, we fix 𝑞 ∈ ℒ𝜂10 (𝐸) + and 𝑔 ∈ ℒ𝐻1 1 (𝑀 (𝐸) ◦ ) + . Let {𝑌𝑡 : 𝑡 ≥ 0} be the (𝒢¯ 𝑡+ )-adapted process in 𝑀 (𝐸) defined by

12.2 Predictable Immigration Rates

345

∫ 𝑡 ∫ ∫ 𝑞𝑠 ( 𝑦) ∫ 𝑤 𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) 𝑌𝑡 = 𝑋𝑡 + ˆ 𝐸 0 0 𝑊 ∫ 𝑡 ∫ d𝑠 𝑞 𝑠 (𝑦)𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) + 0

∫

𝐸 𝑡

∫

∫

𝑔𝑠 (𝜈)

∫

+ 0

𝑀 (𝐸) ◦

ˆ 𝑊

0

𝑤 𝑡−𝑠 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤).

(12.15)

Proposition 12.5 Let {𝑌𝑡 : 𝑡 ≥ 0} be the process defined by (12.15). Then for 𝑡 ≥ 0 and 𝑓 ∈ 𝐵(𝐸) we have  ∫ 𝑡 P[𝑌𝑡 ( 𝑓 )] = P[𝑋0 (𝜋𝑡 𝑓 )] + P Γ𝑠 (𝜋𝑡−𝑠 𝑓 )d𝑠 , 0

where ∫ Γ𝑠 ( 𝑓 ) = 𝜂0 (𝑞 𝑠 𝑓 ) +

𝑀 (𝐸) ◦

𝑔𝑠 (𝜈)𝜈( 𝑓 )𝐻0 (d𝜈)

and (𝜋𝑡 )𝑡 ≥0 is the semigroup given by (2.31) and (2.38). Proof Since (𝑠, 𝑦) ↦→ 𝑞 𝑠 (𝑦) and (𝑠, 𝜈) ↦→ 𝑔𝑠 (𝜈) are predictable, from (12.15) we get  ∫ 𝑡 ∫ P[𝑌𝑡 ( 𝑓 )] = P[𝑋𝑡 ( 𝑓 )] + P d𝑠 𝑞 𝑠 (𝑦)𝜆 𝑡−𝑠 (𝑦, 𝑓 )𝜂0 (d𝑦) 𝐸 ∫ 𝑡 ∫ 0  ∫ +P 𝑤 𝑡−𝑠 ( 𝑓 )L(𝑦, d𝑤) 𝑞 𝑠 (𝑦)𝜂0 (d𝑦) d𝑠 ˆ 𝑊  ∫0 𝑡 ∫𝐸  ∫ d𝑠 +P 𝑤 𝑡−𝑠 ( 𝑓 )Q(𝜈, d𝑤) 𝑔𝑠 (𝜈)𝐻0 (d𝜈) ˆ 𝑀 (𝐸) ◦ 0 𝑊  ∫ ∫ 𝑡 d𝑠 = P[𝑋𝑡 ( 𝑓 )] + P 𝑞 𝑠 (𝑦)𝜆 𝑡−𝑠 (𝑦, 𝑓 )𝜂0 (d𝑦) 𝐸 ∫ 𝑡 ∫ 0  ∫ d𝑠 +P 𝑞 𝑠 (𝑦)𝜂0 (d𝑦) 𝜈( 𝑓 )𝐿 𝑡−𝑠 (𝑦, d𝜈) 𝑀 (𝐸) ◦   ∫0 𝑡 ∫𝐸 𝑔𝑠 (𝜈)𝜈(𝜋𝑡−𝑠 𝑓 )𝐻0 (d𝜈) . +P d𝑠 0

𝑀 (𝐸) ◦

Then we get the result by Corollary 2.28.

□

Corollary 12.6 Let 𝑞 1 , 𝑞 2 ∈ ℒ𝜂10 (𝐸) + and 𝑔1 , 𝑔2 ∈ ℒ𝐻1 1 (𝑀 (𝐸) ◦ ) + . Let {𝑌𝑖 (𝑡) : 𝑡 ≥ 0} be defined by (12.15) with 𝑞 𝑠 (𝑦) = 𝑞 𝑖 (𝑠, 𝑦) and 𝑔𝑠 (𝜈) = 𝑔𝑖 (𝑠, 𝜈) for 𝑖 = 1, 2. Then we have ∫ 𝑡    e𝑐0 (𝑡−𝑠) P ∥𝑞 1 (𝑠, ·) − 𝑞 2 (𝑠, ·) ∥ 𝜂0 P ∥𝑌1 (𝑡) − 𝑌2 (𝑡) ∥ ≤ 0  + ∥𝑔1 (𝑠, ·) − 𝑔2 (𝑠, ·) ∥ 𝐻1 d𝑠, where 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)].

12 State-Dependent Immigration Structures

346

Proof Let 𝑞 ∗ = 𝑞 1 ∧ 𝑞 2 and 𝑞 ∗ = 𝑞 1 ∨ 𝑞 2 . Similarly, let 𝑔∗ = 𝑔1 ∧ 𝑔2 and 𝑔 ∗ = 𝑔1 ∨ 𝑔2 . Then 𝑞 ∗ − 𝑞 ∗ = |𝑞 1 − 𝑞 2 | and 𝑔 ∗ − 𝑔∗ = |𝑔1 − 𝑔2 |. It is easy to see that ∫

𝑞 ∗ (𝑠,𝑦)

∫ ∫

𝑡

∫

𝑌1 (𝑡) − 𝑌2 (𝑡) =

ˆ

𝑞∗ (𝑠,𝑦) 𝑊 𝑡 ∫ ∫ 𝑞∗ (𝑠,𝑦) ∫

0

𝐸

∫

𝑤 𝑡−𝑠 1 {𝑞1 >𝑞2 } (𝑠, 𝑦)𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤)

−

𝑤 𝑡−𝑠 1 {𝑞2 >𝑞1 } (𝑠, 𝑦)𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) ∗ ˆ ∫ 𝑞 (𝑠,𝑦) 𝑊 + d𝑠 [𝑞 1 (𝑠, 𝑦) − 𝑞 2 (𝑠, 𝑦)]𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) ∫0 𝑡 ∫ 𝐸 ∫ 𝑔∗ (𝑠,𝜈) ∫ + 𝑤 𝑡−𝑠 1 {𝑔>𝑔2 } (𝑠, 𝜈)𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤) ◦ ˆ 𝑊 ∫0 𝑡 ∫𝑀 (𝐸) ∫𝑔∗𝑔(𝑠,𝜈) ∫ ∗ (𝑠,𝜈) − 𝑤 𝑡−𝑠 1 {𝑔1 𝑟 𝑖−1 } are still Poisson random measures with intensities d𝑠𝜂0 (d𝑦)d𝑢L(𝑦, d𝑤) and d𝑠𝐻0 (d𝜈)d𝑢Q(𝜈, d𝑤), respectively. For 𝑡 ≥ 0 let ∫ 𝑡∧𝑟𝑖 ∫ ∫ 𝑞𝑠 ( 𝑦) ∫ 𝑤 𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) 𝑌𝑖 (𝑡) = 𝑋𝑡 + ˆ 𝑊 ∫ 𝑡∧𝑟0 𝑖 ∫ 𝐸 0 d𝑠 + 𝑞 𝑠 (𝑦)𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) ∫0 𝑡∧𝑟𝑖 ∫ 𝐸 ∫ 𝑔𝑠 (𝜈) ∫ 𝑤 𝑡−𝑠 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤). + 𝑀 (𝐸) ◦

0

0

ˆ 𝑊

Let 𝒢𝑡𝑖 = 𝜎(ℱ𝑡 ∪ 𝒢𝑡0,𝑖 ∪ 𝒢𝑡1,𝑖 ), where 𝒢𝑡0,𝑖 is the 𝜎-algebra generated by random variables of the form ∫ 𝑟𝑖 ∫ ∫ ∞ ∫ ℎ𝑡 (𝑠, 𝑦, 𝑢, 𝑤)𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) 0

𝐸

ˆ 𝑊

0

with ℎ𝑡 (𝑠, 𝑦, 𝑢, 𝑤) given by (12.3) and 𝒢𝑡1,𝑖 is the 𝜎-algebra generated by random variables of the form ∫ ∞∫ ∫ 𝑟𝑖 ∫ 𝐻𝑡 (𝑠, 𝜈, 𝑢, 𝑤)𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤) 0

𝑀 (𝐸) ◦

0

ˆ 𝑊

12.2 Predictable Immigration Rates

349

with 𝐻𝑡 (𝑠, 𝜈, 𝑢, 𝑤) given by (12.5). Then 𝑌𝑖 (𝑡) = 𝑌𝑡 and 𝒢¯ 𝑡𝑖 = 𝒢¯ 𝑡 for 0 ≤ 𝑡 ≤ 𝑟 𝑖 . We claim that the following properties hold: (i) {(𝑌𝑖 (𝑡), 𝒢¯ 𝑡𝑖 ) : 𝑟 𝑖−1 ≤ 𝑡 ≤ 𝑟 𝑖 } and {(𝑌𝑖 (𝑡), 𝒢¯ 𝑡 ) : 𝑟 𝑖−1 ≤ 𝑡 ≤ 𝑟 𝑖 } are immigration superprocesses with transition semigroup (𝑄 𝑖,𝑡 )𝑡 ≥0 under P(·|𝒢¯ 𝑟𝑖−1 ); (ii) {(𝑌𝑖 (𝑡), 𝒢¯ 𝑡𝑖 ) : 𝑡 ≥ 𝑟 𝑖 } and {(𝑌𝑖 (𝑡), 𝒢¯ 𝑡 ) : 𝑡 ≥ 𝑟 𝑖 } are superprocesses with transition semigroup (𝑄 𝑡 )𝑡 ≥0 under P(·|𝒢¯ 𝑟𝑖−1 ) and P(·|𝒢¯ 𝑟𝑖 ). For 𝑖 = 1 the above properties hold by Theorem 12.3. Now suppose that they hold for some 𝑖 ≥ 1. For 𝑡 ≥ 0 let ∫ 𝑡∧𝑟𝑖+1 ∫ ∫ 𝑞𝑠 ( 𝑦) ∫ 𝑍𝑖 (𝑡) = 𝑤 𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) ˆ 0 𝑊 𝑡∧𝑟 ∫ 𝑖 𝑡∧𝑟𝑖+1 𝐸 ∫ 𝑞 𝑠 (𝑦)𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) + d𝑠 𝑡∧𝑟𝑖 𝐸 ∫ 𝑔𝑠 (𝜈) ∫ ∫ 𝑡∧𝑟𝑖+1 ∫ + 𝑤 𝑡−𝑠 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤) 𝑀 (𝐸) ◦

𝑡∧𝑟𝑖

0

ˆ 𝑊

and let ℋ𝑡𝑖 = 𝜎(ℋ𝑡0,𝑖 ∪ ℋ𝑡1,𝑖 ), where ℋ𝑡0,𝑖 is the 𝜎-algebra generated by random variables of the form ∫ ∞∫ ∫ ∞∫ ℎ𝑡 (𝑠, 𝑦, 𝑢, 𝑤)𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) 𝑟𝑖

𝐸

ˆ 𝑊

0

with ℎ𝑡 (𝑠, 𝑦, 𝑢, 𝑤) given by (12.3) and ℋ𝑡1,𝑖 is the 𝜎-algebra generated by random variables of the form ∫ ∞∫ ∫ ∞∫ 𝐻𝑡 (𝑠, 𝜈, 𝑢, 𝑤)𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤) 𝑟𝑖

𝑀 (𝐸) ◦

0

ˆ 𝑊

with 𝐻𝑡 (𝑠, 𝜈, 𝑢, 𝑤) given by (12.5). Using Theorem 12.3 again we see the following properties hold: • {(𝑍𝑖 (𝑡), ℋ¯𝑡𝑖 ) : 𝑟 𝑖 ≤ 𝑡 ≤ 𝑟 𝑖+1 } and {(𝑍𝑖 (𝑡), 𝒢¯ 𝑡 ) : 𝑟 𝑖 ≤ 𝑡 ≤ 𝑟 𝑖+1 } are immigration superprocesses with transition semigroup (𝑄 𝑖+1,𝑡 )𝑡 ≥0 under P(·|𝒢¯ 𝑟𝑖 ); • {(𝑍𝑖 (𝑡), ℋ¯𝑡𝑖 ) : 𝑡 ≥ 𝑟 𝑖+1 } and {(𝑍𝑖 (𝑡), 𝒢¯ 𝑡 ) : 𝑡 ≥ 𝑟 𝑖+1 } are superprocesses with transition semigroup (𝑄 𝑡 )𝑡 ≥0 under P(·|𝒢¯ 𝑟𝑖 ) and P(·|𝒢¯ 𝑟𝑖+1 ). Clearly, the processes {(𝑌𝑖 (𝑡), 𝒢¯ 𝑡𝑖 ) : 𝑡 ≥ 𝑟 𝑖 } and {(𝑍𝑖 (𝑡), ℋ¯𝑡𝑖 ) : 𝑡 ≥ 𝑟 𝑖 } are independent of each other under P(·|𝒢¯ 𝑟𝑖 ). Observe also that 𝑌𝑖+1 (𝑡) = 𝑌𝑖 (𝑡) + 𝑍𝑖 (𝑡) and 𝒢¯ 𝑡 = 𝜎(𝒢¯ 𝑡𝑖 ∪ ℋ¯𝑡𝑖 ) for 𝑟 𝑖 ≤ 𝑡 ≤ 𝑟 𝑖+1 . By Corollary 9.3 we infer that properties (i) and (ii) also hold when 𝑖 is replaced by 𝑖 + 1. Then they hold for all 𝑖 ≥ 1 by induction. In particular, for 𝑖 ≥ 1 the process {(𝑌𝑡 , 𝒢¯ 𝑡 ) : 𝑟 𝑖−1 ≤ 𝑡 ≤ 𝑟 𝑖 } under P{·|𝒢¯ 𝑟𝑖−1 } is an immigration superprocess with transition semigroup (𝑄 𝑖,𝑡 )𝑡 ≥0 . □ Lemma 12.13 The results of Theorem 12.7 and its corollaries hold for step processes 𝑞 ∈ ℒ𝜂00 (𝐸) + and 𝑔 ∈ ℒ𝐻0 1 (𝑀 (𝐸) ◦ ) + .

350

12 State-Dependent Immigration Structures

Proof Suppose that 𝑞 and 𝑔 are given by (12.16) and (12.17), respectively. By Lemma 12.12 the process {(𝑌𝑡 , 𝒢¯ 𝑡 ) : 𝑟 𝑖−1 ≤ 𝑡 ≤ 𝑟 𝑖 } under P{·|𝒢¯ 𝑟𝑖−1 } is a homogeneous immigration superprocess with transition semigroup (𝑄 𝑖,𝑡 )𝑡 ≥0 defined by (12.18) and (12.19). It is well known that the locally compact separable metric space 𝐸 is topologically complete. By Theorem 1.17, the space 𝑀 (𝐸) is also topologically separable and complete. By Theorems 9.24 and A.7 the process {𝑌𝑡 : 𝑡 ≥ 0} has a càdlàg modification. Then {(𝑌𝑡 , 𝒢¯ 𝑡+ ) : 𝑟 𝑖−1 ≤ 𝑡 ≤ 𝑟 𝑖 } under P{·|𝒢¯ 𝑟𝑖−1 } is an immigration superprocess with transition semigroup (𝑄 𝑖,𝑡 )𝑡 ≥0 by Proposition 9.29. By applying the results in Section 9.3 on the time intervals [𝑟 𝑖−1 , 𝑟 𝑖 ], 𝑖 = 1, 2, . . . □ successively we obtain the theorem and its corollaries. Proof (of Theorem 12.7 and its corollaries) Take sequences {𝑞 𝑘 } ⊂ ℒ𝜂00 (𝐸) + and {𝑔 𝑘 } ⊂ ℒ𝐻0 1 (𝑀 (𝐸) ◦ ) + so that 𝑑1 (𝑞 𝑘 , 𝑞) → 0 and 𝐷 1 (𝑔 𝑘 , 𝑔) → 0 as 𝑘 → ∞. Let {𝑌𝑘 (𝑡) : 𝑡 ≥ 0} be the process defined by (12.15) with 𝑞 𝑠 (𝑦) = 𝑞 𝑘 (𝑠, 𝑦) and 𝑔𝑠 (𝜈) = 𝑔 𝑘 (𝑠, 𝜈). By Lemma 12.13, each {𝑌𝑘 (𝑡) : 𝑡 ≥ 0} has a càdlàg modification, which we shall use in this proof. By Corollary 12.6 we have ∫ 𝑡    P ∥𝑌 𝑗 (𝑡) − 𝑌𝑘 (𝑡) ∥ ≤ e𝑐0 (𝑡−𝑠) P ∥𝑞 𝑗 (𝑠, ·) − 𝑞 𝑘 (𝑠, ·) ∥ 𝜂0 0  + ∥𝑔 𝑗 (𝑠, ·) − 𝑔 𝑘 (𝑠, ·) ∥ 𝐻1 d𝑠. Then there is a process {𝑌 (𝑡) : 𝑡 ≥ 0} in 𝑀 (𝐸) such that   lim sup P ∥𝑌𝑘 (𝑠) − 𝑌 (𝑠) ∥ = 0, 𝑡 ≥ 0. 𝑘→∞ 0≤𝑠 ≤𝑡

For 𝑗, 𝑘 ≥ 1 let 𝑞 𝑗,𝑘 = 𝑞 𝑗 ∧ 𝑞 𝑘 and 𝑞 𝑗,𝑘 = 𝑞 𝑗 ∨ 𝑞 𝑘 . Similarly, let 𝑔 𝑗,𝑘 = 𝑔 𝑗 ∧ 𝑔 𝑘 and 𝑔 𝑗,𝑘 = 𝑔 𝑗 ∨ 𝑔 𝑘 . Then 𝑞 𝑗,𝑘 − 𝑞 𝑗,𝑘 = |𝑞 𝑗 − 𝑞 𝑘 | and 𝑔 𝑗,𝑘 − 𝑔 𝑗,𝑘 = |𝑔 𝑗 − 𝑔 𝑘 |. As in the proof of Corollary 12.6 one can see ∥𝑌 𝑗 (𝑡) − 𝑌𝑘 (𝑡) ∥ ≤ ⟨𝑌 𝑗,𝑘 (𝑡), 1⟩, where ∫

𝑡

|𝑞 𝑗 −𝑞𝑘 | (𝑠,𝑦)

∫ ∫

∫

𝑗,𝑘

𝑤 𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤)

𝑌 𝑗,𝑘 (𝑡) = 0∫

+

𝑡

d𝑠 ∫0 𝑡 ∫

+ 0

ˆ 𝑊

0 ∫

𝐸

𝐸

|𝑞 𝑗 − 𝑞 𝑘 |(𝑠, 𝑦)𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) ∫ |𝑔 𝑗 −𝑔𝑘 | (𝑠,𝜈) ∫ 𝑗,𝑘 𝑤 𝑡−𝑠 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤),

𝑀 (𝐸) ◦

ˆ 𝑊

0

𝑗,𝑘

where 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) is a Poisson random measure with intensity d𝑠𝜂0 (d𝑦) 𝑗,𝑘 d𝑢L(𝑦, d𝑤) and 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤) is a Poisson random measure with intensity d𝑠𝐻0 (d𝜈)d𝑢Q(𝜈, d𝑤). Then {𝑌 𝑗,𝑘 (𝑡) : 𝑡 ≥ 0} is an immigration superprocess with immigration rates (|𝑞 𝑗 − 𝑞 𝑘 |, |𝑔 𝑗 − 𝑔 𝑘 |) relative to (𝜂0 , 𝐻0 ). For 𝑡 ≥ 0 let 𝑧 𝑗,𝑘 (𝑡) = e−𝑐0 𝑡 ⟨𝑌 𝑗,𝑘 (𝑡), 1⟩ −

∫

𝑡

 e−𝑐0 𝑠 ∥𝑞 𝑗 (𝑠, ·) − 𝑞 𝑘 (𝑠, ·) ∥ 𝜂0 0  + ∥𝑔 𝑗 (𝑠, ·) − 𝑔 𝑘 (𝑠, ·) ∥ 𝐻1 d𝑠.

12.3 State-Dependent Immigration Rates

351

By Corollary 12.9 and Lemma 12.13, for any 𝜆 ≥ 0 we have ∫ 𝑡 o n  e−𝑐0 𝑠 ∥𝑞 𝑗 (𝑠, ·) − 𝑞 𝑘 (𝑠, ·) ∥ 𝜂0 𝜆P sup |𝑧 𝑗,𝑘 (𝑠)| ≥ 𝜆 ≤ 2P 0≤𝑠 ≤𝑡

0





+ ∥𝑔 𝑗 (𝑠, ·) − 𝑔 𝑘 (𝑠, ·) ∥ 𝐻1 d𝑠 . The right-hand side vanishes as 𝑗, 𝑘 → ∞. Then we can find a subsequence {𝑘 𝑛 } ⊂ {𝑘 } such that h i P sup ∥𝑌𝑘𝑛−1 (𝑠) − 𝑌𝑘𝑛 (𝑠) ∥ ≥ 1/2𝑛 ≤ 1/2𝑛 , 𝑛 ≥ 1. 0≤𝑠 ≤𝑛

By the Borel–Cantelli lemma, there is a càdlàg process {𝑌𝑡 : 𝑡 ≥ 0} in 𝑀 (𝐸) such that a.s. lim sup ∥𝑌𝑘𝑛 (𝑠) − 𝑌𝑠 ∥ = 0,

𝑡 ≥ 0.

𝑛→∞ 0≤𝑠 ≤𝑡

Then {𝑌𝑡 : 𝑡 ≥ 0} is a modification of {𝑌 (𝑡) : 𝑡 ≥ 0} and hence   lim sup P ∥𝑌𝑘 (𝑠) − 𝑌𝑠 ∥ = 0, 𝑡 ≥ 0. 𝑘→∞ 0≤𝑠 ≤𝑡

Clearly, the equality (12.15) holds. Since the theorem and its corollaries hold for each process {𝑌𝑘 (𝑡) : 𝑡 ≥ 0}, they also hold for {𝑌𝑡 : 𝑡 ≥ 0} by approximation arguments. □

12.3 State-Dependent Immigration Rates Suppose that the (𝜉, 𝜙)-superprocess {(𝑋𝑡 , ℱ𝑡 )} and the Poisson random measures {𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤)} and {𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤)} are given as in Section 12.1. In this section, we assume P[𝑋0 (1)] < ∞. Let (𝒢¯ 𝑡 )𝑡 ≥0 be the augmentation of the filtration (𝒢𝑡 )𝑡 ≥0 defined in Theorem 12.3. Let (𝜇, 𝑦) ↦→ 𝑞(𝜇, 𝑦) be a positive Borel function on 𝑀 (𝐸)×𝐸 and let (𝜇, 𝜈) ↦→ 𝑔(𝜇, 𝜈) be a positive Borel function on 𝑀 (𝐸)×𝑀 (𝐸) ◦ . We assume that: • (linear growth condition) there is a constant 𝐾 ≥ 0 such that, for 𝜇 ∈ 𝑀 (𝐸), ∫ ⟨𝜂0 , 𝑞(𝜇, ·)⟩ + 𝑔(𝜇, 𝜈)⟨𝜈, 1⟩𝐻0 (d𝜈) ≤ 𝐾 (1 + ⟨𝜇, 1⟩); (12.20) 𝑀 (𝐸) ◦

• (local Lipschitz condition) for each 𝑅 > 0 there is a constant 𝐾 𝑅 ≥ 0 such that, for 𝜇1 , 𝜇2 ∈ 𝑀 (𝐸) satisfying ⟨𝜇1 , 1⟩, ⟨𝜇2 , 1⟩ ≤ 𝑅,

352

12 State-Dependent Immigration Structures

∫ ⟨𝜂0 , |𝑞(𝜇1 , ·) − 𝑞(𝜇2 , ·)|⟩ +

𝑀 (𝐸) ◦

|𝑔(𝜇1 , 𝜈) − 𝑔(𝜇2 , 𝜈)|𝐻1 (d𝜈) ≤ 𝐾 𝑅 ∥𝜇1 − 𝜇2 ∥,

(12.21)

where 𝐻1 (d𝜈) = ⟨𝜈, 1⟩𝐻0 (d𝜈). We consider the stochastic integral equation, for 𝑡 ≥ 0, ∫ 𝑡 ∫ ∫ 𝑞 (𝑌𝑠− ,𝑦) ∫ 𝑌𝑡 = 𝑋𝑡 + 𝑤 𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) ˆ 𝑊 ∫ 𝑡 0 ∫𝐸 0 + d𝑠 𝑞(𝑌𝑠− , 𝑦)𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) ∫0 𝑡 ∫ 𝐸 ∫ 𝑔 (𝑌𝑠− ,𝜈) ∫ + 𝑤 𝑡−𝑠 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤). 0

𝑀 (𝐸) ◦

0

(12.22)

ˆ 𝑊

By a solution of (12.22) we mean a càdlàg process {𝑌𝑡 : 𝑡 ≥ 0} in 𝑀 (𝐸) that is adapted to the filtration (𝒢¯ 𝑡+ )𝑡 ≥0 and satisfies the equation with probability one. Theorem 12.14 There is a pathwise unique solution {𝑌𝑡 : 𝑡 ≥ 0} to (12.22) and the solution has the following property: for 𝐺 ∈ 𝐶 2 (R) and 𝑓 ∈ 𝐷 0 ( 𝐴), ∫ 𝑡 𝐺 (𝑌𝑡 ( 𝑓 )) = 𝐺 (𝑌0 ( 𝑓 )) + 𝐺 ′ (𝑌𝑠 ( 𝑓 ))𝑌𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠 0 ∫ 𝑡   ′′ 𝐺 (𝑌𝑠 ( 𝑓 ))𝑌𝑠 (𝑐 𝑓 2 ) + 𝐺 ′ (𝑌𝑠 ( 𝑓 ))⟨𝜂0 , 𝑞(𝑌𝑠 , ·) 𝑓 ⟩ d𝑠 + ∫ ∫0 𝑡 ∫  𝐺 (𝑌𝑠 ( 𝑓 ) + 𝜈( 𝑓 )) − 𝐺 (𝑌𝑠 ( 𝑓 )) d𝑠 𝑌𝑠 (d𝑥) + 0 𝐸 𝑀 (𝐸) ◦ ∫ 𝑡 ∫   − ⟨𝜈, 𝑓 ⟩𝐺 ′ (𝑌𝑠 ( 𝑓 )) 𝐻 (𝑥, d𝜈) + 𝐺 (𝑌𝑠 ( 𝑓 ) + 𝜈( 𝑓 )) d𝑠 0 𝑀 (𝐸) ◦  − 𝐺 (𝑌𝑠 ( 𝑓 )) 𝑔(𝑌𝑠 , 𝜈)𝐻0 (d𝜈) + (𝒢¯ 𝑡+ )-local mart. (12.23) 𝜇

For 𝜇 ∈ 𝑀 (𝐸) let {𝑌𝑡 : 𝑡 ≥ 0} be the solution to (12.22) with 𝑌0 = 𝑋0 = 𝜇 and 𝑞,𝑔 𝜇 let 𝑃𝑡 (𝜇, ·) be the distribution of 𝑌𝑡 on 𝑀 (𝐸). Theorem 12.15 The solution {𝑌𝑡 : 𝑡 ≥ 0} to (12.22) is a strong Markov process 𝑞,𝑔 relative to (𝒢¯ 𝑡+ )𝑡 ≥0 with transition semigroup (𝑃𝑡 )𝑡 ≥0 . In view of (12.22) and (12.23), we may interpret {𝑌𝑡 : 𝑡 ≥ 0} as an immigration superprocess with state-dependent immigration rates {(𝑞(𝑌𝑠− , 𝑦), 𝑔(𝑌𝑠− , 𝜈)) : 𝑠 ≥ 0, 𝑦 ∈ 𝐸, 𝜈 ∈ 𝑀 (𝐸) ◦ } with respect to the reference measures (𝜂0 , 𝐻0 ), where 𝑌0− = 0 by convention. The uniqueness of solution to the martingale problem (12.23) still remains open. Then the above theorem shows the advantage of the stochastic equation (12.22) in constructing the superprocess with state-dependent immigration as a Markov process.

12.3 State-Dependent Immigration Rates

353

Proposition 12.16 There is at most one solution to (12.22). Proof Suppose {𝑌1 (𝑡) : 𝑡 ≥ 0} and {𝑌2 (𝑡) : 𝑡 ≥ 0} are two solutions of (12.22). Let 𝜏𝑅 = inf{𝑡 ≥ 0 : ⟨𝑌1 (𝑡), 1⟩ ≥ 𝑅 or ⟨𝑌2 (𝑡), 1⟩ ≥ 𝑅}. Then we have a.s. 𝜏𝑅 → ∞ as 𝑅 → ∞. For 𝑖 = 1, 2 let ∫ 𝑡∧𝜏𝑅 ∫ ∫ 𝑞 (𝑌𝑖 (𝑠−) ,𝑦) ∫ 𝑍𝑖𝑅 (𝑡) = 𝑋𝑡 + 𝑤 𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) ˆ 0 𝑊 𝐸 0 ∫ 𝑡∧𝜏 ∫ 𝑅 d𝑠 𝑞(𝑌𝑖 (𝑠−), 𝑦)𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) + 0 𝐸 ∫ 𝑡∧𝜏𝑅 ∫ ∫ 𝑔 (𝑌𝑖 (𝑠−) ,𝜈) ∫ 𝑤 𝑡−𝑠 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤). + 0

𝑀 (𝐸) ◦

ˆ 𝑊

0

Then {𝑍𝑖𝑅 (𝑡) : 𝑡 ≥ 0} is an immigration superprocess with predictable immigration rates 1 {𝑠 ≤𝜏𝑅 } (𝑞(𝑌𝑖 (𝑠−), 𝑦), 𝑔(𝑌𝑖 (𝑠−), 𝜈)). It is easy to see that ∥𝑌1 (𝑡) − 𝑌2 (𝑡) ∥1 {𝑡 ≤𝜏𝑅 } ≤ 𝑧 𝑅 (𝑡) := ∥𝑍1𝑅 (𝑡) − 𝑍2𝑅 (𝑡) ∥. By (12.21) and Corollary 12.6 we have  ∫ 𝑡∧𝜏𝑅  P[𝑧 𝑅 (𝑡)] ≤ P e𝑐0 (𝑡−𝑠) ⟨𝜂0 , |𝑞(𝑌1 (𝑠−), ·) − 𝑞(𝑌2 (𝑠−), ·)|⟩   ∫ 0 ∗ + [𝑔 (𝑌1 (𝑠−), 𝜈) − 𝑔∗ (𝑌1 (𝑠−), 𝜈)]𝐻1 (d𝜈) d𝑠 ◦ 𝑀(𝐸)  ∫ 𝑡∧𝜏𝑅 𝑐0 (𝑡−𝑠) ∥𝑌1 (𝑠−) − 𝑌2 (𝑠−) ∥d𝑠 e ≤ 𝐾𝑅 P 0 ∫  𝑡∧𝜏𝑅 + ≤ 𝐾 𝑅 e𝑐0 𝑡 P ∥𝑌1 (𝑠) − 𝑌2 (𝑠) ∥d𝑠 ∫ 𝑡0 + P[𝑧 𝑅 (𝑠)]d𝑠. ≤ 𝐾 𝑅 e𝑐0 𝑡 0

By Gronwall’s inequality we conclude P[𝑧 𝑅 (𝑡)] = 0 for all 𝑡 ≥ 0 and 𝑅 ≥ 0. Then □ {𝑌1 (𝑡) : 𝑡 ≥ 0} and {𝑌2 (𝑡) : 𝑡 ≥ 0} are indistinguishable. Proposition 12.17 Suppose that there is a universal constant 𝐾 ≥ 0 such that, for 𝜇1 , 𝜇2 ∈ 𝑀 (𝐸), ∫ ⟨𝜂0 , |𝑞(𝜇1 , ·) − 𝑞(𝜇2 , ·)|⟩ + |𝑔(𝜇1 , 𝜈) − 𝑔(𝜇2 , 𝜈)|𝐻1 (d𝜈) 𝑀 (𝐸) ◦

≤ 𝐾 ∥𝜇1 − 𝜇2 ∥.

(12.24)

Then the result of Theorem 12.14 holds. Proof By Proposition 12.16 the pathwise uniqueness of solution holds for (12.22). We shall prove the existence of the solution using an iteration argument. Let 𝑌0 (𝑡) = 𝑋𝑡 and inductively define, for 𝑘 ≥ 1,

354

12 State-Dependent Immigration Structures

∫ 𝑡 ∫ ∫ 𝑞 (𝑌𝑘−1 (𝑠−) ,𝑦) ∫ 𝑌𝑘 (𝑡) = 𝑋𝑡 + 𝑤 𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) ˆ 𝑊 ∫ 𝑡 0 ∫𝐸 0 + d𝑠 𝑞(𝑌𝑘−1 (𝑠−), 𝑦)𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) ∫0 𝑡 ∫ 𝐸 ∫ 𝑔 (𝑌𝑘−1 (𝑠−),𝜈) ∫ 𝑤 𝑡−𝑠 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤). (12.25) + 𝑀 (𝐸) ◦

0

ˆ 𝑊

0

By Corollary 12.6 and (12.24) it is not hard to see that ∫ 𝑡  𝑐0 (𝑡−𝑠) ∥𝑌𝑘−1 (𝑠−) − 𝑌𝑘−2 (𝑠−) ∥d𝑠 e P[∥𝑌𝑘 (𝑡) − 𝑌𝑘−1 (𝑡) ∥] ≤ 𝐾P 0 ∫ 𝑡 + P[∥𝑌𝑘−1 (𝑠) − 𝑌𝑘−2 (𝑠) ∥]d𝑠. ≤ 𝐾e𝑐0 𝑡 0

By Corollary 12.6 and (12.20) we have ∫ P[∥𝑌1 (𝑡) − 𝑌0 (𝑡) ∥] ≤ P

𝑡

0

e ∫

𝑐0 (𝑡−𝑠)

+ 𝑀 (𝐸) ◦

∫

∫ 𝑞(𝑋𝑠− , 𝑦)𝜂0 (d𝑦)   𝑔(𝑋𝑠− , 𝜈)𝐻1 (d𝜈) d𝑠 𝐸

𝑡

e𝑐0 (𝑡−𝑠) P[1 + ⟨𝑋𝑠− , 1⟩]d𝑠 ∫ 𝑡 + P[1 + ⟨𝑋𝑠 , 1⟩]d𝑠, = 𝐾e𝑐0 𝑡

≤𝐾

0

0

which is locally bounded. Then a standard argument shows ∞ ∑︁

  sup P ∥𝑌𝑘 (𝑠) − 𝑌𝑘−1 (𝑠) ∥ < ∞.

𝑘=1 0≤𝑠 ≤𝑡

Write 𝑞 𝑘 (𝑠, 𝑦) = 𝑞(𝑌𝑘 (𝑠−), 𝑦) and 𝑔 𝑘 (𝑠, 𝑦) = 𝑔(𝑌𝑘 (𝑠−), 𝑦). In view of (12.24) we have ∞ ∑︁ 

 ∥𝑞 𝑘 − 𝑞 𝑘−1 ∥ 𝜂0 ,𝑡 + ∥𝑔 𝑘 − 𝑔 𝑘−1 ∥ 𝐻1 ,𝑡 < ∞.

𝑘=1

It follows immediately that   lim ∥𝑞 𝑘 − 𝑞 𝑗 ∥ 𝜂0 ,𝑡 + ∥𝑔 𝑘 − 𝑔 𝑗 ∥ 𝐻1 ,𝑡 = 0. 𝑗,𝑘→∞

Then there exist 𝑞 ∈ ℒ𝜂10 (𝐸) + and 𝑔 ∈ ℒ𝐻1 1 (𝑀 (𝐸) ◦ ) + such that 𝑑1 (𝑞 𝑘 , 𝑞) → 0 and 𝐷 1 (𝑔 𝑘 , 𝑔) → 0 as 𝑘 → ∞. By the arguments in the proof of Theorem 12.7, there is a subsequence {𝑘 𝑛 } ⊂ {𝑘 } and a càdlàg process {𝑌𝑡 : 𝑡 ≥ 0} in 𝑀 (𝐸) such that a.s. lim sup ∥𝑌𝑘𝑛 (𝑠) − 𝑌𝑠 ∥ = 0,

𝑘𝑛 →∞ 0≤𝑠 ≤𝑡

𝑡≥0

12.3 State-Dependent Immigration Rates

355

and lim sup P[∥𝑌𝑘 (𝑠) − 𝑌𝑠 ∥] = 0,

𝑡 ≥ 0.

𝑘→∞ 0≤𝑠 ≤𝑡

Clearly, the equality (12.15) holds. By using (12.24) again we see   lim ∥𝑞 𝑘 − 𝑞(𝑌·− , ·) ∥ 𝜂0 ,𝑡 + ∥𝑔 𝑘 − 𝑔(𝑌·− , ·) ∥ 𝐻1 ,𝑡 = 0. 𝑘→∞

It follows that 𝑞 = 𝑞(𝑌·− , ·) in ℒ𝜂10 (𝐸) and 𝑔 = 𝑔(𝑌·− , ·) in ℒ𝐻1 1 (𝑀 (𝐸) ◦ ). Then {𝑌𝑡 : 𝑡 ≥ 0} is a solution of (12.22). The characterization (12.23) of the process □ follows by Theorem 12.7.

Theorem 12.18 Suppose that there is a universal constant 𝐾 ≥ 0 such that (12.24) holds. Then, for 𝜇1 , 𝜇2 ∈ 𝑀 (𝐸), 𝑞,𝑔

𝑊1 (𝑃𝑡

𝑞,𝑔

(𝜇1 , ·), 𝑃𝑡

(𝜇2 , ·)) ≤ ∥𝜇1 − 𝜇2 ∥e (𝑐0 +𝐾)𝑡 ,

𝑡 ≥ 0.

Proof Let 𝑀𝜆 (𝐸) be the subset of 𝑀 (𝐸) consisting of measures absolutely continuous relative to 𝜆 := 𝜇1 + 𝜇2 . Let 𝑁 (d𝑥, d𝑤, d𝑢) be a Poisson random measure on 𝐸 × 𝑊ˆ × (0, ∞) with intensity 𝜆(d𝑥)L(𝑥, d𝑤)d𝑢. We assume 𝑁 (d𝑥, d𝑤, d𝑢) is independent of 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) and 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤). For any 𝜇 ∈ 𝑀𝜆 (𝐸) let 𝜇 𝜇 {𝑋𝑡 : 𝑡 ≥ 0} be defined by (8.63). Then we can construct {𝑌𝑡 : 𝑡 ≥ 0} by the 𝜇 pathwise solution to (12.22) with {𝑋𝑡 : 𝑡 ≥ 0} replaced by {𝑋𝑡 : 𝑡 ≥ 0}. Let 𝜇1 ∨𝜇2 𝜇1 ∧𝜇2 and 𝑋 (𝑡) = 𝑋𝑡 − 𝑋𝑡 𝑞 ∗ (𝑠, 𝑦) = 𝑞(𝑌𝑠−1 , 𝑦) ∧ 𝑞(𝑌𝑠−2 , 𝑦), 𝑞 ∗ (𝑠, 𝑦) = 𝑞(𝑌𝑠−1 , 𝑦) ∨ 𝑞(𝑌𝑠−2 , 𝑦), 𝜇 𝜇 𝜇 𝜇 𝑔∗ (𝑠, 𝜈) = 𝑔(𝑌𝑠−1 , 𝜈) ∧ 𝑔(𝑌𝑠−2 , 𝜈), 𝑔 ∗ (𝑠, 𝑦) = 𝑔(𝑌𝑠−1 , 𝜈) ∨ 𝑔(𝑌𝑠−2 , 𝜈). 𝜇

𝜇

𝜇

𝜇

𝜇

𝜇

𝜇

𝜇

It is not hard to see that ∥ 𝑋𝑡 1 − 𝑋𝑡 2 ∥ ≤ ⟨𝑋 (𝑡), 1⟩ and ∥𝑌𝑡 1 − 𝑌𝑡 2 ∥ ≤ ⟨𝑌 (𝑡), 1⟩, where ∫ 𝑡 ∫ ∫ 𝑞∗ (𝑠,𝑦) ∫ 𝑌 (𝑡) = 𝑋 (𝑡) + 𝑤 𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) ˆ ∫ 𝑡 0∫ 𝐸 𝑞∗ (𝑠,𝑦) 𝑊 d𝑠 [𝑞 ∗ (𝑠, 𝑦) − 𝑞 ∗ (𝑠, 𝑦)]𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) + 0 ∫ 𝑡 ∫ 𝐸 ∫ 𝑔∗ (𝑠,𝜈) ∫ + 𝑤 𝑡−𝑠 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤). 0

𝑀 (𝐸) ◦

𝑔∗ (𝑠,𝜈)

ˆ 𝑊

By Proposition 12.5 we can see as in the proof of Corollary 12.6 that ∫ 𝑡  𝜇 𝜇 P[⟨𝑌 (𝑡), 1⟩] = ⟨|𝜇1 − 𝜇2 |, 𝜋𝑡 1⟩ + P ⟨𝜂0 , |𝑞(𝑌𝑠−1 , ·) − 𝑞(𝑌𝑠−2 , ·)|𝜋𝑡−𝑠 1⟩ 0   ∫ 𝜇 𝜇 |𝑔(𝑌𝑠−1 , 𝜈) − 𝑔(𝑌𝑠−2 , 𝜈)|⟨𝜈, 𝜋𝑡−𝑠 1⟩𝐻0 (d𝜈) d𝑠 + 𝑀 (𝐸) ◦

356

12 State-Dependent Immigration Structures

∫



𝑡

𝜇

𝜇

≤ ⟨|𝜇1 − 𝜇2 |, 1⟩e + P ⟨𝜂0 , |𝑞(𝑌𝑠−1 , ·) − 𝑞(𝑌𝑠−2 , ·)|⟩ 0   ∫ 𝜇1 𝜇2 𝑐0 (𝑡−𝑠) |𝑔(𝑌𝑠− , 𝜈) − 𝑔(𝑌𝑠− , 𝜈)|𝐻1 (d𝜈) e + d𝑠 𝑐0 𝑡

𝑀 (𝐸) ◦

∫

𝑡

e𝑐0 (𝑡−𝑠) P[∥𝑌𝑠 1 − 𝑌𝑠 2 ∥]d𝑠 𝜇

≤ ∥𝜇1 − 𝜇2 ∥e𝑐0 𝑡 + 𝐾

𝜇

0

∫

𝑡

e𝑐0 (𝑡−𝑠) P[⟨𝑌 (𝑠), 1⟩]d𝑠.

≤ ∥𝜇1 − 𝜇2 ∥e𝑐0 𝑡 + 𝐾 0

It follows that e−𝑐0 𝑡 P[⟨𝑌 (𝑡), 1⟩] ≤ ∥𝜇 − 𝜈∥ + 𝐾

∫

𝑡

e−𝑐0 𝑠 P[⟨𝑌 (𝑠), 1⟩]d𝑠.

0

By Gronwall’s inequality we have e−𝑐0 𝑡 P[∥𝑌𝑡 1 − 𝑌𝑡 2 ∥] ≤ e−𝑐0 𝑡 P[⟨𝑌 (𝑡), 1⟩] ≤ ∥𝜇1 − 𝜇2 ∥e𝐾𝑡 , 𝜇

𝜇

which implies the desired estimate.

□

We next use a localization method to establish Theorem 12.14 under the general conditions (12.20) and (12.21). For any integer 𝑛 ≥ 1 it is easy to define a continuously differentiable function 𝑧 ↦→ 𝑎 𝑛 (𝑧) on [0, ∞) such that 𝑎 𝑛 (𝑧) =

for 0 ≤ 𝑧 ≤ 𝑛 − 1, for 𝑧 ≥ 𝑛 + 1.

n1 𝑛/𝑧

In addition, we can also assume 0 ≤ 𝑎 𝑛 (𝑧) ≤ 1 ∧ (𝑛/𝑧) and −1/𝑧 ≤ 𝑎 𝑛′ (𝑧) ≤ 0 for all 𝑧 > 0. Lemma 12.19 Suppose that 𝑞 and 𝑔 satisfy (12.21). For 𝜇 ∈ 𝑀 (𝐸), 𝑦 ∈ 𝐸 and 𝜈 ∈ 𝑀 (𝐸) ◦ let 𝑞 𝑛 (𝜇, 𝑦) = 𝑞(𝑎 𝑛 (⟨𝜇, 1⟩)𝜇, 𝑦),

𝑔𝑛 (𝜇, 𝜈) = 𝑔(𝑎 𝑛 (⟨𝜇, 1⟩)𝜇, 𝜈).

Then we have, for 𝜇1 , 𝜇2 ∈ 𝑀 (𝐸), ∫ ⟨𝜂0 , |𝑞 𝑛 (𝜇1 , ·) − 𝑞 𝑛 (𝜇2 , ·)|⟩ +

𝑀 (𝐸) ◦

|𝑔𝑛 (𝜇1 , 𝜈) − 𝑔𝑛 (𝜇2 , 𝜈)|𝐻1 (d𝜈)⟩ ≤ 2𝐾𝑛 ∥𝜇1 − 𝜇2 ∥.

(12.26)

Proof By the mean-value theorem, for 𝜇1 , 𝜇2 ∈ 𝑀 (𝐸) satisfying ⟨𝜇1 , 1⟩ ≤ ⟨𝜇2 , 1⟩ there exists an 𝑠 ∈ [⟨𝜇1 , 1⟩, ⟨𝜇2 , 1⟩] such that ⟨𝜇1 , 1⟩|𝑎 𝑛 (⟨𝜇1 , 1⟩) − 𝑎 𝑛 (⟨𝜇2 , 1⟩)| ≤ ⟨𝜇1 , 1⟩|𝑎 𝑛′ (𝑠)||⟨𝜇1 , 1⟩ − ⟨𝜇2 , 1⟩| ≤ ⟨𝜇1 , 1⟩𝑠−1 ∥𝜇1 − 𝜇2 ∥ ≤ ∥𝜇1 − 𝜇2 ∥.

12.3 State-Dependent Immigration Rates

357

Let 𝛾 = 𝜇1 +𝜇2 and let 𝜇¤ 1 and 𝜇¤ 2 denote respectively the Radon–Nikodym derivatives of 𝜇1 and 𝜇2 with respect to 𝛾. It follows that l.h.s. of (12.26) ≤ 𝐾𝑛 ∥𝑎 𝑛 (⟨𝜇1 , 1⟩)𝜇1 − 𝑎 𝑛 (⟨𝜇2 , 1⟩)𝜇2 ∥ = 𝐾𝑛 ⟨𝛾, |𝑎 𝑛 (⟨𝜇1 , 1⟩) 𝜇¤ 1 − 𝑎 𝑛 (⟨𝜇2 , 1⟩) 𝜇¤ 2 |⟩ ≤ 𝐾𝑛 |𝑎 𝑛 (⟨𝜇1 , 1⟩) − 𝑎 𝑛 (⟨𝜇2 , 1⟩)|⟨𝛾, 𝜇¤ 1 ⟩ + 𝐾𝑛 𝑎 𝑛 (⟨𝜇2 , 1⟩)⟨𝛾, | 𝜇¤ 1 − 𝜇¤ 2 |⟩ ≤ 𝐾𝑛 |𝑎 𝑛 (⟨𝜇1 , 1⟩) − 𝑎 𝑛 (⟨𝜇2 , 1⟩)|⟨𝜇1 , 1⟩ + 𝐾𝑛 ∥𝜇1 − 𝜇2 ∥ ≤ 2𝐾𝑛 ∥𝜇1 − 𝜇2 ∥. Then we have the desired estimate.

□

Proof (of Theorem 12.14) By Proposition 12.16 the uniqueness of solution holds for (12.22). For each integer 𝑛 ≥ 1 let 𝑞 𝑛 (𝜇, 𝑦) and 𝑔𝑛 (𝜇, 𝜈) be defined as in Lemma 12.19. By Proposition 12.17 there is a unique solution {𝑌𝑛 (𝑡) : 𝑡 ≥ 0} to (12.22) with (𝑞, 𝑔) replaced by (𝑞 𝑛 , 𝑔𝑛 ). Let 𝑦 𝑛 (𝑡) = ⟨𝑌𝑛 (𝑡), 1⟩ and define the stopping time 𝜎𝑛 = inf{𝑡 ≥ 0 : 𝑦 𝑛 (𝑡) ≥ 𝑛 − 1} for 𝑛 ≥ 1. Then {𝑌𝑘 (𝑡 ∧ 𝜎𝑘 ) : 𝑡 ≥ 0} and {𝑌𝑛 (𝑡 ∧ 𝜎𝑘 ) : 𝑡 ≥ 0} are indistinguishable for any 𝑛 ≥ 𝑘 ≥ 1. It is easy to show {𝜎𝑛 } is an increasing sequence. By Corollary 12.9 and Doob’s optional stopping theorem we have   ∫ 𝑡∧𝜎𝑛 P e−𝑐0 (𝑡∧𝜎𝑛 ) 𝑦 𝑛 (𝑡 ∧ 𝜎𝑛 ) − e−𝑐0 𝑠 Γ𝑛 (𝑠, 1)d𝑠 ≤ P[⟨𝑋0 , 1⟩], 0

where ∫ Γ𝑛 (𝑠, 1) = ⟨𝜂0 , 𝑞(𝑌𝑛 (𝑠−), ·)⟩ +

𝑀 (𝐸) ◦

𝑔(𝑌𝑛 (𝑠−), 𝜈)𝐻1 (d𝜈).

Then using (12.20) we obtain P[𝑦 𝑛 (𝑡 ∧ 𝜎𝑛 )] ≤ e

|𝑐0 |𝑡

P[⟨𝑋0 , 1⟩] + e

|𝑐0 |𝑡

∫

−𝑐0 𝑠

e

P

≤ e |𝑐0 |𝑡 P[⟨𝑋0 , 1⟩] + 𝐾e2 |𝑐0 |𝑡 P



𝑡∧𝜎𝑛

0 ∫

𝑡∧𝜎𝑛

Γ𝑛 (𝑠, 1)d𝑠


1 + 𝑦 𝑛 (𝑠−) d𝑠 ,

0

which implies that 𝑡 ↦→ P[𝑦 𝑛 (𝑡 ∧ 𝜎𝑛 )] is locally bounded. It follows that   ∫ 𝑡∧𝜎𝑛    
1 + 𝑦 𝑛 (𝑠) d𝑠 P 1 + 𝑦 𝑛 (𝑡 ∧ 𝜎𝑛 ) ≤ e |𝑐0 |𝑡 P 1 + ⟨𝑋0 , 1⟩ + 𝐾e2 |𝑐0 |𝑡 P ∫ 𝑡0     |𝑐0 |𝑡 2 |𝑐0 |𝑡 ≤e P 1 + ⟨𝑋0 , 1⟩ + 𝐾e P 1 + 𝑦 𝑛 (𝑠 ∧ 𝜎𝑛 ) d𝑠. 0

By Gronwall’s inequality, we get      P 1 + 𝑦 𝑛 (𝑡 ∧ 𝜎𝑛 ) ≤ P 1 + ⟨𝑋0 , 1⟩ exp |𝑐 0 |𝑡 + 𝐾𝑡e2|𝑐0 |𝑡 .

(12.27)

358

12 State-Dependent Immigration Structures

The right continuity of 𝑡 ↦→ 𝑌𝑛 (𝑡) implies 𝑦 𝑛 (𝜎𝑛 ) = ⟨𝑌𝑛 (𝜎𝑛 ), 1⟩ ≥ 𝑛 − 1. From (12.27) it follows that    𝑛P{𝜎𝑛 ≤ 𝑡} ≤ P 1 + ⟨𝑋0 , 1⟩ exp |𝑐 0 |𝑡 + 𝐾𝑡e2 |𝑐0 |𝑡 and so P{𝜎𝑛 ≤ 𝑡} → 0 as 𝑛 → ∞. This implies 𝜎𝑛 → ∞ increasingly. Thus there is a càdlàg process {𝑌𝑡 : 𝑡 ≥ 0} such that a.s. 𝑌𝑛 (𝑡) = 𝑌𝑡 for every 0 ≤ 𝑡 ≤ 𝜎𝑛 . By (12.27) and Fatou’s lemma we get      P 1 + ⟨𝑌𝑡 , 1⟩ ≤ P 1 + ⟨𝑋0 , 1⟩ exp |𝑐 0 |𝑡 + 𝐾𝑡e2|𝑐0 |𝑡 . It is then easy to show that (𝑠, 𝑦) ↦→ 𝑞(𝑌𝑠− , 𝑦) belongs to ℒ𝜂10 (𝐸) and (𝑠, 𝜈) ↦→ 𝑔(𝑌𝑠− , 𝜈) belongs to ℒ𝐻1 1 (𝑀 (𝐸) ◦ ). Clearly the equation (12.22) holds. The charac□ terization (12.23) of {𝑌𝑡 : 𝑡 ≥ 0} follows by Theorem 12.7. 𝑞,𝑔

Proof (of Theorem 12.15) From Theorem 12.18 we know that 𝑃𝑡 (𝜇, d𝜈) is a kernel on 𝑀 (𝐸) if the condition (12.24) is satisfied. This is also true in the general case by the approximation of the solution to (12.22) given in the proof of Theorem 12.14. Let 𝑇 be a (𝒢¯ 𝑡+ )-stopping time. For 𝑡 ≥ 0 we can write ∫

𝑇

∫ ∫

𝑞 (𝑌𝑠− ,𝑦)

∫

𝑍𝑇+𝑡 = 𝑋𝑇+𝑡 + 𝑤 𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) ˆ 𝑊 ∫ 𝑇 0∫ 𝐸 0 + d𝑠 𝑞(𝑌𝑠− , 𝑦)𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) ∫0 𝑇 ∫ 𝐸 ∫ 𝑔 (𝑌𝑠− ,𝜈) ∫ + 𝑤 𝑡−𝑠 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤). 0

𝑀 (𝐸) ◦

0

ˆ 𝑊

By Corollary 12.8 we see that {𝑍𝑇+𝑡 : 𝑡 ≥ 0} is a (𝜉, 𝜙)-superprocess. This process is clearly independent of the Poisson random measures 𝑁0 (𝑇 + d𝑠, d𝑦, d𝑢, d𝑤) and 𝑁1 (𝑇 + d𝑠, d𝜈, d𝑢, d𝑤). It is easy to see that {𝑌𝑇+𝑡 : 𝑡 ≥ 0} solves ∫ 𝑡 ∫ ∫ 𝑞 (𝑌(𝑇+𝑠) − ,𝑦) ∫ 𝑌𝑇+𝑡 = 𝑍𝑇+𝑡 + 𝑤 𝑡−𝑠 𝑁0 (𝑇 + d𝑠, d𝑦, d𝑢, d𝑤) ˆ 𝑊 ∫ 𝑡 0∫ 𝐸 0 𝑞(𝑌 (𝑇+𝑠)− , 𝑦)𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) + d𝑠 ∫0 𝑡 ∫ 𝐸 ∫ 𝑔 (𝑌(𝑇+𝑠) − ,𝜈) ∫ + 𝑤 𝑡−𝑠 𝑁1 (𝑇 + d𝑠, d𝜈, d𝑢, d𝑤). 0

𝑀 (𝐸) ◦

0

ˆ 𝑊

By the uniqueness of solution to this equation, the random measure 𝑌𝑇+𝑡 has distri𝑞,𝑔 bution 𝑃𝑡 (𝑌𝑇 , ·) under the conditional law P(·|𝒢¯ 𝑇+ ). This gives the desired strong Markov property of {𝑌𝑡 : 𝑡 ≥ 0}. □

12.4 Changes of the Branching Mechanism

359

12.4 Changes of the Branching Mechanism Suppose that the (𝜉, 𝜙)-superprocess {(𝑋𝑡 , ℱ𝑡 )} and the Poisson random measures {𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤)} and {𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤)} are given as in Section 12.1. In this section, we assume P[𝑋0 (1)] < ∞. Let (𝒢¯ 𝑡 )𝑡 ≥0 be the augmentation of the filtration (𝒢𝑡 )𝑡 ≥0 defined in Theorem 12.3. Let (𝜇, 𝑥, 𝑦) ↦→ 𝜌(𝜇, 𝑥, 𝑦) and (𝜇, 𝑥, 𝜈) ↦→ 𝛽(𝜇, 𝑥, 𝜈) be positive Borel functions on 𝑀 (𝐸) × 𝐸 2 and 𝑀 (𝐸) × 𝐸 × 𝑀 (𝐸) ◦ , respectively. Suppose that ∫ 𝐶 := sup sup 𝜌(𝜇, 𝑥, 𝑦)𝜂0 (d𝑦) 𝐸 𝜇 ∈𝑀 (𝐸) 𝑥 ∈𝐸  ∫ + 𝛽(𝜇, 𝑥, 𝜈)𝐻1 (d𝜈) < ∞ (12.28) 𝑀 (𝐸) ◦

and there is a constant 𝐾 ≥ 0 such that, for 𝜇1 , 𝜇2 ∈ 𝑀 (𝐸), ∫ |𝜌(𝜇1 , 𝑥, 𝑦) − 𝜌(𝜇2 , 𝑥, 𝑦)|𝜂0 (d𝑦) sup 𝑥 ∈𝐸 𝐸  ∫ |𝛽(𝜇1 , 𝑥, 𝜈) − 𝛽(𝜇2 , 𝑥, 𝜈)|𝐻1 (d𝜈)⟩ + 𝑀 (𝐸) ◦

≤ 𝐾 ∥𝜇1 − 𝜇2 ∥.

(12.29)

For 𝜇 ∈ 𝑀 (𝐸), 𝑦 ∈ 𝐸 and 𝜈 ∈ 𝑀 (𝐸) ◦ let 𝑞(𝜇, 𝑦) = ⟨𝜇, 𝜌(𝜇, ·, 𝑦)⟩,

𝑔(𝜇, 𝜈) = ⟨𝜇, 𝛽(𝜇, ·, 𝜈)⟩.

(12.30)

Theorem 12.20 For the pair (𝑞, 𝑔) defined by (12.30), there is a unique solution {𝑌𝑡 : 𝑡 ≥ 0} to (12.22) and {(𝑌𝑡 , 𝒢¯ 𝑡+ ) : 𝑡 ≥ 0} is a strong Markov process with the following property: for 𝐺 ∈ 𝐶 2 (R) and 𝑓 ∈ 𝐷 0 ( 𝐴) we have ∫ 𝑡 𝐺 (𝑌𝑡 ( 𝑓 )) = 𝐺 (𝑌0 ( 𝑓 )) + 𝐺 ′ (𝑌𝑠 ( 𝑓 ))𝑌𝑠 ( 𝐴 𝑓 + 𝛾 𝑓 − 𝑏 𝑓 )d𝑠 0 ∫ 𝑡h i 𝐺 ′′ (𝑌𝑠 ( 𝑓 ))𝑌𝑠 (𝑐 𝑓 2 ) + 𝐺 ′ (𝑌𝑠 ( 𝑓 ))⟨𝑌𝑠 , 𝜅(𝑌𝑠 , ·, 𝑓 )⟩ d𝑠 + ∫0 𝑡 ∫ ∫ h + 𝑌𝑠 (d𝑥) 𝐺 (𝑌𝑠 ( 𝑓 ) + 𝜈( 𝑓 )) d𝑠 𝐸 0 𝑀 (𝐸) ◦ i − 𝐺 (𝑌𝑠 ( 𝑓 )) − 𝜈( 𝑓 )𝐺 ′ (𝑌𝑠 ( 𝑓 )) 𝐻 (𝑥, d𝜈) ∫ 𝑡 ∫ ∫ h + 𝑌𝑠 (d𝑥) 𝐺 (𝑌𝑠 ( 𝑓 ) + 𝜈( 𝑓 )) d𝑠 𝑀 (𝐸) ◦ 0 𝐸i (12.31) − 𝐺 (𝑌𝑠 ( 𝑓 )) 𝐾 (𝑌𝑠 , 𝑥, d𝜈) + (𝒢¯ 𝑡+ )-local mart., where 𝜅(𝜇, 𝑥, d𝑦) = 𝜌(𝜇, 𝑥, 𝑦)𝜂0 (d𝑦), 𝐾 (𝜇, 𝑥, d𝜈) = 𝛽(𝜇, 𝑥, 𝜈)𝐻0 (d𝜈).

(12.32)

360

12 State-Dependent Immigration Structures

Proof It is sufficient to show the functions 𝑞 and 𝑔 defined by (12.30) satisfy conditions (12.20) and (12.21) so that Theorem 12.14 applies. By (12.30) and (12.32) we have ∫ ∫ 𝜇(d𝑥)𝜅(𝜇, 𝑥, d𝑦) 𝜇(d𝑥) 𝜌(𝜇, 𝑥, 𝑦)𝜂0 (d𝑦) = 𝑞(𝜇, 𝑦)𝜂0 (d𝑦) = 𝐸

𝐸

and

∫

∫ 𝜇(d𝑥) 𝛽(𝜇, 𝑥, 𝜈)𝐻0 (d𝜈) =

𝑔(𝜇, 𝜈)𝐻0 (d𝜈) =

𝜇(d𝑥)𝐾 (𝜇, 𝑥, d𝜈). 𝐸

𝐸

From (12.28) it follows that, for 𝜇 ∈ 𝑀 (𝐸), ∫ 𝑔(𝜇, 𝜈)𝐻1 (d𝜈) ≤ 𝐶 ∥𝜇∥. ⟨𝜂0 , 𝑞(𝜇, ·)⟩ + 𝑀 (𝐸) ◦

For 𝜇1 , 𝜇2 ∈ 𝑀 (𝐸) we use both (12.28) and (12.29) to get ∫ ⟨𝜂0 , |𝑞(𝜇1 , ·) − 𝑞(𝜇2 , ·)|⟩ + |𝑔(𝜇1 , 𝜈) − 𝑔(𝜇2 , 𝜈)|𝐻1 (d𝜈) 𝑀 (𝐸) ◦ ∫ ⟨𝜇1 , 𝜌(𝜇1 , ·, 𝑦)⟩ − ⟨𝜇2 , 𝜌(𝜇2 , ·, 𝑦)⟩ 𝜂0 (d𝑦) = 𝐸∫ ⟨𝜇1 , 𝛽(𝜇1 , ·, 𝜈)⟩ − ⟨𝜇2 , 𝛽(𝜇2 , ·, 𝜈)⟩ 𝐻1 (d𝜈) + ∫ 𝑀 (𝐸) ◦   ≤ |⟨𝜇1 − 𝜇2 , 𝜌(𝜇1 , ·, 𝑦)⟩| + ⟨𝜇2 , 𝜌(𝜇1 , ·, 𝑦) − 𝜌(𝜇2 , ·, 𝑦)⟩ 𝜂0 (d𝑦) 𝐸∫  + |⟨𝜇1 − 𝜇2 , 𝛽(𝜇1 , ·, 𝜈)⟩| ◦ 𝑀 (𝐸)  + ⟨𝜇2 , 𝛽(𝜇1 , ·, 𝜈) − 𝛽(𝜇2 , ·, 𝜈)⟩ 𝐻1 (d𝜈) ≤ (𝐶 + 𝐾 ∥𝜇2 ∥) ∥𝜇1 − 𝜇2 ∥. Then (12.20) and (12.21) are indeed satisfied.

□

Clearly, the martingale problem (12.31) generalizes (7.23). We may regard the process {𝑌𝑡 : 𝑡 ≥ 0} constructed in Theorem 12.20 as a superprocess with statedependent branching mechanism ∫
(𝑥, 𝑓 ) ↦→ 𝜙(𝑥, 𝑓 ) − 𝜅(𝑌𝑠− , 𝑥, 𝑓 ) − 1 − e−𝜈 ( 𝑓 ) 𝐾 (𝑌𝑠− , 𝑥, d𝜈). (12.33) 𝑀 (𝐸) ◦

This gives a change of the branching mechanism.

12.4 Changes of the Branching Mechanism

361

Example 12.1 If 𝜌(𝜇, 𝑥, 𝑦) = 𝜌(𝑥, 𝑦) and 𝛽(𝜇, 𝑥, 𝜈) = 𝛽(𝑥, 𝜈) are independent of 𝜇 ∈ 𝑀 (𝐸), then {𝑌𝑡 : 𝑡 ≥ 0} is a superprocess with decomposable branching mechanism (𝑥, 𝑓 ) ↦→ 𝜙(𝑥, 𝑓 ) − 𝜓(𝑥, 𝑓 ), where ∫ ∫
𝑓 (𝑦) 𝜌(𝑥, 𝑦)𝜂0 (d𝑦) + 1 − e−𝜈 ( 𝑓 ) 𝛽(𝑥, 𝜈)𝐻0 (d𝜈). 𝜓(𝑥, 𝑓 ) = 𝑀 (𝐸) ◦

𝐸

Let 𝛽 ∈ 𝐵(𝐸 × 𝑀 (𝐸) ◦ ) + and let 𝐼 ∈ ℐ(𝐸) be given by (9.12) with 𝜈(1)𝐻 (d𝜈) being a finite measure on 𝑀 (𝐸) ◦ . Suppose that 𝜂(d𝑦) is absolutely continuous with respect to 𝜂0 (d𝑦) and 𝐻 (d𝜈) is absolutely continuous with respect to 𝐻0 (d𝜈). Choose the Radon–Nikodym derivatives 𝜂 ′ (𝑦) =

𝜂(d𝑦) , 𝜂0 (d𝑦)

𝐻 ′ (𝜈) =

𝐻 (d𝜈) . 𝐻0 (d𝜈)

For 𝜇 ∈ 𝑀 (𝐸), 𝑦 ∈ 𝐸 and 𝜈 ∈ 𝑀 (𝐸) ◦ let 𝑞(𝜇, 𝑦) = 𝜂 ′ (𝑦),

𝑔(𝜇, 𝜈) = ⟨𝜇, 𝛽(·, 𝜈)⟩ + 𝐻 ′ (𝜈).

(12.34)

Clearly, the functions (𝑞, 𝑔) satisfy conditions (12.20) and (12.21). As a consequence of Theorems 9.30 and 12.14 we have the following: Theorem 12.21 For the pair (𝑞, 𝑔) defined by (12.34), there is a pathwise unique solution {𝑌𝑡 : 𝑡 ≥ 0} to (12.22) and the solution is an immigration superprocess relative to (𝒢¯ 𝑡+ )𝑡 ≥0 with immigration mechanism 𝐼 and branching mechanism ∫
(𝑥, 𝑓 ) ↦→ 𝜙(𝑥, 𝑓 ) − 1 − e−𝜈 ( 𝑓 ) 𝛽(𝑥, 𝜈)𝐻0 (d𝜈). (12.35) 𝑀 (𝐸) ◦

We next briefly discuss the characterization of some jump times of the immigration superprocesses. For 𝐴 ∈ ℬ(𝑀 (𝐸) ◦ ) let us consider the conditions:   ∫ sup 𝐻 (𝑥, 𝐴) + 𝛽(𝑥, 𝜈)𝐻0 (d𝜈) = 0, 𝐻 ( 𝐴) + sup ℎ 𝐴 (𝑥) < ∞, (12.36) 𝑥 ∈𝐸

𝑥 ∈𝐸

𝐴𝑐

where 𝐴𝑐 = 𝑀 (𝐸) ◦ \ 𝐴 and ∫ ℎ 𝐴 (𝑥) =

𝛽(𝑥, 𝜈)𝐻0 (d𝜈). 𝐴

Let 𝐼 𝐴 ∈ ℐ(𝐸) be defined by ∫


1 − e−𝜈 ( 𝑓 ) 𝐻 (d𝜈),

𝐼 𝐴 ( 𝑓 ) = 𝜂( 𝑓 ) +

𝑓 ∈ 𝐵(𝐸) + .

(12.37)

𝐴𝑐

The next theorem generalizes Theorem 10.13 to the measure-valued setting:

362

12 State-Dependent Immigration Structures

Theorem 12.22 Let {𝑌𝑡 : 𝑡 ≥ 0} be a càdlàg immigration superprocess with initial value 𝑌0 = 𝜇 ∈ 𝑀 (𝐸), immigration mechanism 𝐼 and branching mechanism given by (12.35). Let 𝐴 ∈ ℬ(𝑀 (𝐸) ◦ ) be a set satisfying (12.36) and let 𝜏𝐴 = inf{𝑠 > 0 : 𝑌𝑠 − 𝑌𝑠− ∈ 𝐴}. Then we have   ∫ 𝑡 𝐼 𝐴 (𝑣 𝐴 (𝑠, ·))d𝑠 , (12.38) P(𝜏𝐴 > 𝑡) = exp − ⟨𝜇, 𝑣 𝐴 (𝑡, ·)⟩ − 𝑡𝐻 ( 𝐴) − 0

where 𝐼 𝐴 is given by (12.37) and (𝑡, 𝑥) ↦→ 𝑣 𝐴 (𝑡, 𝑥) is the unique locally bounded positive solution to ∫ 𝑡 ∫ ∫ 𝑡 𝑣 𝐴 (𝑡, 𝑥) = d𝑠 𝜙(𝑦, 𝑣 𝐴 (𝑠, ·))𝑃𝑡−𝑠 (𝑥, d𝑦). (12.39) 𝑃𝑠 ℎ 𝐴 (𝑥)d𝑠 − 0

0

𝐸

Proof Let us consider the immigration superprocess {𝑌𝑡 : 𝑡 ≥ 0} provided by Theorem 12.21 with 𝑌0 = 𝑋0 = 𝜇. By Theorem 12.14 there is also a pathwise unique solution {𝑌 𝐴 (𝑡) : 𝑡 ≥ 0} to ∫

𝑡

∫ ∫

𝑞 (𝑌𝐴 (𝑠−),𝑦)

∫

𝑌 𝐴 (𝑡) = 𝑋𝑡 + 𝑤 𝑡−𝑠 𝑁0 (d𝑠, d𝑦, d𝑢, d𝑤) ˆ 𝑊 ∫ 𝑡 0 ∫𝐸 0 d𝑠 + 𝑞(𝑌 𝐴 (𝑠−), 𝑦)𝜆 𝑡−𝑠 (𝑦, ·)𝜂0 (d𝑦) ∫0 𝑡 ∫ 𝐸∫ 𝑔 (𝑌𝐴 (𝑠−) ,𝜈) ∫ + 𝑤 𝑡−𝑠 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤). 0

𝐴𝑐

(12.40)

ˆ 𝑊

0

The solution {𝑌 𝐴 (𝑡) : 𝑡 ≥ 0} is an immigration superprocess with branching mechanism 𝜙 and immigration mechanism 𝐼 𝐴. Moreover, it is independent of the restricted Poisson random measure 1 𝐴 (𝜈)𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤). By Theorem 12.4, the sets {𝑠 > 0 : 𝑋𝑠 − 𝑋𝑠− ∈ 𝐴} and {𝑠 > 0 : 𝑌 𝐴 (𝑠) − 𝑌 𝐴 (𝑠−) ∈ 𝐴} are a.s. empty. Since 𝑌𝑠− = 𝑌 𝐴 (𝑠−) for 0 < 𝑠 ≤ 𝜏𝐴, from (12.22) and (12.40) it follows that  ∫ 𝑡 ∫ ∫ 𝑔 (𝑌𝑠− ,𝜈) ∫  𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤) = 0 P(𝜏𝐴 > 𝑡) = P ˆ  ∫0 𝑡 ∫𝐴 ∫0 𝑔 (𝑌𝐴 (𝑠−),𝜈)𝑊∫  =P 𝑁1 (d𝑠, d𝜈, d𝑢, d𝑤) = 0 ˆ 0  𝐴∫ 0 𝑊  ∫ 𝑡 d𝑠 = P exp − 𝑔(𝑌 𝐴 (𝑠−), 𝜈)𝐻0 (d𝜈) 𝐴∫ 0   ∫ 𝑡 d𝑠 ⟨𝑌 𝐴 (𝑠), 𝛽(·, 𝜈)⟩𝐻0 (d𝜈) = P exp − 𝑡𝐻 ( 𝐴) − 𝐴  ∫ 𝑡0  −𝑡 𝐻 ( 𝐴) ⟨𝑌 𝐴 (𝑠), ℎ 𝐴⟩d𝑠 . =e P exp − 0

Then we have (12.38) and (12.39) by Corollary 9.23.

□

12.5 Notes and Comments

363

Example 12.2 Let {𝑦(𝑡) : 𝑡 ≥ 0} be a càdlàg 𝑑-dimensional CBI-process with initial state 𝑦(0) = 𝑥 ∈ R+𝑑 and with transition semigroup (𝑄 𝑡𝑁 )𝑡 ≥0 defined in Example 9.1. The branching and immigration mechanisms of the process are given by (2.44) and (9.17), respectively. We assume that ⟨𝑢, 1⟩𝑛(d𝑢) is a finite measure on R+𝑑 \ {0}. Suppose that 𝐴 ∈ ℬ(R+𝑑 \ {0}) and 𝜈( 𝐴) + 𝐻𝑖 ( 𝐴) < ∞ for 𝑖 = 1, · · · , 𝑑. Let 𝜏𝐴 = inf{𝑠 > 0 : 𝑦(𝑠) − 𝑦(𝑠−) ∈ 𝐴}. For 𝜆 ∈ R+𝑑 let ∫
𝜙𝑖𝐴 (𝜆) = 𝜙𝑖 (𝜆) + 1 − e− ⟨𝜆,𝑢⟩ 𝐻𝑖 (d𝑢), 𝐴

and ∫


1 − e− ⟨𝜆,𝑢⟩ 𝑛(d𝑢),

𝜓 𝐴 (𝜆) = ⟨𝛽, 𝜆⟩ + 𝐴𝑐

where 𝐴𝑐 = R+𝑑 \ 𝐴. The branching mechanism 𝜙 𝐴 = (𝜙1𝐴, · · · , 𝜙 𝑑𝐴) clearly satisfies ConditionsÍ7.1 and 7.2. Let 𝜂0 be the counting measure on {1, · · · , 𝑑} and let 𝑑 𝐻𝑖 . The results of Theorems 12.21 and 12.22 apply to this finite𝐻0 = 𝑛 + 𝑖=1 dimensional situation. Then we have   ∫ 𝑡 𝐴 𝐴 𝐴 P(𝜏𝐴 > 𝑡) = exp − ⟨𝑥, 𝑣 (𝑡)⟩ − 𝑡𝜈( 𝐴) − 𝜓 (𝑣 (𝑠))d𝑠 , 0

where 𝑡 ↦→ 𝑣 𝐴 (𝑡) = (𝑣 1𝐴 (𝑡), · · · , 𝑣 𝑑𝐴 (𝑡)) ∈ R+𝑑 is the unique locally bounded vectorvalued solution to the differential equation system d𝑣 𝑖𝐴 (𝑡) = 𝐻𝑖 ( 𝐴) − 𝜙𝑖𝐴 (𝑣 𝐴 (𝑡)), 𝑣 𝑖𝐴 (0) = 0, d𝑡

𝑖 = 1, . . . , 𝑑.

12.5 Notes and Comments The approach of stochastic integral equations in constructing interactive immigration superprocess was suggested by Shiga (1990), who studied a stochastic equation involving Poisson point processes on the space of one-dimensional excursions. However, Shiga (1990) only considered the trivial spatial motion. A generalization of his result to non-trivial spatial motions was given in Fu and Li (2004) for binary local branching. The well-posedness of the martingale problem for a superprocess over R with interactive immigration was established in Mytnik and Xiong (2015) by considering the uniqueness of solution to the corresponding stochastic partial differential equation. The main results in this chapter generalize those in Fu and Li (2004) and Shiga (1990). The local Lipschitz condition (12.21) can be replaced by a Yamada– Watanabe type condition; see Li (2019b) for discussions in the one-dimensional setting. Abraham and Delmas (2009) gave a different method of changing the branching mechanism of a CB-process using immigration.

364

12 State-Dependent Immigration Structures

A class of immigration superprocess with dependent spatial motion were constructed in Li et al. (2005). Using the techniques developed in Kurtz and Xiong (1999), they gave a characterization of the conditional cumulant semigroup of the superprocess in terms of a stochastic partial differential equation driven by a time– space Gaussian white noise. The approach was stimulated by Xiong (2004), who established a similar characterization for the model of Skoulakis and Adler (2001). Let 𝜎 ∈ 𝐶 2 (R) + and ℎ ∈ 𝐶 1 (R) and assume both ℎ and ℎ ′ are square-integrable. Let 𝜂0 ∈ 𝑀 (R) and let 𝜌 be defined by (7.73). Suppose that (𝜈, 𝑥) ↦→ 𝑞(𝜈, 𝑥) is a positive Borel function on 𝑀 (R) × R such that there is a constant 𝐾 > 0 such that ⟨𝜂0 , 𝑞(𝜈, ·)⟩ ≤ 𝐾 (1 + ⟨𝜈, 1⟩),

𝜈 ∈ 𝑀 (R),

and for each 𝑅 > 0 there is a constant 𝐾 𝑅 > 0 such that ⟨𝜂0 , |𝑞(𝜈2 , ·) − 𝑞(𝜈1 , ·)|⟩ ≤ 𝐾 𝑅 ∥𝜈2 − 𝜈1 ∥ for 𝜈1 , 𝜈2 ∈ 𝑀 (R) satisfying ⟨𝜈1 , 1⟩ ≤ 𝑅 and ⟨𝜈2 , 1⟩ ≤ 𝑅. By considering a stochastic equation similar to (12.22) involving one-dimensional excursions carried by a stochastic flow, Dawson and Li (2003) constructed an interactive immigration superprocess with dependent spatial motion, which is a diffusion process {𝑋𝑡 : 𝑡 ≥ 0} in 𝑀 (R) satisfying the martingale problem: For each 𝑓 ∈ 𝐶 2 (R), ∫ 𝑡 1 𝑀𝑡 ( 𝑓 ) = ⟨𝑋𝑡 , 𝑓 ⟩ − ⟨𝑋0 , 𝑓 ⟩ − 𝜌(0) ⟨𝑋𝑠 , 𝑓 ′′⟩d𝑠 2 0 ∫ 𝑡 ∫ − d𝑠 𝑞(𝑌𝑠 , 𝑥) 𝑓 (𝑥)𝜂0 (d𝑥) 0

R

is a continuous martingale with quadratic variation process ∫ 𝑡 ∫ 𝑡 ∫ ⟨𝑀 ( 𝑓 )⟩𝑡 = ⟨𝑋𝑠 , 𝜎 𝑓 2 ⟩d𝑠 + d𝑠 ⟨𝑋𝑠 , ℎ(𝑧 − ·) 𝑓 ′⟩ 2 d𝑧. 0

0

R

Rescaling limits of the superprocess with dependent spatial motion were investigated in Dawson et al. (2004c) and Li et al. (2004), which led to a superprocess with coalescing spatial motion. Zhou (2007) gave a characterization of the conditional Laplace functional of the latter. A lattice branching-coalescing particle system was studied in Athreya and Swart (2005, 2009).

Chapter 13

Generalized Ornstein–Uhlenbeck Processes

Generalized Ornstein–Uhlenbeck processes constitute a large class of explicit examples of Markov processes in infinite-dimensional spaces with rich mathematical structures. Those processes may have non-trivial invariant measures, which make them better candidates for infinite-dimensional reference processes than Lévy processes. In this chapter, we first give a formulation of generalized Ornstein–Uhlenbeck processes in Hilbert spaces using the generalized Mehler semigroups introduced by Bogachev and Röckner (1995) and Bogachev et al. (1996). Then we give a systematic exploration of the structures of the generalized Mehler semigroups. Since such a semigroup can be defined by a linear semigroup and a skew convolution semigroup, we mainly discuss the latter. We shall see that a skew convolution semigroup is always formed with infinitely divisible probability measures. The key result is a characterization of the skew convolution semigroups in terms of infinitely divisible probability entrance laws. For centered skew convolution semigroups with finite second-moments, those entrance laws can be closed by probability measures on an enlarged Hilbert space. We also give some constructions for the generalized Ornstein–Uhlenbeck processes determined by closed entrance laws and study the corresponding Langevin type equations.

13.1 Generalized Mehler Semigroups Suppose that 𝐻 is a real separable Hilbert space with inner product ⟨·, ·⟩. We say a bounded linear operator 𝑆 on this space is symmetric if ⟨𝑆𝑎, 𝑏⟩ = ⟨𝑎, 𝑆𝑏⟩ for all 𝑎, 𝑏 ∈ 𝐻, and say it is positive definite if ⟨𝑆𝑎, 𝑎⟩ ≥ 0 for all 𝑎 ∈ 𝐻. A bounded linear operator 𝑆 on 𝐻 is called a trace class operator if there is an orthonormal basis {𝑒 1 , 𝑒 2 , . . .} of 𝐻 such that ∑︁ ⟨𝑆𝑒 𝑖 , 𝑒 𝑖 ⟩ < ∞. Tr(𝑆) := 𝑖

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_13

365

366

13 Generalized Ornstein–Uhlenbeck Processes

The sum Tr(𝑆) is called the trace of 𝑆, which is independent of the choice of the orthonormal basis {𝑒 1 , 𝑒 2 , . . .}. In most cases, we consider an infinite-dimensional space 𝐻. Given a probability measure 𝜈 on 𝐻, let 𝜈ˆ denote its characteristic functional defined by ∫ 𝜈(𝑎) ˆ (13.1) 𝑎 ∈ 𝐻. e𝑖 ⟨𝑥,𝑎⟩ 𝜈(d𝑥), = 𝐻

It is well known that 𝜈ˆ determines 𝜈 uniquely. The infinite divisibility of the probability measure can be defined as in Section 1.4. If 𝜈 is an infinitely divisible probability ˆ measure on 𝐻, then 𝜈(𝑎) ≠ 0 for all 𝑎 ∈ 𝐻 and there is a unique continuous function for 𝑎 ∈ 𝐻; see, e.g., ˆ log 𝜈ˆ on 𝐻 such that log 𝜈(0) = exp{log 𝜈(𝑎)} ˆ ˆ = 0 and 𝜈(𝑎) Linde (1986, p. 20) and Parthasarathy (1967, p. 171). Let 𝐾0 (𝑥, 𝑎) = e𝑖 ⟨𝑥,𝑎⟩ − 1 − 𝑖⟨𝑥, 𝑎⟩1 { ∥ 𝑥 ∥ ≤1} ,

𝑥, 𝑎 ∈ 𝐻.

Proposition 13.1 A probability measure 𝜈 on 𝐻 is infinitely divisible if and only if 𝜓 := − log 𝜈ˆ is uniquely represented by ∫ 1 𝜓(𝑎) = −𝑖⟨𝑏, 𝑎⟩ + ⟨𝑅𝑎, 𝑎⟩ − 𝐾0 (𝑥, 𝑎) 𝑀 (d𝑥), (13.2) 2 𝐻◦ where 𝑏 ∈ 𝐻, 𝑅 is a symmetric positive definite trace class operator on 𝐻, and 𝑀 is a 𝜎-finite (Lévy) measure on 𝐻 ◦ := 𝐻 \ {0} satisfying ∫ (1 ∧ ∥𝑥∥ 2 ) 𝑀 (d𝑥) < ∞. (13.3) 𝐻◦

Proof See, e.g., Linde (1986, p. 84) and Parthasarathy (1967, p. 181).

□

We write 𝜈 = 𝐼 (𝑏, 𝑅, 𝑀) if 𝜈 is an infinitely divisible probability measure with 𝜓 = − log 𝜈ˆ given by (13.2). Under the stronger moment condition ∫ (∥𝑥∥ ∧ ∥𝑥∥ 2 ) 𝑀 (d𝑥) < ∞, (13.4) 𝐻◦

we can define ∫

∫ 𝑥𝜈(d𝑥) = 𝑏 +

𝛽= 𝐻

𝑥1 { ∥ 𝑥 ∥>1} 𝑀 (d𝑥) 𝐻◦

and rewrite (13.2) as 𝜓(𝑎) = −𝑖⟨𝛽, 𝑎⟩ +

1 ⟨𝑅𝑎, 𝑎⟩ − 2

∫ 𝐻◦

𝐾1 (𝑥, 𝑎) 𝑀 (d𝑥),

where 𝐾1 (𝑥, 𝑎) = e𝑖 ⟨𝑥,𝑎⟩ − 1 − 𝑖⟨𝑥, 𝑎⟩,

𝑥, 𝑎 ∈ 𝐻.

(13.5)

13.1 Generalized Mehler Semigroups

367

In this case, we write 𝜈 = 𝐼1 (𝛽, 𝑅, 𝑀). Let (𝑇𝑡 )𝑡 ≥0 be a strongly continuous semigroup of bounded linear operators on 𝐻. Then the semigroup (𝑇𝑡∗ )𝑡 ≥0 of dual operators of (𝑇𝑡 )𝑡 ≥0 is also strongly continuous on 𝐻; see, e.g., Pazy (1983, p. 41). A family of probability measures (𝛾𝑡 )𝑡 ≥0 on 𝐻 is called a skew convolution semigroup (SC-semigroup) associated with (𝑇𝑡 )𝑡 ≥0 if the following equation is satisfied: 𝛾𝑟+𝑡 = (𝛾𝑟 ◦ 𝑇𝑡−1 ) ∗ 𝛾𝑡 ,

𝑟, 𝑡 ≥ 0.

(13.6)

𝛾

In this case, we can define a transition semigroup (𝑄 𝑡 )𝑡 ≥0 on 𝐻 by 𝛾

𝑄 𝑡 (𝑥, ·) = 𝛿𝑇𝑡 𝑥 (·) ∗ 𝛾𝑡 (·),

(13.7)

which is called a generalized Mehler semigroup associated with (𝑇𝑡 )𝑡 ≥0 . A Markov process in 𝐻 is called a generalized Ornstein–Uhlenbeck process (generalized OU𝛾 process) if it has transition semigroup (𝑄 𝑡 )𝑡 ≥0 . Observe that (13.6) is equivalent to 𝛾ˆ𝑟+𝑡 (𝑎) = 𝛾ˆ𝑟 (𝑇𝑡∗ 𝑎) 𝛾ˆ 𝑡 (𝑎),

𝑟, 𝑡 ≥ 0, 𝑎 ∈ 𝐻.

(13.8)

Proposition 13.2 If (𝛾𝑡 )𝑡 ≥0 is an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 , then each probability measure 𝛾𝑡 is infinitely divisible. Proof This proof is a simplification of that of Schmuland and Sun (2001). By (13.6) we have 𝛾0 = 𝛾0 ∗ 𝛾0 and so 𝛾0 = 𝛿0 , which is certainly infinitely divisible. From (13.8) it follows that | 𝛾ˆ𝑟+𝑡 (𝑎)| = | 𝛾ˆ𝑟 (𝑇𝑡∗ 𝑎)|| 𝛾ˆ 𝑡 (𝑎)|,

𝑟, 𝑡 ≥ 0, 𝑎 ∈ 𝐻.

(13.9)

Then 𝑡 ↦→ | 𝛾ˆ 𝑡 (𝑎)| is decreasing and hence the limit | 𝛾ˆ 0+ (𝑎)| := lim𝑡→0 | 𝛾ˆ 𝑡 (𝑎)| exists. Observe also that lim𝑡→0 𝛾ˆ𝑟 (𝑇𝑡∗ 𝑎) = 𝛾ˆ𝑟 (𝑎). Then we may let 𝑡 → 0 and 𝑟 → 0 in (13.9) to see that | 𝛾ˆ 0+ (𝑎)| = | 𝛾ˆ 0+ (𝑎)| 2 . It thus follows that | 𝛾ˆ 0+ (𝑎)| = 1 or 0. For any 𝑡 > 0 and any integer 𝑘 ≥ 1 we have −1 ) ∗ 𝛾𝑡/𝑘 = 𝛾𝑡 . (𝛾𝑡−𝑡/𝑘 ◦ 𝑇𝑡/𝑘

By Parthasarathy (1967, p. 72), there is a sequence {𝑥 𝑘 } ⊂ 𝐻 such that 𝛾𝑡/𝑘 ∗𝛿 𝑥𝑘 → 𝜈 weakly as 𝑘 → ∞, where 𝜈 is a probability measure on 𝐻. Consequently, | 𝜈(𝑎)| ˆ = lim | 𝛾ˆ 𝑡/𝑘 (𝑎)| = | 𝛾ˆ 0+ (𝑎)| = 1 or 0,

𝑎 ∈ 𝐻.

𝑘→∞

= 1. Then the ˆ ˆ is continuous and | 𝜈(0)| ˆ = 1, we must have | 𝜈(𝑎)| Since 𝛼 ↦→ | 𝜈(𝛼𝑎)| symmetrization of 𝜈 is the unit mass concentrated at zero. It follows that 𝜈 = 𝛿 𝑥 for some 𝑥 ∈ 𝐻; see, e.g., Linde (1986, p. 23). Setting 𝑦 𝑘 = 𝑥 𝑘 −𝑥 we have 𝛾𝑡/𝑘 ∗𝛿 𝑦𝑘 → 𝛿0 as 𝑘 → ∞. Since 𝑡 ↦→ ∥𝑇𝑡 ∥ is a locally bounded function, the probability measures −1 𝜈 𝑘, 𝑗 := (𝛾𝑡/𝑘 ∗ 𝛿 𝑦𝑘 ) ◦ 𝑇𝑡− 𝑗𝑡/𝑘 ,

1 ≤ 𝑗 ≤ 𝑘, 𝑘 ≥ 1,

368

13 Generalized Ornstein–Uhlenbeck Processes

form a uniform infinitesimal triangular array. By applying (13.6) inductively we see that 𝛾𝑡 =

𝑘 Ö

−1 ∗(𝛾𝑡/𝑘 ◦ 𝑇𝑡− 𝑗𝑡/𝑘 )

=

𝑗=1

Ö 𝑘

 ∗𝜈 𝑘, 𝑗 ∗ 𝛿 𝑧𝑘 ,

𝑗=1

Í where 𝑧 𝑘 = − 𝑘𝑗=1 𝑇𝑡− 𝑗𝑡/𝑘 𝑦 𝑘 . It then follows that each 𝛾𝑡 is infinitely divisible; see □ de Acosta et al. (1978) and Parthasarathy (1967, p. 199). By Propositions 13.1 and 13.2, we may write 𝛾𝑡 = 𝐼 (𝑏 𝑡 , 𝑅𝑡 , 𝑀𝑡 ) for 𝑡 ≥ 0. It is not hard to show that (13.8) is satisfied if and only if we have 𝑀𝑟+𝑡 = (𝑀𝑟 ◦ 𝑇𝑡−1 )| 𝐻 ◦ + 𝑀𝑡 ,

(13.10)


1 { ∥𝑇𝑡 𝑥 ∥ ≤1} − 1 { ∥ 𝑥 ∥ ≤1} 𝑇𝑡 𝑥𝑀𝑟 (d𝑥)

(13.11)

𝑅𝑟+𝑡 = 𝑇𝑡 𝑅𝑟 𝑇𝑡∗ + 𝑅𝑡 , and ∫ 𝑏𝑟+𝑡 = 𝑏 𝑡 + 𝑇𝑡 𝑏𝑟 + 𝐻◦

for all 𝑟, 𝑡 ≥ 0. If the stronger moment condition (13.4) is satisfied for each 𝑀𝑡 , we write 𝛾𝑡 = 𝐼1 (𝛽𝑡 , 𝑅𝑡 , 𝑀𝑡 ), where ∫ ∫ 𝛽𝑡 = 𝑥1 { ∥ 𝑥 ∥>1} 𝑀𝑡 (d𝑥). 𝑥𝜈𝑡 (d𝑥) = 𝑏 𝑡 + 𝐻◦

𝐻

In this case, the property (13.8) holds if and only if we have (13.10) and 𝛽𝑟+𝑡 = 𝛽𝑡 + 𝑇𝑡 𝛽𝑟 ,

𝑟, 𝑡 ≥ 0.

(13.12)

Suppose that 𝜈0 is a probability measure on 𝐻 and let 𝜓0 := − log 𝜈ˆ0 . Since the dual semigroup (𝑇𝑡∗ )𝑡 ≥0 is also strongly continuous, for each 𝑡 ≥ 0 we can define an infinitely divisible probability measure 𝛾𝑡 on 𝐻 by   ∫ 𝑡 ∗ 𝜓0 (𝑇𝑠 𝑎)d𝑠 , 𝑎 ∈ 𝐻. (13.13) 𝛾ˆ 𝑡 (𝑎) = exp − 0

It is easy to show that (𝛾𝑡 )𝑡 ≥0 is an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 , which is called a regular SC-semigroup. The corresponding generalized Mehler semigroup is given by   ∫ 𝑡 ∫ 𝛾 𝜓0 (𝑇𝑠∗ 𝑎)d𝑠 . (13.14) e𝑖 ⟨𝑦,𝑎⟩ 𝑄 𝑡 (𝑥, d𝑦) = exp 𝑖⟨𝑥, 𝑇𝑡∗ 𝑎⟩ − 0

𝐻

Example 13.1 Let 𝑏 > 0 and 𝑐 > 0 be constants and let {𝐵(𝑡) : 𝑡 ≥ 0} be a standard Brownian motion. A classical OU-process is the solution of the Langevin equation d𝑋 (𝑡) = 𝑐d𝐵(𝑡) − 𝑏𝑋 (𝑡)d𝑡,

𝑡 ≥ 0.

13.2 Gaussian Type Semigroups

369

Given the initial value 𝑋 (0) = 𝑥, we have ∫ 𝑡 −𝑏𝑡 𝑋 (𝑡) = e 𝑥 + 𝑐 e−𝑏 (𝑡−𝑠) d𝐵(𝑠),

𝑡 ≥ 0.

0

The second term on the right-hand side has Gaussian distribution 𝛾𝑡 := 𝑁 (0, 𝜎𝑡2 ) with ∫ 𝑡 𝑐2 e−2𝑏 (𝑡−𝑠) d𝑠 = 𝜎𝑡2 = 𝑐2 (1 − e−2𝑏𝑡 ). 2𝑏 0 𝛾

Then {𝑋 (𝑡) : 𝑡 ≥ 0} has transition semigroup (𝑄 𝑡 )𝑡 ≥0 given by ∫ 𝛾 𝑄 𝑡 𝑓 (𝑥) = 𝑓 (e−𝑏𝑡 𝑥 + 𝑦)𝛾𝑡 (d𝑦), 𝑡 ≥ 0, 𝑥 ∈ R. R

Setting 𝜇 = 𝑁 (0, 𝑐2 /2𝑏) we obtain the classical Mehler formula ∫ √︁
𝛾 𝑄 𝑡 𝑓 (𝑥) = 𝑓 e−𝑏𝑡 𝑥 + 1 − e−2𝑏𝑡 𝑦 𝜇(d𝑦). R

13.2 Gaussian Type Semigroups We say an SC-semigroup (𝛾𝑡 )𝑡 ≥0 is of the Gaussian type if each 𝛾𝑡 is a centered Gaussian probability measure. By the discussions of Section 13.1, a Gaussian type SC-semigroup (𝛾𝑡 )𝑡 ≥0 associated with (𝑇𝑡 )𝑡 ≥0 is given by 1 log 𝛾ˆ 𝑡 (𝑎) = − ⟨𝑅𝑡 𝑎, 𝑎⟩, 2

𝑡 ≥ 0, 𝑎 ∈ 𝐻,

(13.15)

where 𝑅𝑡 is a symmetric positive definite trace class operator on 𝐻 satisfying the first equation in (13.10). Theorem 13.3 A family of centered Gaussian probability measures (𝛾𝑡 )𝑡 ≥0 on 𝐻 given by (13.15) form an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 if and only if (𝑅𝑡 )𝑡 ≥0 is given by ∫ 𝑡 ⟨𝑅𝑡 𝑎, 𝑎⟩ = ⟨𝑈𝑠 𝑎, 𝑎⟩d𝑠, 𝑡 ≥ 0, 𝑎 ∈ 𝐻, (13.16) 0

where (𝑈𝑠 ) 𝑠>0 is a family of symmetric positive definite trace class operators on 𝐻 satisfying 𝑈𝑠+𝑡 = 𝑇𝑡 𝑈𝑠 𝑇𝑡∗ for all 𝑠, 𝑡 > 0 and ∫ 𝑡 Tr 𝑈𝑠 d𝑠 < ∞, 𝑡 ≥ 0. 0

370

13 Generalized Ornstein–Uhlenbeck Processes

Lemma 13.4 If the family (𝛾𝑡 )𝑡 ≥0 given by (13.15) is an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 , then the function 𝑡 ↦→ ⟨𝑅𝑡 𝑎, 𝑏⟩ is absolutely continuous in 𝑡 ≥ 0 for all 𝑎, 𝑏 ∈ 𝐻. Proof Since (𝑇𝑡 )𝑡 ≥0 is strongly continuous, there are constants 𝐵 ≥ 1 and 𝑐 ≥ 0 such that ∥𝑇𝑡 ∥ ≤ 𝐵e𝑐𝑡 . From (13.6) we have ∫ ∫ ∫ ∥𝑥∥ 2 𝛾𝑟+𝑡 (d𝑥) = ∥𝑥∥ 2 𝛾𝑡 (d𝑥). (13.17) ∥𝑇𝑡 𝑥∥ 2 𝛾𝑟 (d𝑥) + 𝐻

𝐻

𝐻

It follows that ∫

∥𝑥∥ 2 𝛾𝑡 (d𝑥),

𝑡≥0

(13.18)

[𝑔(𝑡 𝑗 ) − 𝑔(𝑟 𝑗 )] ≤ 𝐵2 e2𝑐𝑙 𝑔(𝜎𝑛 )

(13.19)

𝑔(𝑡) := 𝐻

is an increasing function. We claim 𝑛 ∑︁ 𝑗=1

Í for 0 < 𝑟 1 < 𝑡 1 < · · · < 𝑟 𝑛 < 𝑡 𝑛 ≤ 𝑙, where 𝜎𝑛 = 𝑛𝑗=1 (𝑡 𝑗 − 𝑟 𝑗 ). When 𝑛 = 1, this follows from (13.17). Now let us assume (13.19) holds for 𝑛 − 1. By applying (13.17) twice we have ∫ 𝑛 ∑︁ 2 2𝑐𝑙 [𝑔(𝑡 𝑗 ) − 𝑔(𝑟 𝑗 )] ≤ 𝐵 e ∥𝑥∥ 2 𝛾 𝜎𝑛−1 (d𝑥) + [𝑔(𝑡 𝑛 ) − 𝑔(𝑟 𝑛 )] 𝐻

𝑗=1 2 2𝑐𝑙

∫

2

∥𝑥∥ 𝛾 𝜎𝑛−1 (d𝑥) +

=𝐵 e

≤ 𝐵2 e2𝑐𝑙

∫

∫𝐻

∥𝑇𝑟𝑛 𝑥∥ 2 𝛾𝑡𝑛 −𝑟𝑛 (d𝑥)

𝐻

∥𝑥∥ 2 𝛾 𝜎𝑛−1 (d𝑥)

𝐻 ∫

+ 𝐵2 e2𝑐𝑙 ∥𝑇𝜎𝑛−1 𝑥∥ 2 𝛾𝑡𝑛 −𝑟𝑛 (d𝑥) ∫ 𝐻 = 𝐵2 e2𝑐𝑙 ∥𝑥∥ 2 𝛾 𝜎𝑛 (d𝑥), 𝐻

which gives (13.19). Letting 𝑟 → 0 and 𝑡 → 0 in (13.17) and using the fact that 𝑔 is an increasing function one sees that 𝑔(𝑡) → 0 as 𝑡 → 0. Then (13.19) implies that 𝑔 is absolutely continuous in 𝑡 ≥ 0. By the first equality in (13.10) we see 𝑡 ↦→ ⟨𝑅𝑡 𝑎, 𝑎⟩ is increasing for any 𝑎 ∈ 𝐻. For 𝑡 ≥ 𝑟 ≥ 0 we use (13.10) again to see ∫ ∗ ∗ ⟨𝑥, 𝑇𝑟∗ 𝑎⟩ 2 𝛾𝑡−𝑟 (d𝑥) ⟨𝑅𝑡 𝑎, 𝑎⟩ − ⟨𝑅𝑟 𝑎, 𝑎⟩ = ⟨𝑅𝑡−𝑟 𝑇𝑟 𝑎, 𝑇𝑟 𝑎⟩ = 𝐻 ∫ ≤ ∥𝑎∥ 2 ∥𝑇𝑟 𝑥∥ 2 𝛾𝑡−𝑟 (d𝑥) = ∥𝑎∥ 2 [𝑔(𝑡) − 𝑔(𝑟)]. 𝐻

Then ⟨𝑅𝑡 𝑎, 𝑎⟩ is absolutely continuous in 𝑡 ≥ 0. A polarization argument shows □ ⟨𝑅𝑡 𝑎, 𝑏⟩ is absolutely continuous in 𝑡 ≥ 0 for all 𝑎, 𝑏 ∈ 𝐻.

13.2 Gaussian Type Semigroups

371

Lemma 13.5 If the family (𝛾𝑡 )𝑡 ≥0 given by (13.15) is an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 , then there is a family of symmetric positive definite trace class operators (𝑈𝑠 ) 𝑠>0 on 𝐻 such that (13.16) holds. Proof Let {𝑒 𝑛 : 𝑛 = 1, 2, . . . } be an orthonormal basis of the space 𝐻. By Lemma 13.4, there are locally integrable Borel functions 𝐴𝑚,𝑛 on [0, ∞) such that ∫ 𝑡 ⟨𝑅𝑡 𝑒 𝑚 , 𝑒 𝑛 ⟩ = 𝐴𝑚,𝑛 (𝑠)d𝑠, 𝑡 ≥ 0, 𝑚, 𝑛 ≥ 1. (13.20) 0

The symmetry of 𝑅𝑡 implies ∫ 𝑡

∫ 𝐴𝑚,𝑛 (𝑠)d𝑠 =

𝑡

𝐴𝑛,𝑚 (𝑠)d𝑠.

0

(13.21)

0

Since 𝑅𝑡 is positive definite, for any 𝑎 ∈ span{𝑒 1 , 𝑒 2 , . . . } we have ∫ ⟨𝑅𝑡 𝑎, 𝑎⟩ = 0

𝑡

∞ ∑︁

𝐴𝑚,𝑛 (𝑠)⟨𝑎, 𝑒 𝑚 ⟩⟨𝑎, 𝑒 𝑛 ⟩d𝑠 ≥ 0.

(13.22)

𝑚,𝑛=1

(The sum only contains finitely many nontrivial terms!) In addition, since 𝑅𝑡 is a trace class operator, we get ∫ 0

𝑡

 ∑︁ ∞

 𝐴𝑛,𝑛 (𝑠) d𝑠 =

𝑛=1

∞ ∑︁

⟨𝑅𝑡 𝑒 𝑛 , 𝑒 𝑛 ⟩ = Tr(𝑅𝑡 ) < ∞.

(13.23)

𝑛=1

Let 𝐹 be the subset of [0, ∞) consisting of all 𝑠 ≥ 0 such that 𝐴𝑚,𝑛 (𝑠) = 𝐴𝑛,𝑚 (𝑠) for 𝑚, 𝑛 ≥ 1 and ∞ ∑︁

∞ ∑︁

𝐴𝑛,𝑛 (𝑠) < ∞ and

𝑛=1

𝐴𝑚,𝑛 (𝑠)⟨𝑎, 𝑒 𝑚 ⟩⟨𝑎, 𝑒 𝑛 ⟩ ≥ 0

(13.24)

𝑚,𝑛=1

for 𝑎 ∈ span{𝑒 1 , 𝑒 2 , . . . } with rational coefficients. As observed in the proof of Lemma 13.4, the function 𝑡 ↦→ ⟨𝑅𝑡 𝑎, 𝑎⟩ is increasing. From (13.21), (13.22) and (13.23) we conclude that 𝐹 has full Lebesgue measure. For any 𝑠 ∈ 𝐹, 𝑈𝑠 𝑎 =

∞ ∑︁

𝐴𝑚,𝑛 (𝑠)⟨𝑎, 𝑒 𝑚 ⟩𝑒 𝑛

(13.25)

𝑚,𝑛=1

defines a symmetric positive definite linear operator on span{𝑒 1 , 𝑒 2 , . . . }. Taking 𝑏 = 𝑥𝑒 𝑚 + 𝑦𝑒 𝑛 with rational 𝑥 and 𝑦, we get    𝐴𝑚,𝑚 (𝑠) 𝐴𝑚,𝑛 (𝑠) 𝑥 ⟨𝑈𝑠 𝑏, 𝑏⟩ = (𝑥, 𝑦) ≥ 0, 𝐴𝑛,𝑚 (𝑠) 𝐴𝑛,𝑛 (𝑠) 𝑦 so that the 2 × 2 matrix above is positive definite. Therefore, its determinant is positive, that is,

372

13 Generalized Ornstein–Uhlenbeck Processes

𝐴𝑚,𝑛 (𝑠) 2 ≤ 𝐴𝑚,𝑚 (𝑠) 𝐴𝑛,𝑛 (𝑠).

(13.26)

This combined with Schwarz’s inequality gives ∞  ∑︁ ∞ ∑︁

2

∥𝑈𝑠 𝑎∥ =

2 𝐴𝑚,𝑛 (𝑠)⟨𝑎, 𝑒 𝑚 ⟩

𝑛=1 𝑚=1 ∞ ∑︁ ∞ ∑︁

𝐴𝑚,𝑛 (𝑠) 2

≤

∞ ∑︁

⟨𝑎, 𝑒 𝑚 ⟩ 2

𝑚=1

𝑛=1 𝑚=1  ∑︁ ∞

2

𝐴𝑛,𝑛 (𝑠) ∥𝑎∥ 2

≤

𝑛=1

for 𝑎 ∈ span{𝑒 1 , 𝑒 2 , . . . }. Then 𝑈𝑠 is a bounded operator and can be extended to the entire space 𝐻. In fact, 𝑈𝑠 is a trace class operator since Tr(𝑈𝑠 ) =

∞ ∑︁

⟨𝑈𝑠 𝑒 𝑛 , 𝑒 𝑛 ⟩ =

∞ ∑︁

𝐴𝑛,𝑛 (𝑠) < ∞.

𝑛=1

𝑛=1

For 𝑠 ∉ 𝐹 we let 𝑈𝑠 = 0. By (13.22) and (13.25), for 𝑎 ∈ span{𝑒 1 , 𝑒 2 , . . . } we have ∫ 𝑡 (13.27) ⟨𝑈𝑠 𝑎, 𝑎⟩d𝑠. ⟨𝑅𝑡 𝑎, 𝑎⟩ = 0

Since 𝑠 ↦→ Tr(𝑈𝑠 ) is locally integrable, by dominated convergence we see that □ (13.27) holds for all 𝑎 ∈ 𝐻. Proof (of Theorem 13.3) If (𝛾𝑡 )𝑡 ≥0 is a family of probability measures given by (13.15) and (13.16), it is clearly an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 . For the converse, suppose the family (𝛾𝑡 )𝑡 ≥0 given by (13.15) form an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 . Let (𝑈𝑠 ) 𝑠>0 be provided by Lemma 13.5. Note that (13.16) and the first equation of (13.10) imply ∫ 𝑟 ∫ 𝑟 ⟨𝑈𝑠 𝑇𝑡∗ 𝑎, 𝑇𝑡∗ 𝑎⟩d𝑠, 𝑟, 𝑡 ≥ 0, 𝑎 ∈ 𝐻. ⟨𝑈𝑠+𝑡 𝑎, 𝑎⟩d𝑠 = 0

0

Since 𝐻 is separable, by Fubini’s theorem, there are subsets 𝐵 and 𝐵𝑠 of [0, ∞) with full Lebesgue measure such that 𝑈𝑠+𝑡 = 𝑇𝑡 𝑈𝑠 𝑇𝑡∗ ,

𝑠 ∈ 𝐵, 𝑡 ∈ 𝐵𝑠 .

As in the proof of Theorem 8.10, we can choose a decreasing sequence 𝑠 𝑛 ∈ 𝐵 with 𝑠 𝑛 → 0 and redefine (𝑈𝑡 )𝑡 >0 by ∗ 𝑈𝑡 := 𝑇𝑡−𝑠𝑛 𝑈𝑠𝑛 𝑇𝑡−𝑠 , 𝑛

𝑡 > 𝑠𝑛 .

With this modification, the family of operators (𝑈𝑡 )𝑡 >0 satisfy 𝑈𝑟+𝑡 = 𝑇𝑡 𝑈𝑟 𝑇𝑡∗ for all □ 𝑟, 𝑡 > 0 while (13.16) remains unchanged.

13.3 Non-Gaussian Type Semigroups

373

13.3 Non-Gaussian Type Semigroups Suppose that (𝛾𝑡 )𝑡 ≥0 is a family of infinitely divisible probability measures on 𝐻 such that 𝛾𝑡 = 𝐼 (𝑏 𝑡 , 𝑅𝑡 , 𝑀𝑡 ). We say the linear part (𝑏 𝑡 )𝑡 ≥0 is absolutely continuous if there exists an 𝐻-valued path (𝑐 𝑠 ) 𝑠>0 such that ∫ 𝑡 ⟨𝑏 𝑡 , 𝑎⟩ = ⟨𝑐 𝑠 , 𝑎⟩d𝑠, (13.28) 𝑡 ≥ 0, 𝑎 ∈ 𝐻. 0

Proposition 13.6 If (𝛾𝑡 )𝑡 ≥0 is an SC-semigroup given by 𝛾𝑡 = 𝐼 (𝑏 𝑡 , 𝑅𝑡 , 𝑀𝑡 ), we can write ∫ 𝑡 ∫ ∫ 𝐾0 (𝑥, 𝑎)𝐿 𝑠 (d𝑥), 𝑡 ≥ 0, 𝑎 ∈ 𝐻, (13.29) d𝑠 𝐾0 (𝑥, 𝑎) 𝑀𝑡 (d𝑥) = 𝐻◦

0

𝐻◦

where 𝐿 𝑠 (d𝑥) is a 𝜎-finite kernel from (0, ∞) to 𝐻 ◦ satisfying 𝐿 𝑟+𝑡 = (𝐿 𝑟 ◦ 𝑇𝑡−1 )| 𝐻 ◦ for 𝑟, 𝑡 > 0 and ∫ 𝑡 ∫ d𝑠 (1 ∧ ∥𝑥∥ 2 )𝐿 𝑠 (d𝑥) < ∞, 𝑡 ≥ 0. (13.30) 0

𝐻

Proof Let 𝑐 ≥ 1 and 𝑏 ≥ 0 be as in the proof of Lemma 13.4. From the second equation in (13.10) we see that 𝑡 ↦→ 𝑀𝑡 is increasing. Let ∫ ℎ(𝑡) = (1 ∧ ∥𝑥∥ 2 )𝑀𝑡 (d𝑥), 𝑡 ≥ 0. 𝐻◦

By (13.10) for 𝑟, 𝑡 ≥ 0 we have ∫ ℎ(𝑟 + 𝑡) − ℎ(𝑟) =

(1 ∧ ∥𝑇𝑟 𝑥∥ 2 ) 𝑀𝑡 (d𝑥),

𝐻◦

which is bounded above by 𝑐2 e2𝑏𝑟 ℎ(𝑡). As in the proof of Lemma 13.4, one sees that ℎ(𝑡) is absolutely continuous in 𝑡 ≥ 0. Observe that 𝑡 ↦→ 𝜈𝑡 (d𝑥) := (1∧ ∥𝑥∥ 2 )𝑀𝑡 (d𝑥) defines an increasing family of finite measures, so 𝑡 ↦→ 𝜈𝑡 (𝐵) determines a Radon measure 𝜈(d𝑠, 𝐵) on [0, ∞) for each 𝐵 ∈ ℬ(𝐻 ◦ ). A monotone class argument shows that 𝜈( 𝐴, ·) is a Borel measure on 𝐻 ◦ for each 𝐴 ∈ ℬ[0, ∞), so that 𝜈(·, ·) is a bimeasure. By Ethier and Kurtz (1986, p. 502), there is a probability kernel 𝐽𝑠 (d𝑥) from [0, ∞) to 𝐻 ◦ such that ∫ ∫ ∫ ◦ 𝐽𝑠 (𝐵)ℎ ′ (𝑠)d𝑠, 𝜈( 𝐴, 𝐵) = 𝐽𝑠 (𝐵)dℎ(𝑠) = 𝐽𝑠 (𝐵)𝜈(d𝑠, 𝐻 ) = 𝐴

𝐴

𝐴

where ℎ ′ (𝑠) is a Radon–Nikodym derivative of dℎ(𝑠) relative to the Lebesgue measure. Defining the 𝜎-finite kernel 𝐿 𝑠 (d𝑥) = (1 ∧ ∥𝑥∥ 2 ) −1 ℎ ′ (𝑠)𝐽𝑠 (d𝑥) we obtain (13.29). By the second equation of (13.10) one can modify the definition of (𝐿 𝑡 )𝑡 >0 □ so that 𝐿 𝑟+𝑡 = (𝐿 𝑟 ◦ 𝑇𝑡−1 )| 𝐻 ◦ is satisfied for all 𝑟, 𝑡 > 0.

374

13 Generalized Ornstein–Uhlenbeck Processes

We say a family of 𝜎-finite measures (𝜈𝑠 ) 𝑠>0 on 𝐻 is an entrance law for the semigroup (𝑇𝑡 )𝑡 ≥0 if it satisfies 𝜈𝑟+𝑡 = 𝜈𝑟 ◦ 𝑇𝑡−1 for all 𝑟, 𝑡 > 0. In fact, this means (𝜈𝑠 ) 𝑠>0 is an entrance law for the deterministic Markov process {𝑇𝑡 𝑥 : 𝑡 ≥ 0} according to the standard definition. Theorem 13.7 Let (𝛾𝑡 )𝑡 ≥0 be a family of infinitely divisible probability measures on 𝐻 with absolutely continuous linear part. Then (𝛾𝑡 )𝑡 ≥0 is an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 if and only if there is an infinitely divisible probability entrance law (𝜈𝑠 ) 𝑠>0 for (𝑇𝑡 )𝑡 ≥0 such that  ∫ 𝑡 𝛾ˆ 𝑡 (𝑎) = exp (13.31) log 𝜈ˆ 𝑠 (𝑎)d𝑠 , 𝑡 ≥ 0, 𝑎 ∈ 𝐻. 0

Proof If (𝛾𝑡 )𝑡 ≥0 is given by (13.31), it is clearly an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 . Conversely, suppose that (𝛾𝑡 )𝑡 ≥0 is an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 and write 𝛾𝑡 = 𝐼 (𝑏 𝑡 , 𝑅𝑡 , 𝑀𝑡 ) for 𝑡 ≥ 0. It is easy to see that 1 𝑔 log 𝛾ˆ 𝑡 (𝑎) = − ⟨𝑅𝑡 𝑎, 𝑎⟩, 2

𝑡 ≥ 0, 𝑎 ∈ 𝐻,

(13.32)

𝑔

defines a Gaussian type SC-semigroup (𝛾𝑡 )𝑡 ≥0 . Let (𝑈𝑠 ) 𝑠>0 and (𝐿 𝑠 ) 𝑠>0 be provided by Theorem 13.3 and Proposition 13.6, respectively. Suppose that ⟨𝑏 𝑡 , 𝑎⟩ = ∫𝑡 ⟨𝑐 , 𝑠 𝑎⟩d𝑠. By (13.11), we can modify the definition of (𝑐 𝑠 ) 𝑠>0 so that 0 ∫


1 { ∥𝑇𝑡 𝑥 ∥ ≤1} − 1 { ∥ 𝑥 ∥ ≤1} 𝑇𝑡 𝑥𝐿 𝑟 (d𝑥),

𝑐𝑟+𝑡 = 𝑇𝑡 𝑐𝑟 +

𝑟, 𝑡 > 0.

𝐻◦

Then we have (13.31) with 𝜈𝑠 = 𝐼 (𝑐 𝑠 , 𝑈𝑠 , 𝐿 𝑠 ) for 𝑠 > 0.

□

Example 13.2 Let 𝑡 ↦→ 𝑏 𝑡 be a real-valued discontinuous function satisfying 𝑏𝑟+𝑡 = 𝑏𝑟 + 𝑏 𝑡 for all 𝑟, 𝑡 ≥ 0; see, e.g., Sato (1999, p. 37). It is simple to check that (𝛿 𝑏𝑡 )𝑡 ≥0 is a classical convolution semigroup, which cannot be represented in the form (13.31). This example shows that some condition on the linear part 𝑡 ↦→ 𝑏 𝑡 has to be imposed to get the representation (13.31) of the SC-semigroup. Example 13.3 Let 𝜇 be the uniform distribution on [0, 2𝜋) and consider the Hilbert space 𝐿 2 ( [0, 2𝜋), 𝜇) equipped with the inner product ⟨·, ·⟩ defined by ⟨ 𝑓 , ℎ⟩ =

1 2𝜋

∫

2𝜋

𝑓 (𝑥)ℎ(𝑥)d𝑥. 0

For 𝑡 ≥ 0 and 𝑓 ∈ 𝐿 2 ( [0, 2𝜋), 𝜇) let 𝑇𝑡 𝑓 (𝑥) = 𝑓 (𝑥 + 𝑡) (mod 2𝜋). By using approximation by continuous functions it is not hard to show that ∫ lim 𝑡→0

0

2𝜋

| 𝑓 (𝑥 + 𝑡) − 𝑓 (𝑥)| 2 d𝑥 = 0.

13.3 Non-Gaussian Type Semigroups

375

Then (𝑇𝑡 )𝑡 ≥0 is a strongly continuous semigroup on 𝐿 2 ( [0, 2𝜋), 𝜇). Set 𝑏 𝑡 = 𝑓 −𝑇𝑡 𝑓 . It is easy to show that (𝛿 𝑏𝑡 )𝑡 ≥0 is an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 . For any 𝑓 ∈ 𝐿 2 ( [0, 2𝜋), 𝜇) we have the Fourier expansion ∞ ∑︁

𝑓 (𝑥) =

𝑓ˆ(𝑛)e𝑖𝑛𝑥 ,

𝑥 ∈ [0, 2𝜋),

(13.33)

𝑛 = 0, ±1, ±2, . . . ;

(13.34)

𝑛=−∞

where 1 𝑓ˆ(𝑛) = 2𝜋

∫

2𝜋

𝑓 (𝑥)e−𝑖𝑛𝑥 d𝑥,

0

see, e.g., Conway (1990, p. 21). Clearly, the 𝑛-th Fourier coefficient of 𝑇𝑡 𝑓 is 𝑓ˆ(𝑛)e𝑖𝑛𝑡 . Since both 𝑓 and 𝑇𝑡 𝑓 are real functions, from (13.33) and (13.34) we obtain ∫ 2𝜋 1 ⟨ 𝑓 , 𝑏𝑡 ⟩ = ∥ 𝑓 ∥ − 𝑓 (𝑥)𝑇𝑡 𝑓 (𝑥)d𝑥 2𝜋 0 ∞ ∑︁ = ∥ 𝑓 ∥2 − 𝑓ˆ(𝑛) 𝑓ˆ(−𝑛)e−𝑖𝑛𝑡 2

𝑛=−∞ 2

= ∥ 𝑓 ∥ − | 𝑓ˆ(0)| 2 − 2

∞ ∑︁

| 𝑓ˆ(𝑛)| 2 cos(𝑛𝑡).

𝑛=1

Now let us take the particular function 𝑓 ∈ 𝐿 2 ( [0, 2𝜋), 𝜇) given by (13.33) with n −𝑘/2 2 𝑓ˆ(𝑛) = 0

if |𝑛| = 2 𝑘 and 𝑘 ≥ 1, otherwise.

Then we have 𝑓 (𝑥) = 2

∞ ∑︁ 1 cos(2 𝑘 𝑥), 𝑘/2 2 𝑘=1

𝑥 ∈ [0, 2𝜋).

It follows that ∫ 2𝜋 ∞ 2 ∑︁ 1 cos2 (2 𝑘 𝑥)d𝑥 = 2 ∥𝑓∥ = 𝜋 𝑘=1 2 𝑘 0 2

and ⟨ 𝑓 , 𝑏𝑡 ⟩ = 2 − 2

∞ ∑︁

2−𝑘 cos(2 𝑘 𝑡),

𝑘=1

which is Weierstrass’s nowhere differentiable continuous function; see, e.g., Hewitt and Stromberg (1965, p. 258). Therefore (𝛿 𝑏𝑡 )𝑡 ≥0 cannot be represented in the form (13.31).

376

13 Generalized Ornstein–Uhlenbeck Processes

13.4 Extensions of Centered Semigroups In this section, we give some characterizations of centered SC-semigroups with finite second-moments. In particular, we shall see any SC-semigroup of this type can be extended to a regular SC-semigroup on an enlarged Hilbert space. Since (𝑇𝑡 )𝑡 ≥0 is strongly continuous, there are constants 𝐵 ≥ 0 and 𝑐 0 ≥ 0 such that ∥𝑇𝑡 ∥ ≤ 𝐵e𝑐0 𝑡 for every 𝑡 ≥ 0. Let (𝑈 𝛼 ) 𝛼>𝑐0 denote the resolvent of (𝑇𝑡 )𝑡 ≥0 and let 𝐴 denote its generator with domain 𝒟( 𝐴) = 𝑈 𝛼 𝐻 ⊂ 𝐻. : 𝑠 > 0} taking values in 𝐻 is called an entrance path for the A path 𝑥˜ = {𝑥(𝑠) ˜ ˜ + 𝑡) = 𝑇𝑡 𝑥(𝑟) semigroup (𝑇𝑡 )𝑡 ≥0 if it satisfies 𝑥(𝑟 for all 𝑟, 𝑡 > 0. Let 𝐸 denote the ˜ set of all entrance paths for (𝑇𝑡 )𝑡 ≥0 . We say 𝑥˜ ∈ 𝐸 is closable if there is an element for all 𝑠 > 0, and say it is locally square integrable ˜ ˜ ∈ 𝐻 such that 𝑥(𝑠) ˜ 𝑥(0) = 𝑇𝑠 𝑥(0) if ∫ 𝑡 ∥ 𝑥(𝑠) 𝑡 ≥ 0. ˜ ∥ 2 d𝑠 < ∞, (13.35) 0

Lemma 13.8 Let 𝑥˜ ∈ 𝐸. Then (13.35) holds if and only if ∫ ∞ 𝑏 > 𝑐0 . e−2𝑏𝑠 ∥ 𝑥(𝑠) ˜ ∥ 2 d𝑠 < ∞,

(13.36)

0

Proof It is easy to see that (13.36) implies (13.35). For the converse, assume (13.35) holds. Then we have ∫ 1 ∫ ∞ ∞ ∑︁ ˜ ∥ 2 d𝑠 ˜ ∥ 2 d𝑠 = e−2𝑏𝑠 ∥ 𝑥(𝑠) e−2𝑘𝑏 e−2𝑏𝑠 ∥𝑇𝑘 𝑥(𝑠) 0

0

𝑘=0

≤

∞ ∑︁

2 −2𝑘 (𝑏−𝑐0 )

∫

𝐵 e

𝑘=0

The right-hand side is finite for every 𝑏 > 𝑐 0 .

1

e−2𝑏𝑠 ∥ 𝑥(𝑠) ˜ ∥ 2 d𝑠.

0

□

Let 𝐻˜ denote the set of all locally square integrable entrance paths for (𝑇𝑡 )𝑡 ≥0 . We call 𝐻˜ the entrance space for (𝑇𝑡 )𝑡 ≥0 . For any fixed 𝑏 > 𝑐 0 we can define an inner product on 𝐻˜ by ∫ ∞ ˜ ˜ 𝑦˜ ∈ 𝐻. e−2𝑏𝑠 ⟨𝑥(𝑠), (13.37) ⟨𝑥, ˜ 𝑦˜ ⟩∼ = ˜ 𝑦˜ (𝑠)⟩d𝑠, 𝑥, 0

Let ∥ · ∥ ∼ be the norm induced by this inner product. ˜ from 𝐻˜ to 𝐻 is a bounded Lemma 13.9 For every 𝑡 > 0 the projection 𝜋𝑡 : 𝑥˜ ↦→ 𝑥(𝑡) linear operator.

13.4 Extensions of Centered Semigroups

377

Proof The linearity of 𝜋𝑡 is obvious. For any 𝑥˜ ∈ 𝐻˜ we have ∫ 𝑡 ∫ 𝑡 −1 2 2 −1 ˜ =𝑡 ∥ 𝑥(𝑡) ˜ ∥ d𝑠 = 𝑡 ∥𝜋𝑡 𝑥∥ ∥𝑇𝑡−𝑠 𝑥(𝑠) ˜ ∥ 2 d𝑠 0 0 ∫ 𝑡 ≤ 𝐵2 𝑡 −1 e2𝑏𝑡 ˜ ∥ 2 d𝑠 ≤ 𝐵2 𝑡 −1 e2𝑏𝑡 ∥ 𝑥∥ ˜ 2∼ . e−2𝑏𝑠 ∥ 𝑥(𝑠) 0

Then 𝜋𝑡 is a bounded operator.

□

˜ ⟨·, ·⟩∼ ) is a Hilbert space. Lemma 13.10 The norm ∥ · ∥ ∼ is complete and ( 𝐻, Proof Suppose {𝑥˜ 𝑛 } ⊂ 𝐻˜ is a Cauchy sequence under the norm ∥ · ∥ ∼ , that is, ∫ ∞ 2 ∥ 𝑥˜ 𝑛 − 𝑥˜ 𝑚 ∥ ∼ = e−2𝑏𝑠 ∥ 𝑥˜ 𝑛 (𝑠) − 𝑥˜ 𝑚 (𝑠) ∥ 2 d𝑠 → 0 0

as 𝑚, 𝑛 → ∞. By Lemma 13.9, for each 𝑡 > 0 the limit 𝑥(𝑡) ˜ = lim𝑛→∞ 𝑥˜ 𝑛 (𝑡) exists in 𝐻. Since 𝑇𝑠 is a bounded linear operator on 𝐻, for 𝑠 > 0 we have ˜ + 𝑠). 𝑇𝑠 𝑥(𝑡) ˜ = lim 𝑇𝑠 𝑥˜ 𝑛 (𝑡) = lim 𝑥˜ 𝑛 (𝑡 + 𝑠) = 𝑥(𝑡 𝑛→∞

𝑛→∞

˜ Then 𝑥˜ = {𝑥(𝑡) : 𝑡 > 0} is an entrance path for (𝑇𝑡 )𝑡 ≥0 . For 𝜀 > 0 choose 𝑁 ≥ 1 such that ∫ ∞ e−2𝑏𝑠 ∥ 𝑥˜ 𝑛 (𝑠) − 𝑥˜ 𝑚 (𝑠) ∥ 2 d𝑠 < 𝜀, 𝑚, 𝑛 ≥ 𝑁. 0

By Fatou’s lemma we get ∫ ∞ e−2𝑏𝑠 ∥ 𝑥˜ 𝑛 (𝑠) − 𝑥(𝑠) ˜ ∥ 2 d𝑠 ≤ 𝜀,

𝑛 ≥ 𝑁.

0

It follows that ∫

∞

e−2𝑏𝑠 ∥ 𝑥(𝑠) ˜ ∥ 2 d𝑠 ≤

∫

0

∞

˜ − 𝑥˜ 𝑛 (𝑠) ∥ 2 d𝑠 e−2𝑏𝑠 ∥ 𝑥(𝑠)

0∫

+

∞

e−2𝑏𝑠 ∥ 𝑥˜ 𝑛 (𝑠) ∥ 2 d𝑠 < ∞.

0

Then 𝑥˜ ∈ 𝐻˜ and lim𝑛→∞ ∥𝑥 𝑛 − 𝑥∥ 2∼ = 0.

□

Lemma 13.11 The linear operator 𝐽 : 𝑥 ↦→ {𝑇𝑠 𝑥 : 𝑠 > 0} from 𝐻 to 𝐻˜ is a ˜ ∥ · ∥ ∼ ) is separable. continuous dense embedding and ( 𝐻, Proof Since 𝑥 = lim𝑡→0 𝑇𝑡 𝑥, the map 𝐽 is injective. For any 𝑥 ∈ 𝐻, ∫ ∞ ∫ ∞ ∥𝐽𝑥∥ ∼2 = e−2(𝑏−𝑐0 )𝑠 d𝑠. e−2𝑏𝑠 ∥𝑇𝑠 𝑥∥ 2 d𝑠 ≤ 𝐵2 ∥𝑥∥ 2 0

0

Thus 𝐽 is a continuous embedding. For an arbitrary 𝑥˜ ∈ 𝐻˜ we have

378

13 Generalized Ornstein–Uhlenbeck Processes

˜ − 𝑥∥ ˜ 2∼ = ∥𝐽 𝑥(𝑡)

∫

∞

− 𝑥(𝑠) ˜ e−2𝑏𝑠 ∥𝑇𝑡 𝑥(𝑠) ˜ ∥ 2 d𝑠

∫0 𝑟 = 0∫

− 𝑥(𝑠) ˜ e−2𝑏𝑠 ∥𝑇𝑡 𝑥(𝑠) ˜ ∥ 2 d𝑠 ∞

e−2𝑏𝑠 ∥𝑇𝑠−𝑟 [𝑇𝑡 𝑥(𝑟) ˜ − 𝑥(𝑟)] ˜ ∥ 2 d𝑠 ∫ 𝑟 ≤ 2(𝐵 e ˜ ∥ 2 d𝑠 e−2𝑏𝑠 ∥ 𝑥(𝑠) + 1) 0 ∫ ∞ ˜ ˜ ∥2 e−2(𝑏−𝑐0 )𝑠 d𝑠. − 𝑥(𝑟) + 𝐵2 e−2𝑐0 𝑟 ∥𝑇𝑡 𝑥(𝑟) +

𝑟 2 2𝑐0 𝑡

𝑟

Observe that the first integral on the right-hand side goes to zero as 𝑟 → 0 and for fixed 𝑟 > 0 the second term goes to zero as 𝑡 → 0. Then we have ∥𝐽 𝑥(𝑡) ˜ − 𝑥∥ ˜ ∼→0 ˜ Since 𝐻 is separable, so is 𝐻. ˜ □ as 𝑡 → 0, and hence 𝐽𝐻 is dense in 𝐻. ˜ on ( 𝐻, ˜ ∥ · ∥ ∼ ) is also generated by the Lemma 13.12 The Borel 𝜎-algebra ℬ( 𝐻) ˜ projections {𝜋𝑡 : 𝑡 > 0} from 𝐻 to 𝐻. ˜ On Proof By Lemma 13.9, each 𝜋𝑡 is continuous. Then 𝜎({𝜋𝑡 : 𝑡 > 0}) ⊂ ℬ( 𝐻). the other hand, we have 𝑛2

1 ∑︁ −2𝑏𝑖/𝑛 e ∥ 𝑥(𝑖/𝑛) ˜ − 𝑧˜ (𝑖/𝑛) ∥ 2 , 𝑛→∞ 𝑛 𝑖=1

∥ 𝑥˜ − 𝑧˜ ∥ 2∼ = lim

˜ 𝑥, ˜ 𝑧˜ ∈ 𝐻.

˜ the function 𝑥˜ ↦→ ∥ 𝑥˜ − 𝑧˜ ∥ ∼ on 𝐻˜ is measurable with respect Then for any fixed 𝑧˜ ∈ 𝐻, to 𝜎({𝜋𝑡 : 𝑡 > 0}). Consequently, every open ball 𝐵( 𝑧˜, 𝜀) := {𝑥˜ ∈ 𝐻˜ : ∥ 𝑥˜ − 𝑧˜ ∥ ∼ < 𝜀} belongs to 𝜎({𝜋𝑡 : 𝑡 > 0}). Since 𝐻˜ is separable, all open sets in 𝐻˜ are contained in ˜ ⊂ 𝜎({𝜋𝑡 : 𝑡 > 0}). 𝜎({𝜋𝑡 : 𝑡 > 0}) and hence ℬ( 𝐻) □ Theorem 13.13 A family (𝛾𝑡 )𝑡 ≥0 of centered probability measures on 𝐻 satisfying the second-moment condition ∫ ∥𝑥∥ 2 𝛾𝑡 (d𝑥) < ∞, 𝑡 ≥ 0 (13.38) 𝐻◦

is an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 if and only if its characteristic functionals are given by (13.31) with (𝜈𝑠 ) 𝑠>0 being a centered infinitely divisible probability entrance law for (𝑇𝑡 )𝑡 ≥0 satisfying ∫ 𝑡 ∫ d𝑠 ∥𝑥∥ 2 𝜈𝑠 (d𝑥) < ∞, 𝑡 ≥ 0. (13.39) 0

𝐻◦

Proof It is well known that the second-moment of a centered infinitely divisible probability measure only involves the Gaussian covariance operator and the Lévy measure. If the centered infinitely divisible probability measures (𝛾𝑡 )𝑡 ≥0 and (𝜈𝑠 ) 𝑠>0 are related by (13.31), the Gaussian covariance operators and Lévy measures of (𝛾𝑡 )𝑡 ≥0 can be represented as integrals of those of (𝜈𝑠 ) 𝑠>0 . This observation yields

13.4 Extensions of Centered Semigroups

∫

⟨𝑥, 𝑎⟩ 2 𝛾𝑡 (d𝑥) =

𝐻◦

∫

379

∫

𝑡

d𝑠

⟨𝑥, 𝑎⟩ 2 𝜈𝑠 (d𝑥),

𝑡 ≥ 0, 𝑎 ∈ 𝐻.

𝐻◦

0

Let {𝑒 𝑛 : 𝑛 = 1, 2, . . . } be an orthonormal basis of 𝐻. Applying the above equation to each 𝑒 𝑛 and taking the summation we see ∫ 𝑡 ∫ ∫ d𝑠 ∥𝑥∥ 2 𝜈𝑠 (d𝑥), 𝑡 ≥ 0. ∥𝑥∥ 2 𝛾𝑡 (d𝑥) = (13.40) 𝐻◦

𝐻◦

0

In particular, conditions (13.38) and (13.39) are equivalent for the infinitely divisible probability measures (𝛾𝑡 )𝑡 ≥0 and (𝜈𝑠 ) 𝑠>0 related by (13.31). Then the desired result □ follows by Theorem 13.7.

Theorem 13.14 A family (𝛾𝑡 )𝑡 ≥0 of centered probability measures on 𝐻 satisfying (13.38) is an SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 if and only if its characteristic functionals are given by ∫ 𝑡    ∫ log e𝑖 ⟨ 𝑥˜ (𝑠),𝑎⟩ 𝜆0 (d𝑥) ˜ d𝑠 , 𝑎 ∈ 𝐻, 𝛾ˆ 𝑡 (𝑎) = exp (13.41) 𝐻˜

0

where 𝜆0 is a centered infinitely divisible probability measure on 𝐻˜ satisfying ∫ ˜ < ∞. ˜ ∼2 𝜆0 (d𝑥) ∥ 𝑥∥ (13.42) 𝐻˜

Proof Suppose that (𝛾𝑡 )𝑡 ≥0 is a family of centered probability measures on 𝐻 defined by (13.41) and (13.42). Let 𝜈𝑠 be the image of 𝜆0 induced by the projection 𝜋 𝑠 from 𝐻˜ to 𝐻. Then (𝜈𝑠 ) 𝑠>0 is a centered infinitely divisible probability entrance law for (𝑇𝑡 )𝑡 ≥0 satisfying (13.39) and the relation (13.31) holds. By Theorem 13.13, (𝛾𝑡 )𝑡 ≥0 is a centered SC-semigroup satisfying (13.38). Conversely, suppose that (𝛾𝑡 )𝑡 ≥0 is a centered SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 satisfying (13.38). Let (𝜈𝑠 ) 𝑠>0 be the entrance law given by Theorem 13.13. Then (𝜈𝑠 ) 𝑠>0 is a probability entrance law for the continuous Markov process {𝑇𝑡 𝑥 : 𝑡 ≥ 0} with deterministic motion. Let 𝐸 0 be the set of continuous paths {𝑤(𝑡) : 𝑡 > 0} from (0, ∞) to 𝐻. We endow 𝐸 0 with the 𝜎-algebra ℰ0 generated by the coordinate process. Then there is a unique probability measure 𝜆0 on (𝐸 0 , ℰ0 ) under which {𝑤(𝑡) : 𝑡 > 0} is a Markov process with the same transition semigroup as the process {𝑇𝑡 𝑥 : 𝑡 ≥ 0} and 𝜈𝑠 is the image of 𝜆0 under 𝑤 ↦→ 𝑤(𝑠). It follows that ∫ 𝑡    ∫ 𝑖 ⟨𝑤 (𝑠),𝑎⟩ (13.43) 𝛾ˆ 𝑡 (𝑎) = exp 𝜆0 (d𝑤) d𝑠 , 𝑎 ∈ 𝐻. e log 0

𝐸0

Because of the special deterministic motion mechanism of the process {𝑇𝑡 𝑥 : 𝑡 ≥ 0} we may assume that 𝜆0 is supported by the space 𝐸 of the entrance paths. Let ℰ0 (𝐸) ˜ Since 𝑤 ↦→ ∥𝑤(𝑠) ∥ 2 is ˜ denote respectively the traces of ℰ0 on 𝐸 and 𝐻. and ℰ0 ( 𝐻) clearly a non-negative ℰ0 (𝐸)-measurable function on 𝐸,

380

13 Generalized Ornstein–Uhlenbeck Processes

𝑤 ↦→ ∥𝑤∥ 2∼ :=

∫

∞

e−2𝑏𝑠 ∥𝑤(𝑠) ∥ 2 d𝑠

0

is an ℰ0 (𝐸)-measurable function on 𝐸 taking values in [0, ∞]. Since (𝜈𝑠 ) 𝑠>0 satisfies (13.39), we have ∫ ∫ ∞ ∫ ∥𝑤∥ 2∼ 𝜆0 (d𝑤) = 𝜆 0 (d𝑤) e−2𝑏𝑠 ∥𝑤(𝑠) ∥ 2 d𝑠 𝐸 0 𝐸 ∫ ∞ ∫ d𝑠 e−2𝑏𝑠 ∥𝑥∥ 2 𝜈𝑠 (d𝑥) = 0 𝐻 ∫ ∞ ∫ 1 ∑︁ e−2𝑏 (𝑛+𝑠) ∥𝑇𝑛 𝑥∥ 2 𝜈𝑠 (d𝑥) = d𝑠 𝑛=0

≤ 𝐵2

0

∞ ∑︁ 𝑛=0

𝐻

e−2(𝑏−𝑐0 )𝑛

∫

1

∫

e−2𝑏𝑠 ∥𝑥∥ 2 𝜈𝑠 (d𝑥) < ∞.

d𝑠 0

𝐻

Then 𝜆0 is actually supported by 𝐻˜ and (13.42) holds. By Lemma 13.12 we have ˜ so we can regard 𝜆0 as a probability measure on ( 𝐻, ˜ ˜ = ℰ0 ( 𝐻), ˜ ℬ( 𝐻)). ℬ( 𝐻) Now we get (13.41) from (13.43). Because each 𝜈𝑠 is a centered infinitely divisible □ probability measure, so is 𝜆0 . Theorem 13.15 All centered SC-semigroups associated with (𝑇𝑡 )𝑡 ≥0 satisfying (13.38) are regular if and only if all of its locally square integrable entrance paths are closable. Proof Suppose that all entrance paths 𝑥˜ ∈ 𝐻˜ are closable and (𝛾𝑡 )𝑡 ≥0 is an SCsemigroup given by (13.41). To each 𝑥˜ ∈ 𝐻˜ there corresponds some 𝑥(0) ˜ ∈ 𝐻 such that 𝑥(𝑠) ˜ = 𝑇𝑠 𝑥(0) ˜ for all 𝑠 > 0. This element is apparently determined by 𝑥˜ uniquely. Letting 𝜈0 be the image of 𝜆0 under the map 𝑥˜ ↦→ 𝑥(0), ˜ we get (13.31). Conversely, if 𝑥˜ = {𝑥(𝑠) ˜ : 𝑠 > 0} ∈ 𝐻˜ is not closable, then   ∫ 1 𝑡 2 ⟨𝑥(𝑠), ˜ 𝛾ˆ 𝑡 (𝑎) = exp − 𝑎⟩ d𝑠 , 𝑡 ≥ 0, 𝑎 ∈ 𝐻, 2 0 defines an irregular SC-semigroup for (𝑇𝑡 )𝑡 ≥0 .

□

We now discuss how to extend a centered SC-semigroup on 𝐻 to a regular one ˜ Given the semigroup (𝑇𝑡 )𝑡 ≥0 , we can define a semigroup on the entrance space 𝐻. ˜ ˜ It of linear operators (𝑇𝑡 )𝑡 ≥0 on 𝐻˜ by 𝑇˜0 𝑥˜ = 𝑥˜ and 𝑇˜𝑡 𝑥˜ = 𝐽 𝑥(𝑡) ˜ for 𝑡 > 0 and 𝑥˜ ∈ 𝐻. follows that (𝑇˜𝑡 𝑥) ˜ (𝑠) = 𝑥(𝑡 ˜ + 𝑠) = 𝑇𝑡 ( 𝑥(𝑠)), ˜

𝑠, 𝑡 > 0.

In view of (13.37) we have ∫ ∞ ∫ 2 −2𝑏𝑠 2 2 ˜ ∥𝑇𝑡 𝑥∥ ˜ ∼= e ∥ 𝑥(𝑡 ˜ + 𝑠) ∥ d𝑠 ≤ ∥𝑇𝑡 ∥ 0

0

∞

e−2𝑏𝑠 ∥ 𝑥(𝑠) ˜ ∥ 2 d𝑠.

(13.44)

13.4 Extensions of Centered Semigroups

381

Then ∥𝑇˜𝑡 ∥ ∼ ≤ ∥𝑇𝑡 ∥ for every 𝑡 ≥ 0. Let (𝑈˜ 𝛼 ) 𝛼>𝑐0 denote the resolvent of (𝑇˜𝑡 )𝑡 ≥0 ˜ = 𝑈˜ 𝛼 𝐻˜ ⊂ 𝐻. ˜ and let 𝐴˜ denote its generator with domain 𝒟( 𝐴) Lemma 13.16 Let 𝐽 be defined as in Lemma 13.11. Then 𝐽𝑇𝑡 𝑥 = 𝑇˜𝑡 𝐽𝑥 for all 𝑡 ≥ 0 ˜ and 𝑥 ∈ 𝐻 and (𝑇˜𝑡 )𝑡 ≥0 is a strongly continuous semigroup of linear operators on 𝐻. Proof For 𝑡 ≥ 0 and 𝑥 ∈ 𝐻 we have 𝐽𝑇𝑡 𝑥 = {𝑇𝑠 𝑇𝑡 𝑥 : 𝑠 > 0} = {𝑇𝑡 𝑇𝑠 𝑥 : 𝑠 > 0} = 𝑇˜𝑡 𝐽𝑥, giving the first assertion. By the proof of Lemma 13.11 we have lim ∥𝑇˜𝑡 𝑥˜ − 𝑥∥ ˜ − 𝑥∥ ˜ ∼ = lim ∥𝐽 𝑥(𝑡) ˜ ∼ = 0. 𝑡→0

𝑡→0

Then (𝑇˜𝑡 )𝑡 ≥0 is strongly continuous.

□

: 𝑠 > 0} and 𝐴˜ 𝑈˜ 𝛼 𝑥˜ = {𝐴𝑈 𝛼 𝑥(𝑠) ˜ : 𝑠 > 0} Lemma 13.17 We have 𝑈˜ 𝛼 𝑥˜ = {𝑈 𝛼 𝑥(𝑠) ˜ ˜ for all 𝑥˜ ∈ 𝐻. Proof The first assertion follows as we observe that, for 𝛼 > 𝑐 0 , ∫ ∞ ∫ ∞ e−𝛼𝑠 𝑇𝑡 𝑥(𝑠)d𝑡 ˜ = 𝑈 𝛼 𝑥(𝑠). ˜ 𝑈˜ 𝛼 𝑥(𝑠) = = e−𝛼𝑠 𝑇˜𝑡 𝑥(𝑠)d𝑡 ˜ ˜ 0

0

The second follows from the equality 𝐴˜ 𝑈˜ 𝛼 𝑥˜ = 𝛼𝑈˜ 𝛼 𝑥˜ − 𝑥. ˜

□

Theorem 13.18 All entrance paths for (𝑇˜𝑡 )𝑡 ≥0 are closable. Proof Suppose that 𝑥¯ = {𝑥˜𝑢 : 𝑢 > 0} is an entrance path for (𝑇˜𝑡 )𝑡 ≥0 , where each 𝑥˜𝑢 = {𝑥˜𝑢 (𝑠) : 𝑠 > 0} ∈ 𝐻˜ is an entrance path for (𝑇𝑡 )𝑡 ≥0 . In view of (13.44) we have {𝑥˜𝑡+𝑢 (𝑠) : 𝑠 > 0} = 𝑥˜𝑡+𝑢 = 𝑇˜𝑡 𝑥˜𝑢 = {𝑥˜𝑢 (𝑡 + 𝑠) : 𝑠 > 0}.

(13.45)

Set 𝑥˜0 = {𝑥˜ 𝑠/2 (𝑠/2) : 𝑠 > 0}. By (13.45) we have 𝑇𝑡 ( 𝑥˜ 𝑠/2 (𝑠/2)) = 𝑥˜ 𝑠/2 (𝑡 + 𝑠/2) = 𝑥˜ (𝑠+𝑡)/2 ((𝑠 + 𝑡)/2). Then 𝑥˜0 is an entrance path for (𝑇𝑡 )𝑡 ≥0 . Moreover, (𝑇˜𝑢 𝑥˜0 ) (𝑠) = 𝑇𝑢 ( 𝑥˜ 𝑠/2 (𝑠/2)) = 𝑥˜ 𝑠/2 (𝑢 + 𝑠/2) = 𝑥˜𝑢 (𝑠), and hence 𝑇˜𝑢 𝑥˜0 = 𝑥˜𝑢 . Thus 𝑥¯ = {𝑥˜𝑢 : 𝑢 > 0} is closed by 𝑥˜0 .

□

Theorem 13.19 Let (𝛾𝑡 )𝑡 ≥0 be a centered SC-semigroup given by (13.41) and (13.42). Let 𝛾˜ 𝑡 = 𝛾𝑡 ◦ 𝐽 −1 for 𝑡 ≥ 0. Then ( 𝛾˜ 𝑡 )𝑡 ≥0 is a regular centered SC-semigroup associated with (𝑇˜𝑡 )𝑡 ≥0 and ∫ 𝑡 h ∫ ∫ i  ˜ 𝑎⟩ ˜ ∼ ˜ 𝑇˜𝑠∗ 𝑎⟩ ˜ ∼ e𝑖 ⟨ 𝑥, ˜ d𝑠 ˜ = exp (13.46) 𝜆0 (d𝑥) 𝛾˜ 𝑡 (d𝑥) e𝑖 ⟨ 𝑥, log 𝐻˜

˜ for every 𝑡 ≥ 0 and 𝑎˜ ∈ 𝐻.

0

𝐻˜

382

13 Generalized Ornstein–Uhlenbeck Processes

Proof It is not hard to show ( 𝛾˜ 𝑡 )𝑡 ≥0 is an SC-semigroup associated with (𝑇˜𝑡 )𝑡 ≥0 . : 𝑠 > 0} ∈ 𝐻˜ Since (𝑇𝑡 )𝑡 ≥0 is a strongly continuous semigroup, for any 𝑎˜ = { 𝑎(𝑠) ˜ we can use the dominated convergence theorem and (13.41) to see   ∫ ∞ ∫ −2𝑏𝑠 e exp 𝑖 ⟨𝑇𝑠 𝑥, 𝑎(𝑠)⟩d𝑠 ˜ 𝛾𝑡 (d𝑥) 0 𝐻   ∑︁ ∫ ∞ ˜ 𝛾𝑡 (d𝑥) = lim exp 𝑖 𝑛−1 e−2𝑏𝑘/𝑛 ⟨𝑇𝑘/𝑛 𝑥, 𝑎(𝑘/𝑛)⟩ 𝑛→∞ 𝐻 𝑘=1 ∫ e𝑖 ⟨𝑥,𝑎𝑛 ⟩ 𝛾𝑡 (d𝑥) = lim 𝑛→∞ 𝐻   ∫ 𝑡  ∫ e𝑖 ⟨ 𝑥˜ (𝑠),𝑎𝑛 ⟩ 𝜆0 (d𝑥) log ˜ d𝑠 , = lim exp 𝑛→∞

𝐻˜

0

where 𝑎𝑛 =

∞ ∑︁

∗ ˜ 𝑎(𝑘/𝑛). 𝑛−1 e−2𝑏𝑘/𝑛𝑇𝑘/𝑛

𝑘=1

By the strong continuity of (𝑇𝑡∗ )𝑡 ≥0 we have ∫ ∞ 𝑇𝑟∗ 𝑎(𝑟)⟩d𝑟 lim ⟨𝑥(𝑠), ˜ ˜ 𝑎𝑛⟩ = ˜ e−2𝑏𝑟 ⟨𝑥(𝑠), 𝑛→∞ ∫0 ∞ = e−2𝑏𝑟 ⟨𝑥(𝑟), 𝑇𝑠∗ 𝑎(𝑟)⟩d𝑟. ˜ ˜ 0

Then another application of the dominated convergence theorem gives  ∫ ∞  ∫ exp 𝑖 ˜ 𝛾𝑡 (d𝑥) e−2𝑏𝑠 ⟨𝑇𝑠 𝑥, 𝑎(𝑠)⟩d𝑠 𝐻    ∫0 𝑡  ∫ ∫ o n ∞ exp 𝑖 e−2𝑏𝑟 ⟨𝑥(𝑟), ˜ log = exp ˜ 𝜆0 (d𝑥) 𝑇𝑠∗ 𝑎(𝑟)⟩d𝑟 ˜ d𝑠 0  ∫0 𝑡 h ∫𝐻˜ i  ˜ ∼ ˜ 𝑇˜𝑠∗ 𝑎⟩ 𝑖 ⟨ 𝑥, log ˜ d𝑠 . = exp e 𝜆0 (d𝑥) 0

𝐻˜

This proves (13.46).

□

Theorem 13.20 Let ( 𝛾˜ 𝑡 )𝑡 ≥0 be a centered SC-semigroup associated with (𝑇˜𝑡 )𝑡 ≥0 satisfying ∫ ∥ 𝑥∥ ˜ 2 𝛾˜ 𝑡 (d𝑥) ˜ < ∞, 𝑡 ≥ 0. (13.47) 𝐻˜ ◦

Then there is a centered SC-semigroup (𝛾𝑡 )𝑡 ≥0 associated with (𝑇𝑡 )𝑡 ≥0 satisfying (13.38) and 𝛾˜ 𝑡 = 𝛾𝑡 ◦ 𝐽 −1 for each 𝑡 ≥ 0.

13.5 Construction of the Processes

383

Proof By Theorems 13.15 and 13.18, any centered SC-semigroup associated with (𝑇˜𝑡 )𝑡 ≥0 is regular, so ( 𝛾˜ 𝑡 )𝑡 ≥0 has the expression (13.46) for an infinitely divisible ˜ Then we get (𝛾𝑡 )𝑡 ≥0 by Theorem 13.14, which clearly satisfies probability 𝜆0 on 𝐻. the requirements. □ By Theorems 13.19 and 13.20, centered SC-semigroups associated with (𝑇𝑡 )𝑡 ≥0 are in one-to-one correspondence with centered regular SC-semigroups associated with (𝑇˜𝑡 )𝑡 ≥0 . Therefore we may reduce some analysis of irregular SC-semigroups to those of regular ones. The consideration of centered SC-semigroups is not a serious restriction. In fact, if (𝛾𝑡 )𝑡 ≥0 is an arbitrary SC-semigroup satisfying condition (13.38), we can define ∫ 𝑏𝑡 = 𝑥𝛾𝑡 (d𝑥) 𝐻◦

and 𝛾𝑡𝑐 = 𝛿−𝑏𝑡 ∗ 𝛾𝑡 for 𝑡 ≥ 0. It is easy to check that both (𝛿 𝑏𝑡 )𝑡 ≥0 and (𝛾𝑡𝑐 )𝑡 ≥0 are SCsemigroups associated with (𝑇𝑡 )𝑡 ≥0 . Therefore (𝛾𝑡 )𝑡 ≥0 can always be decomposed as the convolution of a degenerate SC-semigroup and a centered one.

13.5 Construction of the Processes In this section, we prove that the generalized OU-processes corresponding to a regular SC-semigroup has a càdlàg realization in a suitable extension of the space. Let (𝑇𝑡 )𝑡 ≥0 be a strongly continuous semigroup on 𝐻 with generator ( 𝐴, 𝒟( 𝐴)). Then 𝒟( 𝐴) with the inner product norm ∥ · ∥ 𝐴 defined by ∥𝑥∥ 2𝐴 = ∥𝑥∥ 2 + ∥ 𝐴𝑥∥ 2 ,

𝑥 ∈ 𝒟( 𝐴)

(13.48)

is a Hilbert space and 𝒟( 𝐴) ⊂ 𝐻 is a continuous embedding. ¯ ∥ · ∥ − ) and a strongly continuous Proposition 13.21 There is a Hilbert space ( 𝐻, ¯ 𝒟( 𝐴)) ¯ such that: ¯ ∥ · ∥ − ) with generator ( 𝐴, semigroup (𝑇¯𝑡 )𝑡 ≥0 on ( 𝐻, (1) 𝐻 ⊂ 𝐻¯ with dense continuous embedding; (2) each 𝑇𝑡 is the restriction of 𝑇¯𝑡 to 𝐻; ¯ with continuous embedding. (3) 𝐻 ⊂ 𝒟( 𝐴) Proof Recall that there are constants 𝐵 ≥ 0 and 𝑐 0 ≥ 0 such that ∥𝑇𝑡 ∥ ≤ 𝐵e𝑐0 𝑡 for every 𝑡 ≥ 0. Let (𝑈 𝛼 ) 𝛼>𝑐0 denote the resolvent of (𝑇𝑡 )𝑡 ≥0 . Fix 𝑏 > 𝑐 0 and define an inner product on 𝐻 by ⟨𝑥, 𝑦⟩− = ⟨𝑈 𝑏 𝑥, 𝑈 𝑏 𝑦⟩,

𝑥, 𝑦 ∈ 𝐻.

(13.49)

Let ∥ · ∥ − be the corresponding norm and let 𝐻¯ be the completion of 𝐻 with respect to this norm. From (13.49) we get

384

13 Generalized Ornstein–Uhlenbeck Processes

∥𝑥∥ − ≤ ∥𝑈 𝑏 ∥ ∥𝑥∥,

𝑥 ∈ 𝐻,

(13.50)

¯ ∥ · ∥ − ) is a continuous dense so the identity mapping 𝐼 from (𝐻, ∥ · ∥) to ( 𝐻, embedding. Consequently, the linear semigroup (𝑇𝑡 )𝑡 ≥0 can be uniquely extended ¯ which satisfies ∥𝑇¯𝑡 ∥ ≤ 𝐵e𝑏𝑡 for to a strongly continuous semigroup (𝑇¯𝑡 )𝑡 ≥0 on 𝐻, ¯ ⊂ 𝐻¯ with continuous embedding. For any every 𝑡 ≥ 0. In addition, we have 𝒟( 𝐴) ¯ we have 𝑥 = 𝑈 𝑏 𝑦 for some 𝑦 ∈ 𝐻. By (13.49) we have 𝑥 ∈ 𝒟( 𝐴) ⊂ 𝒟( 𝐴) ¯ − = ∥ 𝐴𝑈 𝑏 𝑈 𝑏 𝑦∥ = ∥ (𝑏𝑈 𝑏 − 1)𝑈 𝑏 𝑦∥ ≤ (𝑏∥𝑈 𝑏 ∥ + 1) ∥𝑥∥. ∥ 𝐴𝑥∥

(13.51)

Since 𝒟( 𝐴) is a dense subset of (𝐻, ∥ · ∥), for 𝑥 ∈ 𝐻 we can find {𝑥 𝑛 } ⊂ 𝒟( 𝐴) such ¯ Then {𝑥 𝑛 } is a Cauchy sequence in both that lim𝑛→∞ 𝑥 𝑛 = 𝑥 in 𝐻 and hence in 𝐻. ¯ 𝐻 and 𝐻. From (13.48), (13.50) and (13.51) we see {𝑥 𝑛 } is also a Cauchy sequence ¯ Since 𝒟( 𝐴) ¯ is complete and 𝒟( 𝐴) ¯ ⊂ 𝐻¯ is a continuous embedding, we in 𝒟( 𝐴). ¯ Since 𝒟( 𝐴) is dense in ¯ also have lim𝑛→∞ 𝑥 𝑛 = 𝑥 in 𝒟( 𝐴). This proves 𝐻 ⊂ 𝒟( 𝐴). 𝑏 ¯ ¯ 𝐻, from (13.51) we have ∥ 𝐴𝑥∥ − ≤ (𝑏∥𝑈 ∥ + 1) ∥𝑥∥ for all 𝑥 ∈ 𝐻. Then 𝐻 ⊂ 𝒟( 𝐴) □ is a continuous embedding. Now let 𝜈0 be an infinitely divisible probability measure on 𝐻 and let 𝜓0 (𝑎) = − log 𝜈ˆ0 (𝑎) for 𝑎 ∈ 𝐻. A convolution semigroup (𝜇𝑡 )𝑡 ≥0 on 𝐻 is given by 𝜇ˆ 𝑡 (𝑎) = exp{−𝑡𝜓0 (𝑎)},

𝑎 ∈ 𝐻.

Let (𝑃𝑡 )𝑡 ≥0 be the transition semigroup on 𝐻 defined by ∫ 𝑃𝑡 𝑓 (𝑥) = 𝑓 (𝑥 + 𝑦)𝜇𝑡 (d𝑦), 𝑥 ∈ 𝐻, 𝑓 ∈ 𝐶 (𝐻).

(13.52)

(13.53)

𝐻

A càdlàg Markov process in 𝐻 with transition semigroup (𝑃𝑡 )𝑡 ≥0 is called a Lévy process. In view of (13.53), a Lévy process is translation invariant and has independent increments. The existence of such a process is given by the following: Proposition 13.22 There is a Lévy process in 𝐻 with transition semigroup (𝑃𝑡 )𝑡 ≥0 defined by (13.53). Proof By (13.52) and Parthasarathy (1967, p. 189) it is easy to see that lim𝑡→0 𝜇𝑡 = 𝛿0 by the weak convergence. In particular, we have lim sup 𝑃𝑡 (𝑥, 𝐵(𝑥, 𝜀) 𝑐 ) = lim 𝜇𝑡 (𝐵(0, 𝜀) 𝑐 ) = 0, 𝑡→0 𝑥 ∈𝐻

(13.54)

𝑡→0

where 𝐵(𝑥, 𝜀) 𝑐 denotes the complement of the open ball centered at 𝑥 ∈ 𝐻 with radius 𝜀 > 0. Then the result follows by the general theory of stochastic processes; see, e.g., Wentzell (1981, p. 170). □ ¯ ∥ · ∥ − ) is an extension of (𝐻, ∥ · ∥) with the three properties in Suppose that ( 𝐻, Proposition 13.21. Let (𝛾𝑡 )𝑡 ≥0 be the regular SC-semigroup associated with (𝑇𝑡 )𝑡 ≥0 defined by (13.13). We can certainly regard 𝜈0 and 𝛾𝑡 as infinitely divisible probability

13.5 Construction of the Processes

385

¯ Then (𝛾𝑡 )𝑡 ≥0 is also an SC-semigroup associated measures on the enlarged space 𝐻. ¯ with (𝑇𝑡 )𝑡 ≥0 . Let ∫ ¯ 𝑎⟩ ¯ − (13.55) ¯ = − log 𝑎¯ ∈ 𝐻¯ ∗ ⊂ 𝐻. 𝜈0 (d𝑥), 𝜓¯ 0 ( 𝑎) e𝑖 ⟨ 𝑥, ¯ 𝐻¯

From (13.13) it is not hard to show that  ∫ ∫ 𝑖 ⟨ 𝑥, ¯ 𝑎⟩ ¯ − ¯ = exp − 𝛾𝑡 (d𝑥) e 𝐻¯

𝑡

 ∗ ¯ ¯ 𝜓0 (𝑇𝑠 𝑎)d𝑠 , ¯

𝑎¯ ∈ 𝐻¯ ∗ ,

(13.56)

0

𝛾 where (𝑇¯𝑡∗ )𝑡 ≥0 denotes the dual semigroup of (𝑇¯𝑡 )𝑡 ≥0 . Let (𝑄¯ 𝑡 )𝑡 ≥0 be the generalized ¯ Mehler semigroup defined by (13.14) from (𝑇𝑡 )𝑡 ≥0 and (𝛾𝑡 )𝑡 ≥0 . By Proposition 13.22, the transition semigroup (𝑃𝑡 )𝑡 ≥0 defined by (13.53) has a ¯ with 𝑌0 = 0. Since 𝑠 ↦→ 𝐴𝑌 ¯ 𝑠 is right càdlàg realization {𝑌𝑡 : 𝑡 ≥ 0} in 𝐻 ⊂ 𝒟( 𝐴) ¯ continuous, for any 𝑥¯ ∈ 𝐻,

𝑍¯ 𝑡 = 𝑇¯𝑡 𝑥¯ + 𝑌𝑡 +

∫

𝑡

¯ 𝑠 d𝑠 𝑇¯𝑡−𝑠 𝐴𝑌

(13.57)

0

¯ defines a càdlàg process { 𝑍¯ 𝑡 : 𝑡 ≥ 0} in 𝐻. Lemma 13.23 For any 𝑡 ≥ 0 the random variable 𝑍¯ 𝑡 defined by (13.57) has distri𝛾 ¯ bution 𝑄¯ 𝑡 ( 𝑥, ¯ ·) on 𝐻. ¯ 𝑠 Proof By the dominated convergence theorem and the right continuity of 𝑠 ↦→ 𝐴𝑌 we get 𝑛 ∫ ∑︁

𝑍¯ 𝑛 (𝑡) := 𝑇¯𝑡 𝑥¯ + 𝑌𝑡 +

𝑘=1

𝑘𝑡/𝑛

¯ 𝑘𝑡/𝑛 d𝑠 → 𝑍¯ 𝑡 𝑇¯𝑡−𝑠 𝐴𝑌

(13.58)

(𝑘−1)𝑡/𝑛

in 𝐻¯ as 𝑛 → ∞. Observe that 𝑍¯ 𝑛 (𝑡) = 𝑇¯𝑡 𝑥¯ + 𝑌𝑡 +

𝑛 ∑︁

(𝑇¯(𝑛−𝑘+1)𝑡/𝑛 − 𝑇¯(𝑛−𝑘)𝑡/𝑛 )𝑌𝑘𝑡/𝑛

𝑘=1

= 𝑇¯𝑡 𝑥¯ +

𝑛 ∑︁

𝑇¯(𝑛−𝑘+1)𝑡/𝑛 (𝑌𝑘𝑡/𝑛 − 𝑌 (𝑘−1)𝑡/𝑛 ),

𝑘=1

and hence   𝑛 
𝑡 ∑︁ ∗ ∗ ¯ ¯ ¯ 𝜓0 𝑇(𝑛−𝑘+1)𝑡/𝑛 𝑎¯ . ¯ − = exp 𝑖⟨𝑥, ¯ 𝑇𝑡 𝑎⟩ E exp 𝑖⟨ 𝑍 𝑛 (𝑡), 𝑎⟩ ¯ − 𝑛 𝑘=1 Since (𝑇¯𝑡 )𝑡 ≥0 is strongly continuous, so is (𝑇¯𝑡∗ )𝑡 ≥0 . Then 𝑠 ↦→ 𝜓0 (𝑇¯𝑠∗ 𝑎) ¯ is continuous ¯ By letting 𝑛 → ∞ in the equality above we obtain on [0, ∞) for each 𝑎¯ ∈ 𝐻.

386

13 Generalized Ornstein–Uhlenbeck Processes



 ¯ − ¯ − = exp 𝑖⟨𝑥, E exp 𝑖⟨ 𝑍¯ 𝑡 , 𝑎⟩ ¯ 𝑇¯𝑡∗ 𝑎⟩

∫

𝑡


∗ ¯ 𝜓0 𝑇𝑡−𝑠 𝑎¯ d𝑠 .

0

This gives the desired result.

□

Theorem 13.24 The process { 𝑍¯ 𝑡 : 𝑡 ≥ 0} defined by (13.57) is a càdlàg strong 𝛾 Markov process in 𝐻¯ with transition semigroup (𝑄¯ 𝑡 )𝑡 ≥0 . Proof In view of (13.57), the process { 𝑍¯ 𝑡 : 𝑡 ≥ 0} is adapted to the filtration (ℱ𝑡 )𝑡 ≥0 generated by {𝑌𝑡 : 𝑡 ≥ 0}. For 𝑟, 𝑡 ≥ 0 we have ∫ 𝑟+𝑡 ¯ 𝑠 d𝑠 ¯ ¯ ¯ ¯ 𝑍𝑟+𝑡 − 𝑇𝑡 𝑍𝑟 = 𝑌𝑟+𝑡 − 𝑇𝑡 𝑌𝑟 + 𝑇¯𝑟+𝑡−𝑠 𝐴𝑌 ∫𝑟 𝑟+𝑡 ¯ 𝑠 − 𝑌𝑟 )d𝑠. 𝑇¯𝑟+𝑡−𝑠 𝐴(𝑌 = (𝑌𝑟+𝑡 − 𝑌𝑟 ) + 𝑟

Since {𝑌𝑟+𝑡 −𝑌𝑟 : 𝑡 ≥ 0} given ℱ𝑟 is a process with independent increments and has the same law as {𝑌𝑡 : 𝑡 ≥ 0}, an application of Lemma 13.23 shows that   ∫ 𝑡 h i  ¯ 𝜓0 (𝑇¯𝑠∗ 𝑎)d𝑠 . ¯ − ℱ𝑟 = exp 𝑖⟨ 𝑍¯ 𝑟 , 𝑇¯𝑡∗ 𝑎⟩ ¯ −− E exp 𝑖⟨ 𝑍¯ 𝑟+𝑡 , 𝑎⟩ 0

𝛾 Therefore { 𝑍¯ 𝑡 : 𝑡 ≥ 0} is a Markov process with transition semigroup (𝑄¯ 𝑡 )𝑡 ≥0 . 𝛾 Since (𝑄¯ 𝑡 )𝑡 ≥0 preserves 𝐶 (𝐻), the strong Markov property follows by a standard □ argument.

From the theorem above we easily obtain a construction of the generalized OU𝛾 process corresponding to the generalized Mehler semigroup (𝑄 𝑡 )𝑡 ≥0 . Indeed, for ¯ ¯ any 𝑥 ∈ 𝐻 we have 𝑇𝑡 𝑥 = 𝑇𝑡 𝑥 ∈ 𝐻 and hence 𝑍𝑡 ∈ 𝐻 a.s. for every 𝑡 ≥ 0. Then 𝛾 { 𝑍¯ 𝑡 : 𝑡 ≥ 0} is also a Markov process with transition semigroup (𝑄 𝑡 )𝑡 ≥0 . However, this process usually does not have a càdlàg version in 𝐻. In other words, to get the sample path regularity, we need to observe the process in the enlarged state space 𝐻¯ with a weaker topology. A similar phenomenon has been observed in Example 9.3. Suppose that 𝐸 is a real separable Hilbert space containing 𝐻 as a subspace and 𝐴 is the generator of a semigroup of bounded linear operators (𝑇𝑡 )𝑡 ≥0 on 𝐸 with domain 𝒟( 𝐴) ⊃ 𝐻. Let {𝑌𝑡 : 𝑡 ≥ 0} be a Lévy process in 𝐻. We say a stochastic process {𝑋𝑡 : 𝑡 ≥ 0} in 𝐻 solves a Langevin type equation provided ∫ 𝑡 𝑋𝑡 = 𝑋0 + (13.59) 𝐴𝑋𝑠 d𝑠 + 𝑌𝑡 , 𝑡 ≥ 0. 0

As usual, we may write the equation in the differential form d𝑋𝑡 = 𝐴𝑋𝑡 d𝑡 + d𝑌𝑡 ,

𝑡 ≥ 0.

(13.60)

13.5 Construction of the Processes

387

Theorem 13.25 Let {𝑌𝑡 : 𝑡 ≥ 0} be a Lévy process in 𝐻 with transition semigroup (𝑃𝑡 )𝑡 ≥0 defined by (13.53). Then the generalized OU-process { 𝑍¯ 𝑡 : 𝑡 ≥ 0} defined by (13.57) satisfies the stochastic equation  ∫ 𝑡 ¯ ¯ 𝑍𝑡 = 𝑥¯ + 𝑌𝑡 + 𝐴 𝑍¯ 𝑠 d𝑠 , 𝑡 ≥ 0. (13.61) 0

Proof From (13.57) we have ∫ 𝑡 ∫ 𝑡 ∫ 𝑢 ∫ 𝑡 ∫ 𝑡 ¯ 𝑠 d𝑠 𝑍¯ 𝑠 d𝑠 = d𝑢 𝑇¯𝑢−𝑠 𝐴𝑌 ¯ + 𝑇¯𝑠 𝑥d𝑠 𝑌𝑠 d𝑠 + 0 0 ∫0 𝑡 ∫0 𝑡 ∫0 𝑡 (𝑇¯𝑡−𝑠𝑌𝑠 − 𝑌𝑠 )d𝑠 = 𝑇¯𝑠 𝑥d𝑠 𝑌𝑠 d𝑠 + ¯ + 0 ∫0 𝑡 ∫0 𝑡 𝑇¯𝑡−𝑠𝑌𝑠 d𝑠. = 𝑇¯𝑠 𝑥d𝑠 ¯ + 0

0

¯ → 𝐻¯ is a bounded operator, so the above From (13.48) we see that 𝐴¯ : 𝒟( 𝐴) equation implies 𝐴¯

∫

𝑡

∫  𝑍¯ 𝑠 d𝑠 = 𝐴¯

0

𝑡

∫ 𝑛  ∑︁ ¯ 𝑇¯𝑠 𝑥d𝑠 + lim 𝐴¯ 𝑛→∞

0

𝑘=1 𝑘𝑡/𝑛

𝑘𝑡/𝑛

𝑇¯𝑡−𝑠𝑌𝑘𝑡/𝑛 d𝑠 (𝑘−1)𝑡/𝑛

𝑛 ∫ ∑︁ ¯ 𝐴¯ 𝑇¯𝑡−𝑠𝑌𝑘𝑡/𝑛 d𝑠 = 𝑇𝑡 𝑥¯ − 𝑥¯ + lim 𝑛→∞ (𝑘−1)𝑡/𝑛 ∫ 𝑡 𝑘=1 ¯ 𝑠 d𝑠 ¯ = 𝑇𝑡 𝑥¯ − 𝑥¯ + 𝑇¯𝑡−𝑠 𝐴𝑌 0

= 𝑍¯ 𝑡 − 𝑥¯ − 𝑌𝑡 . This proves (13.61).

□

One naturally wishes to exchange the order of the integral and the operation of the generator in (13.61). To do so, we need a further extension of the domain of ¯ ∥ · ∥ − ) with the properties in ˜ ∥ · ∥ ∼ ) be an extension of ( 𝐻, the generator. Let ( 𝐻, Proposition 13.21. Let (𝑇˜𝑡 )𝑡 ≥0 and 𝐴˜ be the corresponding extensions of (𝑇¯𝑡 )𝑡 ≥0 and ¯ respectively. 𝐴, Theorem 13.26 The generalized OU-process { 𝑍¯ 𝑡 : 𝑡 ≥ 0} defined by (13.57) is ˜ and satisfies the Langevin type equation càdlàg in 𝒟( 𝐴) ∫ 𝑡 𝐴˜ 𝑍¯ 𝑠 d𝑠, 𝑡 ≥ 0. (13.62) 𝑍¯ 𝑡 = 𝑥¯ + 𝑌𝑡 + 0

¯ with continuous embedding. Proof By Proposition 13.21, we have 𝐻 ⊂ 𝒟( 𝐴) ¯ ¯ ¯ 𝑠 is càdlàg in 𝐻. ¯ ¯ Since 𝐴 is a bounded operator from 𝒟( 𝐴) to 𝐻, the process 𝑠 ↦→ 𝐴𝑌 ¯ 𝑠 is càdlàg in 𝐻. ˜ By Theorem 13.24 the process 𝑡 ↦→ 𝑍¯ 𝑡 Similarly we find 𝑠 ↦→ 𝐴˜ 𝐴𝑌 ˜ and ¯ so it is càdlàg in 𝒟( 𝐴) is càdlàg in 𝐻,

388

13 Generalized Ornstein–Uhlenbeck Processes

¯ 𝑡+ 𝐴˜ 𝑍¯ 𝑡 = 𝐴˜ 𝑇¯𝑡 𝑥¯ + 𝐴𝑌

∫

𝑡

¯ 𝑠 d𝑠. 𝑇˜𝑡−𝑠 𝐴˜ 𝐴𝑌

0

Moreover, we have 𝐴¯

∫

𝑡

 ∫ 𝑍¯ 𝑠 d𝑠 = 𝐴˜

𝑡

 ∫ 𝑍¯ 𝑠 d𝑠 =

0

0

𝑡

𝐴˜ 𝑍¯ 𝑠 d𝑠.

0

Then (13.62) follows from (13.61).

□

˜ satisfying (13.62), then Theorem 13.27 If { 𝑍¯ 𝑡 : 𝑡 ≥ 0} is a càdlàg process in 𝒟( 𝐴) it is given by (13.57). Consequently, the pathwise uniqueness holds for the equation (13.62). Proof From (13.62) we have ∫ 𝑡 ∫ 𝑡 ∫ 𝑡 ∫ 𝑢 ∫ 𝑡 𝑇¯𝑡−𝑠𝑌𝑠 d𝑠 + d𝑢 ¯ + 𝑇˜𝑡−𝑢 𝐴˜ 𝑍¯ 𝑠 d𝑠 𝑇¯𝑡−𝑠 𝑥d𝑠 𝑇˜𝑡−𝑠 𝑍¯ 𝑠 d𝑠 = 0 0 ∫0 𝑡 ∫0 𝑡 ∫0 𝑡 𝑇¯𝑡−𝑠𝑌𝑠 d𝑠 + ¯ + = 𝑇¯𝑡−𝑠 𝑥d𝑠 (𝑇˜𝑡−𝑠 𝑍¯ 𝑠 − 𝑍¯ 𝑠 )d𝑠, 0

0

0

and hence ∫

𝑡

𝑍¯ 𝑠 d𝑠 =

∫

𝑡

𝑇¯𝑡−𝑠 𝑥d𝑠 ¯ +

0

0

∫

𝑡

𝑇¯𝑡−𝑠𝑌𝑠 d𝑠.

0

It follows that ∫ 0

𝑡

 ∫ 𝑡 ¯ 𝑠 d𝑠 ¯ + 𝑇¯𝑡−𝑠 𝐴𝑌 𝑇𝑡−𝑠 𝑥d𝑠 ¯ 0 0 ∫ 𝑡 ¯ 𝑠 d𝑠. = 𝑇¯𝑡 𝑥¯ − 𝑥¯ + 𝑇¯𝑡−𝑠 𝐴𝑌

𝐴˜ 𝑍¯ 𝑠 d𝑠 = 𝐴¯

∫

𝑡

0

By using (13.62) again we obtain (13.57).

□

13.6 Notes and Comments The concept of the generalized Mehler semigroup was introduced by Bogachev and Röckner (1995) and Bogachev et al. (1996) as a generalization of the classical Mehler formula; see, e.g., Malliavin (1997, p. 25). The subject has become a very interesting field of research. Proposition 13.2 was first proved by Schmuland and Sun (2001). The current form of Theorem 13.3 is due to Dawson et al. (2004b), which extends an earlier result of Bogachev et al. (1996) in the setting of cylindrical measures. By a theorem of Keller-Ressel et al. (2011), every stochastically continuous Ornstein–Uhlenbeck

13.6 Notes and Comments

389

process in a finite-dimensional space is regular. Theorem 13.7 was first proved in Dawson et al. (2004b). The main reference of Sections 13.4 and 13.5 is Dawson and Li (2004). See also Fuhrman and Röckner (2000) for the construction of the process. A set of generalized Ornstein–Uhlenbeck processes were defined using Langevin type equations in Chojnowska-Michalik (1987), where the following mild form of (13.59) was considered: ∫ 𝑡 𝑇𝑡−𝑠 d𝑌𝑠 , 𝑡 ≥ 0. (13.63) 𝑋𝑡 = 𝑇𝑡 𝑋0 + 0

If the Lévy process {𝑌𝑡 : 𝑡 ≥ 0} has transition semigroup given by (13.52) and 𝛾 (13.53), then {𝑋𝑡 : 𝑡 ≥ 0} has transition semigroup (𝑄 𝑡 )𝑡 ≥0 given by (13.14); see, e.g., Applebaum (2007). Let us consider the regular SC-semigroup defined by (13.13) with 𝑎 ↦→ 𝜓0 (𝑎) given by the right-hand side of (13.2). It was proved in Fuhrman and Röckner 𝛾 (2000) that the corresponding generalized Mehler semigroup (𝑄 𝑡 )𝑡 ≥0 is weakly continuous on the space of uniformly continuous bounded functions. The notion of weak continuity was introduced in Cerrai (1994), where it was shown that the 𝛾 strong continuity fails even in the Gaussian case. The generator of (𝑄 𝑡 )𝑡 ≥0 was defined in Fuhrman and Röckner (2000) by the resolvent. Lescot and Röckner (2002) characterized the generator as a pseudo-differential operator. The existence and uniqueness of invariant measures for generalized OU-processes were studied in Chojnowska-Michalik (1987) and Fuhrman and Röckner (2000). The mixed topology on 𝐶 (𝐻) is by definition the finest locally convex topology that agrees on bounded sets with the uniform convergence on compact sets in 𝐻. The 𝛾 semigroup (𝑄 𝑡 )𝑡 ≥0 is strongly continuous on 𝐶 (𝐻) with this topology. Applebaum 𝛾 (2007) gave an explicit representation of the generator of (𝑄 𝑡 )𝑡 ≥0 as a semigroup on 𝐶 (𝐻), which is closable and has a convenient invariant core of cylinder functions. The mixed topology was already used to study Gaussian type Mehler semigroups in Goldys and Kocan (2001) and Goldys and van Neerven (2003). The mild form (13.63) of the Langevin type equation makes sense even when {𝑌𝑡 : 𝑡 ≥ 0} is a Lévy process in some larger space 𝐸 ⊃ 𝐻. Priola and Zabczyk (2011) considered the case where {𝑌𝑡 : 𝑡 ≥ 0} is a cylindrical stable process. Suppose that 𝐴 : 𝒟( 𝐴) → 𝐻 is a self-adjoint operator and {𝑒 1 , 𝑒 2 , . . .} is an orthonormal basis of 𝐻 such that 𝐴𝑒 𝑛 = 𝛾𝑛 𝑒 𝑛 for every 𝑛 ≥ 1 with 𝛾𝑛 > 0 and 𝛾𝑛 → ∞ as 𝑛 → ∞. Then each 𝑒 𝑛 is an eigenvector of 𝐴. Let {𝑌𝑡 : 𝑡 ≥ 0} be a cylindrical stable process given by 𝑌𝑡 =

∞ ∑︁

𝛽𝑛 𝑦 𝑛 (𝑡)𝑒 𝑛 ,

𝑛=1

where {𝑦 𝑛 (𝑡) : 𝑡 ≥ 0}, 𝑛 = 1, 2, . . . are i.i.d. one-dimensional 𝛼-stable processes with 0 < 𝛼 < 2 and 𝛽1 , 𝛽2 , . . . are strictly positive constants. It was proved in Priola and Zabczyk (2011) that for any 𝑋0 = 𝑥 ∈ 𝐻 the generalized OU-process {𝑋𝑡 : 𝑡 ≥ 0} defined by (13.63) takes values in 𝐻 if and only if

390

13 Generalized Ornstein–Uhlenbeck Processes ∞ ∑︁ 𝛽𝑛𝛼 < ∞, 𝛾 𝑛=1 𝑛

and in this case {𝑋𝑡 : 𝑡 ≥ 0} is a stochastically continuous Markov process. Suppose that {𝑌𝑡 : 𝑡 ≥ 0} is a Lévy process in 𝐻 and 𝑥 ↦→ 𝑏(𝑥) is an operator on 𝐻. A generalization of the Langevin type equation (13.59) is the following: d𝑋𝑡 = 𝐴𝑋𝑡 d𝑡 + 𝑏(𝑋𝑡 )d𝑡 + d𝑌𝑡 ,

𝑡 ≥ 0.

(13.64)

This equation was studied in Lescot and Röckner (2004) under certain regularity conditions. Their approach was to construct the transition semigroup of the solution by applying the perturbation theory to the generalized Mehler semigroup in the space 𝐿 2 (𝐻, 𝜇), where 𝜇 is the invariant measure for the solution of the equation with 𝑏 = 0. Priola and Zabczyk (2011) studied the Markov property, irreducibility and strong Feller property of the solution to (13.64) for a cylindrical stable noise. Some powerful Harnack type and functional inequalities for generalized Mehler semigroups were established in Röckner and Wang (2003) and Wang (2005). The reader may refer to Applebaum (2015) for a review of probabilistic properties of generalized Ornstein–Uhlenbeck processes. A systematical study of time inhomogeneous generalized Mehler semigroups and skew convolution semigroups on Hilbert spaces was carried out in Ouyang and Röckner (2016).

Chapter 14

Small-Branching Fluctuation Limits

A typical class of generalized OU-processes arise as small-branching fluctuation limits of subcritical immigration superprocesses around their equilibrium means. In this chapter, we first establish such a fluctuation limit theorem in the space of Schwartz distributions. A stronger result is then proved which shows that the convergence actually holds in a suitable weighted Sobolev space. To avoid complicated regularity assumptions, we only consider the case where the spatial motion is a Brownian motion with killing.

14.1 The Brownian Immigration Superprocess We first introduce the Brownian immigration superprocess to be considered. Let 𝑏 > 0 be a constant and let 𝜉 be a killed Brownian motion in R𝑑 with generator 𝐴 := Δ/2 − 𝑏 and transition semigroup (𝑃𝑡𝑏 )𝑡 ≥0 . Then 𝜉 has finite potential operator 𝑈 given by ∫ ∞ ∫ ∞ 𝑏 e−𝑏𝑡 𝑃𝑡 𝑓 (𝑥)d𝑡, 𝑓 ∈ 𝐵(R𝑑 ), 𝑃𝑡 𝑓 (𝑥)d𝑡 = 𝑈 𝑓 (𝑥) = 0

0

where (𝑃𝑡 )𝑡 ≥0 is the transition semigroup of the standard Brownian motion in R𝑑 . Let 𝜙 be a critical local branching mechanism on R𝑑 given by ∫ ∞ (14.1) (e−𝑢𝜆 − 1 + 𝑢𝜆)𝑚(𝑥, d𝑢), 𝜙(𝑥, 𝜆) = 𝑐(𝑥)𝜆2 + 0

where 𝑐 ∈ 𝐶 (R𝑑 ) + and 𝑢 2 𝑚(𝑥, d𝑢) is a bounded kernel from R𝑑 to (0, ∞). We assume 𝑥 ↦→ 𝑢 2 𝑚(𝑥, d𝑢) is continuous by weak convergence on (0, ∞). The cumulant semigroup (𝑉𝑡 )𝑡 ≥0 of the (𝜉, 𝜙)-superprocess is defined by ∫ 𝑡 𝑏 𝑏 𝑉𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) − 𝑃𝑡−𝑠 (14.2) 𝜙(𝑉𝑠 𝑓 ) (𝑥)d𝑠, 𝑡 ≥ 0, 𝑥 ∈ R𝑑 . 0

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5_14

391

392

14 Small-Branching Fluctuation Limits

Clearly, the actual branching mechanism of the (𝜉, 𝜙)-superprocess is strictly subcritical because of the killing rate 𝑏 > 0 in the underlying spatial motion. We fix a constant 𝑝 > 𝑑 and let ℎ 𝑝 (𝑥) = (1 + |𝑥| 2 ) − 𝑝/2 for 𝑥 ∈ R𝑑 , where | · | denotes the Euclidean norm. It is easy to find a constant 𝛼 > 0 such that ℎ 𝑝 is 𝛼-excessive relative to (𝑃𝑡𝑏 )𝑡 ≥0 . Let 𝐶 𝑝 (R𝑑 ) denote the set of continuous functions 𝑓 ∈ 𝐶0 (R𝑑 ) satisfying | 𝑓 | ≤ const. · ℎ 𝑝 . Let 𝑀 𝑝 (R𝑑 ) be the space of 𝜎-finite measures 𝜇 on R𝑑 satisfying ⟨𝜇, ℎ 𝑝 ⟩ < ∞. We endow 𝑀 𝑝 (R𝑑 ) with the topology defined by the convention: 𝜇 𝑛 → 𝜇 in 𝑀 𝑝 (R𝑑 ) if and only if ⟨𝜇 𝑛 , 𝑓 ⟩ → ⟨𝜇, 𝑓 ⟩ for all 𝑓 ∈ 𝐶 𝑝 (R𝑑 ). In this chapter, we denote the Lebesgue measure on R𝑑 by 𝜆, which clearly belongs 𝜂 to 𝑀 𝑝 (R𝑑 ). Given 𝜂 ∈ 𝑀 𝑝 (R𝑑 ) we define the transition semigroup (𝑄 𝑡 )𝑡 ≥0 on 𝑀 𝑝 (R𝑑 ) by   ∫ ∫ 𝑡 − ⟨𝜈, 𝑓 ⟩ 𝜂 𝑄 𝑡 (𝜇, d𝜈) = exp − ⟨𝜇, 𝑉𝑡 𝑓 ⟩ − e ⟨𝜂, 𝑉𝑠 𝑓 ⟩d𝑠 . (14.3) 𝑀 𝑝 (R𝑑 )

0 𝜂

By Theorem 9.39 there is a càdlàg realization 𝑌 = (𝑊, 𝒢, 𝒢𝑡 , 𝑌𝑡 , Q 𝜇 ) of the immi𝜂 gration superprocess in 𝑀 𝑝 (R𝑑 ) with transition semigroup (𝑄 𝑡 )𝑡 ≥0 . The results of Propositions 9.15 and 9.20 extend immediately to the present case. Then for 𝑡 ≥ 0 and 𝑓 ∈ 𝐶 𝑝 (R𝑑 ) we have ∫

𝜂

𝑡

⟨𝜂, 𝑃𝑠𝑏 𝑓 ⟩d𝑠,

Q 𝜇 [⟨𝑌𝑡 , 𝑓 ⟩] = ⟨𝜇, 𝑃𝑡𝑏 𝑓 ⟩ +

(14.4)

0

and 𝜂 Q 𝜇 [⟨𝑌𝑡 ,

2

𝑓⟩ ] =



∫ ⟨𝜇, 𝑃𝑡𝑏

2

𝑡

⟨𝜂, 𝑃𝑠𝑏

𝑓⟩ +

𝑓 ⟩d𝑠

0

∫ + ∫0 +

𝑡 𝑏 ⟨𝜇, 𝑃𝑡−𝑠 [𝜙 ′′ (·, 0) (𝑃𝑠𝑏 𝑓 ) 2 ]⟩d𝑠 ∫ 𝑢 𝑡 𝑏 [𝜙 ′′ (·, 0) (𝑃𝑠𝑏 𝑓 ) 2 ]⟩d𝑠, d𝑢 ⟨𝜂, 𝑃𝑢−𝑠

0

(14.5)

0

where 𝜙 ′′ (𝑥, 0) is defined by (2.69). Proposition 14.1 Suppose that 𝜂(d𝑥) is absolutely continuous with respect to the 𝜂 Lebesgue measure 𝜆(d𝑥) with bounded density 𝑥 ↦→ 𝜂 ′ (𝑥). Then (𝑄 𝑡 )𝑡 ≥0 has a stationary distribution 𝐹 𝜂 defined by  ∫ ∞  ∫ (14.6) e− ⟨𝜈, 𝑓 ⟩ 𝐹 𝜂 (d𝜈) = exp − ⟨𝜂, 𝑉𝑠 𝑓 ⟩d𝑠 , 𝑀 𝑝 (R𝑑 )

0

where 𝑓 ∈ 𝐶 𝑝 (R𝑑 ) + . Moreover, we have 𝑄 𝑡 (0, ·) → 𝐹 𝜂 by weak convergence as 𝑡 → ∞. 𝜂

14.2 Stochastic Processes in Nuclear Spaces

393

Proof Clearly, the mapping 𝜈(d𝑥) ↦→ ℎ 𝑝 (𝑥)𝜈(d𝑥) induces a homeomorphism be𝜂 𝜂 tween 𝑀 𝑝 (R𝑑 ) and 𝑀 (R𝑑 ). Let 𝐺 𝑡 (d𝜈) denote the image of 𝑄 𝑡 (0, d𝜈) under the above mapping. For any 𝑓 ∈ 𝐵(R𝑑 ) + we have 𝑉𝑡 ( 𝑓 ℎ 𝑝 ) (𝑥) ≤ 𝑃𝑡𝑏 ( 𝑓 ℎ 𝑝 ) (𝑥) ≤ ∥ 𝑓 ∥𝑃𝑡𝑏 ℎ 𝑝 (𝑥),

𝑡 ≥ 0, 𝑥 ∈ R𝑑 ,

where ∥ · ∥ denotes the supremum norm. Since 𝜆(d𝑥) is an invariant measure for the Brownian motion, we have ∫ ∞ ∫ ∞ ⟨𝜂, 𝑃𝑠𝑏 ℎ 𝑝 ⟩d𝑠 ⟨𝜂, 𝑉𝑠 ( 𝑓 ℎ 𝑝 )⟩d𝑠 ≤ ∥ 𝑓 ∥ 0 0∫ ∞ ≤ ∥ 𝑓 𝜂′ ∥ e−𝑏𝑠 ⟨𝜆, ℎ 𝑝 ⟩d𝑠 < ∞. 0

It follows that ∫ 𝑡→∞

∫

e− ⟨𝜈, 𝑓 ⟩ 𝐺 𝑡 (d𝜈) = lim 𝜂

lim

𝑡→∞

𝑀 (R𝑑 )



e− ⟨𝜈, 𝑓 ℎ 𝑝 ⟩ 𝑄 𝑡 (0, d𝜈)  ⟨𝜂, 𝑉𝑠 ( 𝑓 ℎ 𝑝 )⟩d𝑠 , 𝜂

𝑀 𝑝 (R𝑑 ) ∫ ∞

= exp −

(14.7)

0

and the right-hand side is continuous in 𝑓 ∈ 𝐵(R𝑑 ) + with respect to bounded pointwise convergence. By Theorem 1.21, it is the Laplace functional of a probability 𝜂 measure 𝐺 𝜂 on 𝑀 (R𝑑 ) and 𝐺 𝑡 → 𝐺 𝜂 weakly as 𝑡 → ∞. Then (14.6) defines a 𝜂 𝜂 probability measure 𝐹 on 𝑀 𝑝 (R𝑑 ) and 𝑄 𝑡 (0, ·) → 𝐹 𝜂 weakly as 𝑡 → ∞. It is 𝜂 𝜂 □ easily seen that 𝐹 is a stationary distribution of (𝑄 𝑡 )𝑡 ≥0 . Under the condition of Proposition 14.1, the measure potential 𝜁 := 𝜂𝑈 is the mean of 𝐹 𝜂 . In fact, from (14.6) we have ∫ ∞ ∫ ⟨𝜂, 𝑃𝑠𝑏 𝑓 ⟩d𝑠 = ⟨𝜁, 𝑓 ⟩. ⟨𝜈, 𝑓 ⟩𝐹 𝜂 (d𝜈) = 𝑀 𝑝 (R𝑑 )

0

Observe also that ∫ ⟨𝜁, 𝑓 ⟩ =

⟨𝜁, 𝑃𝑡𝑏

𝑡

⟨𝜂, 𝑃𝑠𝑏 𝑓 ⟩d𝑠,

𝑓⟩ +

𝑡 ≥ 0.

(14.8)

0

14.2 Stochastic Processes in Nuclear Spaces Suppose that 𝐸 is an infinite-dimensional real linear space and ∥ · ∥ 0 ≤ ∥ · ∥ 1 ≤ ∥ · ∥ 2 ≤ · · · is a sequence of Hilbertian norms on 𝐸. Let 𝐸 𝑛 be the completion of 𝐸 relative to ∥ · ∥ 𝑛 and let ⟨·, ·⟩𝑛 denote the inner product in 𝐸 𝑛 . Then we have 𝐸 0 ⊃ 𝐸 1 ⊃ 𝐸 2 ⊃ · · · . The sequence of norms ∥ · ∥ 0 ≤ ∥ · ∥ 1 ≤ ∥ · ∥ 2 ≤ · · · induces a Ñ topology on the set 𝐸 ∞ := ∞ 𝑛=0 𝐸 𝑛 , which is compatible with the metric 𝜌 defined by

394

14 Small-Branching Fluctuation Limits

𝜌(𝑥, 𝑦) =

∞ ∑︁ 𝑘=0

∥𝑦 − 𝑥∥ 𝑘 , 2 𝑘 (1 + ∥𝑦 − 𝑥∥ 𝑘 )

𝑥, 𝑦 ∈ 𝐸 ∞ .

(14.9)

Proposition 14.2 The metric space (𝐸, 𝜌) is complete if and only if 𝐸 = 𝐸 ∞ . Proof Suppose that (𝐸, 𝜌) is complete. If 𝑥 ∈ 𝐸 ∞ , then for each 𝑛 ≥ 0 there is an 𝑥 𝑛 ∈ 𝐸 such that ∥𝑥 𝑛 − 𝑥∥ 𝑛 < 1/2𝑛+1 implying ∥𝑥 𝑛 − 𝑥∥ 𝑘 < 1/2𝑛+1 for 0 ≤ 𝑘 ≤ 𝑛. It follows that 𝜌(𝑥 𝑛 , 𝑥) ≤

∞ 𝑛 ∑︁ ∑︁ 1 2 1 1 1 ∥𝑥 − 𝑥∥ + < 𝑛+1 + 𝑛 = 𝑛−1 . 𝑛 𝑘 𝑘 𝑘 2 2 2 2 2 𝑘=𝑛+1 𝑘=0

Then we have 𝑥 ∈ 𝐸, proving 𝐸 = 𝐸 ∞ . For the converse, suppose that 𝐸 = 𝐸 ∞ . If {𝑥 𝑘 } is a Cauchy sequence in (𝐸, 𝜌), it is a Cauchy sequence relative to each norm ∥ · ∥ 𝑛 . Then there is a 𝑦 𝑛 ∈ 𝐸 𝑛 such that ∥𝑥 𝑘 − 𝑦 𝑛 ∥ 𝑛 → 0 as 𝑘 → ∞. By the relations ∥ · ∥ 0 ≤ ∥ · ∥ 1 ≤ ∥ · ∥ 2 ≤ · · · , we must have 𝑦 𝑛 = 𝑦 0 for every 𝑛 ≥ 0. Then ∥𝑥 𝑘 − 𝑦 0 ∥ 𝑛 → 0 as 𝑘 → ∞ for every 𝑛 ≥ 0. From (14.9) it follows that 𝜌(𝑥 𝑘 , 𝑦 0 ) → 0 as 𝑘 → ∞. Thus (𝐸, 𝜌) is complete. □ Proposition 14.3 Let 𝑓 be a linear map of 𝐸 into a normed linear space (𝐹, ||| · |||). Then 𝑓 is continuous relative to the metric defined by (14.9) if and only if it is continuous relative to one of the norms ∥ · ∥ 𝑛 . Proof If 𝑓 is continuous relative to one of the norms ∥ · ∥ 𝑛 , it is clearly continuous relative to the metric 𝜌 defined by (14.9). Conversely, suppose that 𝑓 is a continuous linear map of (𝐸, 𝜌) into (𝐹, ||| · |||). Then there is a neighborhood 𝐺 of zero such that ||| 𝑓 (𝑦)||| < 1 for all 𝑦 ∈ 𝐺. Consequently, there exists 𝑛 ≥ 0 and 𝛿 > 0 such that {𝑥 ∈ 𝐸 : ∥𝑥∥ 𝑛 < 𝛿} ⊂ 𝐺. Therefore ∥𝑥∥ 𝑛 < 𝛿 implies ||| 𝑓 (𝑥)||| < 1, and so ∥𝑥∥ 𝑛 < 𝛿𝜀 implies ||| 𝑓 (𝑥)||| < 𝜀 for every 𝜀 > 0. This gives the continuity of 𝑓 relative to ∥ · ∥ 𝑛 . □ Ð By Proposition 14.3 the space 𝐸 has dual 𝐸 ′ := ∞ 𝑛=0 𝐸 −𝑛 , where 𝐸 −𝑛 denotes the dual space of 𝐸 𝑛 . Let ⟨·, ·⟩ denote the duality between 𝐸 and 𝐸 ′. A subset 𝐵 of 𝐸 is said to be bounded if it is bounded in each norm ∥ · ∥ 𝑛 , that is, sup 𝑥 ∈𝐵 ∥𝑥∥ 𝑛 < ∞ for each 𝑛 ≥ 0. For any bounded set 𝐵 ⊂ 𝐸 define the semi-norm 𝑝 𝐵 on 𝐸 ′ by 𝑝 𝐵 ( 𝑓 ) = sup{| 𝑓 (𝑥)| : 𝑥 ∈ 𝐵},

𝑓 ∈ 𝐸 ′.

(14.10)

We endow 𝐸 ′ with the topology generated by the collection of semi-norms {𝑝 𝐵 : 𝐵 ⊂ 𝐸 is bounded}, which is called the strong topology.

14.2 Stochastic Processes in Nuclear Spaces

395

For every 𝑛 ≥ 0 let {𝑒 1𝑛 , 𝑒 2𝑛 , . . .} be an orthonormal basis of 𝐸 𝑛 and let ∥ · ∥ −𝑛 be the norm of 𝐸 −𝑛 defined by ∥ 𝑓 ∥ 2−𝑛 =

∞ ∑︁

⟨ 𝑓 , 𝑒 𝑛𝑘 ⟩ 2 ,

𝑓 ∈ 𝐸 −𝑛 .

𝑘=1

We identify 𝐸 −0 with 𝐸 0 , but not 𝐸 −𝑛 with 𝐸 𝑛 for 𝑛 ≥ 1. We call 𝐸 or (𝐸, 𝜌) a countably Hilbert nuclear space or simply a nuclear space if the following conditions are satisfied: (1) 𝐸 is separable with respect to ∥ · ∥ 𝑛 for every 𝑛 ≥ 0; (2) for every 𝑚 ≥ 0 there exists 𝑛 > 𝑚 and an orthonormal basis {𝑒 1𝑛 , 𝑒 2𝑛 , . . .} of 𝐸 𝑛 such that ∞ ∑︁

∥𝑒 𝑛𝑘 ∥ 2𝑚 < ∞;

(14.11)

𝑘=1

(3) the metric space (𝐸, 𝜌) is complete. It is well known that the above property (2) is equivalent to the embedding operator 𝜋 𝑛,𝑚 of 𝐸 𝑛 into 𝐸 𝑚 being Hilbert–Schmidt; see, e.g., Kallianpur and Xiong (1995, p. 18). For any orthonormal basis { 𝑓1𝑚 , 𝑓2𝑚 , . . .} of 𝐸 𝑚 we have ∞ ∑︁

∥𝑒 𝑛𝑘 ∥ 2𝑚 =

∞ ∑︁ ∞ ∑︁

2 ⟨𝜋 𝑛,𝑚 𝑒 𝑛𝑘 , 𝑓𝑖𝑚 ⟩𝑚

𝑘=1 𝑖=1

𝑘=1

=

∞ ∞ ∑︁ ∑︁

⟨𝑒 𝑛𝑘 , 𝜋−𝑚,−𝑛 𝑓𝑖𝑚 ⟩𝑛2

𝑖=1 𝑘=1

=

∞ ∑︁

2 . ∥ 𝑓𝑖𝑚 ∥ −𝑛

𝑖=1

Then the value on the left-hand side of (14.11) does not depend on the choice of the orthonormal basis {𝑒 1𝑛 , 𝑒 2𝑛 , . . .}. If 𝐸 is a nuclear space, we have 𝐸′ =

∞ Ø 𝑛=0

𝐸 −𝑛 ⊃ · · · ⊃ 𝐸 −2 ⊃ 𝐸 −1 ⊃ 𝐸 0 ⊃ 𝐸 1 ⊃ 𝐸 2 ⊃ · · · ⊃

∞ Ù

𝐸𝑛 = 𝐸 .

𝑛=0

The following two theorems were established in Mitoma (1983); see also Walsh (1986, pp. 361–365). Theorem 14.4 Let 𝐸 be a nuclear space with strong dual 𝐸 ′. Let {(𝑌𝑘 (𝑡))𝑡 ≥0 : 𝑘 ≥ 1} be a sequence of processes with sample paths in the space 𝐷 ( [0, ∞), 𝐸 ′). If for each 𝑥 ∈ 𝐸 the sequence of real processes {(⟨𝑌𝑘 (𝑡), 𝑥⟩)𝑡 ≥0 : 𝑘 ≥ 1} is tight in 𝐷 ( [0, ∞), R), then {(𝑌𝑘 (𝑡))𝑡 ≥0 : 𝑘 ≥ 1} is tight in 𝐷 ( [0, ∞), 𝐸 ′).

396

14 Small-Branching Fluctuation Limits

Theorem 14.5 Let {(𝑌𝑘 (𝑡))𝑡 ≥0 : 𝑘 ≥ 1} be a sequence of processes satisfying the conditions of Theorem 14.4. Suppose that 𝑛 > 𝑚 ≥ 0 and 𝐸 𝑛 ⊂ 𝐸 𝑚 is a Hilbert– Schmidt embedding. If for every 𝑡 ≥ 0, 𝜌 > 0 and 𝜀 > 0 there exists a 𝛿 > 0 such that 𝑥 ∈ 𝐸 and ∥𝑥∥ 𝑚 ≤ 𝛿 imply o n sup P sup |⟨𝑌𝑘 (𝑠), 𝑥⟩| ≥ 𝜌 ≤ 𝜀, 𝑘 ≥1

0≤𝑠 ≤𝑡

then each process (𝑌𝑘 (𝑡))𝑡 ≥0 has sample paths a.s. in 𝐷 ( [0, ∞), 𝐸 −𝑛 ) and the sequence {(𝑌𝑘 (𝑡))𝑡 ≥0 : 𝑘 ≥ 1} is tight in 𝐷 ( [0, ∞), 𝐸 −𝑛 ). We next consider a typical example of the nuclear space. Let N = {0, 1, 2, . . .}. Let 𝐶 ∞ (R𝑑 ) be the set of bounded infinitely differentiable functions on R𝑑 with bounded derivatives. Let 𝒮(R𝑑 ) ⊂ 𝐶 ∞ (R𝑑 ) denote the Schwartz space of rapidly decreasing functions. That is, a function 𝑓 ∈ 𝒮(R𝑑 ) is infinitely differentiable and for every 𝑘 ≥ 0 and every 𝛼 = (𝛼1 , . . . , 𝛼𝑑 ) ∈ N𝑑 we have lim |𝑥| 𝑘 |𝜕 𝛼 𝑓 (𝑥)| = 0,

(14.12)

| 𝑥 |→∞

where | · | denotes the Euclidean norm and 𝜕 𝛼 𝑓 (𝑥) =

𝜕 𝛼1 +···+𝛼𝑑 𝑓 (𝑥1 , . . . , 𝑥 𝑑 ). 𝜕𝑥1𝛼1 · · · 𝜕𝑥 𝑑𝛼𝑑

We first define an increasing sequence of norms {𝑝 0 , 𝑝 1 , 𝑝 2 , . . .} on 𝒮(R𝑑 ) by ∑︁ 𝑝𝑛 ( 𝑓 ) = sup (1 + |𝑥| 2 ) 𝑛/2 |𝜕 𝛼 𝑓 (𝑥)|, (14.13) 0≤ 𝛼¯ ≤𝑛 𝑥 ∈R

𝑑

where 𝛼¯ = 𝛼1 + · · · + 𝛼𝑑 . The norms {𝑝 0 , 𝑝 1 , 𝑝 2 , . . .} are not Hilbertian. We also define the Hilbertian norms {𝑞 0 , 𝑞 1 , 𝑞 2 , . . .} on 𝒮(R𝑑 ) by ∑︁ ∫ 𝑞𝑛 ( 𝑓 )2 = (1 + |𝑥| 2 ) 𝑛 |𝜕 𝛼 𝑓 (𝑥)| 2 d𝑥. (14.14) 0≤ 𝛼¯ ≤𝑛

R𝑑

Proposition 14.6 For every 𝑛 ≥ 0 there is a constant 𝑏(𝑛) > 0 such that 𝑞 𝑛 ( 𝑓 ) ≤ 𝑏(𝑛) 𝑝 𝑛+𝑑 ( 𝑓 ) and 𝑝 𝑛 ( 𝑓 ) ≤ 𝑏(𝑛)𝑞 𝑛+𝑑 ( 𝑓 ),

𝑓 ∈ 𝒮(R𝑑 ).

Proof For any 𝑛 ≥ 0 and 𝛼 ∈ N𝑑 satisfying 0 ≤ 𝛼¯ ≤ 𝑛 we have   (1 + |𝑥| 2 ) 𝑛 |𝜕 𝛼 𝑓 (𝑥)| 2 ≤ sup (1 + |𝑦| 2 ) 𝑛+𝑑 |𝜕 𝛼 𝑓 (𝑦)| 2 𝑦 ∈R𝑑

≤ 𝑝 𝑛+𝑑 ( 𝑓 ) 2

1 . (1 + |𝑥| 2 ) 𝑑

1 (1 + |𝑥| 2 ) 𝑑

14.2 Stochastic Processes in Nuclear Spaces

397

It follows that 𝑞 𝑛 ( 𝑓 ) ≤ 𝑏 1 (𝑛) 𝑝 𝑛+𝑑 ( 𝑓 ) for a constant 𝑏 1 (𝑛) > 0. On the other hand, for 𝑑 = 1 and 0 ≤ 𝑘 ≤ 𝑛 we have ∫ 𝑥   ′ (1 + 𝑥 2 ) 𝑛/2 | 𝑓 (𝑘) (𝑥)| = (1 + 𝑦 2 ) 𝑛/2 𝑓 (𝑘) (𝑦) d𝑦 ∫ −∞ (1 + 𝑦 2 ) 𝑛/2 𝑓 (𝑘+1) (𝑦) d𝑦 ≤ R ∫ 𝑛 (1 + 𝑦 2 ) 𝑛/2 𝑓 (𝑘) (𝑦) d𝑦 + 2 R ∫  1  ∫  21 d𝑦 2 2 𝑛+1 (𝑘+1) 2 ≤ (1 + 𝑦 ) 𝑓 (𝑦) d𝑦 2 R 1+𝑦 R ∫  12  𝑛 + (1 + 𝑦 2 ) 𝑛+1 𝑓 (𝑘) (𝑦) 2 d𝑦 2 R  √ 𝑛 ≤ 𝜋 1 + 𝑞 𝑛+1 ( 𝑓 ). 2 Then there is a constant 𝑏 2 (𝑛) > 0 such that 𝑝 𝑛 ( 𝑓 ) ≤ 𝑏 2 (𝑛)𝑞 𝑛+1 ( 𝑓 ). The inequality for higher dimensions follows similarly. □ By Proposition 14.6 the sequences of norms {𝑝 𝑛 } and {𝑞 𝑛 } induce the same topology on 𝒮(R𝑑 ). To show this is a nuclear space let us introduce another sequence of Hilbertian norms. The Hermite polynomials on R are given by 𝑔 𝑘 (𝑥) = (−1) 𝑘 e 𝑥

2

d 𝑘 −𝑥 2 e , d𝑥 𝑘

𝑘 = 0, 1, 2, . . . .

Based on those we define the Hermite functions 2 1 e−𝑥 /2 𝑔 𝑘 (𝑥), ℎ 𝑘 (𝑥) = √ √ 4 𝑘 𝜋 2 𝑘!

𝑘 = 0, 1, 2, . . . .

For 𝑥 ∈ R𝑑 and 𝛼 ∈ N𝑑 let ℎ 𝛼 (𝑥) = ℎ 𝛼1 (𝑥 1 ) · · · ℎ 𝛼𝑑 (𝑥 𝑑 ). Then ℎ 𝛼 ∈ 𝒮(R𝑑 ) and {ℎ 𝛼 : 𝛼 ∈ N𝑑 } is an orthonormal basis of 𝐿 2 (R𝑑 ). Let ⟨·, ·⟩ denote the inner product of 𝐿 2 (R𝑑 ). For 𝑓 ∈ 𝒮(R𝑑 ) we write ∑︁ 𝑓 (𝑥) = ⟨ 𝑓 , ℎ 𝛼 ⟩ℎ 𝛼 (𝑥), 𝑥 ∈ R𝑑 (14.15) 𝛼∈N𝑑

and define ∥ 𝑓 ∥ 2𝑛 =

∑︁

(2𝛼¯ + 𝑑) 2𝑛 ⟨ 𝑓 , ℎ 𝛼 ⟩ 2

𝛼∈N𝑑

for 𝑛 = 0, ±1, ±2, . . .. For any 𝑔, 𝑓 ∈ 𝒮(R𝑑 ) we have ∑︁ ⟨𝑔, 𝑓 ⟩ = ⟨𝑔, ℎ 𝛼 ⟩⟨ 𝑓 , ℎ 𝛼 ⟩, 𝛼∈N𝑑

(14.16)

398

14 Small-Branching Fluctuation Limits

and ⟨𝑔, 𝑓 ⟩ ≤ ∥𝑔∥ −𝑛 ∥ 𝑓 ∥ 𝑛 by Schwarz’s inequality. Let 𝐻𝑛 (R𝑑 ) be the completion of 𝒮(R𝑑 ) with respect to ∥ · ∥ 𝑛 , which we refer to as a weighted Sobolev space. By approximation we can extend ⟨·, ·⟩ to a bilinear form between 𝐻−𝑛 (R𝑑 ) and 𝐻𝑛 (R𝑑 ). Let ⟨·, ·⟩𝑛 denote the inner product of 𝐻𝑛 (R𝑑 ). For 𝑔, 𝑓 ∈ 𝐻𝑛 (R𝑑 ) we have ∑︁ ⟨𝑔, 𝑓 ⟩𝑛 = (2𝛼¯ + 𝑑) 2𝑛 ⟨𝑔, ℎ 𝛼 ⟩⟨ 𝑓 , ℎ 𝛼 ⟩ = ⟨𝜋 𝑛 𝑔, 𝑓 ⟩, 𝛼∈N𝑑

where 𝜋𝑛 𝑔 =

∑︁

(2𝛼¯ + 𝑑) 2𝑛 ⟨𝑔, ℎ 𝛼 ⟩ℎ 𝛼 ∈ 𝐻−𝑛 (R𝑑 ).

𝛼∈N𝑑

Then 𝐻−𝑛 (R𝑑 ) and 𝐻𝑛 (R𝑑 ) are dual spaces with the duality ⟨·, ·⟩.

Proposition 14.7 For every 𝑛 ≥ 0 there is a constant 𝑐(𝑛) > 0 such that 𝑞 𝑛 ( 𝑓 ) ≤ 𝑐(𝑛) ∥ 𝑓 ∥ 𝑛 and ∥ 𝑓 ∥ 𝑛 ≤ 𝑐(𝑛)𝑞 2𝑛 ( 𝑓 ),

𝑓 ∈ 𝒮(R𝑑 ).

Proof We only give the proof for the case 𝑑 = 1. The proof in the general case is based on similar ideas with more complicated calculations. It is easy to show that √︂ √︂ 𝑘 𝑘 +1 ′ ℎ 𝑘 (𝑥) = ℎ 𝑘−1 (𝑥) − ℎ 𝑘+1 (𝑥) (14.17) 2 2 and √︂ 𝑥ℎ 𝑘 (𝑥) =

𝑘 ℎ 𝑘−1 (𝑥) + 2

√︂

𝑘 +1 ℎ 𝑘+1 (𝑥) 2

(14.18)

with ℎ−1 (𝑥) = 0 by convention. For 𝑓 ∈ 𝒮(R) we have 𝑓 (𝑥) =

∞ ∑︁

⟨ 𝑓 , ℎ 𝑘 ⟩ℎ 𝑘 (𝑥),

𝑥 ∈ R.

(14.19)

𝑘=0

For 0 ≤ 𝑘, 𝑙 ≤ 𝑛 one can use (14.17) and (14.18) to see 𝑥 𝑙 𝑓 (𝑘) (𝑥) =

∞ ∑︁

𝑖+2𝑛 ∑︁

𝑎 𝑛 (𝑖, 𝑗, 𝑘, 𝑙) (𝑖 + 2𝑛) 𝑛 ⟨ 𝑓 , ℎ 𝑗 ⟩ℎ𝑖 (𝑥),

𝑖=0 𝑗=(𝑖−2𝑛) +

where (𝑖 − 2𝑛) + = 0 ∨ (𝑖 − 2𝑛) and {𝑎 𝑛 (𝑖, 𝑗, 𝑘, 𝑙) : 𝑖, 𝑗, 𝑘, 𝑙 ≥ 0} is a countable set bounded by some 𝑏 0 (𝑛) > 0. Then there are constants 𝑏 𝑖 (𝑛) > 0 such that ∫ R

(1 + 𝑥 2 ) 𝑛 𝑓 (𝑘) (𝑥) 2 d𝑥 ≤ 𝑏 1 (𝑛)

∞ ∑︁

𝑖+2𝑛 ∑︁

𝑖=0 𝑗=(𝑖−2𝑛) +

(𝑖 + 2𝑛) 2𝑛 ⟨ 𝑓 , ℎ 𝑗 ⟩ 2

14.2 Stochastic Processes in Nuclear Spaces

399

= 𝑏 1 (𝑛) ≤ 𝑏 2 (𝑛)

∞ ∑︁

𝑗+2𝑛 ∑︁

𝑗=0

𝑖=( 𝑗−2𝑛) +

∞ ∑︁

(𝑖 + 2𝑛) 2𝑛 ⟨ 𝑓 , ℎ 𝑗 ⟩ 2

( 𝑗 + 4𝑛) 2𝑛 ⟨ 𝑓 , ℎ 𝑗 ⟩ 2

𝑗=0

≤ 𝑏 3 (𝑛)

∞ ∑︁

(2 𝑗 + 1) 2𝑛 ⟨ 𝑓 , ℎ 𝑗 ⟩ 2 .

𝑗=0

This gives the first inequality. Using (14.17) and (14.18) one can show 𝑥 2 ℎ 𝑘 (𝑥) − ℎ ′′ 𝑘 (𝑥) = (2𝑘 + 1)ℎ 𝑘 (𝑥).

(14.20)

For 𝑓 ∈ 𝒮(R) given by (14.19) we have 𝑥 2 𝑓 (𝑥) − 𝑓 ′′ (𝑥) =

∞ ∑︁

(2𝑘 + 1)⟨ 𝑓 , ℎ 𝑘 ⟩ℎ 𝑘 (𝑥).

𝑘=0 ′′ (𝑥) Let us denote the above function by 𝑓1 (𝑥) and define 𝑓𝑛 (𝑥) = 𝑥 2 𝑓𝑛−1 (𝑥) − 𝑓𝑛−1 for 𝑛 ≥ 2 inductively. It is easy to see that

𝑓𝑛 (𝑥) =

∞ ∑︁

(2𝑘 + 1) 𝑛 ⟨ 𝑓 , ℎ 𝑘 ⟩ℎ 𝑘 (𝑥).

𝑘=0

Then there is a constant 𝑐(𝑛) > 0 such that ∫ ∥ 𝑓 ∥𝑛 =

2

 21

𝑓𝑛 (𝑥) d𝑥

≤ 𝑐(𝑛)𝑞 2𝑛 ( 𝑓 ).

R

This gives the second inequality.

□

Proposition 14.8 For 𝑛 > 𝑚 + 𝑑/2 the embedding 𝐻𝑛 (R𝑑 ) ⊂ 𝐻𝑚 (R𝑑 ) is Hilbert– Schmidt. Proof It is easy to see that {(2𝛼¯ + 𝑑) −𝑛 ℎ 𝛼 : 𝛼 ∈ N𝑑 } is an orthonormal basis of 𝒮(R𝑑 ) with respect to the norm ∥ · ∥ 𝑛 and ∑︁ ∑︁ ∥ (2𝛼¯ + 𝑑) −𝑛 ℎ 𝛼 ∥ 2𝑚 = (2𝛼¯ + 𝑑) 2(𝑚−𝑛) < ∞ 𝛼∈N𝑑

𝛼∈N𝑑

for any 𝑛 > 𝑚 + 𝑑/2.

□

Let 𝒮(R𝑑 ) be endowed with the metric 𝜌 defined by (14.9) and let 𝒮 ′ (R𝑑 ) denote its dual endowed with the strong topology. The elements of 𝒮 ′ (R𝑑 ) are called Schwartz distributions. Theorem 14.9 Both 𝒮(R𝑑 ) and 𝒮 ′ (R𝑑 ) are nuclear spaces.

400

14 Small-Branching Fluctuation Limits

Proof By Propositions 14.6 and 14.7 the two families of norms {𝑝 0 , 𝑝 1 , 𝑝 2 , . . .} and {∥ · ∥ 0 , ∥ · ∥ 1 , ∥ · ∥ 2 , . . .} are equivalent. Then it is easily seen that 𝒮(R𝑑 ) is complete under the metric 𝜌 defined by (14.9). Let 𝒢 be the collection of functions 𝑓 ∈ 𝒮(R𝑑 ) having the decomposition

𝑓 (𝑥) =

∑︁

𝑟 𝛼 ℎ 𝛼 (𝑥),

𝑥 ∈ R𝑑

𝛼¯ ≤𝑛

for all possible finite sets of rational coefficients {𝑟 𝛼 : 𝛼¯ ≤ 𝑛}. Clearly, 𝒢 is dense in 𝐻𝑛 for every 𝑛 ≥ 0. In other words, each 𝐻𝑛 is separable. By Proposition 14.8 the embedding 𝐻𝑛 (R𝑑 ) ⊂ 𝐻𝑚 (R𝑑 ) is Hilbert–Schmidt for 𝑛 > 𝑚 + 𝑑/2, so 𝒮(R𝑑 ) is a nuclear space. Since 𝒮(R𝑑 ) is clearly a Fréchet space, its strong dual 𝒮 ′ (R𝑑 ) is also a nuclear space; see, e.g., Treves (1967, p. 523). □

14.3 Fluctuation Limits in the Schwartz Space Let (𝜉, 𝜙, 𝜂) be the parameters given as in Section 14.1 and assume 𝜂(d𝑥) is absolutely continuous with respect to the Lebesgue measure 𝜆(d𝑥) with a bounded density. For any integer 𝑘 ≥ 1 let 𝜙 𝑘 (𝑥, 𝑧) = 𝜙(𝑥, 𝑧/𝑘) and suppose that {𝑌𝑘 (𝑡) : 𝑡 ≥ 0} is a càdlàg immigration superprocess in 𝑀 𝑝 (R𝑑 ) with parameters (𝜉, 𝜙 𝑘 , 𝜂). Then each {𝑌𝑘 (𝑡) : 𝑡 ≥ 0} has equilibrium mean 𝜁 := 𝜂𝑈. We are interested in the asymptotic fluctuating behavior of the immigration processes around this mean. For simplicity, we assume 𝑌𝑘 (0) = 𝜁, so (14.4) and (14.8) imply E⟨𝑌𝑘 (𝑡), 𝑓 ⟩ = ⟨𝜁, 𝑓 ⟩,

𝑡 ≥ 0, 𝑓 ∈ 𝐶 𝑝 (R𝑑 ).

Then we define the centered 𝒮 ′ (R𝑑 )-valued process {𝑍 𝑘 (𝑡) : 𝑡 ≥ 0} by 𝑍 𝑘 (𝑡) = 𝑘 [𝑌𝑘 (𝑡) − 𝜁],

𝑡 ≥ 0.

(14.21)

Since 𝑡 ↦→ ⟨𝑍 𝑘 (𝑡), 𝑓 ⟩ is càdlàg for every 𝑓 ∈ 𝒮(R𝑑 ), the process 𝑡 ↦→ 𝑍 𝑘 (𝑡) is càdlàg in the strong topology of 𝒮 ′ (R𝑑 ); see, e.g., Treves (1967, p. 358). Recall that 𝐶 2 (R) denotes the set of bounded continuous real functions on R with bounded continuous derivatives up to the second order. Lemma 14.10 For any 𝐺 ∈ 𝐶 2 (R) and 𝑓 ∈ 𝒮(R𝑑 ) we have ∫ 𝑡 𝐺 ′ (⟨𝑍 𝑘 (𝑠), 𝑓 ⟩)⟨𝑍 𝑘 (𝑠), 𝐴 𝑓 ⟩d𝑠 𝐺 (⟨𝑍 𝑘 (𝑡), 𝑓 ⟩) = 0∫ 𝑡 𝐺 ′′ (⟨𝑍 𝑘 (𝑠), 𝑓 ⟩)⟨𝑌𝑘 (𝑠), 𝑐 𝑓 2 ⟩d𝑠 + ∫0 𝑡 ∫ d𝑠 + 𝑙 (𝑥, 𝑍 𝑘 (𝑠))𝑌𝑘 (𝑠, d𝑥) + mart., 0

R𝑑

14.3 Fluctuation Limits in the Schwartz Space

401

where ∫

∞

𝑙 (𝑥, 𝜇) = 0

h


𝐺 ⟨𝜇, 𝑓 ⟩ + 𝑢 𝑓 (𝑥) − 𝐺 (⟨𝜇, 𝑓 ⟩) i − 𝐺 ′ (⟨𝜇, 𝑓 ⟩)𝑢 𝑓 (𝑥) 𝑚(𝑥, d𝑢).

Proof Let 𝐹𝑘 (𝜈) = 𝐺 (⟨𝜈, 𝑘 𝑓 ⟩ − ⟨𝜁, 𝑘 𝑓 ⟩) for 𝜈 ∈ 𝑀 𝑝 (R𝑑 ). Then 𝐺 (⟨𝑍 𝑘 (𝑡), 𝑓 ⟩) = 𝐹𝑘 (𝑌𝑘 (𝑡)). By Theorem 9.38, ∫ 𝑡 𝐺 (⟨𝑍 𝑘 (𝑡), 𝑓 ⟩) = 𝐺 ′ (⟨𝑌𝑘 (𝑠), 𝑘 𝑓 ⟩ − ⟨𝜁, 𝑘 𝑓 ⟩)⟨𝑌𝑘 (𝑠), 𝑘 𝐴 𝑓 ⟩d𝑠 0∫ 𝑡 + 𝐺 ′ (⟨𝑌𝑘 (𝑠), 𝑘 𝑓 ⟩ − ⟨𝜁, 𝑘 𝑓 ⟩)⟨𝜂, 𝑘 𝑓 ⟩d𝑠 0 ∫ 𝑡 𝐺 ′′ (⟨𝑌𝑘 (𝑠), 𝑘 𝑓 ⟩ − ⟨𝜁, 𝑘 𝑓 ⟩)⟨𝑌𝑘 (𝑠), 𝑐 𝑓 2 ⟩d𝑠 + 0 ∫ 𝑡 ∫ d𝑠 𝑙 (𝑥, 𝑍 𝑘 (𝑠))𝑌𝑘 (𝑠, d𝑥) + local mart. + 0

R𝑑

Using (14.5) one can see the local martingale above is actually a square-integrable martingale. Observe that ⟨𝜂, 𝑓 ⟩ = −⟨𝜂, 𝑈 𝐴 𝑓 ⟩ = −⟨𝜁, 𝐴 𝑓 ⟩.

(14.22)

Then we obtain the desired equality.

□

Lemma 14.11 Let 𝜙 ′′ (𝑥, 0) be given by (2.69). Then for any 𝑡 ≥ 0 and 𝑓 ∈ 𝐶 𝑝 (R𝑑 ) we have ∫ 𝑡 ⟨𝜁, 𝜙 ′′ (·, 0) (𝑃𝑠𝑏 𝑓 ) 2 ⟩d𝑠. E[⟨𝑍 𝑘 (𝑡), 𝑓 ⟩ 2 ] = (14.23) 0

Proof In view of (14.4) and (14.5) we have  ∫ 𝑡 2   E ⟨𝑍 𝑘 (𝑡), 𝑓 ⟩ 2 = E ⟨𝑌𝑘 (𝑡), 𝑘 𝑓 ⟩ − ⟨𝜁, 𝑘 𝑃𝑡𝑏 𝑓 ⟩ − ⟨𝜂, 𝑘 𝑃𝑠𝑏 𝑓 ⟩d𝑠 0 ∫ 𝑡 𝑏 ′′ 𝑏 2 = ⟨𝜁, 𝑃𝑡−𝑠 [𝜙 (·, 0) (𝑃𝑠 𝑓 ) ]⟩d𝑠 0∫ ∫ 𝑡−𝑠 𝑡 d𝑠 ⟨𝜂, 𝑃𝑢𝑏 [𝜙 ′′ (·, 0) (𝑃𝑠𝑏 𝑓 ) 2 ]⟩d𝑢 + 0 0 ∫ 𝑡 = ⟨𝜁, 𝜙 ′′ (·, 0) (𝑃𝑠𝑏 𝑓 ) 2 ⟩d𝑠, 0

where for the last equality we also used (14.8) to the function 𝜙 ′′ (·, 0) (𝑃𝑠𝑏 𝑓 ) 2 .

□

402

14 Small-Branching Fluctuation Limits

Lemma 14.12 Let 𝜁 ′ be a bounded density of 𝜁 (d𝑥) with respect to the Lebesgue measure. Then for any 𝑡 ≥ 0 and 𝑓 ∈ 𝒮(R𝑑 ) we have h i h i sup E sup ⟨𝑍 𝑘 (𝑠), 𝑓 ⟩ 2 ≤ 𝑡 ∥𝜁 ′ ∥ ∥𝜙 ′′ (·, 0) ∥ 8⟨𝜆, 𝑓 2 ⟩ + 𝑡 2 ⟨𝜆, ( 𝐴 𝑓 ) 2 ⟩ . 𝑘 ≥1

0≤𝑠 ≤𝑡

Proof By Theorem 9.38 we have ∫

𝑡



⟨𝑌𝑘 (𝑡), 𝑓 ⟩ = ⟨𝜁, 𝑓 ⟩ + 𝑀𝑘 (𝑡, 𝑓 ) +

 ⟨𝑌𝑘 (𝑠), 𝐴 𝑓 ⟩ + ⟨𝜂, 𝑓 ⟩ d𝑠,

(14.24)

0

where {𝑀𝑘 (𝑡, 𝑓 ) : 𝑡 ≥ 0} is a càdlàg martingale with quadratic variation process ∫ 𝑡 1 ⟨𝑀𝑘 ( 𝑓 )⟩𝑡 = 2 ⟨𝑌𝑘 (𝑠), 𝜙 ′′ (·, 0) 𝑓 2 ⟩d𝑠. 𝑘 0 From (14.22) and (14.24) we get ∫ ⟨𝑍 𝑘 (𝑡), 𝑓 ⟩ = 𝑘 𝑀𝑘 (𝑡, 𝑓 ) +

𝑡

⟨𝑍 𝑘 (𝑠), 𝐴 𝑓 ⟩d𝑠. 0

Then by (14.5) and (14.23), h i E sup ⟨𝑍 𝑘 (𝑠), 𝑓 ⟩ 2 0≤𝑠 ≤𝑡    ∫ 𝑡 2 2 2 ≤ 2𝑘 E sup |𝑀𝑘 (𝑠, 𝑓 )| + 2E |⟨𝑍 𝑘 (𝑠), 𝐴 𝑓 ⟩|d𝑠 0∫ ∫ 𝑡 0≤𝑠 ≤𝑡 𝑡   ≤ 8 E[⟨𝑌𝑘 (𝑠), 𝜙 ′′ (·, 0) 𝑓 2 ⟩]d𝑠 + 2𝑡 E ⟨𝑍 𝑘 (𝑠), 𝐴 𝑓 ⟩ 2 d𝑠 0 ∫ 0𝑡 ∫ 𝑠 ′′ 2 ′′ ≤ 8𝑡 ∥𝜙 (·, 0) ∥⟨𝜁, 𝑓 ⟩ + 2𝑡 ∥𝜙 (·, 0) ∥ d𝑠 ⟨𝜁, (𝑃𝑢𝑏 𝐴 𝑓 ) 2 ⟩d𝑢 0 0∫ ∫ 𝑠 𝑡 ≤ 8𝑡 ∥𝜁 ′ ∥ ∥𝜙 ′′ (·, 0) ∥⟨𝜆, 𝑓 2 ⟩ + 2𝑡 ∥𝜁 ′ ∥ ∥𝜙 ′′ (·, 0) ∥ d𝑠 ⟨𝜆, (𝑃𝑢𝑏 𝐴 𝑓 ) 2 ⟩d𝑢 0 0 h i ≤ 𝑡 ∥𝜁 ′ ∥ ∥𝜙 ′′ (·, 0) ∥ 8⟨𝜆, 𝑓 2 ⟩ + 𝑡 2 ⟨𝜆, ( 𝐴 𝑓 ) 2 ⟩ . This gives the desired estimate.

□

Lemma 14.13 The sequence {(𝑍 𝑘 (𝑡))𝑡 ≥0 : 𝑘 ≥ 1} is tight in 𝐷 ( [0, ∞), 𝒮 ′ (R𝑑 )).

Proof By Theorem 14.4 we only need to prove the sequence {⟨𝑍 𝑘 (𝑡), 𝑓 ⟩ : 𝑡 ≥ 0; 𝑘 ≥ 1} is tight in 𝐷 ( [0, ∞), R) for every 𝑓 ∈ 𝒮(R𝑑 ). By Lemma 14.12 and Chebyshev’s inequality we have h i sup P sup |⟨𝑍 𝑘 (𝑠), 𝑓 ⟩| ≥ 𝛼 → 0 𝑘 ≥1

0≤𝑠 ≤𝑡

14.3 Fluctuation Limits in the Schwartz Space

403

as 𝛼 → ∞. Then {⟨𝑍 𝑘 (𝑡), 𝑓 ⟩ : 𝑡 ≥ 0} satisfies the compact containment condition of Ethier and Kurtz (1986, p. 142). Let 𝐺 ∈ 𝐶 ∞ (R) and let 𝑙 (𝑥, 𝜇) be defined as in Lemma 14.10. By Taylor’s expansion we have ∫ ∞ |𝑙 (𝑥, 𝜇)| ≤ ∥𝐺 ′′ ∥ 𝑢 2 𝑚(𝑥, d𝑢)| 𝑓 (𝑥)| 2 . 0

Then it is easy to show ∫ 𝑡 ′ sup E 𝐺 (⟨𝑍 𝑘 (𝑠), 𝑓 ⟩)⟨𝑍 𝑘 (𝑠), 𝐴 𝑓 ⟩ + 𝐺 ′′ (⟨𝑍 𝑘 (𝑠), 𝑓 ⟩)⟨𝑌𝑘 (𝑠), 𝑐 𝑓 2 ⟩ 𝑘 ≥1 0 ∫  2 𝑙 (𝑥, 𝑍 𝑘 (𝑠))𝑌𝑘 (𝑠, d𝑥) d𝑠 < ∞. + R𝑑

By Lemma 14.10 and Ethier and Kurtz (1986, p. 145) we infer that {𝐺 (⟨𝑍 𝑘 (𝑡), 𝑓 ⟩) : 𝑡 ≥ 0; 𝑘 ≥ 1} is tight. The tightness of {⟨𝑍 𝑘 (𝑡), 𝑓 ⟩ : 𝑡 ≥ 0; 𝑘 ≥ 1} then follows by □ Ethier and Kurtz (1986, p. 142). Lemma 14.14 Let {𝑍0 (𝑡) : 𝑡 ≥ 0} be any limit point of {𝑍 𝑘 (𝑡) : 𝑡 ≥ 0; 𝑘 ≥ 1} in the sense of distributions on 𝐷 ( [0, ∞), 𝒮 ′ (R𝑑 )). Then for 𝐺 ∈ 𝐶 ∞ (R) and 𝑓 ∈ 𝒮(R𝑑 ) we have ∫ 𝑡h 𝐺 (⟨𝑍0 (𝑡), 𝑓 ⟩) = 𝐺 ′ (⟨𝑍0 (𝑠), 𝑓 ⟩)⟨𝑍0 (𝑠), 𝐴 𝑓 ⟩ + 𝐺 ′′ (⟨𝑍0 (𝑠), 𝑓 ⟩)⟨𝜁, 𝑐 𝑓 2 ⟩ 0∫ i 𝑙 (𝑥, 𝑍0 (𝑠))𝜁 (d𝑥) d𝑠 + mart. + R𝑑

Proof By passing to a subsequence and using the Skorokhod representation, we may assume {𝑍 𝑘 (𝑡) : 𝑡 ≥ 0} and {𝑍0 (𝑡) : 𝑡 ≥ 0} are defined on the same probability space and {𝑍 𝑘 (𝑡) : 𝑡 ≥ 0} converges a.s. to {𝑍0 (𝑡) : 𝑡 ≥ 0} in the topology of 𝐷 ( [0, ∞), 𝒮 ′ (R𝑑 )). Then {⟨𝑍 𝑘 (𝑡), 𝑓 ⟩ : 𝑡 ≥ 0} converges a.s. to {⟨𝑍0 (𝑡), 𝑓 ⟩ : 𝑡 ≥ 0} in the topology of 𝐷 ( [0, ∞), R). Consequently, we have a.s. ⟨𝑍 𝑘 (𝑡), 𝑓 ⟩ → ⟨𝑍0 (𝑡), 𝑓 ⟩ for a.e. 𝑡 ≥ 0; see, e.g., Ethier and Kurtz (1986, p. 118). Note also that {𝑌𝑘 (𝑡) : 𝑡 ≥ 0} converges a.s. to the deterministic constant process {𝑌0 (𝑡) = 𝜁 : 𝑡 ≥ 0} in the topology of 𝐷 ( [0, ∞), 𝒮 ′ (R𝑑 )). From Lemma 14.10 we have ∫ 𝑡 𝐺 ′ (⟨𝑍 𝑘 (𝑠), 𝑓 ⟩)⟨𝑍 𝑘 (𝑠), 𝐴 𝑓 ⟩d𝑠 𝐺 (⟨𝑍 𝑘 (𝑡), 𝑓 ⟩) = 0∫ 𝑡 + 𝐺 ′′ (⟨𝑍 𝑘 (𝑠), 𝑓 ⟩)⟨𝑌𝑘 (𝑠), 𝑐 𝑓 2 ⟩d𝑠 0 ∫ 𝑡 + ⟨𝑌𝑘 (𝑠), 𝑙 (·, 𝑍0 (𝑠)) + 𝑙 𝑘 (𝑠, ·)⟩d𝑠 + mart., (14.25) 0

where 𝑙 𝑘 (𝑠, 𝑥) = 𝑙 (𝑥, 𝑍 𝑘 (𝑠)) − 𝑙 (𝑥, 𝑍0 (𝑠)). By applying the mean-value theorem to the function 𝑧 ↦→ 𝐻 (𝑥, 𝑢, 𝑧) := 𝐺 (𝑧 + 𝑢 𝑓 (𝑥)) − 𝐺 (𝑧) − 𝐺 ′ (𝑧)𝑢 𝑓 (𝑥)

404

14 Small-Branching Fluctuation Limits

we get ∫

∞

𝐻 𝑧′ (𝑥, 𝑢, 𝜃 𝑠 )𝑚(𝑥, d𝑢),

𝑙 𝑘 (𝑠, 𝑥) = ⟨𝑍 𝑘 (𝑠) − 𝑍0 (𝑠), 𝑓 ⟩ 0

where ⟨𝑍 𝑘 (𝑠), 𝑓 ⟩ ∧ ⟨𝑍0 (𝑠), 𝑓 ⟩ ≤ 𝜃 𝑠 ≤ ⟨𝑍 𝑘 (𝑠), 𝑓 ⟩ ∨ ⟨𝑍0 (𝑠), 𝑓 ⟩. By Taylor’s expansion, |𝐻 𝑧′ (𝑥, 𝑢, 𝜃 𝑠 )| = |𝐺 ′ (𝜃 𝑠 + 𝑢 𝑓 (𝑥)) − 𝐺 ′ (𝜃 𝑠 ) − 𝐺 ′′ (𝜃 𝑠 )𝑢 𝑓 (𝑥)| 1 ≤ ∥𝐺 (3) ∥𝑢 2 𝑓 (𝑥) 2 . 2 It follows that |𝑙 𝑘 (𝑠, 𝑥)| ≤

1 (3) ∥𝐺 ∥ 𝑓 (𝑥) 2 |⟨𝑍 𝑘 (𝑠) − 𝑍0 (𝑠), 𝑓 ⟩| 2

∫

∞

𝑢 2 𝑚(𝑥, d𝑢).

0

Then we have ⟨|𝑙 𝑘 (𝑠, ·)|, 𝑌𝑘 (𝑠)⟩ ≤ 𝐶 |⟨𝑍 𝑘 (𝑠) − 𝑍0 (𝑠), 𝑓 ⟩|⟨𝑌𝑘 (𝑠), 𝑓 2 ⟩,

(14.26)

where 𝐶=

1 (3) ∥𝐺 ∥ sup 2 𝑥 ∈R𝑑

∫

∞

𝑢 2 𝑚(𝑥, d𝑢).

0

For 𝑛 ≥ 1 let ∫

n

𝑡

𝜏𝑛 = inf 𝑡 ≥ 0 : sup 𝑘 ≥1

o ⟨𝑍 𝑘 (𝑠) − 𝑍0 (𝑠), 𝑓 ⟩ 2 d𝑠 ≥ 𝑛 .

0

Then 𝜏𝑛 → ∞ as 𝑛 → ∞. By (14.26) and Schwarz’s inequality, i 2 ⟨|𝑙 𝑘 (𝑠, ·)|, 𝑌𝑘 (𝑠)⟩d𝑠 0 h ∫ 𝑡∧𝜏𝑛 i ≤ 𝐶 𝑘 (𝑡)E ⟨𝑍 𝑘 (𝑠) − 𝑍0 (𝑠), 𝑓 ⟩ 2 d𝑠 ,

 h∫ E

𝑡∧𝜏𝑛

0

where 𝐶 𝑘 (𝑡) = 𝐶

2

∫

𝑡

E[⟨𝑌𝑘 (𝑠), 𝑓 2 ⟩ 2 ]d𝑠.

0

By (14.5) it is easy to show sup 𝑘 ≥1 𝐶 𝑘 (𝑡) < ∞. Now (14.27) implies

(14.27)

14.3 Fluctuation Limits in the Schwartz Space

∫

𝑡∧𝜏𝑛

⟨|𝑙 𝑘 (𝑠, ·)|, 𝑌𝑘 (𝑠)⟩d𝑠 = 0.

lim E 𝑘→∞

405



0

From (14.5) and (14.23) it is easy to show that the sequences {⟨𝑍 𝑘 (𝑠), 𝐴 𝑓 ⟩}, {⟨𝑌𝑘 (𝑠), 𝑐 𝑓 2 ⟩}, {⟨𝑌𝑘 (𝑠), 𝑙 (·, 𝑍0 (𝑠))⟩} are all uniformly integrable on Ω × [0, 𝑡] relative to the product measure P(d𝜔)d𝑠. Then letting 𝑘 → ∞ in (14.25) we obtain ∫ 𝑡h 𝐺 (⟨𝑍0 (𝑡), 𝑓 ⟩) = 𝐺 ′ (⟨𝑍0 (𝑠), 𝑓 ⟩)⟨𝑍0 (𝑠), 𝐴 𝑓 ⟩ + 𝐺 ′′ (⟨𝑍0 (𝑠), 𝑓 ⟩)⟨𝜁, 𝑐 𝑓 2 ⟩ 0∫ i 𝑙 (𝑥, 𝑍0 (𝑠))𝜁 (d𝑥) d𝑠 + local mart. + R𝑑

Here the local martingale is clearly a square-integrable martingale.

□

Proposition 14.15 For every 𝜇 ∈ 𝒮 ′ (R𝑑 ) there is a process {𝑍 (𝑡) : 𝑡 ≥ 0} with sample paths in 𝐷 ( [0, ∞), 𝒮 ′ (R𝑑 )) such that for 𝐺 ∈ 𝐶 ∞ (R) and 𝑓 ∈ 𝒮(R𝑑 ) we have ∫ 𝑡 𝐺 (⟨𝑍 (𝑡), 𝑓 ⟩) = 𝐺 (⟨𝜇, 𝑓 ⟩) + 𝐺 ′ (⟨𝑍 (𝑠), 𝑓 ⟩)⟨𝑍 (𝑠), 𝐴 𝑓 ⟩d𝑠 0 ∫ 𝑡 ′′ 𝐺 (⟨𝑍 (𝑠), 𝑓 ⟩)⟨𝜁, 𝑐 𝑓 2 ⟩d𝑠 + ∫0 𝑡 ∫ d𝑠 + 𝑙 (𝑥, 𝑍 (𝑠))𝜁 (d𝑥) + mart. (14.28) 0

R𝑑

Proof Let {𝑍0 (𝑡) : 𝑡 ≥ 0} be as in Lemma 14.14 and let 𝑍 (𝑡) = 𝑃𝑡𝑏 𝜇 + 𝑍0 (𝑡). Then (14.28) clearly holds. □ Proposition 14.16 Suppose that {𝑍 (𝑡) : 𝑡 ≥ 0} is a process that has sample paths in 𝐷 ( [0, ∞), 𝒮 ′ (R𝑑 )) and solves the martingale problem given by (14.28). Then 𝜁 {𝑍 (𝑡) : 𝑡 ≥ 0} is a Markov process with transition semigroup (𝑄 𝑡 )𝑡 ≥0 defined by ∫ 𝜁 e𝑖 ⟨𝜈, 𝑓 ⟩ 𝑄 𝑡 (𝜇, d𝜈) 𝒮′ (R𝑑 )   ∫ 𝑡 ⟨𝜁, 𝜙(−𝑖𝑃𝑠𝑏 𝑓 )⟩d𝑠 , (14.29) = exp 𝑖⟨𝜇, 𝑃𝑡𝑏 𝑓 ⟩ + 0

where 𝑓 ∈ 𝒮(R𝑑 ). Proof Let (𝑡, 𝑧) ↦→ 𝐺 (𝑡, 𝑧) be a function on [0, ∞) ×R such that 𝑧 ↦→ 𝐺 (𝑡, 𝑧) belongs to 𝐶 ∞ (R) for every 𝑡 ≥ 0 and 𝑡 ↦→ 𝐺 (𝑡, 𝑧) is continuously differentiable for every 𝑧 ∈ R. Let (𝑡, 𝑥) ↦→ 𝑓𝑡 (𝑥) be a function on [0, ∞) × R𝑑 such that 𝑥 ↦→ 𝑓𝑡 (𝑥) belongs to 𝒮(R𝑑 ) for every 𝑡 ≥ 0 and 𝑡 ↦→ 𝑓𝑡 (𝑥) is continuously differentiable for every 𝑥 ∈ R𝑑 . Using Proposition 14.15 one can show as in the proof of Theorem 7.16 that

406

14 Small-Branching Fluctuation Limits

∫

𝑡

𝐺 (𝑡, ⟨𝑍 (𝑡), 𝑓𝑡 ⟩) = 𝐺 (0, ⟨𝜇, 𝑓0 ⟩) + 𝐺 ′𝑧 (𝑠, ⟨𝑍 (𝑠), 𝑓𝑠 ⟩)⟨𝑍 (𝑠), 𝐴 𝑓𝑠 ⟩d𝑠 0 ∫ 𝑡 ′′ (𝑠, ⟨𝑍 (𝑠), 𝑓𝑠 ⟩)⟨𝜁, 𝑐 𝑓𝑠2 ⟩d𝑠 + 𝐺 𝑧𝑧 0 ∫ 𝑡h i 𝐺 ′𝑧 (𝑠, ⟨𝑍 (𝑠), 𝑓𝑠 ⟩)⟨𝑍 (𝑠), 𝑓𝑠′⟩ + 𝐺 𝑠′ (𝑠, ⟨𝑍 (𝑠), 𝑓𝑠 ⟩) d𝑠 + ∫0 𝑡 ∫ 𝑙 𝑠 (𝑥, 𝑍 (𝑠))𝜁 (d𝑥) + mart., d𝑠 + 0

𝐸

where 𝑓𝑠′ (𝑥) = (d/d𝑠) 𝑓𝑠 (𝑥) and 𝑙 𝑠 (𝑥, 𝜇) is defined as in Lemma 14.10 with 𝑓 and 𝐺 replaced by 𝑓𝑠 and 𝐺 (𝑠, ·), respectively. Clearly, the equality above remains valid 𝑏 𝑓 and when (𝑡, 𝑧) ↦→ 𝐺 (𝑡, 𝑧) is a complex function. By applying this to 𝑓𝑡 = 𝑃𝑇−𝑡  ∫ 𝐺 (𝑡, 𝑧) = exp 𝑖𝑧 +



𝑇−𝑡

⟨𝜁, 𝜙(−𝑖𝑃𝑠𝑏 𝑓 )⟩d𝑠

0

one sees that  ∫ 𝑏 𝑡 ↦→ exp 𝑖⟨𝑍 (𝑡), 𝑃𝑇−𝑡 𝑓 ⟩ +



𝑇−𝑡

⟨𝜁, 𝜙(−𝑖𝑃𝑠𝑏

𝑓 )⟩d𝑠

0

is a complex martingale on [0, 𝑇]. Then {𝑍 (𝑡) : 𝑡 ≥ 0} is a Markov process with 𝜁 □ transition semigroup (𝑄 𝑡 )𝑡 ≥0 defined by (14.29). By Propositions 14.15 and 14.16 there is a unique solution to the martingale problem given by (14.28) and the solution is a Markov process with transition 𝜁 semigroup (𝑄 𝑡 )𝑡 ≥0 . This process gives a description of the asymptotic fluctuations of the immigration superprocesses as the branching mechanisms are small. More precisely, we have the following: Theorem 14.17 As 𝑘 → ∞, the process {𝑍 𝑘 (𝑡) : 𝑡 ≥ 0} converges weakly in 𝐷 ( [0, ∞), 𝒮 ′ (R𝑑 )) to the unique solution {𝑍 (𝑡) : 𝑡 ≥ 0} of the martingale problem given by (14.28) with 𝑍 (0) = 0. Proof By Lemma 14.13 the sequence {𝑍 𝑘 (𝑡) : 𝑡 ≥ 0; 𝑘 ≥ 1} is tight in the space 𝐷 ( [0, ∞), 𝒮 ′ (R𝑑 )). Then we get the result by Lemma 14.14 and Proposition 14.16.□ Example 14.1 Suppose that 𝜂 ∈ 𝑀 (R𝑑 ) is a finite measure with 𝜂(R𝑑 ) = 𝑏 and 𝜙(𝑧) is a local branching mechanism given by (14.1) with (𝑐, 𝑚) independent of 𝑥 ∈ R𝑑 . Let 𝑧 𝑘 (𝑡) = 𝑘 [⟨𝑌𝑘 (𝑡), 1⟩ − 1] for 𝑡 ≥ 0. A modification of the arguments in this section shows {𝑧 𝑘 (𝑡) : 𝑡 ≥ 0} converges weakly in 𝐷 ( [0, ∞), R) to a Markov process {𝑧(𝑡) : 𝑡 ≥ 0} with transition semigroup (𝑄 𝑡𝑏 )𝑡 ≥0 defined by   ∫ 𝑡 ∫ 𝑖𝑢𝑦 𝑏 −𝑏𝑡 −𝑏𝑡 e 𝑄 𝑡 (𝑥, d𝑦) = exp e 𝑖𝑢𝑥 + 𝑢 ∈ R. 𝜙(−e 𝑖𝑢)d𝑠 , R

0

This is a one-dimensional OU-type process; see, e.g., Sato (1999, pp. 106–108).

14.4 Fluctuation Limits in Sobolev Spaces

407

14.4 Fluctuation Limits in Sobolev Spaces In this section, we show the fluctuation limit theorem actually holds in a suitable weighted Sobolev space. Recall that the weighted Sobolev space 𝐻𝑛 (R𝑑 ) with index 𝑛 ≥ 0 is the completion of 𝒮(R𝑑 ) with respect to the norm ∥·∥ 𝑛 defined in (14.16) and 𝐻−𝑛 (R𝑑 ) denotes the dual space of 𝐻𝑛 (R𝑑 ) with duality ⟨·, ·⟩. Let {𝑍 𝑘 (𝑡) : 𝑡 ≥ 0} and {𝑍 (𝑡) : 𝑡 ≥ 0} be defined as in Section 14.3. Theorem 14.18 For any integer 𝑛 > 2 + 𝑑/2 the processes {𝑍 𝑘 (𝑡) : 𝑡 ≥ 0} and {𝑍 (𝑡) : 𝑡 ≥ 0} live in the weighted Sobolev space 𝐻−𝑛 (R𝑑 ) and {𝑍 𝑘 (𝑡) : 𝑡 ≥ 0} converges as 𝑘 → ∞ to {𝑍 (𝑡) : 𝑡 ≥ 0} weakly in 𝐷 ( [0, ∞), 𝐻−𝑛 (R𝑑 )). Proof For any 𝑓 ∈ 𝒮(R𝑑 ) we have ⟨𝜆, 𝑓 2 ⟩ = ∥ 𝑓 ∥ 20 ≤ ∥ 𝑓 ∥ 22 . By Proposition 14.7 there is a constant 𝐶 > 0 such that ⟨𝜆, ( 𝐴 𝑓 ) 2 ⟩ ≤ ⟨𝜆, (Δ 𝑓 ) 2 ⟩ + 2𝑏 2 ⟨𝜆, 𝑓 2 ⟩ ≤ (1 + 2𝑏 2 )𝑞 2 ( 𝑓 ) 2 ≤ 𝐶 ∥ 𝑓 ∥ 22 . Then Lemma 14.12 implies h i sup E sup ⟨𝑍 𝑘 (𝑠), 𝑓 ⟩ 2 ≤ 𝐶 (𝑡) ∥ 𝑓 ∥ 22 𝑘 ≥1

0≤𝑠 ≤𝑡

for a locally bounded function 𝑡 ↦→ 𝐶 (𝑡). Thus for 𝑡 ≥ 0 and 𝜌 > 0 we have o n sup P sup |⟨𝑍 𝑘 (𝑠), 𝑓 ⟩| ≥ 𝜌 ≤ 𝐶 (𝑡) ∥ 𝑓 ∥ 22 /𝜌 2 . 𝑘 ≥1

0≤𝑠 ≤𝑡

By Theorem 14.5 and Proposition 14.8 the sequence {𝑍 𝑘 (𝑡) : 𝑡 ≥ 0; 𝑘 ≥ 1} is tight in 𝐷 ( [0, ∞), 𝐻−𝑛 (R𝑑 )). Then the result follows by Theorem 14.17. □ Example 14.2 Let us consider the case 𝑑 = 1. Suppose that {𝑊 (d𝑠, d𝑥)} is a time– space Gaussian white noise on (0, ∞) × R with covariance measure 2𝑐(𝑥)d𝑠𝜁 (d𝑥) and {𝑁 (d𝑠, d𝑥, d𝑢)} is a Poisson random measure on (0, ∞) × R × (0, ∞) with intensity d𝑠𝜁 (d𝑥)𝑚(𝑥, d𝑢). We assume {𝑁 (d𝑠, d𝑥, d𝑢)} and {𝑊 (d𝑠, d𝑥)} are defined on some filtered probability space (Ω, ℱ, ℱ𝑡 , P) and are independent of each other. Let 𝑝 𝑡 (𝑥, 𝑦) denote the transition density of the killed Brownian motion generated by 𝐴 = Δ/2−𝑏. Given 𝑋0 ∈ 𝐻0 (R) we can define an 𝐻0 (R)-valued process {𝑋𝑡 : 𝑡 ≥ 0} by ∫ 𝑡∫ 𝑋𝑡 (𝑦) = 𝑃𝑡𝑏 𝑋0 (𝑦) + 𝑝 𝑡−𝑠 (𝑥, 𝑦)𝑊 (d𝑠, d𝑥) ∫ 𝑡 ∫ ∫ ∞0 R + 𝑢 𝑝 𝑡−𝑠 (𝑥, 𝑦) 𝑁˜ (d𝑠, d𝑥, d𝑢), (14.30) 0

R

0

where 𝑁˜ (d𝑠, d𝑥, d𝑢) = 𝑁 (d𝑠, d𝑥, d𝑢) − d𝑠𝜁 (d𝑥)𝑚(𝑥, d𝑢). In fact, it is easily seen that ∫ ∫ 𝑡 ∫ 2 2 𝑝 𝑡−𝑠 (𝑥, 𝑦) 2 𝜙 ′′ (𝑥, 0)𝜁 (d𝑥) < ∞. E[∥ 𝑋𝑡 ∥ 0 ] ≤ 3∥ 𝑋0 ∥ 0 + 3 d𝑦 d𝑠 R

0

R

408

14 Small-Branching Fluctuation Limits

For 𝑓 ∈ 𝒮(R) we have  E exp 𝑖⟨𝑋𝑡 , 𝑓 ⟩   ∫ 𝑡∫ 𝑏 𝑏 = E exp 𝑖⟨𝑋0 , 𝑃𝑡 𝑓 ⟩ + 𝑖𝑃𝑡−𝑠 𝑓 (𝑥)𝑊 (d𝑠, d𝑥) 0 R  ∫ 𝑡∫ ∫ ∞ 𝑏 𝑓 (𝑥) 𝑁˜ (d𝑠, d𝑥, d𝑢) + 𝑖𝑢𝑃𝑡−𝑠 0 R 0 ∫ 𝑡 ∫ 𝑏 = exp 𝑖⟨𝑋0 , 𝑃𝑡𝑏 𝑓 ⟩ − d𝑠 [𝑃𝑡−𝑠 𝑓 (𝑥)] 2 𝑐(𝑥)𝜁 (d𝑥)  ∫ 0∞  R ∫ 𝑡 ∫  𝑏 𝑓 ( 𝑥) 𝑖𝑢𝑃𝑡−𝑠 𝑏 + − 1 − 𝑖𝑢𝑃𝑡−𝑠 𝑓 (𝑥) 𝑚(𝑥, d𝑢) d𝑠 𝜁 (d𝑥) e 0 R  ∫0 𝑡 𝑏 = exp 𝑖⟨𝑋0 , 𝑃𝑡𝑏 𝑓 ⟩ + 𝑓 )⟩d𝑠 . ⟨𝜁, 𝜙(−𝑖𝑃𝑡−𝑠 0

A comparison of this equality with (14.29) shows that for any 𝜇 ∈ 𝐻0 (R) the 𝜁 probability measure 𝑄 𝑡 (𝜇, ·) is actually supported by 𝐻0 (R) and   ∫ ∫ 𝑡 𝜁 ⟨𝜁, 𝜙(−𝑖𝑃𝑠𝑏 𝑓 )⟩d𝑠 . (14.31) e𝑖 ⟨𝜈, 𝑓 ⟩ 𝑄 𝑡 (𝜇, d𝜈) = exp 𝑖⟨𝜇, 𝑃𝑡𝑏 𝑓 ⟩ + 𝐻0 (R)

0

By considering an approximating sequence from 𝒮(R) we see that the above formula holds for every 𝑓 ∈ 𝐻0 (R). This gives a special case of the transition semigroup defined by (13.7) and (13.31). A similar calculation based on the property of independent increments of {𝑊 (d𝑠, d𝑥)} and {𝑁 (d𝑠, d𝑥, d𝑢)} shows that {𝑋𝑡 : 𝑡 ≥ 0} is 𝜁 a Markov process with transition semigroup (𝑄 𝑡 )𝑡 ≥0 given by (14.31). Therefore {𝑋𝑡 : 𝑡 ≥ 0} have identical finite-dimensional distributions with the limit process {𝑍 (𝑡) : 𝑡 ≥ 0} in Theorems 14.17 and 14.18. In other words, the limiting fluctuation process is a generalized OU-process with state space 𝐻0 (R) in the terminology of the last chapter.

14.5 Notes and Comments A general reference for nuclear spaces is Treves (1967). For the theory of classical Sobolev spaces see Adams and Fournier (2003). There are several references for stochastic processes in nuclear spaces; see, e.g., Kallianpur and Xiong (1995) and Walsh (1986). The equilibrium distributions of super-stable processes without immigration were characterized in Dawson (1977). A simplified approach to the asymptotic behavior of superprocesses was given in Wang (1997b, 1998b). Fluctuation limits of branching particle systems and superprocesses, which usually give rise to time-inhomogeneous OU-processes, have been studied extensively; see, e.g., Bojdecki and Gorostiza (1986, 1991, 2002), Dawson et al. (1989a) and the references therein. Engländer and Winter (2006) proved a law of large numbers for super-diffusions which improves

14.5 Notes and Comments

409

an earlier result of Engländer and Turaev (2002). Chen et al. (2008) proved an almost sure scaling limit theorem for Dawson–Watanabe superprocesses. In Méléard (1996) fluctuation limits of McKean–Vlasov interacting particle systems were studied, where the limiting OU-process was characterized as the unique solution of a Langevin type equation in a weighted Sobolev space. The fluctuation limit theorems given in this chapter are modifications of those in Gorostiza and Li (1998) and Li and Zhang (2006); see also Dawson et al. (2004b). Three different kinds of fluctuation limits (high-density fluctuation, small-branching fluctuation and large-scale fluctuation) of immigration superprocess with binary branching were studied in Li (1999), which led to generalized OU-diffusions. Some Gaussian processes with long-range dependence arising from occupation time fluctuations of immigration particle systems with or without branching were studied in Gorostiza et al. (2005). A construction of the two-dimensional regular affine process in 𝐷 = R+ × R was given in Dawson and Li (2006) as the strong solution of a system of stochastic equations. Let {(𝑥(𝑡), 𝑧(𝑡)) : 𝑡 ≥ 0} be a realization of the affine process. Then the first coordinator {𝑥(𝑡) : 𝑡 ≥ 0} is a one-dimensional CBI-process. In fact, Dawson and Li (2006) showed that the second coordinator {𝑧(𝑡) : 𝑡 ≥ 0} may arise as the fluctuation limit of a generalized CBI-process with branching rate depending on the first one. A similar limit theorem for discrete-state branching processes with immigration was proved in Li and Ma (2008).

Appendix A

Markov Processes

For the convenience of the reader, in this appendix we give a summary of some of the concepts and results for general stochastic processes and Markov processes that are used in the main text. Many of them can be found in Sharpe (1988); see also Ethier and Kurtz (1986) and Getoor (1975).

A.1 Measurable Spaces Given a class ℱ of functions on a non-empty set 𝐸, we define bℱ = { 𝑓 ∈ ℱ : 𝑓 is bounded} and pℱ = { 𝑓 ∈ ℱ : 𝑓 is positive}. We say ℱ separates points if for every 𝑥 ≠ 𝑦 ∈ 𝐸 there exists an 𝑓 ∈ ℱ such that 𝑓 (𝑥) ≠ 𝑓 (𝑦). For a class 𝒢 of functions on (or subsets of) 𝐸, we use 𝜎(𝒢) to denote the 𝜎-algebra on 𝐸 generated by 𝒢, that is, 𝜎(𝒢) = ∩{ℱ : ℱ is a 𝜎-algebra on 𝐸 and all elements of 𝒢 are ℱ-measurable}. If (𝐸, ℰ) is a measurable space, we also use ℰ to denote the class of real ℰ-measurable functions on 𝐸. We write 𝜇( 𝑓 ) for the integral of a function 𝑓 ∈ ℰ with respect to a measure 𝜇 on (𝐸, ℰ) if the integral exists. Let R denote the one-dimensional Euclidean space. Let ∥ · ∥ denote the supremum/uniform norm of functions. We say a sequence { 𝑓𝑛 } of functions on 𝐸 converges uniformly to a function 𝑓 on 𝐸 if ∥ 𝑓𝑛 − 𝑓 ∥ → 0 as 𝑛 → ∞. We say { 𝑓𝑛 } converges boundedly and pointwise to 𝑓 if there is a constant 𝐶 ≥ 0 such that ∥ 𝑓𝑛 ∥ ≤ 𝐶 for all 𝑛 ≥ 1 and 𝑓𝑛 (𝑥) → 𝑓 (𝑥) as 𝑛 → ∞ for all 𝑥 ∈ 𝐸. A monotone vector space ℒ on the set 𝐸 is defined to be a collection of bounded real functions on 𝐸 satisfying the conditions: (i) ℒ is a vector space over R; (ii) ℒ contains the constant function 1𝐸 ; (iii) if { 𝑓𝑛 } ⊂ pℒ and 𝑓𝑛 → 𝑓 increasingly for a bounded function 𝑓 , then 𝑓 ∈ ℒ. Proposition A.1 (Monotone Class Theorem; Sharpe, 1988, p. 364) Let 𝒦 be a collection of bounded real functions on the set 𝐸 which is closed under multiplication. If ℒ is a monotone vector space containing 𝒦, then ℒ ⊃ b𝜎(𝒦). © Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5

411

A Markov Processes

412

Proposition A.2 (Modified Monotone Class Theorem) Let 𝒦 be a vector space of bounded real functions on the set 𝐸 which contains 1𝐸 and is closed under multiplication. If another collection of bounded real functions 𝒢 contains 𝒦 and is closed under bounded pointwise convergence, then 𝒢 ⊃ b𝜎(𝒦). Proof Let ℒ be the intersection of all classes of bounded real functions that contain 𝒦 and are closed under bounded pointwise convergence. Then ℒ is closed under bounded pointwise convergence and 𝒦 ⊂ ℒ ⊂ 𝒢. For 𝑓 ∈ ℒ let ℒ𝑓 = {𝑔 ∈ ℒ : 𝑎 𝑓 + 𝑏𝑔 ∈ ℒ for all 𝑎, 𝑏 ∈ R}. It is easy to see that ℒ𝑓 is closed under bounded pointwise convergence. For 𝑓 ∈ 𝒦 we have 𝒦 ⊂ ℒ𝑓 and so ℒ𝑓 = ℒ. If 𝑓 ∈ ℒ, for every 𝑔 ∈ 𝒦 we have 𝑓 ∈ ℒ𝑔 and so 𝑔 ∈ ℒ𝑓 . It follows that 𝒦 ⊂ ℒ𝑓 , yielding ℒ𝑓 = ℒ. Therefore ℒ is a vector space. By the monotone class theorem we have ℒ ⊃ b𝜎(𝒦), which implies the □ desired result. Let us consider a measurable space (𝐸, ℰ). Suppose that 𝜇 is a 𝜎-finite measure on (𝐸, ℰ). A set 𝑁 ⊂ 𝐸 is called a 𝜇-null set if there is an 𝑁0 ∈ ℰ such that 𝑁 ⊂ 𝑁0 and 𝜇(𝑁0 ) = 0. For 𝐴, 𝐵 ⊂ 𝐸 we define the symmetric difference 𝐴△𝐵 := ( 𝐴 \ 𝐵) ∪ (𝐵 \ 𝐴).

(A.1)

ℰ 𝜇 := { 𝐴 ⊂ 𝐸 : 𝐴△𝐵 is a 𝜇-null set for some 𝐵 ∈ ℰ}

(A.2)

It is easy to show that

is a 𝜎-algebra, which is called the 𝜇-completion of ℰ. We can let 𝜇( 𝐴) = 𝜇(𝐵) for 𝐵 ∈ ℰ such that 𝐴△𝐵 is a 𝜇-null set to extend 𝜇 uniquely to a 𝜎-finite measure on (𝐸, ℰ 𝜇 ). The measure space (𝐸, ℰ, 𝜇) is said to be complete if ℰ = ℰ 𝜇 . The universal completion of ℰ is the 𝜎-algebra ℰ 𝑢 defined to be the intersection of the 𝜇-completions of ℰ as 𝜇 runs over all finite measures on (𝐸, ℰ). Proposition A.3 If ℰ1 and ℰ2 are 𝜎-algebras on the set 𝐸 such that ℰ1 ⊂ ℰ2 ⊂ ℰ1𝑢 , then ℰ2𝑢 = ℰ1𝑢 . 𝜇

Proof Let 𝐴 ∈ ℰ1𝑢 and let 𝜇 be a finite measure on ℰ2 . Since 𝐴 ∈ ℰ1 , it is easy to 𝜇 find 𝐴1 , 𝐴2 ∈ ℰ1 ⊂ ℰ2 such that 𝐴1 ⊂ 𝐴 ⊂ 𝐴2 and 𝜇( 𝐴1 ) = 𝜇( 𝐴2 ). Then 𝐴 ∈ ℰ2 , 𝑢 𝑢 𝑢 implying ℰ1 ⊂ ℰ2 . To show the reverse inclusion, let 𝐴 ∈ ℰ2 and let 𝜇 be a finite 𝜇 measure on ℰ1 . Then 𝜇 extends uniquely to ℰ2 ⊂ ℰ1𝑢 and 𝐴 ∈ ℰ2 . Consequently, 𝜇 𝑢 there are 𝐴1 , 𝐴2 ∈ ℰ2 ⊂ ℰ1 ⊂ ℰ1 such that 𝐴1 ⊂ 𝐴 ⊂ 𝐴2 and 𝜇( 𝐴1 ) = 𝜇( 𝐴2 ). This yields the existence of 𝐵1 , 𝐵2 ∈ ℰ1 such that 𝐵1 ⊂ 𝐴1 , 𝐴2 ⊂ 𝐵2 and 𝜇(𝐵1 ) = 𝜇(𝐵2 ). 𝜇 □ Then 𝐴 ∈ ℰ1 , which implies ℰ2𝑢 ⊂ ℰ1𝑢 . Let (𝐸, ℰ) be a measurable space. The trace or restriction of ℰ on a subset 𝐴 ⊂ 𝐸 is defined to be the 𝜎-algebra ℰ𝐴 := {𝐵 ∩ 𝐴 : 𝐵 ∈ ℰ}. For a measure 𝜇 on (𝐸, ℰ), the (outer) trace or restriction 𝜇 𝐴 of 𝜇 on ( 𝐴, ℰ𝐴) is defined by

A.1 Measurable Spaces

413

𝜇 𝐴 (𝐶) = inf{𝜇(𝐵) : 𝐶 = 𝐵 ∩ 𝐴, 𝐵 ∈ ℰ}. The trace 𝜇 𝐴 can be realized as follows. Choose 𝐴0 ∈ ℰ with 𝐴0 ⊃ 𝐴 having minimal 𝜇-measure. Then for 𝐶 ∈ ℰ𝐴 of the form 𝐶 = 𝐵 ∩ 𝐴 with 𝐵 ∈ ℰ we have 𝜇 𝐴 (𝐶) = 𝜇(𝐵 ∩ 𝐴0 ); see Sharpe (1988, p. 367). Proposition A.4 (Sharpe, 1988, p. 368) Let 𝐴 ⊂ 𝐸 and let ℰ𝐴 be the trace of ℰ on 𝐴. Then we have: (1) given a finite measure 𝜇 on ( 𝐴, ℰ𝐴), the formula 𝜇(𝐵) := 𝜇(𝐵 ∩ 𝐴) for 𝐵 ∈ ℰ ¯ defines a finite measure 𝜇¯ on (𝐸, ℰ) whose trace on 𝐴 is 𝜇; (2) (ℰ 𝑢 ) 𝐴 ⊂ (ℰ𝐴) 𝑢 and these two coincide if 𝐴 ∈ ℰ 𝑢 . Suppose that (𝐸, ℰ) and (𝐹, ℱ) are measurable spaces. A 𝜎-finite kernel from (𝐸, ℰ) to (𝐹, ℱ) is a function 𝐾 = 𝐾 (·, ·) on 𝐸 × ℱ having values in [0, ∞] such that: (1) for each 𝐴 ∈ ℱ the mapping 𝑥 ↦→ 𝐾 (𝑥, 𝐴) is ℰ-measurable; (2) for each 𝑥 ∈ 𝐸 the mapping 𝐴 ↦→ 𝐾 (𝑥, 𝐴) is a 𝜎-finite measure on (𝐹, ℱ). A kernel 𝐾 is said to be finite or bounded if 𝑥 ↦→ 𝐾 (𝑥, 𝐹) is respectively a finite or bounded function on 𝐸. The kernel 𝐾 is called Markov or sub-Markov if 𝐾 (𝑥, 𝐹) = 1 or 𝐾 (𝑥, 𝐹) ≤ 1, respectively, for each 𝑥 ∈ 𝐸. A kernel from (𝐸, ℰ) to (𝐸, ℰ) is simply called a kernel on (𝐸, ℰ). Given a bounded kernel 𝐾 from (𝐸, ℰ) to (𝐹, ℱ), for any 𝑓 ∈ bℱ we can define 𝐾 𝑓 ∈ bℰ by ∫ 𝑓 (𝑦)𝐾 (𝑥, d𝑦), 𝑥 ∈ 𝐸, 𝐾 𝑓 (𝑥) = 𝐾 (𝑥, 𝑓 ) = 𝐹

and for any finite measure 𝜇 on (𝐸, ℰ) we can define a finite measure 𝜇𝐾 on (𝐹, ℱ) by ∫ 𝐾 (𝑥, 𝐵)𝜇(d𝑥), 𝜇𝐾 (𝐵) = 𝐵 ∈ ℱ. 𝐸

Proposition A.5 (Sharpe, 1988, p. 376) A bounded kernel 𝐾 from (𝐸, ℰ) to (𝐹, ℱ) extends in a unique way to a bounded kernel 𝐾 from (𝐸, ℰ 𝑢 ) to (𝐹, ℱ 𝑢 ). For a metrizable topological space 𝐸 with a metric 𝑑 compatible with its topology, let 𝒞(𝐸) := 𝒞(𝐸, 𝑑) denote the space of 𝑑-continuous real functions on (𝐸, 𝑑) and let 𝒞𝑢 (𝐸) := 𝒞𝑢 (𝐸, 𝑑) denote the space of 𝑑-uniformly continuous real functions on 𝐸. The advantage of 𝒞𝑢 (𝐸) is that if (𝐸, 𝑑) is separable and totally bounded, then b𝒞𝑢 (𝐸) with the supremum norm is separable, whereas b𝒞(𝐸) is not. The Borel 𝜎-algebra ℬ(𝐸) = ℬ(𝐸, 𝑑) on 𝐸 is defined to be the 𝜎-algebra generated by b𝒞(𝐸) or, equivalently, by all open subsets of 𝐸. If 𝐸 is locally compact, we let 𝐶0 (𝐸) denote the space of continuous real functions on 𝐸 vanishing at infinity. A topological space is called a Radon topological space or Lusin topological space if it is homeomorphic to a universally measurable subset or a Borel subset, respectively, of a compact metric space. A measurable space (𝐹, ℱ) is called a Radon measurable space or Lusin measurable space if it is measurably isomorphic to (𝐸, ℬ(𝐸)) with 𝐸 being a Radon or Lusin topological space, respectively.

414

A Markov Processes

A.2 Stochastic Processes Let (Ω, 𝒢, P) be a probability space. We shall use either E(𝑋) or P(𝑋) to denote the expectation of a random variable 𝑋 defined on this space. A collection (𝒢𝑡 )𝑡 ∈𝐼 of sub-𝜎-algebras of 𝒢 indexed by an interval 𝐼 ⊂ R is called a filtration of (Ω, 𝒢) if 𝒢𝑟 ⊂ 𝒢𝑡 for every 𝑟 ≤ 𝑡 ∈ 𝐼. If a filtration (𝒢𝑡 )𝑡 ∈𝐼 is defined on (Ω, 𝒢, P), we call (Ω, 𝒢, 𝒢𝑡 , P)𝑡 ∈𝐼 a filtered probability space. Suppose that (Ω, 𝒢, 𝒢𝑡 , P)𝑡 ∈𝐼 is a filtered probability space. A random variable 𝑇 taking values in 𝐼 ∪ {∞} is called a stopping time or an optional time over the filtration (𝒢𝑡 )𝑡 ∈𝐼 if {𝜔 ∈ Ω : 𝑇 (𝜔) ≤ 𝑡} ∈ 𝒢𝑡 for all 𝑡 ∈ 𝐼. Given a stopping time 𝑇 over (𝒢𝑡 )𝑡 ∈𝐼 , we can define a 𝜎-algebra 𝒢𝑇 := {𝐴 ∈ 𝒢(𝐼) : 𝐴 ∩ {𝑇 ≤ 𝑡} ∈ 𝒢𝑡 for every 𝑡 ∈ 𝐼},

(A.3)

where 𝒢(𝐼) = 𝜎(∪𝑡 ∈𝐼 𝒢𝑡 ). Let 𝜏 = sup 𝐼 and let 𝒢𝑡+ = ∩{𝒢𝑠 : 𝑡 < 𝑠 ∈ 𝐼} for 𝑡 ∈ 𝐼 \ {𝜏}. We say (𝒢𝑡 )𝑡 ∈𝐼 is right continuous if 𝒢𝑡+ = 𝒢𝑡 for every 𝑡 ∈ 𝐼 \ {𝜏}. Let 𝒢𝜏+ = 𝒢𝜏 if 𝜏 ∈ 𝐼. If 𝑇 is a stopping time over (𝒢𝑡+ )𝑡 ∈𝐼 , we define 𝒢𝑇+ by (A.3) with 𝒢𝑡 replaced by 𝒢𝑡+ . The special case 𝐼 = [0, ∞) is often considered. Suppose that (Ω, 𝒢, 𝒢𝑡 , P)𝑡 ≥0 is a filtered probability space. Let 𝒢¯ be the P-completion of 𝒢 and let 𝒩¯ = {𝐴 ∈ 𝒢¯ : ¯ 𝒢¯ 𝑡 )𝑡 ≥0 the augmentation ¯ for 𝑡 ≥ 0. We call (𝒢, P( 𝐴) = 0}. Let 𝒢¯ 𝑡 = 𝜎(𝒢𝑡 ∪ 𝒩) ¯ of (𝒢, 𝒢𝑡 )𝑡 ≥0 by the probability P. If 𝒢 = 𝒢 and 𝒢𝑡 = 𝒢¯ 𝑡 for every 𝑡 ≥ 0, we say (𝒢, 𝒢𝑡 )𝑡 ≥0 are augmented. We say a filtered probability space (Ω, 𝒢, 𝒢𝑡 , P)𝑡 ≥0 satisfies the usual hypotheses if (𝒢, 𝒢𝑡 )𝑡 ≥0 are augmented and (𝒢𝑡 )𝑡 ≥0 is right continuous. Proposition A.6 Suppose that (𝒢, 𝒢𝑡 )𝑡 ≥0 are augmented. If 𝑆 and 𝑇 are stopping times over (𝒢𝑡 )𝑡 ≥0 such that P{𝑆 ≠ 𝑇 } = 0, then 𝒢𝑆 = 𝒢𝑇 . Proof For any 𝐴 ∈ 𝒢𝑆 we have 𝐴 ∈ 𝒢∞ and 𝐴 ∩ {𝑆 ≤ 𝑡} ∈ 𝒢𝑡 for 𝑡 ≥ 0. Since (𝒢, 𝒢𝑡 )𝑡 ≥0 are augmented and P{𝑆 ≠ 𝑇 } = 0, we have 𝐴 ∩ {𝑇 ≤ 𝑡} ∈ 𝒢𝑡 for 𝑡 ≥ 0. □ Then 𝐴 ∈ 𝒢𝑇 . This proves 𝒢𝑆 ⊂ 𝒢𝑇 . Similarly we have 𝒢𝑇 ⊂ 𝒢𝑆 . We say the filtration (𝒢𝑡 )𝑡 ≥0 is quasi-left continuous if for every increasing sequence of stopping times {𝑇𝑛 } with limit 𝑇 we have 𝒢𝑇 = 𝜎(∪∞ 𝑛=1𝒢𝑇𝑛 ). A stopping time 𝑇 is called a predictable time if there is an announcing sequence of stopping times {𝑇𝑛 } such that lim𝑛→∞ 𝑇𝑛 = 𝑇 and 𝑇𝑛 < 𝑇 on {𝑇 > 0} for each 𝑛 ≥ 1. A stopping time 𝑇 is said to be totally inaccessible if for every predictable time 𝑆 we have 𝑆 ≠ 𝑇 a.s. on {𝑇 < ∞}. Suppose that (Ω, 𝒢, P) is a given probability space. Let 𝐼 ⊂ R be an interval and let (𝐸, ℰ) be a measurable space. A collection (𝑋𝑡 )𝑡 ∈𝐼 of measurable maps of (Ω, 𝒢) into (𝐸, ℰ) is called a stochastic process with state space (𝐸, ℰ). For fixed 𝜔 ∈ Ω, the map 𝑡 ↦→ 𝑋𝑡 (𝜔) from 𝐼 to 𝐸 is called a sample path of (𝑋𝑡 )𝑡 ∈𝐼 . The ℰ-natural 𝜎-algebras ℱ and (ℱ𝑡 )𝑡 ∈𝐼 of (𝑋𝑡 )𝑡 ∈𝐼 are defined by ℱ = 𝜎({ 𝑓 (𝑋𝑠 ) : 𝑠 ∈ 𝐼, 𝑓 ∈ bℰ})

A.2 Stochastic Processes

415

and ℱ𝑡 = 𝜎({ 𝑓 (𝑋𝑠 ) : 𝑠 ∈ 𝐼 ≤𝑡 , 𝑓 ∈ bℰ}), where 𝐼 ≤𝑡 = (−∞, 𝑡] ∩ 𝐼. The process (𝑋𝑡 )𝑡 ∈𝐼 is ℰ-adapted relative to a filtration (𝒢𝑡 )𝑡 ∈𝐼 if ℱ𝑡 ⊂ 𝒢𝑡 for every 𝑡 ∈ 𝐼. It is ℰ-progressive relative to (𝒢𝑡 )𝑡 ∈𝐼 if the mapping (𝜔, 𝑠) ↦→ 𝑋𝑠 (𝜔) restricted to Ω × 𝐼 ≤𝑡 is (𝒢𝑡 × ℬ(𝐼 ≤𝑡 ))-measurable for every 𝑡 ∈ 𝐼. Clearly, an ℰ-progressive process is ℰ-adapted. Let (𝑋𝑡 )𝑡 ∈𝐼 be a stochastic process taking values in (𝐸, ℰ). For any 𝑡 1 < · · · < 𝑡 𝑛 ∈ 𝐼 let 𝑃𝑡1 ,...,𝑡𝑛 be the probability measure on (𝐸 𝑛 , ℰ 𝑛 ) induced by the mapping 𝜔 ↦→ (𝑋𝑡1 (𝜔), . . . , 𝑋𝑡𝑛 (𝜔)). We call {𝑃𝑡1 ,...,𝑡𝑛 : 𝑡 1 < · · · < 𝑡 𝑛 ∈ 𝐼, 𝑛 = 1, 2, . . .} the family of finite-dimensional distributions of (𝑋𝑡 )𝑡 ∈𝐼 . If another process (𝑌𝑡 )𝑡 ∈𝐼 has identical finite-dimensional distributions as (𝑋𝑡 )𝑡 ∈𝐼 , we say it is a realization of (𝑋𝑡 )𝑡 ∈𝐼 . If the processes (𝑋𝑡 )𝑡 ∈𝐼 and (𝑌𝑡 )𝑡 ∈𝐼 are defined on the same probability space and if P{𝑋𝑡 = 𝑌𝑡 } = 1 for every 𝑡 ∈ 𝐼, we say (𝑌𝑡 )𝑡 ∈𝐼 is a modification of (𝑋𝑡 )𝑡 ∈𝐼 . In the case where 𝐸 is a topological space, we say a process (𝑋𝑡 )𝑡 ∈𝐼 is continuous or right continuous if all its sample paths 𝑡 ↦→ 𝑋𝑡 (𝜔) are continuous or right continuous on 𝐼, respectively. A path or process (𝑋𝑡 )𝑡 ≥0 is said to be càdlàg (continu à droite avec limites à gauche) if it is right continuous at every 𝑡 ≥ 0 and possesses left limit at every 𝑡 > 0. Now consider a metric 𝑑 on the space 𝐸. Suppose that 𝑇 is a subset of [0, ∞) and 𝑡 ↦→ 𝑥(𝑡) is a path from 𝑇 to 𝐸. For any 𝜀 > 0 the number of 𝜀-oscillations of 𝑡 ↦→ 𝑥(𝑡) on 𝑇 is defined as 𝑚(𝜀) = sup{𝑛 ≥ 0 : there are 𝑡0 < 𝑡1 < · · · < 𝑡 𝑛 ∈ 𝑇 such that 𝑑 (𝑥(𝑡 𝑖−1 ), 𝑥(𝑡𝑖 )) ≥ 𝜀 for all 1 ≤ 𝑖 ≤ 𝑛}. An earlier version of the proof of the following result was suggested to the author by Thomas G. Kurtz. Theorem A.7 Suppose that 𝑑 is a complete metric on 𝐸 such that the metric function (𝑥, 𝑦) ↦→ 𝑑 (𝑥, 𝑦) is ℰ 2 -measurable. Then a stochastic process (𝑋𝑡 )𝑡 ≥0 in (𝐸, ℰ) has a 𝑑-càdlàg modification if and only if it has a 𝑑-càdlàg realization. Proof Only one direction demands proof. Suppose that (𝑋𝑡 )𝑡 ≥0 has a 𝑑-càdlàg realization (𝜉𝑡 )𝑡 ≥0 . Take a countable dense subset 𝑇 = {𝑟 1 , 𝑟 2 , . . .} of [0, ∞) and let 𝑇𝑛 = {𝑟 1 , . . . , 𝑟 𝑛 }. For 𝜀 > 0 and 𝑎 > 0 let 𝑚 𝑎 (𝜀) and 𝑚 𝑛𝑎 (𝜀) denote the numbers of 𝜀-oscillations of 𝑡 ↦→ 𝑋𝑡 on 𝑇 𝑎 := 𝑇 ∩ [0, 𝑎] and 𝑇𝑛𝑎 := 𝑇𝑛 ∩ [0, 𝑎], respectively. Let 𝜇 𝑎 (𝜀) and 𝜇 𝑛𝑎 (𝜀) denote respectively those numbers of 𝑡 ↦→ 𝜉𝑡 . Then 𝑚 𝑛𝑎 (𝜀) → 𝑚 𝑎 (𝜀) and 𝜇 𝑛𝑎 (𝜀) → 𝜇 𝑎 (𝜀) increasingly as 𝑛 → ∞. For any integer 𝑘 ≥ 0 we have {𝜇 𝑛𝑎 (𝜀) ≥ 𝑘 } = {𝜔 ∈ Ω : there are 𝑡0 < 𝑡1 < · · · < 𝑡 𝑘 ∈ 𝑇𝑛𝑎 such that 𝑑 (𝜉𝑡𝑖−1 (𝜔), 𝜉𝑡𝑖 (𝜔)) ≥ 𝜀 for 1 ≤ 𝑖 ≤ 𝑘 }. (A.4)

416

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By the ℰ 2 -measurability of the metric function, we have {(𝑥, 𝑦) ∈ 𝐸 2 : 𝑑 (𝑥, 𝑦) ≥ 𝜀} ∈ ℰ 2 . Using (A.4) one can show each 𝜇 𝑛𝑎 (𝜀) is a random variable with distribution determined by that of the random vector (𝜉𝑟1 , · · · , 𝜉𝑟𝑛 ). Similarly, each 𝑚 𝑛𝑎 (𝜀) is a random variable with distribution determined by that of the random vector (𝑋𝑟1 , · · · , 𝑋𝑟𝑛 ). Since (𝜉𝑡 )𝑡 ≥0 is a realization of (𝑋𝑡 )𝑡 ≥0 , the random variables 𝜇 𝑎 (𝜀) = lim𝑛→∞ 𝜇 𝑛𝑎 (𝜀) and 𝑚 𝑎 (𝜀) = lim𝑛→∞ 𝑚 𝑛𝑎 (𝜀) are identically distributed. Since (𝜉𝑡 )𝑡 ≥0 is a 𝑑-càdlàg process, we get P{𝑚 𝑎 (𝜀) < ∞} = P{𝜇 𝑎 (𝜀) < ∞} = 1. 𝑗 Let Ω1 = ∩∞ 𝑗=1 {𝑚 (1/ 𝑗) < ∞}. Then P(Ω1 ) = 1. It is simple to show that for 𝜔 ∈ Ω1 the limit 𝑌𝑡 (𝜔) := lim𝑇 ∋𝑠↓𝑡 𝑋𝑠 (𝜔) exists at 𝑡 ≥ 0 and 𝑍𝑡 (𝜔) := lim𝑇 ∋𝑠↑𝑡 𝑋𝑠 (𝜔) exists at 𝑡 > 0. Fix 𝑥 0 ∈ 𝐸 and let 𝑌𝑡 (𝜔) = 𝑥 0 for all 𝑡 ≥ 0 and 𝜔 ∈ Ω \ Ω1 . Then (𝑌𝑡 )𝑡 ≥0 is a 𝑑-càdlàg process. Since (𝑋𝑡 )𝑡 ≥0 is clearly right continuous in probability, we have 𝑌𝑡 = 𝑋𝑡 a.s. for every 𝑡 ≥ 0. Therefore (𝑌𝑡 )𝑡 ≥0 is a 𝑑-càdlàg modification of □ (𝑋𝑡 )𝑡 ≥0 .

We remark that if (𝐸, 𝑑) is a separable and complete metric space, then ℬ(𝐸) 2 = ℬ(𝐸 2 ) and the metric function (𝑥, 𝑦) ↦→ 𝑑 (𝑥, 𝑦) is ℬ(𝐸) 2 -measurable.

A.3 Right Markov Processes Let 𝐸 be a Radon topological space. For clarity we may also write ℬ0 (𝐸) for the Borel 𝜎-algebra ℬ(𝐸). Let ℬ𝑢 (𝐸) be the universal completion of ℬ0 (𝐸) and let ℬ• (𝐸) be a 𝜎-algebra such that ℬ0 (𝐸) ⊂ ℬ• (𝐸) ⊂ ℬ𝑢 (𝐸). Then ℬ𝑢 (𝐸) is also the universal completion of ℬ• (𝐸) by Proposition A.3. A family of Markov or subMarkov kernels (𝑃𝑡 )𝑡 ≥0 on (𝐸, ℬ• (𝐸)) is called a transition semigroup if it satisfies the following Chapman–Kolmogorov equation: ∫ (A.5) 𝑃𝑟+𝑡 (𝑥, 𝐵) = 𝑃𝑟 (𝑥, d𝑦)𝑃𝑡 (𝑦, 𝐵) 𝐸

for all 𝑟, 𝑡 ≥ 0, 𝑥 ∈ 𝐸 and 𝐵 ∈ ℬ• (𝐸). By Proposition A.5, we can always regard (𝑃𝑡 )𝑡 ≥0 as kernels on (𝐸, ℬ𝑢 (𝐸)). A Borel transition semigroup (𝑃𝑡 )𝑡 ≥0 is a transition semigroup on a Lusin topological space 𝐸 such that 𝑃𝑡 𝑓 ∈ bℬ0 (𝐸) for each 𝑡 ≥ 0 and 𝑓 ∈ bℬ0 (𝐸). We say the transition semigroup (𝑃𝑡 )𝑡 ≥0 is Markov or conservative if each 𝑃𝑡 is a Markov kernel. We say (𝑃𝑡 )𝑡 ≥0 is normal if 𝑃0 (𝑥, ·) = 𝛿 𝑥 for every 𝑥 ∈ 𝐸. Let us consider a transition semigroup (𝑃𝑡 )𝑡 ≥0 on (𝐸, ℬ• (𝐸)). Let 𝐼 (𝛼) = (𝛼, ∞) for 𝛼 ∈ [−∞, ∞) or [𝛼, ∞) for 𝛼 ∈ R = (−∞, ∞). A family (𝜇𝑡 )𝑡 ∈𝐼 ( 𝛼) of 𝜎-finite measures on (𝐸, ℬ• (𝐸)) is called an entrance rule at 𝛼 for (𝑃𝑡 )𝑡 ≥0 if 𝜇 𝑠 𝑃𝑡−𝑠 ≤ 𝜇𝑡 for all 𝑠 ≤ 𝑡 ∈ 𝐼 (𝛼). It is called a regular entrance rule if 𝜇 𝑠 𝑃𝑡−𝑠 → 𝜇𝑡 increasingly as 𝑠 ↑ 𝑡 ∈ 𝐼 (𝛼). The family is called an entrance law if 𝜇 𝑠 𝑃𝑡−𝑠 = 𝜇𝑡 for all 𝑠 ≤ 𝑡 ∈ 𝐼 (𝛼). Clearly, an entrance rule (𝜇𝑡 )𝑡 ∈𝐼 ( 𝛼) can be extended to an entrance

A.3 Right Markov Processes

417

rule (𝜇𝑡 )𝑡 ∈R by setting 𝜇𝑡 = 0 for 𝑡 ∈ R \ 𝐼 (𝛼). We say an entrance rule or law (𝜇𝑡 )𝑡 ∈𝐼 ( 𝛼) is bounded if 𝑡 ↦→ 𝜇𝑡 (𝐸) is a bounded function on 𝐼 (𝛼). A probability entrance law is an entrance law (𝜇𝑡 )𝑡 ∈𝐼 ( 𝛼) where each 𝜇𝑡 is a probability measure. We say an entrance law (𝜇𝑡 )𝑡 ∈𝐼 ( 𝛼) is minimal or extremal if every entrance law dominated by (𝜇𝑡 )𝑡 ∈𝐼 ( 𝛼) is proportional to it. For 𝛼 ∈ R we say an entrance law (𝜇𝑡 )𝑡 ≥ 𝛼 is closed. We say an entrance law (𝜇𝑡 )𝑡 > 𝛼 is closable if it can be extended to a closed entrance law (𝜇𝑡 )𝑡 ≥ 𝛼 . The concepts of entrance rules and entrance laws can obviously be extended to semigroups of bounded kernels. We sometimes make use of those extensions. Let 𝒦 1 (𝑃) denote the set of all probability entrance laws (𝜇𝑡 )𝑡 >0 at zero for (𝑃𝑡 )𝑡 ≥0 endowed with the 𝜎-algebra generated by all mappings {𝜇 ↦→ 𝜇𝑡 ( 𝑓 ) : 𝑡 > 0, 𝑓 ∈ bℬ• (𝐸)}. Let 𝒦𝑚1 (𝑃) be the set of minimal probability entrance laws in 𝒦 1 (𝑃). From Dynkin (1978, Theorems 3.1 and 9.1) we know 𝒦 1 (𝑃) is a simplex, that is, 𝒦𝑚1 (𝑃) is a measurable subset of 𝒦 1 (𝑃) and for each 𝜇 ∈ 𝒦 1 (𝑃) there is a unique probability measure 𝑄 𝜇 on 𝒦𝑚1 (𝑃) such that ∫ 𝜈𝑡 (·)𝑄 𝜇 (d𝜈), 𝑡 > 0. 𝜇𝑡 (·) = 1 ( 𝑃) 𝒦𝑚

A 𝜎-finite measure 𝑚 on (𝐸, ℬ• (𝐸)) is called an excessive measure for (𝑃𝑡 )𝑡 ≥0 if 𝑚𝑃𝑡 ≤ 𝑚 for every 𝑡 ≥ 0. The measure 𝑚 is called a purely excessive measure if 𝑚𝑃𝑡 ≤ 𝑚 for every 𝑡 ≥ 0 and 𝑚𝑃𝑡 → 0 as 𝑡 → ∞, and it is called an invariant measure if 𝑚𝑃𝑡 = 𝑚 for every 𝑡 ≥ 0. A stationary distribution means an invariant probability measure. A Markov process with transition semigroup (𝑃𝑡 )𝑡 ≥0 is said to be ergodic if it has a unique stationary distribution 𝑚 and 𝜇𝑃𝑡 → 𝑚 by weak convergence as 𝑡 → ∞ for every probability measure 𝜇 on (𝐸, ℬ• (𝐸)). For 𝛼 ≥ 0 we say a function 𝑓 ∈ pℬ𝑢 (𝐸) is 𝛼-super-mean-valued for (𝑃𝑡 )𝑡 ≥0 if e−𝛼𝑡 𝑃𝑡 𝑓 ≤ 𝑓 for all 𝑡 ≥ 0, and it is called an 𝛼-excessive function for (𝑃𝑡 )𝑡 ≥0 if e−𝛼𝑡 𝑃𝑡 𝑓 → 𝑓 increasingly as 𝑡 → 0. In the special case with 𝛼 = 0, we simply say 𝑓 is super-meanvalued or excessive, respectively. Let 𝒮 𝛼 denote the set of 𝛼-excessive functions for (𝑃𝑡 )𝑡 ≥0 . A family of bounded kernels (𝑈 𝛼 ) 𝛼>0 on (𝐸, ℬ• (𝐸)) is called a resolvent if the resolvent equation 𝑈 𝛼 𝑓 (𝑥) − 𝑈 𝛽 𝑓 (𝑥) = (𝛽 − 𝛼)𝑈 𝛼𝑈 𝛽 𝑓 (𝑥)

(A.6)

is satisfied for all 𝛼, 𝛽 > 0, 𝑥 ∈ 𝐸 and 𝑓 ∈ bℬ• (𝐸). A resolvent (𝑈 𝛼 ) 𝛼>0 is called Markov or conservative if 𝛼𝑈 𝛼 is a Markov kernel for all 𝛼 > 0. A function 𝑓 ∈ pℬ𝑢 (𝐸) is called 𝛼-supermedian for the resolvent (𝑈 𝛼 ) 𝛼>0 if 𝛽𝑈 𝛼+𝛽 𝑓 ≤ 𝑓 for all 𝛽 > 0. Let 𝒮˜ 𝛼 denote the class of all 𝛼-supermedian functions for (𝑈 𝛼 ) 𝛼>0 . If (𝑃𝑡 )𝑡 ≥0 is a transition semigroup on (𝐸, ℬ• (𝐸)) such that (𝑡, 𝑥) ↦→ 𝑃𝑡 𝑓 (𝑥) is measurable with respect to ℬ[0, ∞) × ℬ• (𝐸) for every 𝑓 ∈ bℬ• (𝐸), then the operators (𝑈 𝛼 ) 𝛼>0 defined by ∫ ∞ 𝛼 (A.7) e−𝛼𝑡 𝑃𝑡 𝑓 (𝑥)d𝑡, 𝑓 ∈ bℬ• (𝐸), 𝑈 𝑓 (𝑥) = 0

418

A Markov Processes

constitute a resolvent, which is called the resolvent of (𝑃𝑡 )𝑡 ≥0 . We also call 𝑈 𝛼 the 𝛼-potential operator of (𝑃𝑡 )𝑡 ≥0 . The potential operator 𝑈 of (𝑃𝑡 )𝑡 ≥0 is defined by ∫ ∞ 𝑓 ∈ pℬ• (𝐸). 𝑈 𝑓 (𝑥) = (A.8) 𝑃𝑡 𝑓 (𝑥)d𝑡, 0

However, this kernel may not be 𝜎-finite. It is easy to show that if 𝑓 is 𝛼-supermean-valued for (𝑃𝑡 )𝑡 ≥0 , it is 𝛼-supermedian for (𝑈 𝛼 ) 𝛼>0 . Suppose that (𝑃𝑡 )𝑡 ≥0 is a Markov transition semigroup on (𝐸, ℬ• (𝐸)) and (𝜉𝑡 )𝑡 ∈𝐼 is a stochastic process in (𝐸, ℬ• (𝐸)) indexed by an interval 𝐼 ⊂ R. We assume that (𝜉𝑡 )𝑡 ∈𝐼 is defined on the probability space (Ω, 𝒢, P) and is ℬ• (𝐸)-adapted to a filtration (𝒢𝑡 )𝑡 ∈𝐼 . We say {(𝜉𝑡 , 𝒢𝑡 ) : 𝑡 ∈ 𝐼} has the simple ℬ• (𝐸)-Markov property with transition semigroup (𝑃𝑡 )𝑡 ≥0 if   P 𝑓 (𝜉𝑡 )|𝒢𝑟 = 𝑃𝑡−𝑟 𝑓 (𝜉𝑟 ), 𝑟 ≤ 𝑡 ∈ 𝐼, 𝑓 ∈ bℬ• (𝐸). (A.9) If {(𝜉𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} satisfies the simple ℬ• (𝐸)-Markov property with transition semigroup (𝑃𝑡 )𝑡 ≥0 , the distribution 𝜇0 of 𝜉0 is called the initial law of (𝜉𝑡 )𝑡 ≥0 . In this case, we necessarily have   P 𝑓1 (𝜉𝑡1 ) 𝑓2 (𝜉𝑡2 ) · · · 𝑓𝑛−1 (𝜉𝑡𝑛−1 ) 𝑓𝑛 (𝜉𝑡𝑛 )
= 𝜇0 𝑃𝑡1 ( 𝑓1 · · · 𝑃𝑡𝑛−1 −𝑡𝑛−2 ( 𝑓𝑛−1 𝑃𝑡𝑛 −𝑡𝑛−1 𝑓𝑛 )) (A.10) for 0 ≤ 𝑡 1 ≤ 𝑡2 ≤ · · · ≤ 𝑡 𝑛 and 𝑓1 , 𝑓2 , . . . , 𝑓𝑛 ∈ bℬ• (𝐸), which is a simple consequence of (A.9) by an induction argument. Consequently, the restriction of P on the ℬ• (𝐸)-natural 𝜎-algebra ℱ • := 𝜎({ 𝑓 (𝑋𝑠 ) : 𝑠 ∈ 𝐼, 𝑓 ∈ bℬ• (𝐸)}) is determined uniquely by (A.10). Proposition A.8 Suppose that {(𝜉𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} satisfies the simple ℬ• (𝐸)-Markov property (A.9). Let (𝒢∗ , 𝒢𝑡∗ ) denote the augmentations of (𝒢, 𝒢𝑡 ) with respect to P. Then {(𝜉𝑡 , 𝒢𝑡∗ ) : 𝑡 ≥ 0} satisfies the simple ℬ𝑢 (𝐸)-Markov property. Proof Let 𝜇𝑡 denote the distribution of 𝜉𝑡 on (𝐸, ℬ• (𝐸)). For 𝑓 ∈ bℬ𝑢 (𝐸) we can choose 𝑓1 , 𝑓2 ∈ bℬ• (𝐸) so that 𝑓1 ≤ 𝑓 ≤ 𝑓2 and 𝜇𝑡 ( 𝑓2 − 𝑓1 ) = 0. Then 𝑓1 (𝜉𝑡 ), 𝑓2 (𝜉𝑡 ) ∈ b𝒢𝑡 and   (A.11) P 𝑓2 (𝜉𝑡 ) − 𝑓1 (𝜉𝑡 ) = 𝜇𝑡 ( 𝑓2 − 𝑓1 ) = 0. It follows that 𝑓 (𝜉𝑡 ) ∈ b𝒢𝑡∗ , and so (𝜉𝑡 )𝑡 ≥0 is ℬ𝑢 (𝐸)-adapted relative to (𝒢𝑡∗ )𝑡 ≥0 . Then to get the desired result it suffices to show     P 𝑓 (𝜉𝑡 )1 𝐴 = P 𝑃𝑡−𝑟 𝑓 (𝜉𝑟 )1 𝐴 (A.12) for 𝑡 ≥ 𝑟 ≥ 0, 𝐴 ∈ 𝒢𝑟∗ and 𝑓 ∈ bℬ𝑢 (𝐸). Let 𝒩 = {𝑁 ∈ 𝒢∗ : P(𝑁) = 0}. Then there is an 𝐴0 ∈ 𝒢𝑟 such that 𝐴△𝐴0 ∈ 𝒩. By (A.9) we have (A.12) for 𝑓 ∈ bℬ• (𝐸). For 𝑓 ∈ bℬ𝑢 (𝐸) we can take 𝑓1 , 𝑓2 ∈ bℬ• (𝐸) so that 𝑓1 ≤ 𝑓 ≤ 𝑓2 and (A.11) holds. □ Since (A.12) holds for both 𝑓1 and 𝑓2 , it also holds for 𝑓 .

A.3 Right Markov Processes

419

Corollary A.9 (Sharpe, 1988, p. 6) Suppose that (𝑃𝑡 )𝑡 ≥0 preserves ℬ• (𝐸) and ℬ ⋄ (𝐸) with ℬ0 (𝐸) ⊂ ℬ• (𝐸) ⊂ ℬ ⋄ (𝐸) ⊂ ℬ𝑢 (𝐸). Let {(𝜉𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} satisfy the simple ℬ• (𝐸)-Markov property (A.9). If (𝜉𝑡 )𝑡 ≥0 is ℬ ⋄ (𝐸)-adapted to (𝒢𝑡 )𝑡 ≥0 , then {(𝜉𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} satisfies the simple ℬ ⋄ (𝐸)-Markov property. Proof By Proposition A.8 we infer {(𝜉𝑡 , 𝒢𝑡∗ ) : 𝑡 ≥ 0} satisfies the simple ℬ𝑢 (𝐸)Markov property. Then {(𝜉𝑡 , 𝒢𝑡 ) : 𝑡 ≥ 0} satisfies the simple ℬ ⋄ (𝐸)-Markov □ property. Definition A.10 (Sharpe, 1988, p. 7) Suppose that (𝑃𝑡 )𝑡 ≥0 is a normal Markov transition semigroup on (𝐸, ℬ• (𝐸)). The collection 𝜉 = (Ω, 𝒢, 𝒢𝑡 , 𝜉𝑡 , 𝜃 𝑡 , P 𝑥 ) is called a ℬ• (𝐸)-Markov process with transition semigroup (𝑃𝑡 )𝑡 ≥0 if 𝜉 satisfies the following conditions: (1) (Ω, 𝒢, 𝒢𝑡 )𝑡 ≥0 is a filtered measurable space, and (𝜉𝑡 )𝑡 ≥0 is an 𝐸-valued process ℬ• (𝐸)-adapted to (𝒢𝑡 )𝑡 ≥0 . (2) (𝜃 𝑡 )𝑡 ≥0 is a collection of shift operators for 𝜉, that is, maps of Ω into itself satisfying 𝜃 𝑠 ◦ 𝜃 𝑡 = 𝜃 𝑠+𝑡 and 𝜉 𝑠 ◦ 𝜃 𝑡 = 𝜉 𝑠+𝑡 identically for 𝑡, 𝑠 ≥ 0. (3) For every 𝑥 ∈ 𝐸, P 𝑥 is a probability measure on (Ω, 𝒢) and 𝑥 ↦→ P 𝑥 (𝐻) is ℬ• (𝐸)-measurable for each 𝐻 ∈ b𝒢. (4) For every 𝑥 ∈ 𝐸, we have P 𝑥 {𝜉0 = 𝑥} = 1 and the process (𝜉𝑡 )𝑡 ≥0 has the simple Markov property (A.9) relative to (𝒢𝑡 , P 𝑥 ) with transition semigroup (𝑃𝑡 )𝑡 ≥0 . We say 𝜉 is right continuous if 𝑡 ↦→ 𝜉𝑡 (𝜔) is right continuous for every 𝜔 ∈ Ω. If the above conditions (1)–(4) are satisfied, we also say that 𝜉 is a realization of the semigroup (𝑃𝑡 )𝑡 ≥0 . In this case, for any finite measure 𝜇 on (𝐸, ℬ• (𝐸)) we may define the finite measure P 𝜇 on (Ω, 𝒢) by ∫ P 𝑥 (𝐻)𝜇(d𝑥), 𝐻 ∈ b𝒢. (A.13) P 𝜇 (𝐻) = 𝐸

In the sequel, we always assume 𝜇 is a probability measure unless stated otherwise. It is easy to verify that (𝜉𝑡 )𝑡 ≥0 has the simple ℬ• (𝐸)-Markov property relative to (𝒢𝑡 , P 𝜇 ) with initial law 𝜇. We mention that the measurability of 𝑥 ↦→ P 𝑥 (𝐻) is used in the definition (A.13) of the measure P 𝜇 on (Ω, 𝒢). Of course, this measurability follows automatically if (𝒢, 𝒢𝑡 ) are the natural 𝜎-algebras of {𝜉𝑡 : 𝑡 ≥ 0}. Consider a ℬ• (𝐸)-Markov process 𝜉 = (Ω, 𝒢, 𝒢𝑡 , 𝜉𝑡 , 𝜃 𝑡 , P 𝑥 ) with transition semigroup (𝑃𝑡 )𝑡 ≥0 and resolvent (𝑈 𝛼 ) 𝛼>0 on (𝐸, ℬ• (𝐸)). Let 𝒢 𝜇 denote the P 𝜇 completion of 𝒢 and let 𝒩 𝜇 (𝒢) denote the family of P 𝜇 -null sets in 𝒢 𝜇 . Then define: 𝒢¯ = ∩{𝒢 𝜇 : 𝜇 is an initial law on 𝐸 }; 𝒩(𝒢) = ∩{𝒩 𝜇 (𝒢) : 𝜇 is an initial law on 𝐸 }; 𝜇 𝒢𝑡 = 𝜎(𝒢𝑡 ∪ 𝒩 𝜇 (𝒢)); 𝜇 𝒢¯ 𝑡 = ∩{𝒢𝑡 : 𝜇 is an initial law on 𝐸 }.

420

A Markov Processes 𝜇

Therefore (𝒢 𝜇 , 𝒢𝑡 ) is the augmentation of (𝒢, 𝒢𝑡 ) by the probability P 𝜇 . We call ¯ 𝒢¯ 𝑡 ) the augmentation of (𝒢, 𝒢𝑡 ) by the system of probabilities {P 𝜇 : 𝜇 is (𝒢, a probability on 𝐸 }. It is easy to see that (𝜉𝑡 )𝑡 ≥0 is ℬ𝑢 (𝐸)-adapted relative to ¯ Moreover, for any 𝐻 ∈ b𝒢¯ the mapping (𝒢¯ 𝑡 )𝑡 ≥0 and each P 𝜇 extends uniquely to 𝒢. 𝑢 𝑥 ↦→ P 𝑥 (𝐻) is ℬ (𝐸)-measurable and the equality in (A.13) remains true. Using Proposition A.8 and Corollary A.9 one can see (𝜉𝑡 )𝑡 ≥0 has the simple ℬ𝑢 (𝐸)-Markov 𝜇 property relative to (𝒢𝑡 , P 𝜇 ) and (𝒢¯ 𝑡 , P 𝜇 ). We say (𝒢, 𝒢𝑡 )𝑡 ≥0 are augmented with respect to the system {P 𝜇 : 𝜇 is a probability on 𝐸 } provided 𝒢 = 𝒢¯ and 𝒢𝑡 = 𝒢¯ 𝑡 for all 𝑡 ≥ 0. As observed in Sharpe (1988, p. 25), further application of the augmentation procedure to (𝒢¯ 𝑡 )𝑡 ≥0 is fruitless in 𝜇 the sense that [𝒢¯ 𝑡 ] 𝜇 = 𝒢𝑡 and [𝒢¯ 𝑡 ] − = 𝒢¯ 𝑡 . Let ℱ 𝑢 be the ℬ𝑢 (𝐸)-natural 𝜎-algebra of {𝜉𝑡 : 𝑡 ≥ 0}. Let ℱ 𝜇 denote the P 𝜇 -completion of ℱ 𝑢 and let ℱ = ∩{ℱ 𝜇 : 𝜇 is an initial law on 𝐸 }. As in Sharpe (1988, p. 25), one can show that    𝜇 (A.14) P 𝜇 𝐹 ◦ 𝜃 𝑡 |𝒢¯ 𝑡 = P 𝜇 𝐹 ◦ 𝜃 𝑡 |𝒢𝑡 = P 𝜉𝑡 (𝐹) for every 𝑡 ≥ 0, 𝐹 ∈ bℱ and initial law 𝜇. In the sequel, we assume 𝜉 = (Ω, 𝒢, 𝒢𝑡 , 𝜉𝑡 , 𝜃 𝑡 , P 𝑥 ) is right continuous. Writing 𝜇 𝜇 𝒢𝑡+ = ∩𝑠>𝑡 𝒢𝑠 , we have [𝒢𝑡 ] + = [𝒢𝑡+ ] 𝜇 , which will be denoted simply by 𝒢𝑡+ . It is 𝜇 𝜇 easy to see that for any initial law 𝜇 on 𝐸 the filtered space (Ω, 𝒢 , 𝒢𝑡+ , P 𝜇 ) satisfies the usual hypotheses. 𝜇

Proposition A.11 Suppose that 𝑇 is a stopping time over (𝒢𝑡+ ). Then 𝒢𝑇+ = 𝜎(𝒢𝑇+ ∪ 𝒩 𝜇 (𝒢)). 𝜇

Proof Since 𝒢𝑇+ ⊃ 𝜎(𝒢𝑇+ ∪𝒩 𝜇 (𝒢)) is obvious, we only need to verify the inclusion 𝜇 𝜇 𝒢𝑇+ ⊂ 𝜎(𝒢𝑇+ ∪ 𝒩 𝜇 (𝒢)). It suffices to show for every 𝐴 ∈ 𝒢𝑇+ there is a 𝐵 ∈ 𝒢𝑇+ such that 𝐴△𝐵 ∈ 𝒩 𝜇 (𝒢), where “△” denotes the symmetric difference defined by (A.1). For each 𝑛 ≥ 1 define the stopping time 𝑇𝑛 over (𝒢𝑡 ) by 𝑇𝑛 (𝜔) =

n 𝑘/2𝑛 ∞

if (𝑘 − 1)/2𝑛 ≤ 𝑇 (𝜔) < 𝑘/2𝑛 , if 𝑇 (𝜔) = ∞.

(A.15)

Then 𝑇𝑛 → 𝑇 decreasingly as 𝑛 → ∞. In view of (A.15) we have 𝐴 ∩ {𝑇𝑛 = 𝑘/2𝑛 } ∈ 𝜇 𝒢𝑘/2𝑛 for 1 ≤ 𝑛 < ∞ and 1 ≤ 𝑘 = ∞. Then there exists 𝐴𝑛,𝑘 ∈ 𝒢𝑘/2𝑛 such that ( 𝐴 ∩ {𝑇𝑛 = 𝑘/2𝑛 })△𝐴𝑛,𝑘 ∈ 𝒩 𝜇 (𝒢). Since {𝑇𝑛 = 𝑘/2𝑛 } ∈ 𝒢𝑘/2𝑛 , we have 𝐵𝑛,𝑘 := 𝐴𝑛,𝑘 ∩ {𝑇𝑛 = 𝑘/2𝑛 } ∈ 𝒢𝑘/2𝑛 . Observe also that ( 𝐴 ∩ {𝑇𝑛 = 𝑘/2𝑛 })△𝐵𝑛,𝑘 ∈ 𝒩 𝜇 (𝒢). 𝜇 Let 𝐵𝑛 = (∪∞ 𝑘=1 𝐵 𝑛,𝑘 ) ∪ 𝐵 𝑛,∞ . Then 𝐴△𝐵 𝑛 ∈ 𝒩 (𝒢) and

A.3 Right Markov Processes

421

𝐵𝑛 ∩ {𝑇𝑛 = 𝑘/2𝑛 } = 𝐵𝑛,𝑘 ∈ 𝒢𝑘/2𝑛 . It follows that 𝐵𝑛 ∈ 𝒢𝑇𝑛 ⊂ 𝒢𝑇𝑘 for 𝑛 ≥ 𝑘. By the right continuity of (𝒢𝑡+ ), 𝐵 :=

∞ ∞ Ø Ù

𝐵𝑛 ∈

𝑘=1 𝑛=𝑘

∞ Ù

𝒢𝑇𝑘 + = 𝒢𝑇+ .

𝑘=1

Moreover, we have 𝐴△𝐵 ∈ 𝒩 𝜇 (𝒢). This gives the desired result.

□ 𝜇

Corollary A.12 For any initial law 𝜇 on 𝐸 and any stopping time 𝑇 for (𝒢𝑡+ ), there is a stopping time 𝑆 for (𝒢𝑡+ ) such that {𝑇 ≠ 𝑆} ∈ 𝒩 𝜇 (𝒢). In this case, we have 𝜇

𝜇

𝒢𝑇+ = 𝒢𝑆+ = 𝜎(𝒢𝑆+ ∪ 𝒩 𝜇 (𝒢)).

(A.16)

Proof The first assertion was proved in Sharpe (1988, p. 25). Then (A.16) follows □ by Propositions A.6 and A.11. Proposition A.13 (Sharpe, 1988, p. 26) Let 𝑓 ∈ ℬ𝑢 (𝐸) and let 𝜇 be an initial law on 𝐸. Then we have: 𝜇

𝜇

(1) If 𝑇 is an stopping time over (𝒢𝑡+ ), then 𝑓 (𝜉𝑇 )1 {𝑇 0 and 𝛼𝑈 𝛼 𝑓 → 𝑓 pointwise as 𝛼 → ∞ for all 𝑓 ∈ 𝐶0 (𝐸). Then there is a right process 𝜉 with state space 𝐸 having resolvent (𝑈 𝛼 ) 𝛼>0 such that: (1) 𝜉 is quasi-left continuous; (2) for all 𝑡 > 0 the set {𝜉 𝑠 (𝜔) : 0 ≤ 𝑠 ≤ 𝑡} a.s. has compact closure in 𝐸; (3) a.s. the left limit 𝜉𝑡− := lim𝑠↑𝑡 𝜉 𝑠 exists in 𝐸 for all 𝑡 > 0. Corollary A.26 A Feller semigroup has a realization as a Hunt process. Now suppose we are given a general Radon topological space 𝐸 with a totally bounded metric 𝑑 for its topology. Let (𝑈 𝛼 ) 𝛼>0 be a Markov resolvent on (𝐸, ℬ𝑢 (𝐸)) satisfying ℬ(𝐸) ⊂ 𝜎({𝑈 𝛼 𝑓 : 𝛼 > 0, 𝑓 ∈ 𝒞𝑢 (𝐸, 𝑑)}).

(A.22)

If (𝑈 𝛼 ) satisfies (A.22), the family {𝑈 𝛼 𝑓 : 𝑓 ∈ 𝒞𝑢 (𝐸, 𝑑)} clearly separates the points of 𝐸. The condition is satisfied if (𝑈 𝛼 ) is generated by a conservative right semigroup. For, in this case, we have 𝛼𝑈 𝛼 𝑓 → 𝑓 pointwise as 𝛼 → ∞ for every 𝑓 ∈ 𝒞𝑢 (𝐸, 𝑑). A set 𝒴 ⊂ pbℬ𝑢 (𝐸) is called a rational cone if it is closed under positive rational linear combinations. For 𝒟 ⊂ pbℬ𝑢 (𝐸), we denote by 𝑞(𝒟) the rational cone generated by 𝒟, that is, the smallest rational cone containing 𝒟. For a rational cone 𝒴 ⊂ pbℬ𝑢 (𝐸), set 𝜆(𝒴) = { 𝑓1 ∧ · · · ∧ 𝑓𝑛 : 𝑛 ≥ 1, 𝑓𝑖 ∈ 𝒴} and 𝑢(𝒴) = {𝑈 𝛼1 𝑓1 + · · · + 𝑈 𝛼𝑛 𝑓𝑛 : 𝑛 ≥ 1, 𝑓𝑖 ∈ 𝒴 and strictly positive rationals 𝛼𝑖 }. It is obvious that 𝑢(𝒴) is a rational cone contained in the cone ∪ 𝛼>0 b𝒮˜ 𝛼 . That 𝜆(𝒴) is also a rational cone comes from the trivial identities (∧𝑖 𝑎 𝑖 ) + 𝑏 = ∧𝑖 (𝑎 𝑖 + 𝑏) and (∧𝑖 𝑎 𝑖 ) + (∧ 𝑗 𝑏 𝑗 ) = ∧𝑖, 𝑗 (𝑎 𝑖 + 𝑏 𝑗 ). For a given function class 𝒟 ⊂ pbℬ𝑢 (𝐸), we set ℛ0 = 𝑢(𝑞(𝒟)) and set ℛ𝑛 = 𝜆(ℛ𝑛−1 + 𝑢(ℛ𝑛−1 )) for 𝑛 ≥ 1 inductively, where ℛ𝑛−1 + 𝑢(ℛ𝑛−1 ) = { 𝑓 + 𝑔 : 𝑓 ∈ ℛ𝑛−1 and 𝑔 ∈ 𝑢(ℛ𝑛−1 )}. The set ℛ(𝒟) := ∪𝑛≥0 ℛ𝑛 is called the rational

A.4 Ray–Knight Completion

427

Ray cone generated by (𝑈 𝛼 ) 𝛼>0 and 𝒟; see Getoor (1975, p. 58) and Sharpe (1988, p. 90). The rational Ray cone ℛ = ℛ(𝒟) generated by (𝑈 𝛼 ) 𝛼>0 and 𝒟 ⊂ pbℬ𝑢 (𝐸) is the smallest rational cone contained in pbℬ𝑢 (𝐸) such that: (1) 𝑈 𝛼 (ℛ) ⊂ ℛ for all rationals 𝛼 > 0; (2) 𝑓 , 𝑔 ∈ ℛ implies 𝑓 ∧ 𝑔 ∈ ℛ; (3) ℛ contains 𝑢(𝑞(𝒟)). Clearly, for each 𝑓 ∈ ℛ(𝒟) there is a constant 𝛽 = 𝛽( 𝑓 ) > 0 such that 𝑓 is a 𝛽-supermedian function for (𝑈 𝛼 ) 𝛼>0 . Furthermore, if (𝑈 𝛼 ) 𝛼>0 is the resolvent associated with a conservative right semigroup (𝑃𝑡 )𝑡 ≥0 on (𝐸, 𝑑), for each 𝑓 ∈ ℛ(𝒟) there exists a 𝛽 = 𝛽( 𝑓 ) > 0 such that 𝑓 is 𝛽-excessive for (𝑃𝑡 )𝑡 ≥0 . Proposition A.27 (Sharpe, 1988, p. 90) If 𝒟 is a countable uniformly dense subset of p𝒞𝑢 (𝐸, 𝑑) and contains the constant function 1𝐸 , then the rational Ray cone ℛ = ℛ(𝒟) is countable, contains the positive rational constant functions, and separates the points of 𝐸. In the remainder of this section, we assume 𝒟 ⊂ p𝒞𝑢 (𝐸, 𝑑) satisfies the conditions of Proposition A.27. Recall that ∥ · ∥ denotes the supremum norm. We give the rational Ray cone ℛ = ℛ(𝒟) an enumeration {𝑔0 , 𝑔1 , 𝑔2 , . . .} with 𝑔0 = 0. Clearly, 𝜌(𝑥, 𝑦) =

∞ ∑︁ |𝑔𝑛 (𝑥) − 𝑔𝑛 (𝑦)| , 2𝑛 ∥𝑔𝑛 ∥ 𝑛=1

𝑥, 𝑦 ∈ 𝐸,

(A.23)

¯ 𝜌) defines a metric 𝜌 on 𝐸, and each 𝑔𝑛 is 𝜌-uniformly continuous. Let ( 𝐸, ¯ denote the completion of (𝐸, 𝜌). Observe that the map 𝑥 → ↦ (𝑔 (𝑥)) of 𝐸 into 𝐾 := 𝑛 𝑛≥1 Î∞ 𝑛=1 [0, ∥𝑔 𝑛 ∥] with the metric 𝑞 defined by 𝑞(𝑎, 𝑏) =

∞ ∑︁ |𝑎 𝑛 − 𝑏 𝑛 | , 2𝑛 ∥𝑔𝑛 ∥ 𝑛=1

𝑎, 𝑏 ∈ 𝐾,

¯ 𝜌) is an isometry. It follows that the completion ( 𝐸, ¯ is compact. The topology on 𝐸 induced by the metric 𝜌 is called the Ray topology of (𝑈 𝛼 ) 𝛼>0 . Proposition A.28 (Sharpe, 1988, p. 91) Each function 𝑓 ∈ 𝒞𝑢 (𝐸, 𝜌) extends to a ¯ 𝜌). unique 𝑓¯ ∈ 𝒞𝑢 ( 𝐸, ¯ For each 𝛼 > 0, we have 𝑈 𝛼 (𝒞𝑢 (𝐸, 𝜌)) ⊂ 𝒞𝑢 (𝐸, 𝜌) and 𝛼 𝑈 (𝒞𝑢 (𝐸, 𝑑)) ⊂ 𝒞𝑢 (𝐸, 𝜌), and 𝒞𝑢 (𝐸, 𝜌) is the uniform closure of ℛ − ℛ := { 𝑓 − 𝑔 : 𝑓 , 𝑔 ∈ ℛ}. Proposition A.29 (Sharpe, 1988, p. 91) If 𝑈 𝛼 (𝒞𝑢 (𝐸, 𝑑)) ⊂ 𝒞𝑢 (𝐸, 𝑑) for all 𝛼 > 0, then the Ray topology is coarser than the original topology. Proposition A.30 (Sharpe, 1988, p. 92) Let ℬ𝑟 (𝐸) denote the 𝜎-algebra on 𝐸 generated by the Ray topology. Then ℬ(𝐸) ⊂ ℬ𝑟 (𝐸) ⊂ ℬ𝑢 (𝐸) and 𝑈 𝛼 ℬ𝑟 (𝐸) ⊂ ℬ𝑟 (𝐸) for every 𝛼 > 0.

428

A Markov Processes

¯ so (𝐸, 𝜌) is Proposition A.31 (Sharpe, 1988, pp. 92–93) We have 𝐸 ∈ ℬ𝑢 ( 𝐸), 𝛼 a Radon space. If (𝐸, 𝑑) is Lusin and if (𝑈 ) 𝛼>0 maps bℬ(𝐸) into itself, then ¯ so (𝐸, 𝜌) is a Lusin space. ℬ(𝐸) = ℬ𝑟 (𝐸) and 𝐸 ∈ ℬ( 𝐸), ¯ 𝜌) ¯ we clearly have 𝑓 := 𝑓¯| 𝐸 ∈ 𝒞𝑢 (𝐸, 𝜌). Then 𝑈 𝛼 𝑓 ∈ For every 𝑓¯ ∈ 𝒞𝑢 ( 𝐸, 𝒞𝑢 (𝐸, 𝜌) by Proposition A.28 and so 𝑈 𝛼 𝑓 extends continuously to some (𝑈 𝛼 𝑓 ) − ∈ ¯ 𝜌). ¯ 𝜌) ¯ by ¯ Define the operators (𝑈¯ 𝛼 ) 𝛼>0 on 𝒞𝑢 ( 𝐸, 𝒞𝑢 ( 𝐸, 𝑈¯ 𝛼 𝑓¯ = (𝑈 𝛼 𝑓 ) − ,

¯ 𝜌). 𝛼 > 0, 𝑓¯ ∈ 𝒞𝑢 ( 𝐸, ¯

¯ ℬ( 𝐸)) ¯ such that By the Riesz representation theorem, there is a kernel 𝑈¯ 𝛼 on ( 𝐸, ∫ ¯ 𝜌). ¯ 𝑓¯ ∈ 𝒞𝑢 ( 𝐸, ¯ 𝑈¯ 𝛼 𝑓¯(𝑥) = 𝑓¯(𝑦)𝑈¯ 𝛼 (𝑥, d𝑦), 𝑥 ∈ 𝐸, 𝐸¯

A continuity argument shows that (𝑈¯ 𝛼 ) 𝛼>0 satisfies the resolvent equation (A.6). Theorem A.32 (Sharpe, 1988, p. 93) For 𝛼 > 0 and 𝑥 ∈ 𝐸 the measure 𝑈¯ 𝛼 (𝑥, ·) ¯ and its restriction to 𝐸 is 𝑈 𝛼 (𝑥, ·). Moreover, the family is carried by 𝐸 ∈ ℬ𝑢 ( 𝐸) 𝛼 ¯ ¯ (𝑈 ) 𝛼>0 is a Ray resolvent on the space 𝐸. ¯ 𝜌) ¯ constructed We call (𝑈¯ 𝛼 ) 𝛼>0 the Ray extension of (𝑈 𝛼 ) 𝛼>0 . The space ( 𝐸, above is called the Ray–Knight completion of (𝐸, 𝜌) with respect to (𝑈 𝛼 ) 𝛼>0 . It depends not only on 𝐸, 𝑑 and (𝑈 𝛼 ) 𝛼>0 but also on the choice of the family 𝒟 ⊂ p𝒞𝑢 (𝐸, 𝑑). If (𝑈 𝛼 ) 𝛼>0 is the resolvent associated with a conservative right ¯ 𝜌) ¯ the Ray–Knight completion of (𝐸, 𝜌) with semigroup (𝑃𝑡 )𝑡 ≥0 , we also call ( 𝐸, respect to (𝑃𝑡 )𝑡 ≥0 . In this case, the Ray semigroup ( 𝑃¯𝑡 )𝑡 ≥0 associated with (𝑈¯ 𝛼 ) 𝛼 ≥0 is called the Ray extension of (𝑃𝑡 )𝑡 ≥0 . Theorem A.33 (Sharpe, 1988, p. 94) Let (𝑃𝑡 )𝑡 ≥0 be a conservative right semigroup on 𝐸. Then there is a realization 𝜉 = (Ω, 𝒢, 𝒢𝑡 , 𝜉𝑡 , 𝜃 𝑡 , P 𝑥 ) of (𝑃𝑡 )𝑡 ≥0 which is a right process in both (𝐸, 𝑑) and (𝐸, 𝜌) and the left limit 𝜉𝑡− := lim𝑠↑𝑡 𝜉 𝑠 taken in the Ray topology exists in 𝐸¯ for all 𝑡 > 0. Theorem A.34 Suppose that (𝑃𝑡 )𝑡 ≥0 is a conservative Borel right semigroup on a Lusin topological space 𝐸. Then every right continuous realization of (𝑃𝑡 )𝑡 ≥0 with the augmented natural 𝜎-algebras is a right process. In particular, the semigroup can be realized canonically on the space of right continuous paths from [0, ∞) to 𝐸. Proof Let 𝜉 be a right process with semigroup (𝑃𝑡 )𝑡 ≥0 . Then each 𝑓 ∈ 𝒮 𝛼 is a nearly Borel function of 𝜉 relative to the Ray topology; see Sharpe (1988, p. 95). By Proposition A.31, we have ℬ(𝐸) = ℬ𝑟 (𝐸), so each 𝑓 ∈ 𝒮 𝛼 is nearly Borel in the original topology. Then the result follows by Sharpe (1988, p. 98). □ If (𝑃𝑡 )𝑡 ≥0 is a right semigroup on (𝐸, 𝑑) not necessarily conservative, the associated resolvent (𝑈 𝛼 ) 𝛼>0 is not necessarily Markov. In this case, we let 𝐸˜ = 𝐸 ∪{𝜕} for ˜ be a topological extension of (𝐸, 𝑑) with 𝜕 being ˜ 𝑑) an abstract point 𝜕 ∉ 𝐸. Let ( 𝐸, an isolated point and let ( 𝑃˜𝑡 )𝑡 ≥0 denote the conservative extension of (𝑃𝑡 )𝑡 ≥0 on 𝐸˜

A.5 Entrance Space and Entrance Laws

429

with 𝜕 being a cemetery. Let (𝑈˜ 𝛼 ) 𝛼>0 denote the resolvent associated with ( 𝑃˜𝑡 )𝑡 ≥0 . ˜ which is a count˜ be the rational Ray cone for (𝑈˜ 𝛼 ) 𝛼>0 constructed from 𝒟, Let ℛ ˜ ˜ able uniformly dense subset of p𝒞𝑢 ( 𝐸, 𝑑) and contains the constant function 1𝐸˜ . ˜ 𝑈˜ 𝛼 , 𝑃˜𝑡 ). ¯ 𝜌, ˜ 𝑑, ¯ 𝑈¯ 𝛼 , 𝑃¯𝑡 ) be the corresponding Ray–Knight completion of ( 𝐸, Let ( 𝐸, Proposition A.35 In the situation described above, if there are constants 𝛼 > 0 and ¯ 𝜀 > 0 such that 𝑈 𝛼 1𝐸 (𝑥) ≥ 𝜀 for all 𝑥 ∈ 𝐸, then 𝜕 is an isolated point of 𝐸. ˜ such ˜ is uniformly dense in p𝒞𝑢 ( 𝐸, ˜ there is a function 𝑔˜ ∈ 𝒟 ˜ 𝑑), Proof Since 𝒟 ≥ 1 for every 𝑥 ∈ 𝐸. Fix a rational 𝛽 ∈ (𝛼/2, 𝛼). Then that 𝑔(𝜕) ˜ ˜ < 𝛼𝜀/2 and 𝑔(𝑥) ˜ by the construction of ℛ. ˜ Since 𝜕 is a cemetery for ( 𝑃˜𝑡 )𝑡 ≥0 , we 𝑓˜ := 𝑈˜ 𝛽 𝑔˜ ∈ ℛ −1 ˜ ˜ have 𝑓 (𝜕) = 𝛽 𝑔(𝜕) < 𝜀. However, for every 𝑥 ∈ 𝐸 we have 𝑓˜(𝑥) = 𝑈˜ 𝛽 𝑔(𝑥) ≥ ˜ 𝛼 ¯ ¯ It follows that 𝑈 1𝐸 (𝑥) ≥ 𝜀. Let 𝑓 be the unique continuous extension of 𝑓˜ to 𝐸. ¯ 𝑓¯(𝑥) ≥ 𝜀 for every 𝑥 ∈ 𝐸¯ \ {𝜕}. Then the point 𝜕 must be isolated in 𝐸. □ By Proposition A.35, if (𝑃𝑡 )𝑡 ≥0 is a conservative right semigroup, then 𝜕 is an ¯ In that case, the topology of 𝐸 inherited from 𝐸¯ coincides with isolated point of 𝐸. its Ray topology defined directly by (𝑃𝑡 )𝑡 ≥0 . In the general case, we also call the inherited topology of 𝐸 the Ray topology of (𝑃𝑡 )𝑡 ≥0 .

A.5 Entrance Space and Entrance Laws Let 𝜉 = (Ω, 𝒢, 𝒢𝑡 , 𝜉𝑡 , 𝜃 𝑡 , P 𝑥 ) be a right process on the Radon topological space 𝐸 with transition semigroup (𝑃𝑡 )𝑡 ≥0 and resolvent (𝑈 𝛼 ) 𝛼>0 . We first assume (𝑃𝑡 )𝑡 ≥0 is conservative. Let 𝑑 be a metric for the topology of 𝐸 such that the 𝑑-completion ¯ 𝜌, of 𝐸 is compact. Let ( 𝐸, ¯ 𝑈¯ 𝛼 ) be a Ray–Knight completion of (𝐸, 𝑑, 𝑈 𝛼 ) and let ( 𝑃¯𝑡 )𝑡 ≥0 be the Ray extension of (𝑃𝑡 )𝑡 ≥0 . Set 𝐸 𝑅 = {𝑥 ∈ 𝐸¯ : 𝑈¯ 1 (𝑥, ·) is carried by 𝐸 }, which is called the Ray space for 𝜉 or (𝑃𝑡 )𝑡 ≥0 . It was proved in Sharpe (1988, ¯ is a Radon topological space and 𝐸 ⊂ 𝐸 𝑅 . By the resolvent p. 191) that 𝐸 𝑅 ∈ ℬ𝑢 ( 𝐸) equation we have 𝐸 𝑅 = {𝑥 ∈ 𝐸¯ : 𝑈¯ 𝛼 (𝑥, ·) is carried by 𝐸 } for each 𝛼 > 0. Theorem A.36 (Sharpe, 1988, p. 191) Let ( 𝐸¯ 1 , 𝜌¯ 1 , 𝑈¯ 1𝛼 ) and ( 𝐸¯ 2 , 𝜌¯ 2 , 𝑈¯ 2𝛼 ) be Ray– Knight completions of (𝐸, 𝑑1 , 𝑈 𝛼 ) and (𝐸, 𝑑2 , 𝑈 𝛼 ) respectively, where 𝑑1 and 𝑑2 are totally bounded metrics for the original topology of 𝐸. Then the corresponding Ray spaces 𝐸 𝑅1 and 𝐸 𝑅2 are homeomorphic under a mapping 𝜓 : 𝐸 𝑅1 → 𝐸 𝑅2 satisfying 𝑈¯ 1𝛼 (𝑥, 𝐵) = 𝑈¯ 2𝛼 (𝜓(𝑥), 𝐵) for all 𝛼 > 0 and 𝐵 ∈ ℬ(𝐸). Therefore, the Ray space 𝐸 𝑅 together with the resolvent (𝑈¯ 𝛼 ) 𝛼>0 restricted to 𝐸 𝑅 is uniquely determined, up to homeomorphism, by the original topology on 𝐸 and (𝑈 𝛼 ) 𝛼>0 . This makes the Ray space a natural object. Let 𝐷 denote the set of ¯ and let 𝐸 𝐷 = 𝐷 ∩ 𝐸 𝑅 = {𝑥 ∈ 𝐸 𝑅 : 𝑃¯0 (𝑥, ·) = non-branch points of ( 𝑃¯𝑡 )𝑡 ≥0 on 𝐸, ¯ we have 𝛿 𝑥 (·)} which is called the entrance space for (𝑃𝑡 )𝑡 ≥0 . Since 𝐷 ∈ ℬ( 𝐸), 𝐸 𝐷 ∈ ℬ(𝐸 𝑅 , 𝜌).

430

A Markov Processes

Proposition A.37 (Sharpe, 1988, pp. 192–193) We have: (1) For 𝑥 ∈ 𝐸 𝑅 and 𝑡 > 0, 𝑃¯𝑡 (𝑥, ·) is carried by 𝐸. (2) For 𝑥 ∈ 𝐸 𝑅 , 𝑃¯0 (𝑥, ·) is carried by 𝐸 𝐷 . ¯ (3) For 𝑥 ∈ 𝐵, 𝑃¯0 (𝑥, ·) is not concentrated at any point of 𝐸. The restriction (𝑄 𝑡 )𝑡 ≥0 of ( 𝑃¯𝑡 )𝑡 ≥0 to (𝐸 𝐷 , 𝜌) is a right semigroup, and 𝐸 𝐷 \ 𝐸 is quasi-polar for any realization 𝑌 of (𝑄 𝑡 )𝑡 ≥0 as a right process, that is, for every initial law 𝜇 on (𝐸 𝐷 , 𝜌) the Q 𝜇 -outer measure of {𝜔 : 𝑌𝑡 (𝜔) ∈ 𝐸 for all 𝑡 > 0} is equal to one; see Sharpe (1988, p. 193). The following theorem gives a complete characterization of probability entrance laws for a conservative right semigroup. Theorem A.38 (Sharpe, 1988, p. 196) For every probability entrance law (𝜂𝑡 )𝑡 >0 for (𝑃𝑡 )𝑡 ≥0 on 𝐸, there is a unique probability measure 𝜂0 on ℬ𝑢 (𝐸 𝐷 , 𝜌) such that 𝜂𝑡 = 𝜂0 𝑃¯𝑡 for every 𝑡 > 0. Corollary A.39 If 𝐸 is a locally compact separable metric space and (𝑃𝑡 )𝑡 ≥0 is a Feller semigroup on 𝐸, then all probability entrance laws for (𝑃𝑡 )𝑡 ≥0 are closable. Proof We assume 𝐸 is not compact, for otherwise the proof is easier. Let 𝐸¯ = 𝐸 ∪{𝜕} be a one-point compactification of 𝐸. Then 𝐸¯ is compact and separable, so it is metrizable. Let 𝑑¯ be a metric on 𝐸¯ compatible with its topology and let 𝑑 be the restriction of 𝑑¯ to 𝐸. It is easy to see that the Ray–Knight completion of 𝐸 given by 𝑑 and (𝑃𝑡 )𝑡 ≥0 coincides with 𝐸¯ and the entrance space is just 𝐸. Then the result □ follows from Theorem A.38. In the remainder of this section, let 𝐸 be a Lusin topological space and consider a Borel right semigroup (𝑃𝑡 )𝑡 ≥0 on 𝐸, which is not necessarily conservative. Let 𝐸 𝜕 = 𝐸 ∪ {𝜕} be the topological extension of 𝐸 with 𝜕 being an isolated point. Let ˆ be the space of paths 𝑤 : R → 𝐸 𝜕 such that 𝑡 ↦→ 𝑤 𝑡 takes values in 𝐸 and is right Ω continuous in a nonvoid interval (𝛼(𝑤), 𝛽(𝑤)) or [𝛼(𝑤), 𝛽(𝑤)) ⊂ R and takes the ˆ generated by value 𝜕 elsewhere. Let (ℱ 0 , ℱ𝑡0 )𝑡 ∈R be the natural 𝜎-algebras on Ω ˆ : 𝛼(𝑤) = 𝑠}. ˆ 𝑠 = {𝑤 ∈ Ω the coordinate process. For 𝑠 ∈ [−∞, ∞) let Ω Theorem A.40 (Dellacherie et al., 1992; Getoor and Glover, 1987) For an entrance rule (𝜂𝑡 )𝑡 ∈R for (𝑃𝑡 )𝑡 ≥0 there exists a Radon measure 𝜌(d𝑠) on R and a countable set 𝑇 ⊂ R such that ∫ ∑︁ −∞ (A.24) 𝜂𝑡 = 𝜈𝑡 + 𝜈𝑡𝑠 𝜌(d𝑠) + 𝜇𝑡𝑠 , 𝑡 ∈ R, (−∞,𝑡)

𝑠 ∈𝑇∩(−∞,𝑡 ]

where (𝜈𝑡𝑠 )𝑡 >𝑠 is an entrance law at 𝑠 ∈ [−∞, ∞) and (𝜇𝑡𝑠 )𝑡 ≥𝑠 is a closed entrance law at 𝑠 ∈ 𝑇. Proof In the special case where (𝜂𝑡 )𝑡 ∈R is a regular entrance rule, the representation (A.24) with 𝑇 = ∅ was established in Theorem 2.33 of Getoor and Glover (1987, p. 57). In the general case, since 𝑠 ↦→ 𝜂 𝑠 𝑃𝑡−𝑠 is increasing, we have

A.5 Entrance Space and Entrance Laws

431

𝜂ˆ𝑡 := lim 𝜂 𝑠 𝑃𝑡−𝑠 = sup 𝜂 𝑠 𝑃𝑡−𝑠 ≤ 𝜂𝑡 . 𝑠↑𝑡

𝑠 𝜂ˆ𝑟 implies 𝑟 ∈ 𝑇. It follows that ∫ ∑︁ 𝜂𝑡 = (𝜂𝑡 − 𝜂ˆ𝑡 ) + 𝜈𝑡−∞ + 𝜆({𝑟 })𝛾𝑡𝑟 𝛾𝑡𝑠 𝑚(d𝑠) + (−∞,𝑡)

=

𝜈𝑡−∞

𝑟 ∈𝑇∩(−∞,𝑡)

∫ 𝛾𝑡𝑠 𝑚(d𝑠)

+

∑︁

+

(−∞,𝑡)

𝑟 ∈𝑇∩(−∞,𝑡)

∑︁

𝜅 𝑡𝑟 +

𝜇𝑟𝑡 ,

𝑟 ∈𝑇∩(−∞,𝑡 ]

where 𝜇𝑟𝑡 = (𝜂𝑟 − 𝜂ˆ𝑟 )𝑃𝑡−𝑟 . Then by setting 𝜈𝑡𝑠 = 1𝑇 𝑐 (𝑠)𝛾𝑡𝑠 + 1𝑇 (𝑠)𝜅 𝑡𝑠

and

𝜌(d𝑠) = 𝑚(d𝑠) +

∑︁

𝛿𝑟 (d𝑠)

𝑟 ∈𝑇

we obtain (A.24).

□

Theorem A.41 (Dellacherie et al., 1992; Getoor and Glover, 1987) To an entrance ˆ ℱ0) rule (𝜂𝑡 )𝑡 ∈R for (𝑃𝑡 )𝑡 ≥0 there corresponds a unique 𝜎-finite measure Q 𝜂 on ( Ω, such that Q 𝜂 {𝑤 𝑡1 ∈ d𝑥 1 , 𝑤 𝑡2 ∈ d𝑥2 , . . . , 𝑤 𝑡𝑛 ∈ d𝑥 𝑛 } = 𝜂𝑡1 (d𝑥1 )𝑃𝑡2 −𝑡1 (𝑥1 , d𝑥 2 ) · · · 𝑃𝑡𝑛 −𝑡𝑛−1 (𝑥 𝑛−1 , d𝑥 𝑛 )

(A.27)

for all {𝑡1 < · · · < 𝑡 𝑛 } ⊂ R and {𝑥1 , . . . , 𝑥 𝑛 } ⊂ 𝐸. The measure Q 𝜂 defined by (A.27) is called the Kuznetsov measure corresponding to the entrance rule (𝜂𝑡 )𝑡 ∈R . The property roughly means that {𝑤 𝑡 : 𝑡 ∈ R} is a Markov process under Q 𝜂 with transition semigroup (𝑃𝑡 )𝑡 ≥0 and one-dimensional distributions (𝜂𝑡 )𝑡 ∈R . The existence of this type of measure was first studied by Kuznetsov (1973). By Theorem A.40 we can assume (𝜂𝑡 )𝑡 ∈R is given by (A.24). From the closed entrance law (𝜇𝑡𝑠 )𝑡 ≥𝑠 at 𝑠 ∈ 𝑇 it is easy to construct the corresponding ˆ 𝑠 : 𝑤 𝑠 ∈ 𝐸 }. In fact, we have P𝑠 (𝑤 𝑠 ∈ Kuznetsov measure P𝑠 carried by {𝑤 ∈ Ω 𝑠 d𝑥) = 𝜇 𝑠 (d𝑥) for 𝑥 ∈ 𝐸. By Proposition 3.5 in Getoor and Glover (1987, p. 61), the ˆ −∞ Kuznetsov measure Q−∞ of the entrance law (𝜈𝑡−∞ )𝑡 >−∞ at −∞ is carried by Ω 𝑠 𝑠 and the Kuznetsov measure Q of the entrance law (𝜈𝑡 )𝑡 >𝑠 at 𝑠 ∈ R is carried by ˆ 𝑠 : 𝑤 𝑠 = 𝜕}. A representation of the measure Q 𝜂 is given by {𝑤 ∈ Ω ∫ ∑︁ (A.28) Q 𝜂 (d𝑤) = Q−∞ (d𝑤) + Q𝑠 (d𝑤) 𝜌(d𝑠) + P𝑠 (d𝑤). R

𝑠 ∈𝑇

We refer to Dellacherie et al. (1992) and Getoor (1990) for the theory of Kuznetsov measures. Example A.2 Let (𝑃𝑡 )𝑡 ≥0 be the transition semigroup of the absorbing-barrier Brownian motion in (0, ∞). For any 𝑡 > 0 the kernel 𝑃𝑡 (𝑥, d𝑦) has density 𝑝 𝑡 (𝑥, 𝑦) = 𝑔𝑡 (𝑥 − 𝑦) − 𝑔𝑡 (𝑥 + 𝑦),

𝑥, 𝑦 > 0,

(A.29)

A.6 Concatenations and Weak Generators

433

where 𝑔𝑡 (𝑧) = √

1 2𝜋𝑡

exp{−𝑧 2 /2𝑡},

𝑡 > 0, 𝑧 ∈ R.

We can define an entrance law (𝜅 𝑡 )𝑡 >0 for (𝑃𝑡 )𝑡 ≥0 by ∫ 2 ∞ d 𝜅𝑡 ( 𝑓 ) = 𝑦𝑔𝑡 (𝑦) 𝑓 (𝑦)d𝑦 = 𝑃𝑡 𝑓 (0+), 𝑡 0 d𝑥

𝑓 ∈ bℬ(0, ∞).

(A.30)

(A.31)

The corresponding Kuznetsov measure n(d𝑤) is called Itô’s excursion law, which is carried by the set of positive continuous paths {𝑤 𝑡 : 𝑡 > 0} such that 𝑤 0+ = 𝑤 𝑡 = 0 for every 𝑡 ≥ 𝜏0 (𝑤) := inf{𝑠 > 0 : 𝑤 𝑠 = 0}; see, e.g., Ikeda and Watanabe (1989, p. 124).

A.6 Concatenations and Weak Generators Suppose that 𝐸 is a Lusin topological space and (𝑃𝑡 )𝑡 ≥0 is a Borel right semigroup on this space. We consider a right process 𝜉 = (Ω, 𝒢, 𝒢𝑡 , 𝜉𝑡 , 𝜃 𝑡 , P 𝑥 ) with transition semigroup (𝑃𝑡 )𝑡 ≥0 . Let (ℱ, ℱ𝑡 ) be the augmentations of the ℬ(𝐸)-natural 𝜎algebras (ℱ 0 , ℱ𝑡0 ) generated by {𝜉𝑡 : 𝑡 ≥ 0}. A right continuous (ℱ𝑡 )-adapted increasing process {𝐾 (𝑡) : 𝑡 ≥ 0} is called an additive functional of 𝜉 if 𝐾0 = 0 and for every bounded (ℱ𝑡 )-stopping time 𝑇 we have a.s. 𝐾𝑇+𝑡 = 𝐾𝑇 + 𝐾𝑡 ◦ 𝜃 𝑇 ,

𝑡 ≥ 0.

(A.32)

Here the lifetime does not enter the formulation. Therefore, an additive functional of a process with possibly finite lifetime is simply an additive functional of its conservative extension in 𝐸 ∪ {𝜕} with 𝜕 being an isolated cemetery. Clearly, an additive functional {𝐾 (𝑡) : 𝑡 ≥ 0} defines a 𝜎-finite random measure 𝐾 (d𝑠) on [0, ∞). For any 𝛽 ∈ bℬ(𝐸) write ∫ 𝐾𝑡 (𝛽) = 𝛽(𝜉 𝑠 )𝐾 (d𝑠), 𝑡 ≥ 0. [0,𝑡 ]

We say a real or complex function (𝑡, 𝑥) ↦→ 𝑓 (𝑡, 𝑥) defined on the product space [0, ∞) × 𝐸 is locally bounded provided sup sup | 𝑓 (𝑠, 𝑥)| < ∞,

𝑡 ≥ 0.

0≤𝑠 ≤𝑡 𝑥 ∈𝐸

A real or complex stochastic process (𝑋𝑡 )𝑡 ≥0 is said to be locally bounded if (𝑡, 𝜔) ↦→ 𝑋𝑡 (𝜔) is a locally bounded function on [0, ∞) × Ω. Given an additive functional {𝐾 (𝑡) : 𝑡 ≥ 0}, we denote by bℰ(𝐾) the set of functions 𝛽 ∈ bℬ(𝐸) such that 𝑡 ↦→ e−𝐾𝑡 (𝛽) is a locally bounded stochastic process. Note that bℰ(𝐾) ⊃ pbℬ(𝐸). We say an additive functional {𝐾 (𝑡) : 𝑡 ≥ 0} is

434

A Markov Processes

admissible if each 𝜔 ↦→ 𝐾𝑡 (𝜔) is measurable with respect to the natural 𝜎-algebra ℱ 0 and   𝑘 (𝑡) := sup P 𝑥 𝐾 (𝑡) → 0, (A.33) 𝑡 → 0. 𝑥 ∈𝐸

In this case, the map (𝑡, 𝜔) ↦→ 𝐾𝑡 (𝜔) is measurable with respect to the product 𝜎-algebra ℬ[0, ∞) × ℱ 0 on [0, ∞) × Ω. In the sequel, we assume {𝐾 (𝑡) : 𝑡 ≥ 0} is a continuous admissible additive functional of 𝜉. Let 𝑏 ∈ bℰ(𝐾) and let 𝛾(𝑥, d𝑦) be a bounded Borel kernel on 𝐸. For 𝑓 ∈ bℬ(𝐸) we consider the linear evolution equation ∫ 𝑡    𝑞 𝑡 (𝑥) = P 𝑥 e−𝐾𝑡 (𝑏) 𝑓 (𝜉𝑡 ) + P 𝑥 e−𝐾𝑠 (𝑏) 𝛾(𝜉 𝑠 , 𝑞 𝑡−𝑠 )𝐾 (d𝑠) , (A.34) 0

where 𝑡 ≥ 0 and 𝑥 ∈ 𝐸. Recall that ∥ · ∥ denotes the supremum norm. Proposition A.42 For every 𝑓 ∈ bℬ(𝐸) there is a unique locally bounded Borel function (𝑡, 𝑥) ↦→ 𝑞 𝑡 (𝑥) on [0, ∞) × 𝐸 solving (A.34), which is given by  ∫ 𝑡   e−𝐾𝑠1 (𝑏) 𝐾 (d𝑠1 )P 𝜇𝑠1 𝑓 ( 𝜉𝑡−𝑠1 ) 𝑞𝑡 ( 𝑥) = P 𝑥 e−𝐾𝑡 (𝑏) 𝑓 ( 𝜉𝑡 ) + P 𝑥 0  ∫ 𝑡−𝜎1 ∫ 𝑡 ∞ ∑︁ −𝐾𝑠1 (𝑏) e−𝐾𝑠2 (𝑏) 𝐾 (d𝑠2 ) · · · e 𝐾 (d𝑠1 )P 𝜇𝑠1 + P𝑥 0 𝑖=2  ∫ 0   𝑡−𝜎𝑖−1   e−𝐾𝑠𝑖 (𝑏) 𝐾 (d𝑠𝑖 )P 𝜇𝑠𝑖 e−𝐾𝑡−𝜎𝑖 (𝑏) 𝑓 ( 𝜉𝑡−𝜎𝑖 ) · · · , P 𝜇𝑠𝑖−1 0

Í𝑖

where 𝜎𝑖 = 𝑗=1 𝑠 𝑗 and 𝜇 𝑠 = 𝛾(𝜉 𝑠 , ·). Moreover, the operators 𝜋𝑡 : 𝑓 ↦→ 𝑞 𝑡 form a locally bounded semigroup (𝜋𝑡 )𝑡 ≥0 . Proof For 𝑟 ≥ 0 it is not hard to see that (𝑡, 𝑥) ↦→ 𝑞 𝑡 (𝑥) satisfies (A.34) for 𝑡 ≥ 0 if and only if it satisfies the equation for 0 ≤ 𝑡 ≤ 𝑟 and (𝑡, 𝑥) ↦→ 𝑞 𝑟+𝑡 (𝑥) satisfies   𝑞𝑟+𝑡 ( 𝑥) = P 𝑥 e−𝐾𝑡 (𝑏) 𝑞𝑟 ( 𝜉𝑡 ) + P 𝑥

∫

𝑡

e−𝐾𝑠 (𝑏) 𝛾 ( 𝜉𝑠 , 𝑞𝑟+𝑡−𝑠 ) 𝐾 (d𝑠)

 (A.35)

0

for 𝑡 ≥ 0. Let 𝑡 ↦→ 𝑙 (𝑡) be an increasing deterministic function such that e−𝐾𝑡 (𝑏) ≤ 𝑙 (𝑡) for all 𝑡 ≥ 0. Fix a constant 𝛿 > 0 such that 𝑘 (𝛿)𝑙 (𝛿) ∥𝛾(·, 1) ∥ < 1. Observe that the 𝑖-th term of the series in the definition of 𝑞 𝑡 (𝑥) is bounded by 𝑘 (𝑡) 𝑖 𝑙 (𝑡) 𝑖 ∥𝛾(·, 1) ∥ 𝑖 ∥ 𝑓 ∥. Then the series converges uniformly on [0, 𝛿] × 𝐸. Since each 𝜔 ↦→ 𝐾𝑡 (𝜔) is measurable with respect to the natural 𝜎-algebra, it is easy to see that (𝑡, 𝑥) ↦→ 𝑞 𝑡 (𝑥) is jointly measurable and satisfies (A.34) on [0, 𝛿] × 𝐸. By the relation of (A.34) and (A.35) we can extend (𝑡, 𝑥) ↦→ 𝑞 𝑡 (𝑥) to a solution of (A.34) on [0, ∞) × 𝐸. Moreover, the operator 𝑓 ↦→ 𝑞 𝑡 determines a bounded Borel kernel 𝜋𝑡 (𝑥, d𝑦) on 𝐸 and (𝜋𝑡 )𝑡 ≥0 form a locally bounded semigroup. To show the uniqueness of the solution of (A.34), suppose that (𝑡, 𝑥) ↦→ 𝑣 𝑡 (𝑥) is a locally bounded solution of (A.34) with 𝑣 0 (𝑥) ≡ 0. It is easily seen that

A.6 Concatenations and Weak Generators

435

∫ ∥𝑣 𝑡 ∥ ≤ 𝑙 (𝑡) ∥𝛾(·, 1) ∥ sup P 𝑥 𝑥 ∈𝐸



𝑡

∥𝑣 𝑡−𝑠 ∥𝐾 (d𝑠) , 0

and hence sup ∥𝑣 𝑠 ∥ ≤ 𝑘 (𝑡)𝑙 (𝑡) ∥𝛾(·, 1) ∥ sup ∥𝑣 𝑠 ∥ 0≤𝑠 ≤𝑡

0≤𝑠 ≤𝑡

for every 𝑡 ≥ 0. Then we must have ∥𝑣 𝑡 ∥ = 0 for 0 ≤ 𝑡 ≤ 𝛿. Using the above procedure and the relation of (A.34) and (A.35) successively we get ∥𝑣 𝑡 ∥ = 0 for all 𝑡 ≥ 0. Since (A.34) is a linear equation, this gives the uniqueness of the solution. □ Proposition A.43 Let 𝑓 ∈ bℬ(𝐸) and let (𝑡, 𝑥) ↦→ 𝜋𝑡 𝑓 (𝑥) be defined by (A.34). Then 𝑡 ↦→ 𝜋𝑡 𝑓 (𝑥) is right continuous pointwise on 𝐸 if and only if so is 𝑡 ↦→ 𝑃𝑡 𝑓 (𝑥). Proof Clearly, the second term on the right-hand side of (A.34) tends to zero as 𝑡 → 0. Moreover, by (A.33) we have   lim P 𝑥 [(1 − e−𝐾𝑡 (𝑏) ) 𝑓 (𝜉𝑡 )] ≤ lim ∥ 𝑓 ∥P 𝑥 |1 − e−𝐾𝑡 (𝑏) | = 0. 𝑡→0

𝑡→0

It follows that   lim 𝜋𝑡 𝑓 (𝑥) = lim P 𝑥 e−𝐾𝑡 (𝑏) 𝑓 (𝜉𝑡 ) = lim 𝑃𝑡 𝑓 (𝑥), 𝑡→0

𝑡→0

𝑡→0

which means if one of the limits exists, so do the other two and the equalities hold. Then we get the result by the semigroup properties of (𝑃𝑡 )𝑡 ≥0 and (𝜋𝑡 )𝑡 ≥0 . □ Now suppose that 𝑏(𝑥) ≥ 𝛾(𝑥, 1) for every 𝑥 ∈ 𝐸. Let (𝜋𝑡 )𝑡 ≥0 be defined by (A.34). Since (𝑃𝑡 )𝑡 ≥0 is not conservative in general, we can only understand 𝜉 = (Ω, 𝒢, 𝒢𝑡 , 𝜉𝑡 , P 𝑥 ) as a right process in the extended state space 𝐸 ∪ {𝜕} with 𝜕 being an isolated cemetery. Let 𝐸¯ be a Ray–Knight completion of 𝐸 ∪{𝜕} relative to 𝜉. Then ¯ By Theorem A.33 we have P 𝑥 {the left limit Proposition A.31 implies 𝐸 ∈ ℬ( 𝐸). ˆ d𝑦) 𝜉𝑡− := lim𝑠↑𝑡 𝜉 𝑠 taken in the Ray topology exists in 𝐸¯ for all 𝑡 > 0} = 1. Let 𝛾(𝑥, be a sub-Markov kernel on 𝐸 satisfying 𝛾(𝑥, d𝑦) = 𝑏(𝑥) 𝛾(𝑥, ˆ d𝑦) ˆ d𝑦). We extend 𝛾(𝑥, ˆ {𝜕}) = 1 − 𝛾(𝑥, ˆ 𝐸). Fix 𝑥0 ∈ 𝐸 and to a Markov kernel from 𝐸 to 𝐸¯ by setting 𝛾(𝑥, ˆ 𝒢ˆ 𝑡 , 𝜉ˆ𝑡 , Pˆ 𝑥 ) be the let 𝑏(𝑥) = 𝑏(𝑥0 ) and 𝛾(𝑥, ˆ ·) = 𝛾(𝑥 ˆ 0 , ·) for 𝑥 ∈ 𝐸¯ \ 𝐸. Let 𝜉ˆ = (Ω, 𝒢, subprocess with lifetime 𝜁 constructed from 𝜉 and the strictly positive multiplicative functional 𝑡 ↦→ exp{−𝐾𝑡 (𝑏)}. Then 𝜉ˆ is also a right process; see Sharpe (1988, ˜ 𝒢˜ 𝑡 , 𝜉˜𝑡 , P˜ 𝑥 ) be the concatenation defined from an infinite ˜ 𝒢, p. 287). Let 𝜉˜ = ( Ω, ˆ 𝜁 ( 𝜔)− (𝜔), d𝑦) as in sequence of copies of 𝜉ˆ and the transfer kernel 𝜂(𝜔, d𝑦) := 𝛾(𝜉 Sharpe (1988, p. 79 and p. 82). The intuitive idea of this concatenation is described as follows. The process 𝜉˜ evolves as 𝜉 until time 𝜁, it is then revived by means of the kernel 𝜂, and evolves again as 𝜉 and so on. It is known that 𝜉˜ is also a right process; see Sharpe (1988, pp. 82–83). Suppose that every 𝑓 ∈ bℬ(𝐸) is extended trivially to 𝐸¯ \ 𝐸. Then we have the renewal equation ˜ P˜ 𝑥 [ 𝑓 ( 𝜉˜𝑡 )] = P 𝑥 [1 {𝑡 0 be the resolvent of the process 𝜉. We denote by b𝒞 𝜉 (𝐸) the set of functions 𝑓 ∈ bℬ(𝐸) that are finely continuous relative to 𝜉. By Theorem A.20 the function 𝑡 ↦→ 𝑃𝑡 𝑓 (𝑥) is right continuous pointwise for every 𝑓 ∈ b𝒞 𝜉 (𝐸). By Theorems A.16 and A.20 we have b𝒞 𝜉 (𝐸) ⊃ 𝑈 𝛼 bℬ(𝐸) for every 𝛼 > 0. Lemma A.45 The set of functions 𝑈 𝛽 b𝒞 𝜉 (𝐸) is independent of 𝛽 > 0. Moreover, if 𝑔1 , 𝑔2 ∈ b𝒞 𝜉 (𝐸) and 𝑈 𝛽 𝑔1 = 𝑈 𝛽 𝑔2 for some 𝛽 > 0, then 𝑔1 = 𝑔2 . Proof Let us consider two constants 𝛼, 𝛽 > 0. If 𝑓 ∈ 𝑈 𝛽 b𝒞 𝜉 (𝐸), we have 𝑓 = 𝑈 𝛽 𝑔 for some 𝑔 ∈ b𝒞 𝜉 (𝐸). Then the resolvent equation implies that 𝑓 = 𝑈 𝛼 𝑔 − (𝛽 − 𝛼)𝑈 𝛼𝑈 𝛽 𝑔 = 𝑈 𝛼 ℎ, where ℎ = 𝑔 − (𝛽 − 𝛼)𝑈 𝛽 𝑔 ∈ b𝒞 𝜉 (𝐸). It follows that 𝑈 𝛽 b𝒞 𝜉 (𝐸) ⊂ 𝑈 𝛼 b𝒞 𝜉 (𝐸). By symmetry we have 𝑈 𝛼 b𝒞 𝜉 (𝐸) ⊂ 𝑈 𝛽 b𝒞 𝜉 (𝐸). This proves the first assertion. Suppose that 𝑔1 , 𝑔2 ∈ b𝒞 𝜉 (𝐸) and 𝑈 𝛽 𝑔1 = 𝑈 𝛽 𝑔2 for some 𝛽 > 0. By the resolvent equation we have 𝑈 𝛼 𝑔1 = 𝑈 𝛼 𝑔2 for every 𝛼 > 0. Since 𝑡 ↦→ 𝑃𝑡 𝑔1 (𝑥) and 𝑡 ↦→ 𝑃𝑡 𝑔2 (𝑥) are right continuous, we have 𝑔1 = 𝑔2 by the uniqueness of Laplace transforms. □ Fix 𝛽 > 0 and let 𝒟( 𝐴) = 𝑈 𝛽 b𝒞 𝜉 (𝐸). For 𝑓 = 𝑈 𝛽 𝑔 ∈ 𝒟( 𝐴) with 𝑔 ∈ b𝒞 𝜉 (𝐸) set 𝐴 𝑓 = 𝛽 𝑓 − 𝑔, which is well-defined by Lemma A.45. We call ( 𝐴, 𝒟( 𝐴)) the weak generator of (𝑃𝑡 )𝑡 ≥0 . By the resolvent equation of (𝑈 𝛼 ) 𝛼>0 it is easy to show that ( 𝐴, 𝒟( 𝐴)) is independent of the choice of 𝛽 > 0. Clearly, for every 𝑓 ∈ 𝒟( 𝐴) we have 𝑓 ∈ b𝒞 𝜉 (𝐸) and 𝐴 𝑓 ∈ b𝒞 𝜉 (𝐸).

A.6 Concatenations and Weak Generators

437

Following Ethier and Kurtz (1986), we also define a multi-valued version of ˜ = 𝑈 𝛽 bℬ(𝐸) and for any 𝑓 ∈ 𝒟( 𝐴) ˜ let the weak generator as follows. Let 𝒟( 𝐴) 𝛽 ˜ ˜ ˜ 𝐴 𝑓 = {𝛽 𝑓 − 𝑔 : 𝑔 ∈ bℬ(𝐸) and 𝑈 𝑔 = 𝑓 }. It is easy to show that ( 𝐴, 𝒟( 𝐴)) is also independent of the choice of 𝛽 > 0. In particular, for any 𝑓 ∈ 𝒟( 𝐴) we have 𝐴 𝑓 ∈ 𝐴˜ 𝑓 . Proposition A.46 Let 𝛼 > 0. Then 𝑈 𝛼 (𝛼 − 𝐴) 𝑓 = 𝑓 for every 𝑓 ∈ 𝒟( 𝐴) and (𝛼 − 𝐴)𝑈 𝛼 𝑓 = 𝑓 for every 𝑓 ∈ b𝒞 𝜉 (𝐸). Proof For any 𝑓 ∈ 𝒟( 𝐴) there is a 𝑔 ∈ b𝒞 𝜉 (𝐸) such that 𝑓 = 𝑈 𝛽 𝑔. Then the definition of 𝐴 𝑓 and the resolvent equation yields 𝑈 𝛼 (𝛼 − 𝐴) 𝑓 = 𝑈 𝛼 (𝛼 𝑓 − 𝛽 𝑓 + 𝑔) = (𝛼 − 𝛽)𝑈 𝛼𝑈 𝛽 𝑔 + 𝑈 𝛼 𝑔 = 𝑈 𝛽 𝑔 = 𝑓 , giving the first assertion. For any 𝑓 ∈ b𝒞 𝜉 (𝐸) we first use the resolvent equation to see 𝑈 𝛼 𝑓 = 𝑈 𝛽 𝑓 + (𝛽 − 𝛼)𝑈 𝛽 𝑈 𝛼 𝑓 = 𝑈 𝛽 ℎ, where ℎ = 𝑓 + (𝛽 − 𝛼)𝑈 𝛼 𝑓 . Therefore (𝛼 − 𝐴)𝑈 𝛼 𝑓 = 𝛼𝑈 𝛼 𝑓 − 𝐴𝑈 𝛽 ℎ = 𝛼𝑈 𝛼 𝑓 − 𝛽𝑈 𝛽 ℎ + ℎ = 𝑓 . This gives the second assertion.

□

Theorem A.47 Let ( 𝐴, 𝒟( 𝐴)) be the weak generator of (𝑃𝑡 )𝑡 ≥0 . Then for 𝑓 ∈ 𝒟( 𝐴) we have ∫ 𝑡 𝑡 ≥ 0, 𝑥 ∈ 𝐸 . 𝑃𝑡 𝑓 (𝑥) = 𝑓 (𝑥) + 𝑃𝑠 𝐴 𝑓 (𝑥)d𝑠, (A.36) 0

Proof Suppose that 𝑓 = 𝑈 𝛽 𝑔 for 𝑔 ∈ b𝒞 𝜉 (𝐸). Then 𝑈 𝛼 𝐴 𝑓 = 𝛼𝑈 𝛼 𝑓 − 𝑓 for 𝛼 > 0 by Proposition A.46. Using this relation it is easy to show ∫ ∞ ∫ 𝑡 ∫ ∞ 1 e−𝛼𝑡 (𝑃𝑡 𝑓 − 𝑓 )d𝑡 = 𝑈 𝛼 𝐴 𝑓 = e−𝛼𝑡 d𝑡 𝑃𝑠 𝐴 𝑓 d𝑠. 𝛼 0 0 0 Since 𝑓 is finely continuous relative to 𝜉, the function 𝑡 ↦→ 𝑃𝑡 𝑓 (𝑥) is right continuous for every 𝑥 ∈ 𝐸. Then (A.36) follows by the uniqueness of Laplace transforms. □ Corollary A.48 Let ( 𝐴, 𝒟( 𝐴)) be the weak generator of (𝑃𝑡 )𝑡 ≥0 . Then for 𝑓 ∈ 𝒟( 𝐴) we have  1 𝑃𝑡 𝑓 (𝑥) − 𝑓 (𝑥) , 𝑡→0 𝑡

𝐴 𝑓 (𝑥) = lim

𝑥 ∈ 𝐸.

(A.37)

Proof Since 𝐴 𝑓 ∈ b𝒞 𝜉 (𝐸), the function 𝑠 ↦→ 𝑃𝑠 𝐴 𝑓 (𝑥) is right continuous pointwise. Then we get (A.37) from (A.36). □

438

A Markov Processes

In the remainder of this section we consider the semigroup (𝜋𝑡 )𝑡 ≥0 defined by (A.34) in the special case with 𝐾 (d𝑠) = d𝑠 being the Lebesgue measure. Given a function 𝑏 ∈ bℬ(𝐸), we define a locally bounded semigroup of Borel kernels (𝑃𝑡𝑏 )𝑡 ≥0 on 𝐸 by the following Feynman–Kac formula: i h ∫𝑡 (A.38) 𝑃𝑡𝑏 𝑓 (𝑥) = P 𝑥 e− 0 𝑏 ( 𝜉𝑠 )d𝑠 𝑓 (𝜉𝑡 ) , 𝑥 ∈ 𝐸, 𝑓 ∈ bℬ(𝐸). Then we can rewrite (A.34) as ∫ 𝜋𝑡 𝑓 (𝑥) =

𝑃𝑡𝑏

𝑡 𝑏 𝛾𝜋 𝑠 𝑓 (𝑥)d𝑠, 𝑃𝑡−𝑠

𝑓 (𝑥) +

𝑡 ≥ 0, 𝑥 ∈ 𝐸 .

(A.39)

0

Lemma A.49 (Gronwall’s inequality) Suppose that 𝑡 ↦→ 𝑔(𝑡) ≥ 0 and 𝑡 ↦→ ℎ(𝑡) are integrable functions on the interval [0, 𝑇]. If there is a constant 𝐶 > 0 such that ∫ 𝑡 (A.40) 𝑔(𝑡) ≤ ℎ(𝑡) + 𝐶 𝑔(𝑠)d𝑠, 0 ≤ 𝑡 ≤ 𝑇, 0

then ∫ 𝑔(𝑡) ≤ ℎ(𝑡) + 𝐶

𝑡

e𝐶 (𝑡−𝑠) ℎ(𝑠)d𝑠,

0 ≤ 𝑡 ≤ 𝑇.

(A.41)

0

Proof Let 𝑓 (𝑡) denote the right-hand side of (A.41). By integration by parts,  ∫ 𝑡 ∫ 𝑢 ∫ 𝑡 ∫ 𝑡 𝑓 (𝑠)d𝑠 = e−𝐶𝑠 ℎ(𝑠)d𝑠 d𝑢 ℎ(𝑠)d𝑠 + 𝐶 e𝐶𝑢 0 0 0∫ ∫ 𝑡 ∫0 𝑡 𝑡 𝐶𝑡 −𝐶𝑠 ℎ(𝑠)d𝑠 − ℎ(𝑠)d𝑠 e ℎ(𝑠)d𝑠 + e = 0 0 ∫0 𝑡 = e𝐶 (𝑡−𝑠) ℎ(𝑠)d𝑠. 0

It follows that ∫

𝑡

𝑓 (𝑠)d𝑠,

𝑓 (𝑡) = ℎ(𝑡) + 𝐶

0 ≤ 𝑡 ≤ 𝑇.

(A.42)

0

Let Δ(𝑡) = 𝑓 (𝑡) − 𝑔(𝑡). From (A.40) and (A.42) we have ∫ 𝑡 ∫ 𝑡 ∫ 𝑠 ∫ 𝑡 2 2 Δ(𝑟)d𝑟 = 𝐶 d𝑠 Δ(𝑠)d𝑠 ≥ 𝐶 (𝑡 − 𝑟)Δ(𝑟)d𝑟 Δ(𝑡) ≥ 𝐶 0 0 ∫ 𝑡 ∫0 𝑡 ∫ 𝑟 0 3 𝐶 (𝑡 − 𝑠) 2 Δ(𝑠)d𝑠 Δ(𝑠)d𝑠 = ≥ 𝐶3 (𝑡 − 𝑟)d𝑟 2 0 0 0 ≥ ··· ∫ 𝑡 𝐶𝑛 (𝑡 − 𝑠) 𝑛−1 Δ(𝑠)d𝑠. ≥ (𝑛 − 1)! 0 The right-hand side goes to zero as 𝑛 → ∞. Then Δ(𝑡) ≥ 0 and (A.41) follows.

□

A.6 Concatenations and Weak Generators

439

Proposition A.50 Let (𝑃𝑡𝑎 )𝑡 ≥0 be defined by (A.38) with 𝑏 replaced by 𝑎 ∈ bℬ(𝐸). Then for any 𝑓 ∈ bℬ(𝐸) the solution to (A.39) is also the unique locally bounded solution to ∫ 𝑡 𝑎 𝑃𝑡−𝑠 (𝛾 + 𝑎 − 𝑏)𝜋 𝑠 𝑓 (𝑥)d𝑠, 𝑡 ≥ 0, 𝑥 ∈ 𝐸 . (A.43) 𝜋𝑡 𝑓 (𝑥) = 𝑃𝑡𝑎 𝑓 (𝑥) + 0

Proof Let (𝑡, 𝑥) ↦→ 𝜋𝑡 𝑓 (𝑥) be the unique locally bounded solution of (A.39). We can use the Markov property of 𝜉 and Fubini’s theorem to write ∫ 𝑡 ∫ 𝑎 d𝑠 [𝑎(𝑦) − 𝑏(𝑦)]𝑃𝑠𝑏 𝑓 (𝑦)𝑃𝑡−𝑠 (𝑥, d𝑦) 0 𝐸 ∫ 𝑡 n ∫ 𝑡−𝑠 o  ∫𝑠 = P 𝑥 e− 0 𝑎 ( 𝜉𝑢 )d𝑢 [𝑎(𝜉𝑡−𝑠 ) − 𝑏(𝜉𝑡−𝑠 )]P 𝜉𝑡−𝑠 e− 0 𝑏 ( 𝜉𝑢 )d𝑢 𝑓 (𝜉 𝑠 ) d𝑠 ∫0 𝑡 n ∫ o ∫𝑡 𝑡−𝑠 = P 𝑥 e− 0 𝑎 ( 𝜉𝑢 )d𝑢 [𝑎(𝜉𝑡−𝑠 ) − 𝑏(𝜉𝑡−𝑠 )]e− 𝑡−𝑠 𝑏 ( 𝜉𝑢 )d𝑢 𝑓 (𝜉𝑡 ) d𝑠 0   ∫ 𝑡 ∫𝑡 ∫𝑡 [𝑎 ( 𝜉𝑢 )−𝑏 ( 𝜉𝑢 ) ]d𝑢 − 0 𝑎 ( 𝜉𝑢 )d𝑢 𝑡−𝑠 [𝑎(𝜉𝑡−𝑠 ) − 𝑏(𝜉𝑡−𝑠 )]e = P𝑥 e 𝑓 (𝜉𝑡 ) d𝑠 0 o n ∫𝑡 ∫𝑡   = P 𝑥 e− 0 𝑎 ( 𝜉𝑢 )d𝑢 e 0 [𝑎 ( 𝜉𝑢 )−𝑏 ( 𝜉𝑢 ) ]d𝑢 − 1 𝑓 (𝜉𝑡 ) = 𝑃𝑡𝑏 𝑓 (𝑥) − 𝑃𝑡𝑎 𝑓 (𝑥). By similar calculations,  ∫ 𝑡 ∫  ∫ 𝑠 𝑏 𝑎 𝑃𝑠−𝑟 𝛾𝜋𝑟 𝑓 (𝑦)d𝑟 𝑃𝑡−𝑠 (𝑥, d𝑦) d𝑠 [𝑎(𝑦) − 𝑏(𝑦)] 0 𝐸 0 ∫ 𝑡 ∫ 𝑠 n ∫ 𝑡−𝑠 P 𝑥 e− 0 𝑎 ( 𝜉𝑢 )d𝑢 [𝑎(𝜉𝑡−𝑠 ) = d𝑠 0 0 o  ∫ 𝑠−𝑟 − 𝑏(𝜉𝑡−𝑠 )]P 𝜉𝑡−𝑠 e− 0 𝑏 ( 𝜉𝑢 )d𝑢 𝛾𝜋𝑟 𝑓 (𝜉 𝑠−𝑟 ) d𝑟 ∫ 𝑡 ∫ 𝑠 h ∫ 𝑡−𝑠 P 𝑥 e− 0 𝑎 ( 𝜉𝑢 )d𝑢 [𝑎(𝜉𝑡−𝑠 ) = d𝑠 0 0 i ∫ 𝑡−𝑟 − 𝑏(𝜉𝑡−𝑠 )]e− 𝑡−𝑠 𝑏 ( 𝜉𝑢 )d𝑢 𝛾𝜋𝑟 𝑓 (𝜉𝑡−𝑟 ) d𝑟 ∫ 𝑡 ∫ 𝑠 h ∫ 𝑡−𝑟 P 𝑥 e− 0 𝑎 ( 𝜉𝑢 )d𝑢 [𝑎(𝜉𝑡−𝑠 ) = d𝑠 0 0 i ∫ 𝑡−𝑟 − 𝑏(𝜉𝑡−𝑠 )]e 𝑡−𝑠 [𝑎 ( 𝜉𝑢 )−𝑏 ( 𝜉𝑢 ) ]d𝑢 𝛾𝜋𝑟 𝑓 (𝜉𝑡−𝑟 ) d𝑟 ∫ 𝑡 ∫ ∫ 𝑡 𝑡−𝑟 − 0 𝑎 ( 𝜉𝑢 )d𝑢 e [𝑎(𝜉𝑡−𝑠 ) = P𝑥 𝛾𝜋𝑟 𝑓 (𝜉𝑡−𝑟 )d𝑟 𝑟 0  ∫ 𝑡−𝑟 ( )−𝑏 ( 𝜉 𝜉 [𝑎 ) ]d𝑢 𝑢 𝑢 − 𝑏(𝜉𝑡−𝑠 )]e 𝑡−𝑠 d𝑠 ∫ 𝑡 ∫  𝑡−𝑟   ∫ 𝑡−𝑟 = P𝑥 e− 0 𝑎 ( 𝜉𝑢 )d𝑢 𝛾𝜋𝑟 𝑓 (𝜉𝑡−𝑟 ) e 0 [𝑎 ( 𝜉𝑢 )−𝑏 ( 𝜉𝑢 ) ]d𝑢 − 1 d𝑟 ∫ 𝑡 ∫ 𝑡 0 𝑏 𝑎 𝛾𝜋𝑟 𝑓 (𝑥)d𝑟. 𝑃𝑡−𝑟 = 𝑃𝑡−𝑟 𝛾𝜋𝑟 𝑓 (𝑥)d𝑟 − 0

0

440

A Markov Processes

Then we can add up the two equations and use (A.39) to get (A.43). The uniqueness of the locally bounded solution of (A.43) is a standard application of Gronwall’s inequality. □

Corollary A.51 Let (𝑃𝑡𝑏 )𝑡 ≥0 be defined by (A.38). Then for any 𝑓 ∈ bℬ(𝐸), (𝑡, 𝑥) ↦→ 𝑃𝑡𝑏 𝑓 (𝑥) is the unique locally bounded solution to ∫ 𝑡 𝑏 𝑃𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) − 𝑃𝑡−𝑠 (𝑏𝑃𝑠𝑏 ) 𝑓 (𝑥)d𝑠, 𝑡 ≥ 0, 𝑥 ∈ 𝐸 . 0 𝛾

Let (𝑃𝑡 )𝑡 ≥0 be the locally bounded semigroup of kernels defined by (A.38) with 𝑏 replaced by 𝛾(·, 1). By Theorem A.44 we can define a Borel right semigroup ( 𝑃˜𝑡 )𝑡 ≥0 on 𝐸 by the unique locally bounded solution to ∫ 𝑡 ∫ 𝛾 𝛾 (A.44) 𝑃˜𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) + 𝛾(𝑦, 𝑃˜ 𝑠 𝑓 )𝑃𝑡−𝑠 (𝑥, d𝑦), d𝑠 0

𝐸

where 𝑥 ∈ 𝐸 and 𝑓 ∈ bℬ(𝐸).

Proposition A.52 Let (𝜋𝑡 )𝑡 ≥0 and ( 𝑃˜𝑡 )𝑡 ≥0 be the semigroups defined by (A.39) and (A.44), respectively. Then we have ∫ 𝑡 ∫ d𝑠 [𝛾(𝑦, 1) − 𝑏(𝑦)]𝜋 𝑠 𝑓 (𝑦) 𝑃˜𝑡−𝑠 (𝑥, d𝑦). (A.45) 𝜋𝑡 𝑓 (𝑥) = 𝑃˜𝑡 𝑓 (𝑥) + 0

𝐸

Proof Let 𝛽(𝑥) = 𝛾(𝑥, 1) − 𝑏(𝑥). By Proposition A.50 we can rewrite (A.39) equivalently as ∫ 𝑡 ∫ 𝛾 𝛾 𝜋𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) + d𝑠 [𝛾(𝑦, 𝜋 𝑠 𝑓 ) + 𝛽(𝑦)𝜋 𝑠 𝑓 (𝑦)]𝑃𝑡−𝑠 (𝑥, d𝑦). (A.46) 0

𝐸

From (A.44) and (A.46) it follows that ∫ ∫ 𝑡 𝛾 𝑃𝑡−𝑠 (𝛽𝜋 𝑠 𝑓 ) (𝑥)d𝑠 + 𝜋𝑡 𝑓 (𝑥) = 𝑃˜𝑡 𝑓 (𝑥) + 0

𝑡

𝑃𝑡−𝑠 𝛾(𝜋 𝑠 𝑓 − 𝑃˜ 𝑠 𝑓 ) (𝑥)d𝑠. 𝛾

0

Using the above relation successively we have ∫ 𝑡 𝛾 ˜ 𝜋𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) + 𝑃𝑡−𝑠1 (𝛽𝜋 𝑠1 𝑓 ) (𝑥)d𝑠1 0 ∫ 𝑡 ∫ 𝑠1 𝛾 𝛾 𝑃𝑡−𝑠1 𝛾𝑃𝑠1 −𝑠2 (𝛽𝜋 𝑠2 𝑓 ) (𝑥)d𝑠2 + d𝑠1 0 0 ∫ 𝑡 ∫ 𝑠1 𝛾 𝛾 𝑃𝑡−𝑠1 𝛾𝑃𝑠1 −𝑠2 𝛾(𝜋 𝑠2 𝑓 − 𝑃˜ 𝑠2 𝑓 ) (𝑥)d𝑠2 d𝑠1 + 0

0

A.6 Concatenations and Weak Generators

= 𝑃˜𝑡 𝑓 (𝑥) +

∫

𝑡 𝛾

𝑃𝑡−𝑠1 (𝛽𝜋 𝑠1 𝑓 ) (𝑥)d𝑠1 +

0

+

𝑛 ∫ ∑︁

441

∫

∫

𝑡

∫

𝑠1

0

𝑖=3 0

𝑡

𝑠1 𝛾

0

0

𝛾

𝑃𝑡−𝑠1 𝛾𝑃𝑠1 −𝑠2 (𝛽𝜋 𝑠2 𝑓 ) (𝑥)d𝑠2

d𝑠1

𝑠𝑖−1 𝛾

𝛾

𝛾

···

d𝑠1

∫

𝑃𝑡−𝑠1 𝛾𝑃𝑠1 −𝑠2 · · · 𝛾𝑃𝑠𝑖−1 −𝑠𝑖 (𝛽𝜋 𝑠𝑖 𝑓 ) (𝑥)d𝑠𝑖

0

+ 𝜀 𝑛 (𝑡, 𝑥),

(A.47)

where ∫ 𝜀 𝑛 (𝑡, 𝑥) =

∫

𝑡

0

∫

𝑠1

𝑠𝑛−1

𝑃𝑡−𝑠1 𝛾𝑃𝑠1 −𝑠2 · · · 𝛾𝑃𝑠𝑛−1 −𝑠𝑛 𝛾(𝜋 𝑠𝑛 𝑓 − 𝑃˜ 𝑠𝑛 𝑓 ) (𝑥)d𝑠 𝑛 .

0

0

𝛾

𝛾

···

d𝑠1

𝛾

Let 𝑡 ↦→ 𝐶 (𝑡, 𝑓 ) be a locally bounded positive function on [0, ∞) so that ∥𝜋𝑡 𝑓 − 𝑃˜𝑡 𝑓 ∥ ≤ 𝐶 (𝑡, 𝑓 ). Then we get ∫ 𝑠1 ∫ 𝑠𝑛−1 ∫ 𝑡 ∥𝜀 𝑛 (𝑡, ·) ∥ ≤ 𝐶 (𝑡, 𝑓 ) ∥𝛾(·, 1) ∥ 𝑛 d𝑠 𝑛 d𝑠1 d𝑠2 · · · ≤ 𝐶 (𝑡, 𝑓 ) ∥𝛾(·, 1) ∥

𝑛𝑡

0 𝑛

0

0

.

𝑛! By Proposition A.42, the unique solution of (A.44) is given by 𝛾 𝑃˜𝑡 𝑓 (𝑥) = 𝑃𝑡 𝑓 (𝑥) +

∞ ∫ ∑︁ 𝑖=1

0

∫

𝑡

∫

𝑠1

𝑠𝑖−1

d𝑠2 · · ·

d𝑠1 0

0

𝛾

𝛾

𝛾

𝑃𝑡−𝑠1 𝛾𝑃𝑠1 −𝑠2 · · · 𝛾𝑃𝑠𝑖 𝑓 (𝑥)d𝑠𝑖 .

Then letting 𝑛 → ∞ in (A.47) we obtain (A.45).

□

Theorem A.53 Let 𝑐 0 = sup 𝑥 ∈𝐸 [𝛾(𝑥, 1) − 𝑏(𝑥)]. Then (e−𝑐0 𝑡 𝜋𝑡 )𝑡 ≥0 is a Borel right semigroup on 𝐸. ˜ ℱ, ˜ ℱ ˜ 𝑡 , 𝜉˜𝑡 , P˜ 𝑥 ) be a right process with transition semigroup Proof Let 𝜉˜ = ( Ω, ( 𝑃˜𝑡 )𝑡 ≥0 . From Corollary A.51 and Proposition A.52 it follows that  ∫ 𝑡  𝜋𝑡 𝑓 (𝑥) = P˜ 𝑥 𝑓 ( 𝜉˜𝑡 ) exp [𝛾( 𝜉˜𝑠 , 1) − 𝑏( 𝜉˜𝑠 )]d𝑠 . 0

Therefore (e−𝑐0 𝑡 𝜋𝑡 )𝑡 ≥0 is the transition semigroup of a subprocess of 𝜉˜ generated by the decreasing multiplicative functional   ∫ 𝑡 ˜ ˜ 𝑡 ↦→ 𝑚 𝑡 := exp − 𝑐 0 𝑡 + [𝛾( 𝜉 𝑠 , 1) − 𝑏( 𝜉 𝑠 )]d𝑠 . 0

Then it is a Borel right semigroup; see Sharpe (1988, p. 287).

□

442

A Markov Processes

We next prove some analytic properties of the semigroup (𝜋𝑡 )𝑡 ≥0 defined by (A.39). By Theorem A.53 we have ∥𝜋𝑡 ∥ ≤ e𝑐0 𝑡 for 𝑡 ≥ 0. Then we can define the operators (𝑅 𝛼 ) 𝛼>𝑐0 on bℬ(𝐸) by ∫ ∞ 𝛼 e−𝛼𝑡 𝜋𝑡 𝑓 (𝑥)d𝑡, 𝑥 ∈ 𝐸, 𝑓 ∈ bℬ(𝐸). (A.48) 𝑅 𝑓 (𝑥) = 0

Proposition A.54 For every 𝛼 > 𝑐 0 and 𝑓 ∈ bℬ(𝐸) we have 𝑅 𝛼 𝑓 (𝑥) = 𝑈 𝛼 𝑓 (𝑥) + 𝑈 𝛼 (𝛾 − 𝑏)𝑅 𝛼 𝑓 (𝑥),

𝑥 ∈ 𝐸.

Proof By taking the Laplace transforms of both sides of (A.43) with 𝑎 = 0 we have ∫ ∞ ∫ 𝑡 𝑅 𝛼 𝑓 (𝑥) = 𝑈 𝛼 𝑓 (𝑥) + e−𝛼𝑡 d𝑡 𝑃𝑡−𝑠 (𝛾 − 𝑏)𝜋 𝑠 𝑓 (𝑥)d𝑠 ∫0 ∞ ∫ ∞ 0 d𝑠 = 𝑈 𝛼 𝑓 (𝑥) + e−𝛼𝑡 𝑃𝑡−𝑠 (𝛾 − 𝑏)𝜋 𝑠 𝑓 (𝑥)d𝑡 𝑠 ∫0 ∞ e−𝛼𝑠 𝑈 𝛼 (𝛾 − 𝑏)𝜋 𝑠 𝑓 (𝑥)d𝑠 = 𝑈 𝛼 𝑓 (𝑥) + 0

= 𝑈 𝛼 𝑓 (𝑥) + 𝑈 𝛼 (𝛾 − 𝑏)𝑅 𝛼 𝑓 (𝑥). This proves the desired equation.

□

˜ 𝛼 𝑓 for 𝛼 > 0 and Proposition A.55 Let 𝑓 ∈ bℬ(𝐸). Then we have 𝑓 ∈ (𝛼 − 𝐴)𝑈 𝛼 𝑓 ∈ (𝛼 − 𝐴˜ − 𝛾 + 𝑏)𝑅 𝑓 for 𝛼 > 𝑐 0 . Proof Let ℎ = 𝑓 +(𝛽−𝛼)𝑈 𝛼 𝑓 . By the resolvent equation we have 𝑈 𝛼 𝑓 = 𝑈 𝛽 ℎ. Then the definition of 𝐴˜ implies 𝑓 = (𝛼 − 𝛽)𝑈 𝛼 𝑓 + ℎ ∈ {𝛼𝑈 𝛼 𝑓 − 𝛽𝑈 𝛼 𝑓 + 𝑔 : 𝑔 ∈ bℬ(𝐸) ˜ 𝛼 𝑓 , which gives the first assertion. For 𝛼 > 𝑐 0 we get and 𝑈 𝛽 𝑔 = 𝑈 𝛼 𝑓 } = (𝛼 − 𝐴)𝑈 from Proposition A.54 that ˜ 𝛼 𝑓 − (𝛾 − 𝑏)𝑅 𝛼 𝑓 (𝛼 − 𝐴˜ − 𝛾 + 𝑏)𝑅 𝛼 𝑓 = (𝛼 − 𝐴)𝑅 ˜ 𝛼 [ 𝑓 + (𝛾 − 𝑏)𝑅 𝛼 𝑓 ] − (𝛾 − 𝑏)𝑅 𝛼 𝑓 . = (𝛼 − 𝐴)𝑈 By the first assertion, the set represented by the first term on the right-hand side includes 𝑓 + (𝛾 − 𝑏)𝑅 𝛼 𝑓 . Then (𝛼 − 𝐴˜ − 𝛾 + 𝑏)𝑅 𝛼 𝑓 includes 𝑓 , proving the second assertion. □ ˜ ˜ Then for any ℎ ∈ (𝛼− 𝐴−𝛾+𝑏) 𝑓 Lemma A.56 Let 𝛼 > 𝑐 1 := ∥𝛾−𝑏∥ and 𝑓 ∈ 𝒟( 𝐴). we have ∥ℎ∥ ≥ (𝛼 − 𝑐 1 ) ∥ 𝑓 ∥. Proof For ℎ ∈ (𝛼 − 𝐴˜ − 𝛾 + 𝑏) 𝑓 we have 𝛼 𝑓 − 𝛾 𝑓 + 𝑏 𝑓 − ℎ ∈ 𝐴˜ 𝑓 . By the definition ˜ there exist 𝑔 ∈ bℬ(𝐸) such that 𝑓 = 𝑈 𝛼 𝑔 and 𝛼 𝑓 − 𝛾 𝑓 + 𝑏 𝑓 − ℎ = 𝛼 𝑓 − 𝑔, so of 𝐴, □ 𝑔 = ℎ + 𝛾 𝑓 − 𝑏 𝑓 . It follows that 𝛼∥ 𝑓 ∥ ≤ ∥𝑔∥ ≤ ∥ℎ∥ + 𝑐 1 ∥ 𝑓 ∥. ˜ then for any 𝛼 > 𝑐 1 the Lemma A.57 If 𝑓1 and 𝑓2 are distinct functions from 𝒟( 𝐴), intersection (𝛼 − 𝐴˜ − 𝛾 + 𝑏) 𝑓1 ∩ (𝛼 − 𝐴˜ − 𝛾 + 𝑏) 𝑓2 is empty.

A.7 Time–Space Processes

443

Proof Suppose that ℎ ∈ (𝛼 − 𝐴˜ − 𝛾 + 𝑏) 𝑓1 ∩ (𝛼 − 𝐴˜ − 𝛾 + 𝑏) 𝑓2 . Then there exist ℎ1 ∈ 𝐴˜ 𝑓1 and ℎ2 ∈ 𝐴˜ 𝑓2 such that ℎ = (𝛼 − 𝛾 + 𝑏) 𝑓1 − ℎ1 = (𝛼 − 𝛾 + 𝑏) 𝑓2 − ℎ2 . By ˜ for any 𝛽 > 0 there exist 𝑔1 , 𝑔2 ∈ bℬ(𝐸) such that 𝑓𝑖 = 𝑈 𝛽 𝑔𝑖 the definition of 𝐴, and ℎ𝑖 = 𝛽 𝑓𝑖 − 𝑔𝑖 for 𝑖 = 1 and 2. It follows that ( 𝑓2 − 𝑓1 ) = 𝑈 𝛽 (𝑔2 − 𝑔1 ) and ˜ 𝑓2 − 𝑓1 ), and so ℎ2 − ℎ1 = 𝛽( 𝑓2 − 𝑓1 ) − (𝑔2 − 𝑔1 ). These imply ℎ2 − ℎ1 ∈ 𝐴( ˜ ( 𝑓2 − 𝑓1 ), 0 = (𝛼 − 𝛾 + 𝑏) ( 𝑓2 − 𝑓1 ) − (ℎ2 − ℎ1 ) ∈ (𝛼 − 𝛾 + 𝑏 − 𝐴) which is in contradiction to Lemma A.56.

□

Lemma A.58 For any 𝛼 > 𝑐 1 and 𝑓 ∈ 𝒟( 𝐴) we have 𝑓 = 𝑅 𝛼 (𝛼 − 𝐴 − 𝛾 + 𝑏) 𝑓 . Proof Clearly, for 𝑓 ∈ 𝒟( 𝐴) the set (𝛼 − 𝐴˜ − 𝛾 + 𝑏) 𝑓 contains the function ℎ := (𝛼 − 𝐴 − 𝛾 + 𝑏) 𝑓 . By Proposition A.55 we have ℎ ∈ (𝛼 − 𝐴˜ − 𝛾 + 𝑏)𝑅 𝛼 ℎ. Then the sets (𝛼 − 𝐴˜ − 𝛾 + 𝑏) 𝑓 and (𝛼 − 𝐴˜ − 𝛾 + 𝑏)𝑅 𝛼 ℎ have a non-empty intersection. □ Thus Lemma A.57 implies that 𝑓 = 𝑅 𝛼 ℎ. Theorem A.59 Let 𝑓 ∈ 𝒟( 𝐴) and let (𝑡, 𝑥) ↦→ 𝜋𝑡 𝑓 (𝑥) be defined by (A.39). Then we have ∫ 𝑡 𝜋𝑡 𝑓 (𝑥) = 𝑓 (𝑥) + 𝜋 𝑠 ( 𝐴 + 𝛾 − 𝑏) 𝑓 (𝑥)d𝑠, 𝑡 ≥ 0, 𝑥 ∈ 𝐸 . 0

Proof By Theorems A.16 and A.20, any 𝑓 ∈ 𝒟( 𝐴) is finely continuous relative to 𝜉, so 𝑡 ↦→ 𝑃𝑡 𝑓 (𝑥) is right continuous pointwise. Then Proposition A.43 implies 𝑡 ↦→ 𝜋𝑡 𝑓 (𝑥) is right continuous pointwise. For 𝛼 > 𝑐 1 it is easy to show that ∫ ∞ ∫ 𝑡 1 𝜋 𝑠 ( 𝐴 + 𝛾 − 𝑏) 𝑓 d𝑠 = 𝑅 𝛼 ( 𝐴 + 𝛾 − 𝑏) 𝑓 . e−𝛼𝑡 d𝑡 𝛼 0 0 Using Lemma A.58 one can see the above value is equal to ∫ ∞ 1 𝛼 e−𝛼𝑡 (𝜋𝑡 𝑓 − 𝑓 )d𝑡. 𝑅 𝑓− 𝑓 = 𝛼 0 Then the desired equation follows by the uniqueness of the Laplace transform.

□

A.7 Time–Space Processes In this section we briefly discuss time–space processes associated with inhomogeneous Markov processes. For simplicity we only consider Borel transition semigroups. Suppose that 𝐼 ⊂ R is an interval and 𝐹 is a Lusin topological space. Let 𝐸˜ be ˜ For 𝑡 ∈ 𝐼 let 𝐸 𝑡 = {𝑥 ∈ 𝐹 : (𝑡, 𝑥) ∈ 𝐸˜ }. a Borel subset of 𝐼 ×𝐹 and let 𝐸˜ 𝑐 = (𝐼 ×𝐹)\ 𝐸. Then each 𝐸 𝑡 is a Lusin topological space. We fix an abstract point 𝜕 ∉ 𝐼 × 𝐹 and assume all functions on 𝐸˜ ⊂ 𝐼 × 𝐹 have been extended trivially to 𝐸˜ 𝑐 ∪ {𝜕}.

444

A Markov Processes

Suppose that for each pair 𝑟 ≤ 𝑡 ∈ 𝐼 there is a Markov kernel 𝑃𝑟 ,𝑡 from (𝐸𝑟 , ℬ(𝐸𝑟 )) to (𝐸 𝑡 , ℬ(𝐸 𝑡 )). The family (𝑃𝑟 ,𝑡 : 𝑟 ≤ 𝑡 ∈ 𝐼) is called an inhomogeneous transition semigroup with global state space 𝐸˜ if it satisfies the Chapman– Kolmogorov equation ∫ 𝑃𝑟 ,𝑡 (𝑥, 𝐵) = 𝑃𝑟 ,𝑠 (𝑥, d𝑦)𝑃𝑠,𝑡 (𝑦, 𝐵), (A.49) 𝐸𝑠

where 𝑟 ≤ 𝑠 ≤ 𝑡 ∈ 𝐼, 𝑥 ∈ 𝐸𝑟 and 𝐵 ∈ ℬ(𝐸 𝑡 ). In this work, we assume for every ˜ the function 𝑓 ∈ bℬ( 𝐸) ∫ 𝑓 (𝑡, 𝑦)𝑃𝑟 ,𝑡 (𝑥, d𝑦) (𝑟, 𝑥, 𝑡) ↦→ 1 {𝑟 ≤𝑡 } 𝐸𝑡

˜ × ℬ(𝐼) = ℬ( 𝐸˜ × 𝐼). is measurable with respect to the 𝜎-algebra ℬ( 𝐸) Definition A.60 Suppose that (𝑃𝑟 ,𝑡 : 𝑟 ≤ 𝑡 ∈ 𝐼) is an inhomogeneous transition ˜ The collection 𝜉 = (Ω, 𝒢, 𝒢𝑟 ,𝑡 , 𝜉𝑡 , P𝑟 , 𝑥 ) is semigroup with global state space 𝐸. called an inhomogeneous Markov process with global state space 𝐸˜ and transition semigroup (𝑃𝑟 ,𝑡 : 𝑟 ≤ 𝑡 ∈ 𝐼) if the following conditions are satisfied: (1) For every 𝑟 ∈ 𝐼, (Ω, 𝒢, 𝒢𝑟 ,𝑡 : 𝑡 ∈ 𝐼 ∩ [𝑟, ∞)) is a filtered measurable space such that 𝒢𝑠,𝑡 ⊂ 𝒢𝑟 ,𝑢 for 𝑟 ≤ 𝑠 ≤ 𝑡 ≤ 𝑢 ∈ 𝐼. (2) For every 𝑟 ≤ 𝑡 ∈ 𝐼, 𝜔 ↦→ 𝜉𝑡 (𝜔) is a measurable mapping from (Ω, 𝒢𝑟 ,𝑡 ) to (𝐸 𝑡 , ℬ(𝐸 𝑡 )). ˜ P𝑟 , 𝑥 is a probability measure on (Ω, 𝒢) such that for every (3) For every (𝑟, 𝑥) ∈ 𝐸, ˜ 𝐻 ∈ b𝒢 the function (𝑟, 𝑥) ↦→ P𝑟 , 𝑥 (𝐻) is ℬ( 𝐸)-measurable. ˜ (4) For every (𝑟, 𝑥) ∈ 𝐸 we have P𝑟 , 𝑥 {𝜉𝑟 = 𝑥} = 1 and the following simple Markov property holds:   P𝑟 , 𝑥 𝑓 (𝜉𝑡 )|𝒢𝑟 ,𝑠 = 𝑃𝑠,𝑡 𝑓 (𝜉 𝑠 ), 𝑟 ≤ 𝑠 ≤ 𝑡 ∈ 𝐼, 𝑓 ∈ bℬ(𝐸 𝑡 ). (A.50) We say 𝜉 is right continuous if 𝑡 ↦→ 𝜉𝑡 (𝜔) is right continuous for every 𝜔 ∈ Ω. In the situation of the above definition, given any probability measure 𝜇 on (𝐸𝑟 , ℬ(𝐸𝑟 )), we define the probability measure P𝑟 , 𝜇 on (Ω, 𝒢) by ∫ (A.51) P𝑟 , 𝜇 (𝐻) = P𝑟 , 𝑥 (𝐻)𝜇(d𝑥), 𝐻 ∈ b𝒢. 𝐸

By (A.50) and a monotone class argument it follows that
P𝑟 , 𝜇 𝐹 |𝒢𝑟 ,𝑠 = P𝑠, 𝜉𝑠 (𝐹), 𝑟 ≤ 𝑠 ∈ 𝐼, 𝐹 ∈ b𝜎({𝜉𝑡 : 𝑡 ≥ 𝑠, 𝑡 ∈ 𝐼}). Given an inhomogeneous Markov transition semigroup (𝑃𝑟 ,𝑡 : 𝑟 ≤ 𝑡 ∈ 𝐼) with ˜ we can define a homogeneous transition semigroup ( 𝑃˜𝑡 )𝑡 ≥0 on global state space 𝐸, 𝐸˜ by

A.7 Time–Space Processes

𝑃˜𝑡 𝑓 (𝑟, 𝑥) = 1𝐼 (𝑟 + 𝑡)

445

∫ 𝑓 (𝑟 + 𝑡, 𝑦)𝑃𝑟 ,𝑟+𝑡 (𝑥, d𝑦),

(A.52)

𝐸𝑟+𝑡

˜ We call ( 𝑃˜𝑡 )𝑡 ≥0 the time–space semigroup where 𝑡 ≥ 0, (𝑟, 𝑥) ∈ 𝐸˜ and 𝑓 ∈ bℬ( 𝐸). associated with (𝑃𝑟 ,𝑡 : 𝑟 ≤ 𝑡 ∈ 𝐼). Suppose that 𝜉 = (Ω, 𝒢, 𝒢𝑟 ,𝑡 , 𝜉𝑡 , P𝑟 , 𝑥 ) is an inhomogeneous Markov process with ˜ = 𝐼 × Ω and 𝒢˜ = ℬ(𝐼) × 𝒢. For transition semigroup (𝑃𝑟 ,𝑡 : 𝑟 ≤ 𝑡 ∈ 𝐼). Let Ω ˜ (𝑣, 𝜔) ∈ Ω define n (𝑣 + 𝑡, 𝜉 (𝜔)) 𝑣+𝑡 𝜉˜𝑡 (𝑣, 𝜔) = 𝜕

if 𝑡 ≥ 0 and 𝑣 + 𝑡 ∈ 𝐼, if 𝑡 ≥ 0 and 𝑣 + 𝑡 ∉ 𝐼.

(A.53)

˜ For (𝑟, 𝑥) ∈ 𝐸˜ ˜ 𝑡 = 𝜎({ 𝜉˜𝑠 : 0 ≤ 𝑠 ≤ 𝑡}) on Ω. For 𝑡 ≥ 0 define the 𝜎-algebra ℱ ˜ ˜ ˜ let P𝑟 , 𝑥 be the probability measure on ( Ω, ℱ) induced by P𝑟 , 𝑥 via the mapping 𝜔 ↦→ (𝑟, 𝜔). ˜ ℱ ˜ 𝒢, ˜ 𝑡 , 𝜉˜𝑡 , P˜ 𝑟 , 𝑥 ) is a Markov process in 𝐸˜ with Theorem A.61 The system 𝜉˜ = ( Ω, ˜ transition semigroup ( 𝑃𝑡 )𝑡 ≥0 . Proof Let (𝑟, 𝑥) ∈ 𝐸˜ and 𝑡 ≥ 𝑠 ≥ 0. Let 𝑓 = 𝑓𝑣 (𝑥) = 𝑓 (𝑣, 𝑥) be a bounded Borel ˜ Since P˜ 𝑟 , 𝑥 is carried by {𝑟} × Ω, if 𝑟 + 𝑡 ∈ 𝐼, we have function on 𝐸.     ˜ 𝑠 = P𝑟 , 𝑥 𝑓 (𝑟 + 𝑡, 𝜉𝑟+𝑡 )|𝜎({𝜉𝑟+𝑣 : 0 ≤ 𝑣 ≤ 𝑠}) P˜ 𝑟 , 𝑥 𝑓 ( 𝜉˜𝑡 )| ℱ   = P𝑟 , 𝑥 𝑓 (𝑟 + 𝑡, 𝜉𝑟+𝑡 )|𝜉𝑟+𝑠 = 𝑃𝑟+𝑠,𝑟+𝑡 𝑓𝑟+𝑡 (𝜉𝑟+𝑠 ) = 𝑃˜𝑡−𝑠 𝑓 (𝑟 + 𝑠, 𝜉𝑟+𝑠 ) = 𝑃˜𝑡−𝑠 𝑓 ( 𝜉˜𝑠 ), where the fourth equality follows by (A.52). If 𝑟 + 𝑡 ∉ 𝐼, both sides of the above equality are equal to zero. Then 𝜉˜ is a Markov process with transition semigroup ( 𝑃˜𝑡 )𝑡 ≥0 . □ Theorem A.62 Suppose that 𝐸˜ = 𝐼 × 𝐸 for a Lusin topological space 𝐸 and there is a Borel right semigroup (𝑃𝑡 )𝑡 ≥0 on 𝐸 such that 𝑃𝑟 ,𝑡 = 𝑃𝑡−𝑟 for 𝑡 ≥ 𝑟 ∈ 𝐼. Then ( 𝑃˜𝑡 )𝑡 ≥0 is a right semigroup if and only if sup 𝐼 ∉ 𝐼. Proof In this case ( 𝑃˜𝑡 )𝑡 ≥0 is the Cartesian product of (𝑃𝑡 )𝑡 ≥0 and the transition semigroup (𝑅𝑡 )𝑡 ≥0 of the uniform motion to the right on 𝐼. As observed in Example A.1, the latter is a right semigroup if and only if sup 𝐼 ∉ 𝐼. Then the result follows by Sharpe (1988, p. 84). □ ˜ ℱ ˜ 𝒢, ˜ 𝑡 , 𝜉˜𝑡 , P˜ 𝑟 , 𝑥 ) the time–space process of 𝜉. By Theorem A.61, We call 𝜉˜ = ( Ω, the study of the inhomogeneous process 𝜉 can be reduced to that of the homogeneous ˜ If 𝜉˜ has a right realization, we call (𝑃𝑟 ,𝑡 : 𝑟 ≤ 𝑡 ∈ 𝐼) time–space process 𝜉. an inhomogeneous right transition semigroup. By Theorem A.62, in this case it is necessary that sup 𝐼 ∉ 𝐼. The following theorem shows that the terminology is consistent with that in the homogeneous case. Theorem A.63 Suppose that 𝐸˜ = [0, ∞) × 𝐸 for a Lusin topological space 𝐸 and there is a homogeneous Markov transition semigroup (𝑃𝑡 )𝑡 ≥0 on 𝐸 such that 𝑃𝑟 ,𝑡 = 𝑃𝑡−𝑟 for 𝑡 ≥ 𝑟 ≥ 0. Then ( 𝑃˜𝑡 )𝑡 ≥0 is a right semigroup if and only if so is (𝑃𝑡 )𝑡 ≥0 .

446

A Markov Processes

˜ 𝒢˜ 𝑡 , (𝛼𝑡 , 𝜉𝑡 ), P˜ 𝑟 , 𝑥 ) ˜ 𝒢, Proof Suppose that ( 𝑃˜𝑡 )𝑡 ≥0 is a right semigroup. Let 𝜉˜ = ( Ω, ˜ ˜ ˜ ˜ be a right realization of ( 𝑃𝑡 )𝑡 ≥0 . It is clear that 𝜉 = ( Ω, 𝒢, 𝒢𝑡 , 𝜉𝑡 , P˜ 0, 𝑥 ) is a right process with transition semigroup (𝑃𝑡 )𝑡 ≥0 . Then (𝑃𝑡 )𝑡 ≥0 is a right semigroup. The □ converse was obtained in Sharpe (1988, p. 86). Starting from a realization of the time–space semigroup, we can also easily reconstruct a realization of the original inhomogeneous transition semigroup. For this purpose, let us consider a realization 𝜉˜ = (Ω, 𝒢, 𝒢˜ 𝑡 , 𝜉˜𝑡 , P𝑟 , 𝑥 ) of ( 𝑃˜𝑡 )𝑡 ≥0 , where 𝜉˜𝑡 = (𝛼𝑡 , 𝑦 𝑡 ). In view of (A.52) we may assume 𝛼𝑡 = 𝛼0 + 𝑡 for 𝑡 ≥ 0. For 𝜔 ∈ Ω define n𝑦 (𝜔) if 𝑡 ∈ 𝐼 ∩ [𝛼0 (𝜔), ∞), 𝜉𝑡 (𝜔) = 𝑡−𝛼0 ( 𝜔) (A.54) if 𝑡 ∈ 𝐼 ∩ (−∞, 𝛼0 (𝜔)). 𝜕 Let ℱ𝑟 ,𝑡 = 𝜎({𝜉 𝑠 : 𝑟 ≤ 𝑠 ≤ 𝑡}) for 𝑟 ≤ 𝑡 ∈ 𝐼. Theorem A.64 The system 𝜉 = (Ω, 𝒢, ℱ𝑟 ,𝑡 , 𝜉𝑡 , P𝑟 , 𝑥 ) is an inhomogeneous Markov process with transition semigroup (𝑃𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ∈ 𝐼). Proof For any (𝑟, 𝑥) ∈ 𝐸˜ the probability measure P𝑟 , 𝑥 is carried by {𝛼0 = 𝑟 }. Given 𝑡 ≥ 0 and 𝑓 ∈ bℬ(𝐸 𝑡 ), write 𝑓˜(𝑠, 𝑦) = 1 {𝑠=𝑡 ,𝑦 ∈𝐸𝑡 } 𝑓 (𝑦),

(𝑠, 𝑦) ∈ 𝐸˜ .

Then for any 𝑟 ≤ 𝑠 ≤ 𝑡 ∈ 𝐼 we have     P𝑟 , 𝑥 𝑓 (𝜉𝑡 )|ℱ𝑟 ,𝑠 = P𝑟 , 𝑥 𝑓 (𝜉𝑡 )|𝜎({𝜉 𝑣 : 𝑟 ≤ 𝑣 ≤ 𝑠})   = P𝑟 , 𝑥 𝑓 (𝑦 𝑡−𝑟 )|𝜎({𝑦 𝑣−𝑟 : 𝑟 ≤ 𝑣 ≤ 𝑠})   = P𝑟 , 𝑥 𝑓˜( 𝜉˜𝑡−𝑟 )|𝜎({ 𝜉˜𝑣−𝑟 : 𝑟 ≤ 𝑣 ≤ 𝑠}) = 𝑃˜𝑡−𝑠 𝑓˜( 𝜉˜𝑠−𝑟 ) = 𝑃𝑠,𝑡 𝑓 (𝜉 𝑠 ). This gives the desired Markov property of 𝜉.

□

Example A.3 Suppose that 𝐸 is a complete separable metric space. Let 𝐷 𝐸 := 𝐷 ( [0, ∞), 𝐸) be the space of càdlàg paths from [0, ∞) to 𝐸 equipped with the usual Skorokhod metric. Then 𝐷 𝐸 is also a complete separable metric space. Suppose that (𝑃𝑡 )𝑡 ≥0 is a Borel right semigroup on 𝐸 with a càdlàg realization. For simplicity we consider the canonical realization 𝜉 = (𝐷 𝐸 , ℱ 0 , ℱ𝑡0 , 𝜉𝑡 , P 𝑥 ), where (ℱ 0 , ℱ𝑡0 ) are the natural 𝜎-algebras of 𝐷 𝐸 and 𝜉𝑡 (𝜔) = 𝜔(𝑡) for 𝑡 ≥ 0 and 𝜔 ∈ 𝐷 𝐸 . Let 𝑦 𝑠 (𝑡) = 𝑦(𝑡 ∧ 𝑠) for 𝑠, 𝑡 ≥ 0 and 𝑦 ∈ 𝐷 𝐸 . Let 𝑆 = {(𝑡, 𝑦) ∈ [0, ∞) × 𝐷 𝐸 : 𝑦 = 𝑦 𝑡 }. For 𝑡 ≥ 0 let 𝐷 𝑡𝐸 = {𝑦 ∈ 𝐷 𝐸 : 𝑦 = 𝑦 𝑡 } = {𝑦 ∈ 𝐷 𝐸 : (𝑡, 𝑦) ∈ 𝑆}. Then we have 𝐷 𝑠𝐸 ⊂ 𝐷 𝑡𝐸 for 𝑡 ≥ 𝑠 ≥ 0. Given 𝑟 ≥ 0 and 𝑦 1 , 𝑦 2 ∈ 𝐷 𝐸 we define 𝑦 1 /𝑟/𝑦 2 ∈ 𝐷 𝐸 by (𝑦 1 /𝑟/𝑦 2 ) (𝑡) =

n 𝑦 (𝑡) 1 𝑦 2 (𝑡 − 𝑟)

if 0 ≤ 𝑡 < 𝑟, if 𝑟 ≤ 𝑡 < ∞.

A.7 Time–Space Processes

447

The operators 𝑦 ↦→ 𝑦 𝑠 and (𝑟, 𝑦 1 , 𝑦 2 ) ↦→ 𝑦 1 /𝑟/𝑦 2 are Borel measurable; see Dellacherie and Meyer (1978, p. 146). We can define an inhomogeneous Borel transition semigroup ( 𝑃¯𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ≥ 0) with global state space 𝑆 by 𝑃¯𝑟 ,𝑡 𝑓 (𝑦) = P 𝑦 (𝑟) [ 𝑓 (𝑦/𝑟/𝜉 𝑡−𝑟 )],

𝑦 ∈ 𝐷 𝑟𝐸 , 𝑓 ∈ bℬ(𝐷 𝑡𝐸 ).

(A.55)

From Proposition 2.1.2 of Dawson and Perkins (1991, p. 14) it follows that ( 𝑃¯𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ≥ 0) is a right transition semigroup. For 𝜔 ∈ 𝐷 𝐸 and 𝑡 ≥ 0 let 𝜉¯𝑡 (𝜔) = 𝜔𝑡 ∈ 𝐷 𝑡𝐸 . It is easy to see ℱ𝑟0,𝑡 := 𝜎({ 𝜉¯𝑠 : 𝑟 ≤ 𝑠 ≤ 𝑡}) = ℱ𝑡0 for 𝑡 ≥ 𝑟 ≥ 0. For 𝑟 ≥ 0 and 𝑦 ∈ 𝐷 𝑟𝐸 define the probability measure P¯ 𝑟 ,𝑦 on (𝐷 𝐸 , ℱ 0 ) by P¯ 𝑟 ,𝑦 ( 𝐴) = P 𝑦 (𝑟) ({𝜔 ∈ 𝐷 𝐸 : 𝑦/𝑟/𝜔 ∈ 𝐴}),

𝐴 ∈ ℱ0 .

(A.56)

Then 𝜉¯ = (𝐷 𝐸 , ℱ 0 , ℱ𝑟0,𝑡 , 𝜉¯𝑡 , P¯ 𝑟 ,𝑦 ) is a càdlàg realization of ( 𝑃¯𝑟 ,𝑡 : 𝑡 ≥ 𝑟 ≥ 0). This process is called the path process of 𝜉; see Dawson and Perkins (1991). Clearly, the path process records all the information of the history of the sample path of 𝜉.

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Subject Index

𝐺 𝛿 set, 8 𝛼-stable CIR-model, 305 absolutely continuous linear part, 373 adapted process, ℰ-, 415 additive functional, 433 admissible additive functional, 36, 434 admissible family of branching mechanisms, 309 affine process, 97 affine semigroup, 97 age-structured superprocess, 150 announcing sequence, 414 augmentation by a probability, 414 augmentation of a filtration, 420 augmented filtration, 414, 420 Bernstein functions, 28 Bernstein polynomials, 22 binary local branching mechanism, 49 Borel 𝜎-algebra, 1, 413 Borel function, 1 Borel measure, 2 Borel right process, 423 Borel transition semigroup, 416 bounded entrance law, 417 bounded entrance rule, 417

bounded kernel, 413 bounded pointwise convergence, 411 bounded set in a nuclear space, 394 branch point, 426 branching mechanism, 46, 76, 158, 159, 255, 256 branching particle system, 100 branching property, 31, 314 Brownian motion with drift, 92 Brownian snake, 𝜉-, 108 càdlàg path, 415 càdlàg process, 415 Campbell measure, 62 canonical entrance rule, 35, 207, 248 catalyst measure, 49 catalytic super-Brownian motion, 49 CB-process, 49 CB-process in Lévy environment, 307 CBI-process, 76 Chapman–Kolmogorov equation, 416, 444 characteristic function, 29 characteristic functional, 366 CIR-model, 78 classical Mehler formula, 369 classical OU-process, 368 closable entrance law, 417 closable entrance path, 376

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5

467

468

closed entrance law, 417 cluster representation, 18 compensated Poisson random measure, 14 complete measure space, 412 completely monotone function, 22 completion, 𝜇-, 412 compound Poisson random measure, 16 conditioned superprocess, 155 conservative extension, 424 conservative process, 424 conservative resolvent, 417 conservative transition semigroup, 416 continuous part of immigration, 289 continuous process, 415 continuous-state branching process, 49 continuous-state branching process with immigration, 76 continuous-state nonlinear branching process, 306 convolution, 16 countably Hilbert nuclear space, 395 coupling, 58 covariance measure of orthogonal martingale measure, 184 covariance measure of worthy martingale measure, 184 Cox–Ingersoll–Ross model, 78 critical CB-process, 66 critical MB-process, 52 critical superprocess, 52 cumulant semigroup, 33, 66 Dawson’s Girsanov transform, 200 Dawson–Watanabe superprocess, 45 decomposable branching mechanism, 50 difference operator, 21 diffuse measure, 2 discontinuous part of immigration, 289 dominating measure, 184

Subject Index

downcrossing, 𝛿-, 92 entrance law, 374, 416 entrance path, 376 entrance rule, 416 entrance space, 376, 429 ergodic process, 417 evanescent process, P 𝜇 (𝒢)-, 422 excessive function, 417 excessive function, 𝛼-, 417 excessive measure, 417 excursion, 227 excursion law, 227 exit measure, 162 exit time, 161 explosion time, 316 exponential distribution, 21 extinction time, 68, 153 extremal entrance law, 417 Feller semigroup, 425 Feller’s branching diffusion, 78 Feynman–Kac formula, 46, 438 filtered probability space, 414 filtration, 414 fine topology, 423 finite kernel, 413 finite-dimensional distributions, 415 Fleming–Viot superprocess, 63 flow of CBI-processes, 324 Galton–Watson branching process, 87 Galton–Watson branching process with immigration, 87 Gamma distribution, 21 Gaussian type SC-semigroup, 369 general branching particle system, 110 generalized CBI-process, 305 generalized Mehler semigroup, 367 generalized Ornstein–Uhlenbeck process, 367 generalized OU-process, 367 global state space, 444 Grey’s condition, 69

Subject Index

Gronwall’s inequality, 438 GW-process, 87 GWI-process, 87 height distribution of a compound Poisson random measure, 16 Hermite functions, 397 Hermite polynomials, 397 Hilbert–Schmidt embedding, 395 historical superprocess, (𝜉, 𝜙)-, 160 Hunt process, 423, 424 immigration mechanism, 76, 246, 255, 256 immigration process, 242 immigration superprocess, 242 immigration superprocess with dependent spatial motion, 364 infinitely divisible distribution, 16 infinitely divisible random measure, 16 inhomogeneous CB-process, 159 inhomogeneous CBI-process, 256 inhomogeneous Dawson–Watanabe superprocess, 158 inhomogeneous immigration rate, 341 inhomogeneous immigration superprocess, 255 inhomogeneous Markov process, 444 inhomogeneous right transition semigroup, 445 inhomogeneous skew-convolution semigroup, 255 inhomogeneous transition semigroup, 444 initial law, 418 intensity of a compound Poisson random measure, 16 intensity of a Poisson random measure, 13 intensity of orthogonal martingale measure, 184 interactive growth rate, 200

469

interactive immigration superprocess with dependent spatial motion, 364 invariant measure, 417 Itô’s excursion law, 433 kernel, 413 killing density, 45 killing functional, 45 Kuznetsov measure, 221 Lévy process, 384 Lévy white noise, 289 Lévy–Khintchine formula, 29 Lamperti transformation, 293 Langevin equation, 368 Langevin type equation, 386 Langevin type equation, mild form, 389 Laplace functional, 9, 13 Laplace transform, 13 Lipschitz constant, 58 local branching mechanism, 49 local part of a branching mechanism, 50 local projection of branching mechanism, 140 locally bounded function, 433 locally bounded stochastic process, 433 locally integrable process, 300 locally square integrable entrance path, 376 Lusin measurable space, 8, 413 Lusin topological space, 8, 413 Markov kernel, 413 Markov process, ℬ• (𝐸)-, 419 Markov resolvent, 417 Markov transition semigroup, 416 martingale functional, 184 martingale measure, 183 MB-process, 32 measure, 2 measure-valued branching process, 32

470

minimal entrance law, 417 modification, 415 monotone vector space, 411 multitype superprocess, 149 natural 𝜎-algebras, ℰ-, 414 nearly Borel function, 422 nearly optional function, 422 Neveu’s CB-process, 96 non-branch point, 426 non-local part of a branching mechanism, 50 normal transition semigroup, 416 nuclear space, 395 null set, 𝜇-, 412 number of 𝜀-oscillations, 415 one-sided 𝛼-stable noise, 201 one-sided stable distribution (with index 0 < 𝛼 < 1), 21 one-sided stable distribution (with index 1 < 𝛼 < 2), 29 optional time, 414 orthogonal martingale measure, 184 OU-type process, 406 Palm distributions, 63 path process, 447 path-valued growing process, 314 pathwise uniqueness of solutions, 280, 285 Poisson random measure, 13 positive definite operator, 365 positive definite signed measure, 183 positive solution, 169 potential operator of a transition semigroup, 418 potential operator, 𝛼-, 418 predictable immigration rate, 347 predictable process, 300 predictable time, 414 predictable two-parameter process, 186 predictable version of a progressive process, 300 probability entrance law, 417

Subject Index

progressive process, ℰ-, 415 progressive two-parameter process, 186 purely atomic, 2 purely excessive measure, 417 quasi-left continuous filtration, 414 quasi-left continuous process, 423 Radon measurable space, 413 Radon topological space, 413 random measure, 13 rational cone, 426 rational Ray cone, 426 Ray extension of a resolvent, 428 Ray extension of a semigroup, 428 Ray resolvent, 425 Ray semigroup, 426 Ray space, 429 Ray topology, 427, 429 Ray–Knight completion, 428 Ray–Knight theorem, 92 realization, 415 realization of a semigroup, 419 regular affine process, 97 regular branching property, 31 regular entrance rule, 416 regular MB-process, 32 regular SC-semigroup, 246, 368 resolvent, 417 resolvent equation, 417 resolvent of a transition semigroup, 418 restriction of a 𝜎-algebra, 412 restriction of a measure, 412 right continuous filtration, 414 right continuous inhomogeneous Markov process, 444 right continuous Markov process, 419 right continuous process, 415 right Markov process, 423 right process, 423, 424 right transition semigroup, 423, 424 sample path, 414

Subject Index

SC-semigroup (on a Hilbert space), 367 SC-semigroup (on a space of measures), 241 Schwartz distribution, 399 Schwartz space, 396 Schwarz’s inequality, 183 separate points, 411 shift operator, 419 simple ℬ• (𝐸)-Markov property, 418 simple Markov property, 444 simplex, 417 skew convolution semigroup (on a Hilbert space), 367 skew convolution semigroup (on a space of measures), 241 solution to a stochastic equation, 280, 285 spatial motion, 45, 158, 255 spatially constant branching mechanism, 46 state space, 414 state-dependent branching mechanism, 360 state-dependent immigration rate, 352 stationary distribution, 417 step process, 186, 300 stochastic Fubini’s theorem, 190 stochastic immigration rate, 305 stochastic integral (with respect to a martingale measure), 189 stochastic process, 414 stopped path, 108 stopping time, 414 strong Feller property, 61, 251 strong Markov property, 421 strong solution to a stochastic equation, 280, 285 strong topology, 394 sub-Markov kernel, 413 subcritical CB-process, 66 subcritical MB-process, 52 subcritical superprocess, 52

471

subprocess of a superprocess, 142, 154 super-Brownian motion, 49 super-mean-valued function, 417 super-mean-valued function, 𝛼-, 417 supercritical CB-process, 66 supercritical MB-process, 52 supercritical superprocess, 52 supermedian function, 𝛼-, 417 superprocess with dependent spatial motion, 203 superprocess, (𝜉, 𝜙)-, 46 superprocess, (𝜉, 𝜙, 𝜓)-, 50 superprocess, (𝜉, 𝐾, 𝜙)-, 45, 144 superprocess, (𝜉, 𝐾, 𝜙, 𝜓)-, 50 superprocesses with coalescing spatial motion, 364 symmetric difference, 412 symmetric operator, 365 symmetric signed measure, 183 tempered measure, 13 the 𝑛-th root of a probability measure, 16 time–space Gaussian white noise, 184 time–space process, 445 time–space semigroup, 445 topology of weak convergence, 3 total population, 134 total variation, 58 total variation norm, 58 totally inaccessible stopping time, 414 trace class operator, 365 trace of a 𝜎-algebra, 412 trace of a bounded linear operator, 366 trace of a measure, 412 transition semigroup, 416 uniform convergence, 411 uniform elliptic condition, 147 uniform motion to the right, 425 universal completion, 412 usual hypotheses, 414

472

Wasserstein distance, 58 weak convergence, 2 weak generator of a semigroup, 436 Weibull distribution, 95 weighted occupation time, 133

Subject Index

weighted Sobolev space, 398 worthy martingale measure, 184 worthy martingale measure, 𝜎-finite, 191

Symbol Index

(𝑁𝑡 )𝑡 ≥0 : SC-semigroup on a space of measures, 241 (𝑄 𝑡𝑁 )𝑡 ≥0 : transition semigroup of an immigration process, 242 𝛾 (𝑄 𝑡 )𝑡 ≥0 : generalized Mehler semigroup, 367 (𝑄 𝑡 )𝑡 ≥0 : transition semigroup of an MB-process, 31 (𝑄 ◦𝑡 )𝑡 ≥0 : restriction of (𝑄 𝑡 )𝑡 ≥0 to 𝑀 (𝐸) ◦ , 34 ∗ (𝑇𝑡 )𝑡 ≥0 : dual semigroup of (𝑇𝑡 )𝑡 ≥0 , 367 (𝑉𝑡 )𝑡 ≥0 : cumulant semigroup of an MB-process, 33 ¯ ( 𝐸, 𝜌) ¯ : Ray–Knight completion of (𝐸, 𝜌), 428 ( 𝑃¯𝑡 )𝑡 ≥0 : (Ray) extension of (𝑃𝑡 )𝑡 ≥0 , 428 (𝑈¯ 𝛼 ) 𝛼>0 : (Ray) extension of (𝑈 𝛼 ) 𝛼>0 , 428 (·, ·) 𝐾 ,𝑇 : a bilinear form, 183 (𝛾𝑡 )𝑡 ≥0 : SC-semigroup on a Hilbert space, 367 (𝑣 𝑡 )𝑡 ≥0 : cumulant semigroup of a CB-process, 66 𝐵(𝐸) = bℬ(𝐸) : space of bounded Borel functions on 𝐸, 1 𝐵(𝐸) + : set of positive functions 𝑓 ∈ 𝐵(𝐸), 1

𝐵 : set of branching points of a Ray semigroup, 426 𝐵 𝑎 (𝐸) : set of functions 𝑓 ∈ 𝐵(𝐸) satisfying ∥ 𝑓 ∥ ≤ 𝑎, 1 𝐵 ℎ (𝐸) : set of Borel functions on 𝐸 bounded by const. · ℎ, 13 𝐶 (𝐸) : space of bounded continuous real functions on 𝐸, 1 𝐶 (𝐸) ++ : set of strictly positive functions 𝑓 ∈ 𝐶 (𝐸), 1 𝐶 2 (R𝑑 ): set of bounded continuous real functions on R𝑑 with bounded continuous derivatives up to the second order, 147 𝐶 2 (R+ ): set of bounded continuous real functions on R+ with bounded continuous derivatives up to the second order, 77 𝐶 ∞ (R𝑑 ) : set of bounded infinitely differentiable functions on R𝑑 with bounded derivatives, 396 𝐶0 (𝐸) : space of continuous real functions on locally compact space 𝐸 vanishing at infinity, 1, 413

© Springer-Verlag GmbH Germany, part of Springer Nature 2022 Z. Li, Measure-Valued Branching Markov Processes, Probability Theory and Stochastic Modelling 103, https://doi.org/10.1007/978-3-662-66910-5

473

474

𝐶ℎ (𝐸) : set of continuous functions on 𝐸 bounded by const. · ℎ, 13 𝐶𝑢 (𝐸) = 𝐶𝑢 (𝐸, 𝑑) : subset of 𝐶 (𝐸) of 𝑑-uniformly continuous real functions, 1 𝐷 : set of non-branch points of a Ray semigroup, 426 𝐸𝐶 : a subset of 𝐸 determined by a cumulant semigroup, 35 𝐸 𝐷 : entrance space of a transition semigroup, 429 𝐸 𝑅 : Ray space of a transition semigroup, 429 𝐸 𝐷𝐶 : a subset of 𝐸 𝐷 determined by a cumulant semigroup, 215 𝐼 (𝜆, 𝐿) : an infinitely divisible probability determined by (𝜆, 𝐿), 18 𝐼 (𝑏, 𝑅, 𝑀) : an infinitely divisible probability determined by (𝑏, 𝑅, 𝑀), 366 𝐼1 (𝛽, 𝑅, 𝑀) : an infinitely divisible probability determined by (𝛽, 𝑅, 𝑀), 367 𝐿 1 (𝐻1 ) : Banach space of 𝐻1 -integrable functions on 𝑀 (𝐸) ◦ , 337 1 𝐿 (𝜂0 ) : Banach space of 𝜂0 -integrable functions on 𝐸, 337 𝐿 𝑄 : Laplace functional of 𝑄, 9 𝐿 var (𝐹) : Lipschitz constant of a function 𝐹 on 𝑀 (𝐸), 58 𝑀 (𝐸) : space of finite Borel measures on 𝐸, 2 𝑀 (𝐸) ◦ : = 𝑀 (𝐸) \ {0}, where 0 is the null measure, 12 𝑀ℎ (𝐸) : space of Borel measures 𝜇 on 𝐸 satisfying 𝜇(ℎ) < ∞, 13 𝑀ℎ (𝐸) ◦ : = 𝑀ℎ (𝐸) \ {0}, where 0 is the null measure, 143 𝑃(𝐸) : space of probability measures on 𝐸, 2

Symbol Index

𝑆(𝐸, 𝑑) : a dense sequence in 𝐶𝑢 (𝐸), 5 𝑆1 (𝐸, 𝑑) : a dense sequence in { 𝑓 ∈ 𝐶𝑢 (𝐸) + : ∥ 𝑓 ∥ ≤ 1}, 5 𝑆2 (𝐸, 𝑟) : a dense sequence in 𝐶 (𝐸, 𝑟) ++ , 10 𝑆¯2 (𝐸, 𝑟) := 𝑆2 (𝐸, 𝑟) ∪ {0}, 10 𝒢¯ : augmentation of 𝒢, 419 𝒢¯ 𝑡 : augmentation of 𝒢𝑡 , 419 𝑣¯ 𝑡 : function on 𝐸 defined by 𝑣¯ 𝑡 (𝑥) = lim𝜆→∞ 𝑉𝑡 𝜆(𝑥), 34 𝛿 𝑥 : unit measure concentrated at point 𝑥, 2 𝜈ˆ : characteristic functional of 𝜈, 366 R : real line or 1-dimensional Euclidean space, 1, 411 R𝑑 : 𝑑-dimensional Euclidean space, 147 R+ = [0, ∞) : positive half line, 1 bℱ : = { 𝑓 ∈ ℱ : 𝑓 is bounded}, 1, 411 pℱ : = { 𝑓 ∈ ℱ : 𝑓 is positive}, 1, 411 ℬ(𝐸) : Borel 𝜎-algebra on 𝐸, 1 𝒞(𝐸) = 𝒞(𝐸, 𝑑) : space of 𝑑-continuous real functions on (𝐸, 𝑑), 413 𝒞𝑢 (𝐸) = 𝒞𝑢 (𝐸, 𝑑) : subset of 𝒞(𝐸) of 𝑑-uniformly continuous real functions, 413 ℰ(𝑄 ◦ ) : a set of excessive measures for (𝑄 ◦𝑡 )𝑡 ≥0 , 269 ∗ ℰ (𝑄) : a set of probabilities on 𝑀 (𝐸), 269 𝑢 ℰ : universal completion of ℰ, 412 ℰ𝐴 : trace/restriction of ℰ on 𝐴, 412 ℱ 𝜇 : P 𝜇 -completion of ℱ 𝑢 , 420 𝒢 𝜇 : P 𝜇 -completion of 𝒢, 419 𝜇 𝒢𝑡 : augmentation of 𝒢𝑡 , 419 ℐ(𝐸) : a convex cone of functionals on 𝐵(𝐸) + , 16 ℐ : a convex cone of functions on [0, ∞), 20

Symbol Index

𝒦(𝑃) : a set of entrance laws for (𝑃𝑡 )𝑡 ≥0 , 207 𝒦(𝑃) ◦ : = 𝒦(𝑃) \ {0}, where 0 is the trivial entrance law of (𝑃𝑡 )𝑡 ≥0 , 210 𝒦(𝑄) : a set of entrance laws for (𝑄 𝑡 )𝑡 ≥0 , 209 𝒦(𝑄 ◦ ) : a set of entrance laws for (𝑄 ◦𝑡 )𝑡 ≥0 , 209 𝒦(𝜋) : a set of entrance laws for (𝜋𝑡 )𝑡 ≥0 , 207 𝒦 𝑎 (𝑄) : a subset of 𝒦(𝑄), 210 𝒦 𝑎 (𝑄 ◦ ) : a subset of 𝒦(𝑄 ◦ ), 210 𝒦𝑚𝑎 (𝑄) : set of minimal elements of 𝒦 𝑎 (𝑄), 210 𝒦𝑚𝑎 (𝑄 ◦ ) : set of minimal elements of 𝒦 𝑎 (𝑄 ◦ ), 210 ℒ 0 : set of step processes in ℒ 1 , 300 ℒ𝐾0 (𝐸) : set of step processes in ℒ𝐾2 (𝐸), 186 1 ℒ : a space of predictable processes, 300 ℒ𝐾2 (𝐸) : a space of two-parameter predictable processes, 186 ¯ 419 𝒩(𝒢) : family of null sets in 𝒢, 𝜇 𝒩 (𝒢) : family of P 𝜇 -null sets in 𝒢 𝜇 , 419

475

ℛ : rational Ray cone of a resolvent, 427 𝒮(R𝑑 ) : Schwartz space on R𝑑 , 396 𝒮 𝛼 : set of 𝛼-excessive functions for a semigroup, 417 𝒮 ′ (R𝑑 ) : dual space of 𝒮(R𝑑 ), 399 𝜇( 𝑓 ) : integral of 𝑓 with respect to 𝜇, 2, 411 𝜇 𝐴 : trace of 𝜇 on 𝐴, 412 𝜎(𝒢) : 𝜎-algebra generated by 𝒢, 1, 411 𝑑2 (·, ·) : a metric on ℒ𝐾2 (𝐸), 186 | · | : Euclidean norm, 147 ∥ · ∥ : supremum/uniform norm of functions, 1, 411 ∥ · ∥ : total variation norm of signed measures, 58 ∥ · ∥ 𝐻1 ,𝑡 : a seminorm on ℒ𝐻1 1 (𝑀 (𝐸) ◦ ), 342 ∥ · ∥ 𝐻1 : the norm of the Banach space 𝐿 1 (𝐻1 ), 337 ∥ · ∥ 𝐾 ,𝑇 : a seminorm on ℒ𝐾2 (𝐸), 186 ∥ · ∥ 𝜂0 ,𝑡 : a seminorm on ℒ𝜂10 (𝐸), 342 ∥ · ∥ 𝜂0 : the norm of the Banach space 𝐿 1 (𝜂0 ), 337