Convolution-like Structures, Differential Operators and Diffusion Processes (Lecture Notes in Mathematics, 2315) [1st ed. 2022] 3031052951, 9783031052958

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Lecture Notes in Mathematics  2315

Rúben Sousa Manuel Guerra Semyon Yakubovich

Convolutionlike Structures, Differential Operators and Diffusion Processes

Lecture Notes in Mathematics Volume 2315

Editors-in-Chief Jean-Michel Morel, CMLA, ENS, Cachan, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Alessio Figalli, ETH Zurich, Zurich, Switzerland Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Mark Policott, Mathematics Institute, University of Warwick, Coventry, UK Sylvia Serfaty, NYU Courant, New York, NY, USA László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig, Germany Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

This series reports on new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes: 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research. Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is sometimes more important than its form, which may be preliminary or tentative. Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews, and zbMATH.

Rúben Sousa • Manuel Guerra • Semyon Yakubovich

Convolution-like Structures, Differential Operators and Diffusion Processes

Rúben Sousa Department of Mathematics, Faculty of Sciences University of Porto Porto, Portugal

Manuel Guerra Department of Mathematics, ISEG Universidade de Lisboa Lisbon, Portugal

Semyon Yakubovich Department of Mathematics, Faculty of Sciences University of Porto Porto, Portugal

This work was supported by Fundação para a Ciência e a Tecnologia (Portugal) (UIDB/05069/2020, PD/BI/128072/2016, PD/BD/135281/2017).

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-031-05295-8 ISBN 978-3-031-05296-5 (eBook) https://doi.org/10.1007/978-3-031-05296-5 Mathematics Subject Classification: 60G53, 60G51, 47F10, 33C10, 33C15, 44A15 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed 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 Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

It is well-known that convolutions, differential operators and diffusion processes are interconnected subjects: the ordinary convolution commutes with the Laplacian, and the law of Brownian motion has a convolution semigroup property with respect to the ordinary convolution. If we seek to generalize this useful connection so as to cover other differential operators and diffusion processes, we are naturally led to the notion of a convolution-like operator—i.e. a bilinear operator with respect to which a given diffusion (other than the Brownian motion) has the convolution semigroup property, and which commutes with the generator of the given diffusion. The study of this and other related concepts in generalized harmonic analysis has, since the 1930s, attracted the attention of many researchers, most notably Delsarte, Levitan, Berezansky, Urbanik, Chebli, Heyer and Trimèche, Zeuner, among others. A few books have been published on the theory of generalized harmonic analysis, but a general, systematic presentation of the problem of constructing convolutionlike operators and of its applications to stochastic processes and differential equations is missing. The goal of this book is to try to fill in this gap, while at the same time providing an accessible introduction to recent developments on this topic. The book also intends to draw attention to a wide range of questions which still remain open in this area of research. We are confident that this book will be a valuable resource for graduate students and researchers interested on the intersections between harmonic analysis, probability theory and differential equations. We are deeply thankful to all our friends and fellow researchers who have contributed to this project with comments, conversations or encouragement. We are also thankful to the organizers of the various conferences where our work has been featured and where many fruitful interactions took place, namely the 12th Vilnius Conference on Probability Theory and Statistics (Vilnius, Lithuania, 2018), the WSMC12 Workshop on Statistics, Mathematics and Computation (Covilhã, Portugal, 2018), the Probability and Analysis Conference (Bedlewo, Poland, 2019), the IWOTA 2019 (Lisbon, Portugal, 2019), the 12th ISAAC Congress (Aveiro, Portugal, 2019) and the ÖMG Conference (Dornbirn, Austria, 2019).

v

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Preface

The work of the first and third authors was partially supported by CMUP, which is financed by national funds through FCT—Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020. The first author was also supported by the grants PD/BI/128072/2016 and PD/BD/135281/2017, under the FCT PhD Programme UC|UP MATH PhD Program. The work of the second author was partially supported by the project CEMAPRE/REM—UIDB/05069/2020— financed by FCT through national funds. Porto, Portugal Lisbon, Portugal Porto, Portugal September 2020

Rúben Sousa Manuel Guerra Semyon Yakubovich

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Motivation and Scope .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Organization of the Book .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 5

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Continuous-Time Markov Processes . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Sturm-Liouville Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Solutions of the Sturm-Liouville Equation . . . . . . . . . . . . . . . . . . . . 2.2.2 Eigenfunction Expansions .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.3 Diffusion Semigroups Generated by Sturm-Liouville Operators .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.4 Remarkable Particular Cases . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Generalized Convolutions and Hypergroups . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Harmonic Analysis with Respect to the Kingman Convolution . . . . . .

9 9 17 18 22 28 30 35 43

3 The Whittaker Convolution.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 A Special Case: The Kontorovich–Lebedev Convolution .. . . . . . . . . . . . 3.2 The Product Formula for the Whittaker Function .. . . . . . . . . . . . . . . . . . . . 3.3 Whittaker Translation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Index Whittaker Transforms .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Whittaker Convolution of Measures . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.1 Infinitely Divisible Distributions .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.2 Lévy–Khintchine Type Representation . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Lévy Processes with Respect to the Whittaker Convolution .. . . . . . . . . 3.6.1 Convolution Semigroups . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.2 Lévy and Gaussian Processes . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.3 Some Auxiliary Results on the Whittaker Translation .. . . . . . . 3.6.4 Moment Functions .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.5 Lévy-Type Characterization of the Shiryaev Process . . . . . . . . . 3.7 Whittaker Convolution of Functions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7.1 Mapping Properties in the Spaces Lp (rα ) . . . . . . . . . . . . . . . . . . . . .

51 52 53 64 71 80 82 85 89 89 93 98 101 108 111 112

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Contents

3.7.2 The Convolution Banach Algebra Lα,ν . . . .. . . . . . . . . . . . . . . . . . . . 116 3.8 Convolution-Type Integral Equations . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 120 4 Generalized Convolutions for Sturm-Liouville Operators . . . . . . . . . . . . . . 4.1 Known Results and Motivation .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Laplace-Type Representation .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 The Existence Theorem for Sturm-Liouville Product Formulas .. . . . . 4.3.1 The Associated Hyperbolic Cauchy Problem . . . . . . . . . . . . . . . . . 4.3.2 The Time-Shifted Product Formula .. . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.3 The Product Formula for wλ as the Limit Case . . . . . . . . . . . . . . . 4.4 Sturm-Liouville Transform of Measures . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Sturm-Liouville Convolution of Measures.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.1 Infinite Divisibility and Lévy-Khintchine Type Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.2 Convolution Semigroups . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.3 Additive and Lévy Processes . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Sturm-Liouville Hypergroups . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6.1 The Nondegenerate Case . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6.2 The Degenerate Case: Degenerate Hypergroups of Full Support .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Harmonic Analysis on Lp Spaces . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7.1 A Family of L1 Spaces . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7.2 Application to Convolution-Type Integral Equations . . . . . . . . .

127 128 130 139 139 145 147 150 153

5 Convolution-Like Structures on Multidimensional Spaces.. . . . . . . . . . . . . 5.1 Convolutions Associated with Conservative Strong Feller Semigroups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Nonexistence of Convolutions: Diffusion Processes on Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.1 Special Cases and Numerical Examples .. .. . . . . . . . . . . . . . . . . . . . 5.2.2 Some Auxiliary Results . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.3 Eigenfunction Expansions, Critical Points and Nonexistence Theorems . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Nonexistence of Convolutions: One-Dimensional Diffusions . . . . . . . . 5.4 Families of Convolutions on Riemannian Structures with Cone-Like Metrics.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.1 The Eigenfunction Expansion of the Laplace–Beltrami Operator . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.2 Product Formulas and Convolutions .. . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.3 Infinitely Divisible Measures and Convolution Semigroups.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.4 Special Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.5 Product Formulas and Convolutions Associated with Elliptic Operators on Subsets of R2 . .. . . . . . . . . . . . . . . . . . . .

183

156 158 159 165 165 171 175 176 180

183 196 197 201 203 207 214 216 224 230 235 239

Contents

ix

A Some Open Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 247 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 249 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 257

List of Symbols

1(·) 1B (·) (a)n Aα ACloc (a, b) Bb (E) B(x, ε) C(E) C0 (E) Cb (E) Cc (E) Ck (E) Ckc (E) C∞ c,even d(·, ·) Dμ (z) D(L) δx e(μ), eα (μ), ek (μ) E, EL , EN Ex 0 F 2 F1 (a, b; c; z) G, G(0) , G(b) , G(2) (z) H k (E) H Iη (z)

function identically equal to one indicator function of the subset B Pochhammer symbol, 55 generator of the Shiryaev process, 51 absolutely continuous}  {f : (a, b) −→ C | f is locally  f : E −→ C measurable  supx∈E |f (x)| < ∞ ball centred at x with radius ε {f : E −→ C | f is continuous} {f ∈ C(E) | f vanishes at infinity} {f ∈ C(E) | f is bounded} {f ∈ C(E) | f has compact support} {f ∈ C(E) | f is k times continuously differentiable} Cc (E) ∩ Ck (E) {f : [0, ∞) −→ C | f is the restriction of an even C∞ c (R) −function} distance function, 11, 187 parabolic cylinder function, 56 domain of the operator L Dirac measure at the point x Poisson(-like) measure associated with μ, 85, 157, 230 sesquilinear forms, 29, 185 expectation for a time-homogeneous Markov process started at x0 Sturm-Liouville integral transform, 23 Gauss hypergeometric function, 31 infinitesimal generators, 10, 13, 185 Gamma function, 44 Sobolev space, 186 Hankel transform, 44 modified Bessel function of the first kind, 44

xi

xii

List of Symbols

Jη (z) J η (z) kα (x, y, ξ ) Kη (z)

, 

Lp (E, μ) L, L(2) Mκ,ν (z) MC (E) M+ (E) μ∗n , μn v μn −→ μ w μn −→ μ g pt,x (dy), p(t, x, y) Px0 P(E) Pid , Pα,id

(a, b; z) qt (x, y, ξ ) (α,β) Rn (x) R+ R+ 0 ρL (p)

Tt , Tt Tx , Tμ ℘ Vk,λ (·), Vλ,η (·) wλ (·) Wα,ν (z) Wα,ν (z) Wα X A,f ZX

Bessel function of the first kind, 44 normalized Bessel function of the first kind, 44 kernel of the Whittaker product formula, 63 modified Bessel function of the second kind, 55 Sturm-Liouville operators    f : E −→ C measurable  E |f |p dμ < ∞ realization of a Sturm-Liouville operator Whittaker function of the first kind, 32 space of finite complex Borel measures on E space of finite positive Borel measures on E n-fold convolution of μ with itself vague convergence of measures, 10 weak convergence of measures, 10 Riemannian volume on a manifold with cone-like metric, 216 transition probabilities, transition density of a diffusion process distribution of a time-homogeneous Markov process started at x0 space of probability measures on E subset of infinitely divisible distributions, 82, 156 confluent hypergeometric function of the second kind, 57 kernel of the regularized product formula, 145, 192, 208 Jacobi polynomials, 31 open half-line (0, ∞) closed half-line [0, ∞) spectral measure for the operator L, 23 strongly continuous one-parameter semigroups generalized translation operator generalized eigenfunctions on spaces with cone-like metrics, 218, 240 solution of the Sturm-Liouville boundary value problem, 18 Whittaker function of the second kind, 32 normalized Whittaker function of the second kind, 54 index Whittaker transform, 71 domain for the Whittaker translation, 98 Dynkin martingale, 95

Chapter 1

Introduction

The goal of this book is to provide an introduction to recent developments in the theory of generalized harmonic analysis and its applications to the study of differential operators and diffusion processes. Throughout the book, we shall provide answers to the following research question, which triggered our interest in this topic: given a diffusion process {Xt }t ≥0 on a metric space E, can we construct a convolution-like operator ∗ on the space of probability measures on E with respect to which the law of Xt has the ∗-convolution semigroup property, i.e. can be written as P [Xt ∈ ·] = μt ∗δX0 , where the measures μt are such that μt +s = μt ∗ μs for all t, s ≥ 0? The significance of this problem stems both from its interpretation as a generalization of classical harmonic analysis and from its probabilistic applications. These motivations are discussed in the next section, where we highlight the connections between the construction of convolution-like structures and disciplines such as stochastic processes, ordinary and partial differential equations, spectral theory, special functions, and integral transforms. In Sect. 1.2 we describe the logical sequence of the chapters and provide a summary of the main results presented in this volume.

1.1 Motivation and Scope Harmonic analysis for elliptic differential operators A first motivation for the problem formulated above comes from the fact that the existence of a generalized convolution structure for the diffusion process generated by a given elliptic differential operator L puts at our disposal a valuable tool for the study of elliptic and parabolic partial differential equations determined by L.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Sousa et al., Convolution-like Structures, Differential Operators and Diffusion Processes, Lecture Notes in Mathematics 2315, https://doi.org/10.1007/978-3-031-05296-5_1

1

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1 Introduction

Indeed, the most straightforward way to investigate the properties of various heattype equations (and their nonlocal counterparts) is, in many cases, by making use of techniques from (standard) harmonic analysis [6, 129, 151, 160]. If the properties of the convolution-like operator are similar to those of the ordinary convolution, then the resulting algebraic structure allows one to develop the basic notions of harmonic analysis in parallel with the standard theory [15, 127]; therefore, it is natural to expect that a positive answer to our problem will lead to a better understanding of the properties of the corresponding differential operators and the associated potential-theoretic objects. We note that the problem of constructing a generalized convolution can be formulated for a large class of operators which includes, in particular, the (Dirichlet, Neumann, Robin) Laplacian on Euclidean domains and Riemannian manifolds. This interplay between convolutions and elliptic operators originates in the observation that the existence of a convolution-like operator for the diffusion {Xt } is closely related to the properties of the generalized eigenfunctions of its (infinitesimal) generator L, i.e. of the solutions of the elliptic equation Lu = λu (λ ∈ C). Indeed, suppose that ∗ is a bilinear operator on the set P(E) of probability measures on E satisfying the conditions C1. (Convolution semigroup property) Px [Xt ∈ ·] = μt ∗ δx (t > 0, x ∈ E), where {μt }t ≥0 ⊂ P(E) is such that μt +s = μt ∗ μs for all t, s ≥ 0, and Px stands for the distribution of {Xt } started at x; C2. There exists a family  of bounded continuous functions such that 



 ϑ dμ · ϑ dν

ϑ d(μ ∗ ν) = E

E

for all ϑ ∈  and μ, ν ∈ P(E).

E

(1.1) Notice that (1.1) holds if ∗ is the ordinary convolution and ϑ(x) = eλx with λ ∈ C; condition C2 can thus be interpreted as a general formulation of a trivialization property similar to that of the Fourier transform with respect to the ordinary convolution. From C1 and C2 it is not difficult to deduce (cf. Chap. 5) that each ϑ ∈  is a generalized eigenfunction of the transition semigroup of {Xt } and, consequently, a generalized eigenfunction of the elliptic operator L. Replacing μ and ν by Dirac measures in (1.1), we find that there exists a family of measures ν x,y ∈ P(E) such that the probabilistic product formula  ϑλ dν x,y = ϑλ (x) ϑλ (y),

x, y ∈ E

(1.2)

E

holds for bounded solutions ϑλ of L ϑλ = λϑλ . (We use the word ‘probabilistic’ in order to emphasize that {ν x,y } is a family of probability measures.) Conversely, if a probabilistic product formula of the form (1.2) holds for a sufficiently large family of generalized eigenfunctions of L, then the generalized convolution operator defined

1.1 Motivation and Scope

3

 as (μ∗ν)(dξ ) = ν x,y (dξ ) μ(dx) ν(dy) is such that the ∗-convolution semigroup property C1 holds for the distribution of {Xt }. The historical development of the topic of generalized harmonic analysis began with the seminal works of Delsarte [45] and Levitan [117], where it was first noticed that product formulas are the key ingredient for the construction of such convolution-like structures. The nontrivial motivating example came from the Bessel differential operator, for which the existence of the product formula (1.2) follows from a classical result on the Bessel function. (An overview on this motivating example will be presented in Sect. 2.4.) This led, on the one hand, to the proposal of axiomatic structures—often referred to as generalized convolutions, generalized translations, hypercomplex systems or hypergroups—which aimed to identify the essential features which allow one to derive analogues of the basic facts of classical harmonic analysis. In Sect. 2.3 we provide some historical background on the development of such axiomatic theories, whose range extends far beyond the particular case of structures associated with diffusion processes or elliptic operators. On the other hand, there has been a continuous interest in finding additional examples of nontrivial product formulas associated with Sturm–Liouville and elliptic operators. Besides the Bessel example, other product formulas have been obtained by exploiting the properties of special functions of hypergeometric type [70, 73, 105]. An alternative strategy relies on the fact that certain differential operators are related with topological groups [105]. Yet another approach, which (unlike the former techniques) is applicable to one-dimensional operators with general coefficients,

is to rely on the  associated hyperbolic PDE: if L is the Sturm–Liouville operator 1r −(pu ) + qu and ϑλ are the generalized eigenfunctions satisfying the boundary condition ϑλ (a) = 1, then the product f (x, y) = ϑλ (x)ϑλ (y) is a solution of the hyperbolic PDE

 1 −∂x p(x) ∂x f (x, y) + q(x)f (x, y) r(x)

 1 −∂y p(y) ∂y f (x, y) + q(y)f (x, y) . = r(y) satisfying the boundary condition f (x, a) = ϑλ (x); studying the properties of this PDE is therefore a natural strategy for proving the existence of a product formula of the form (1.2) and extracting information about the measure ν x,y [30, 38, 120, 210]. The existing theory on convolution-like operators associated with elliptic operators is mostly limited to the one-dimensional (Sturm–Liouville) case. One of the reasons for this is the fact that there is a well-developed spectral theory for Sturm– Liouville operators which, in particular, ensures that (under suitable boundary conditions) the corresponding generalized eigenfunctions are the kernel of a Sturm–  Liouville type integral transform (Fh)(λ) := I h(x) wλ (x) m(dx) which, similar to the Fourier transform, defines an isometric isomorphism between L2 spaces. This class of transformations includes many common integral transforms (Hankel, Kontorovich–Lebedev, Mehler–Fock, Jacobi, Laguerre, etc.). The construction of

4

1 Introduction

Sturm–Liouville convolutions satisfying the trivialization identity F(h ∗ g) = (Fh)·(Fg) triggers a better understanding of the mapping properties of such Sturm– Liouville integral transforms. Due to our probabilistic motivations, the present discussion focuses on convolution-like structures where the convolution is a bilinear operator acting on finite complex measures. We observe, however, that in the theory of integral transforms and special functions it is more common to define a convolution (say, associated with a given integral transform) as an operator acting on suitable spaces of integrable functions [73, 207]. In this context, it usually becomes less of a concern whether or not the convolution preserves properties such as positivity or boundedness. Construction of Lévy-like processes Lévy processes are a very important class of Markov processes. By definition, they are stochastically continuous and have stationary and independent increments. Lévy processes are a versatile class of processes with jumps whose continuous representatives are the drifted Brownian motions (in the sense that any Lévy process with continuous paths is a drifted Brownian motion); therefore, they can be seen as a natural generalization of Brownian motion. Replacing Brownian motions by Lévy processes with jumps is a common strategy for obtaining models with greater flexibility in mathematical finance and other applications [6, 143]. The Brownian motion is the most famous diffusion process, but many other diffusion processes also find diverse applications in a wide range of fields. One such field is mathematical finance, where one-dimensional diffusions such as the Bessel, Ornstein–Uhlenbeck and Shiryaev processes are often used in the modelling of the underlying financial variables, while two-dimensional diffusion processes have been extensively applied in the context of stochastic volatility models [122, 143]. It is relevant to ask whether these other diffusion processes can also be generalized into a class of processes characterized by some analogue of the notions of stationarity and independence. By a well-known characterization, Lévy processes can be equivalently defined as Feller processes whose law satisfies the convolution semigroup property (as stated in condition C1) with respect to the usual convolution. It is thus natural to generalize the notion of Lévy process by replacing the requirement of stationarity and independence by the convolution semigroup property with respect to any convolution-like operator with suitable properties. This provides us with a recipe for defining a class of Lévy-like processes associated with a given diffusion process: as prescribed in the problem above, one should construct a convolution-like operator such that condition C1 holds for the law of the given diffusion. This generalized notion of Lévy process has been proposed in various papers [23, 83, 163]; however, the class of diffusions which have been proved to admit such an associated family of Lévy-like processes is still very limited. The notion of a convolution semigroup is closely related with that of an infinitely divisible distribution. In the case of the usual convolution, a central role is played

1.2 Organization of the Book

5

by the Lévy–Khintchine theorem which provides a complete description of the set of infinitely divisible distributions; in addition, laws of large numbers and other limit theorems have been established for random walks (the discrete analogues of Lévy processes). It is, of course, desirable to determine what are the properties which ensure that analogues of those fundamental results hold for convolution-like operators constructed via the above procedure. Scope of this book Following the motivations above, this book gives an account of the construction of generalized convolution operators and measure algebras that are natural to a given diffusion or equivalently, to a given elliptic operator. This includes presentation of the corresponding analogues of classic harmonic analysis and Lévy processes. However, the presentation of these topics is kept at a fairly basic level. Harmonic analysis is mostly limited to the construction of analogues of Fourier transforms, while the discussion of Lévy processes is centered on Lévy–Khintchine-like representations, the characterization of certain processes playing a role analogous to Brownian motion in classic stochastic calculus, and basic sample paths properties. This is due to two main reasons: First, harmonic analysis and Lévy processes are very rich fields, with many ramifications into other fields of mathematics. Thus, a deeper study of these topics would require a much larger volume. Stark and possibly arbitrary choices in selection of subtopics would be required. Second, and most important, a large part of such study is not yet done. We trust that our presentation of the core building blocks will convince the reader that rather complete analogues to classical theories are not only feasible but also desirable.

1.2 Organization of the Book Unlike previous monographs on generalized harmonic analysis [15, 19, 184], the presentation of the material in this book proceeds from the particular to the general. In our view, this reflects the natural flow of progression in mathematical reasoning, and stimulates the reader to pursue further generalizations of the results stated throughout the text. To make this book accessible to a wider audience, in Chap. 2 we start by presenting a comprehensive summary of background material on stochastic processes, harmonic analysis and Sturm–Liouville theory. Chapter 3 is devoted to the construction of the Whittaker convolution, a convolution-like structure associated with the Shiryaev process, whose infinitesimal d2 d generator is the Sturm–Liouville operator x 2 dx 2 + (1 + 2(1 − α)x) dx , where α is a real parameter. We open the chapter by explaining why this diffusion process is a natural starting point for our exposition. As we will see, this has to do not only with the importance and the numerous applications of this diffusion, but also with the

6

1 Introduction

fact that its connection with the Kontorovich-Lebedev transform provides us with the required product formula for the particular case α = 0. In the general case, the generalized eigenfunctions of the generator are written in terms of the Whittaker W function with first parameter α. In Chap. 3 we state and prove a product formula for the Whittaker W function whose kernel does not depend on the second parameter and is given in closed form in terms of the parabolic cylinder function. Our method is based on special function theory and standard integral transform techniques. Furthermore, we show that if α < 12 then the convolution-like operator induced by the product formula has the property that the convolution of probability measures is a probability measure, and therefore defines a measure algebra in which the Shiryaev process becomes a Lévy-like process. We also provide a Lévy–Khintchine type theorem which describes the general form of an index Whittaker transform of a Lévy-like process, and we show that the Shiryaev process admits a martingale characterization analogous to Lévy’s characterization of Brownian motion. Chapter 3 also contains three other results of independent interest: an integral representation for the Whittaker function as a Fourier transform of a parabolic cylinder function, an analogue of the Wiener-Lévy theorem for the index Whittaker transform, and an existence and uniqueness theorem for a family of convolutiontype integral equations. An example is provided where the existence and uniqueness theorem yields an explicit expression for the solution of an integral equation with the Whittaker function in the kernel. A property of the Whittaker convolution is that the support of the measures of the underlying product formula is the whole half-line. This fact is remarkable because, in contrast, the so-called Sturm–Liouville hypergroups—a class of convolutions associated with one-dimensional diffusions which have been extensively studied in previous literature [19, 163, 210], and whose general construction is based on the associated hyperbolic PDE introduced above—have the property that the measures of the underlying product formula have compact support. This distinction raises a natural question, namely whether it is possible to construct other onedimensional convolutions where the measures of underlying product formula do not have compact support. A positive answer is given in Chap. 4, where we develop a unified approach for constructing Sturm–Liouville convolutions whose supports can be either compact or noncompact. Our technique is based on the hyperbolic PDE approach, which is shown to be extensible to a larger class of Sturm–Liouville operators whose associated hyperbolic equations are possibly degenerate at the initial line. The extension relies on an existence and uniqueness theorem for hyperbolic Cauchy problems which is useful in itself, as it covers many parabolically degenerate cases which are outside the scope of the classical theory and for which it is not even clear whether the Cauchy problem is well-posed or not. We also introduce a regularization method which makes use of the properties of the diffusion semigroup to construct a sequence of regularized product formulas, from which the desired product formula is obtained via a weak convergence argument. Many probabilistic properties of the Whittaker convolution, such as the interpretation of the associated diffusion as a

1.2 Organization of the Book

7

Lévy-like process or the Lévy–Khintchine type theorem, extend to this general family of Sturm–Liouville convolutions. The convolutions constructed in Chap. 4 satisfy the compactness axiom if and only if the hyperbolic equation determined by the Sturm–Liouville operator is uniformly hyperbolic on its domain. If this is the case, then one can check that the convolution satisfies all the axioms of hypergroups; this leads to an improvement of previous existence theorems for Sturm–Liouville hypergroups. In turn, the case where the hyperbolic PDE is parabolically degenerate yields a general family of degenerate Sturm–Liouville hypergroups which includes the Whittaker convolution as a particular case. The results described thus far are restricted to convolution structures for onedimensional diffusions, but our opening discussion makes it clear that the problem of constructing convolution-like operators associated with diffusion processes is meaningful in a much more general framework. In Chap. 5 we study the construction of convolutions for diffusions on a general locally compact separable metric space. We start by identifying the requirements that such a convolution should satisfy in order to allow for the development of the basic notions of probabilistic harmonic analysis, and we then determine necessary and sufficient conditions which relate the existence of the convolution structure with certain properties of the eigenfunctions of the generator. One of the necessary conditions is that the eigenfunctions should have a common maximizer, which is quite restrictive in dimension greater than 1; this explains in part why a significant part of the literature on generalized harmonic analysis is devoted to problems on one-dimensional spaces. Using standard results on spectral theory of differential operators, one can prove that the common maximizer property does not hold for reflected Brownian motions on smooth domains of Rd (d ≥ 2) or on compact Riemannian manifolds; this leads to a nonexistence theorem for convolutions on such domains. Going back to the onedimensional problem, (the failure of) the common maximizer property will also be shown to yield nonexistence theorems for some one-dimensional diffusions which are not covered in the preceding chapter. Another difficulty which arises in the multidimensional setting is that the associated hyperbolic equation becomes ultrahyperbolic, and therefore the PDE approach for the construction of convolutions is only applicable if the elliptic operator admits separation of variables. This is a significant limitation, but it does not hinder the construction of nontrivial multidimensional convolutions, as there are many elliptic operators which admit separation of variables but do not decompose trivially into a product of one-dimensional operators. In the final section of Chap. 5 we discuss the interesting example of the Laplace–Beltrami operator on a general class of two-dimensional manifolds endowed with cone-like metrics. The product formula for the generalized eigenfunctions is shown to depend on one of the two spectral parameters; accordingly, it induces a family of convolution operators (rather than a single convolution). This structure gives rise to a Lévy-like representation for the reflected Brownian motion on the manifold, together with other analogues of the one-dimensional results.

8

1 Introduction

Appendix A collects some open problems which naturally arise from the present work. Part of the results presented in this book have appeared in our papers [172–176], but are here presented in a streamlined and more self-contained way, often with improvements to the original formulation of the theorems.

Chapter 2

Preliminaries

Our opening discussion in the introductory chapter sketched some connections between the problem of constructing generalized convolutions and various fields of mathematics such as harmonic analysis, stochastic processes, differential equations, spectral theory and special functions. In the first three sections of this chapter, we review some prerequisite notions and facts from these fields. Section 2.4 closes the chapter with an example of a generalized convolution (the Kingman convolution), providing motivation and a benchmark for the results in further chapters. For brevity, our treatment is limited to the essential. Proofs are omited, being replaced by appropriate bibliographic references, whenever such references exist and are easily accessible. Since the topics are dispersed among various fields, there is no single work, or a small number of works, covering all the topics in this chapter. To help those readers wishing to study parts of this background in some depth, we provide some guidance: For Markov processes, associated operators and semigroups, the classical book by S. Ethier and T. Kurtz [57] is an excellent source. For one-dimensional diffusions, we suggest M. Fukushima’s paper [65], and for infinitely divisible distributions and Lévy processes K. Sato’s classical book [166]. G. Teschl’s book [185] contains the essentials of Sturm-Liouville theory in a compact and accessible presentation, but readers may need to complement it with papers [53] and [122]. Concerning harmonic analysis on generalized convolution structures, the most important background material for this book can be found in W.R. Bloom and H. Heyer’s monography [19].

2.1 Continuous-Time Markov Processes In what follows we write Px0 for the distribution of a given time-homogeneous Markov process started at the point x0 and Ex0 for the associated expectation operator. A sequence of finite complex measures μn on a locally compact separable © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Sousa et al., Convolution-like Structures, Differential Operators and Diffusion Processes, Lecture Notes in Mathematics 2315, https://doi.org/10.1007/978-3-031-05296-5_2

9

10

2 Preliminaries

metric space E is said to converge weakly (respectively, vaguely) to the finite   complex measure μ if limn E g(ξ )μn (dξ ) = E g(ξ )μ(dξ ) for all g ∈ Cb (E) (respectively, for all g ∈ Cc (E)). It is clear that weak convergence implies vague convergence. The converse is true if the sequence of total variations μn  converges to the total variation of μ (for example, if μn and μ are probability measures), but may fail if μn is a generic sequence. Feller Semigroups and Processes A family {Tt }t ≥0 of linear operators from a Banach space V to itself is said to be a semigroup if T0 = Id (the identity operator on V) and Tt +s = Tt Ts for all t, s ≥ 0. The semigroup is said to be a contraction semigroup if Tt  ≤ 1 for all t and strongly continuous if for all f ∈ V we have Tt f − f  −→ 0 as t ↓ 0. A Feller semigroup on a locally compact separable metric space E is a strongly continuous contraction semigroup of positive operators on the C0 (E) provided with the supremum norm. A time-homogeneous Markov process {Xt }t ≥0 with state space E is called a Feller process if its transition semigroup (Tt f )(x) := Ex [f (Xt )] (f ∈ C0 (E)) is a Feller semigroup. Given a Feller semigroup {Tt }t ≥0 on E, onecan use the Riesz representation theorem [10, §29] to write it as (Tt f )(x) = E f (y) pt,x (dy), where {pt,x (·)} is a uniquely defined family of sub-probability measures on E which is vaguely v continuous in x (i.e. pt,xn (·) −→ pt,x (·) whenever xn → x). We can then use the standard Kolmogorov consistency theorem to construct a Markov process {Xt } on E such that (Tt f )(x) := Ex [f (Xt )] (cf. [9, §36]). Therefore, Feller processes are in one-to-one correspondence with Feller semigroups. Moreover, this representation allows us to define the natural extension of a Feller semigroup to a semigroup of operators on Bb (E) as  (Tt f )(x) =

f (y) pt,x (dy),

f ∈ Bb (E).

E

  If this extension is such that Tt Bb (E) ⊂ Cb (E) for all t > 0, then we say that {Tt } is a strong Feller semigroup and {Xt } is a strong Feller process. A Feller semigroup {Tt }t ≥0 on E (and the associated Feller process) is said to be conservative if Tt 1 = 1 for all t (where 1 denotes the function identically equal to one) or, equivalently, if {pt,x (·)} is a family of probability measures on E. In this case the family {pt,x (·)} is weakly continuous in x. If {Tt }t ≥0 is a conservative Feller semigroup, then the strong continuity on C0 (E) extends to local uniform continuity on Cb (E), i.e. we have limt ↓0 Tt f = f uniformly on compact sets for all f ∈ Cb (E).   The (infinitesimal) generator G, D(G) of a strongly continuous semigroup {Tt }t ≥0 on a Banach space V is the (generally unbounded) operator 

 D(G) := f ∈ V  lim 1t (Tt f − f ) exists in the topology of V , t ↓0

G : D(G) ⊂ V −→ V,

Gf := lim 1t (Tt f − f ). t ↓0

2.1 Continuous-Time Markov Processes

11

In particular, the domain of the infinitesimal generator of a Feller semigroup {Tt }t ≥0 is the set D(G(0) ) of functions f ∈ C0 (E) such that limt ↓0 1t (Tt f (x) − f (x)) exists as a uniform limit. Infinitesimal generators have the following property: Proposition 2.1 ([57, Chapter 1, Corollary 1.6]) The infinitesimal generator of any strongly continuous semigroup on a Banach space is a densely defined closed linear operator. Further, using Theorem 1.24 in Davies’ book [42], it is possible to show that in the case of Feller semigroups, pointwise convergence of 1t (Tt f (x) − f (x)) can be substituted for uniform convergence: Proposition 2.2 Let {Tt } be a Feller semigroup. The domain D(G(0) ) of its infinitesimal generator is the subspace     ∃g ∈ C0 (E) such that    D(G ) = f ∈ C0 (E)  . g(x) = limt ↓0 1t (Tt f )(x) − f (x) for all x ∈ E (0)

Concerning the sample path properties of Feller processes, it is clear that strong continuity of the Feller semigroup in C0 (E) implies stochastic continuity of the corresponding Feller process {Xt }t ≥0, i.e., for every t ≥ 0, we have lim P [d(Xs , Xt ) > ε] = 0

s→t

∀ε > 0,

where d(·, ·) is the distance on E. Further, the following important regularity result holds: Proposition 2.3 ([57, Chapter 4, Theorem 2.7]) Let {Tt }t ≥0 be a Feller semigroup on E. For each μ ∈ P(E), there exists a càdlàg Feller process {Xt }t ≥0 corresponding to {Tt } with initial distribution μ. If {F X t }t ≥0 is the filtration generated by the process {X }, then {X } is strong Markov with respect to the filtration t t   X  F t + = s>t F X s t ≥0 . In particular, every Feller process {Xt } has a càdlàg modification, i.e. there exists t } such that P [X t = Xt ] = 1 for all t ≥ 0, and the sample path a Feller process {X  t → Xt (ω) is for almost every (a.e.) ω right continuous with finite left-hand limits. The following is a useful criterion for continuity of sample paths. Proposition 2.4 ([57, Chapter 4, Proposition 2.9 and Remark 2.10]) Let {Xt }t ≥0 be a càdlàg Feller process on E. If for all ε > 0, x ∈ E we have Px [Xt ∈ E \ B(x, ε)] = o(t)

as t ↓ 0

(2.1)

then the paths t → Xt (ω) are continuous for a.e. ω. In particular, if the domain D(G(0) ) of the infinitesimal generator of {Xt } is such that for all ε > 0, x ∈ E there exists f ∈ D(G(0) ) such that f (x) = f , supy∈E\B(x,ε) f (y) < f  and G(0) f (x) = 0, then (2.1) holds and consequently the Feller process {Xt } has almost surely (a.s.) continuous paths.

12

2 Preliminaries

Conversely, it is possible to prove some properties of infinitesimal generators starting from sample path properties of the corresponding Feller process. We will use the following: Proposition 2.5 ([25, Theorem 1.40]) Let {Xt }t ≥0 be a Feller process on Rd whose paths are a.s. continuous. Then the infinitesimal generator (G(0), D(G(0) )) is a local operator, i.e. we have (G(0)f1 )(x) = (G(0)f2 )(x) whenever f1 , f2 ∈ D(G(0) ) and f1 |B(x,ε) = f2 |B(x,ε) for some ε > 0. The following criterion provides a sufficient condition for a Feller semigroup to have the strong Feller property. Proposition 2.6 ([25, Theorem 1.14]) Let {Tt }t ≥0 be a conservative Feller semigroup, and assume that the kernels pt,x can be written as pt,x (dy) = pt (x, y)μ(dy), where μ is a positive Borel measure on E and the function pt (·, ·) is locally bounded on E × E for each t > 0. Then {Tt }t ≥0 is a strong Feller semigroup. Martingales and Local Martingales An adapted integrable stochastic process {Xt }t ≥0 on a filtered probability space ( , F, {Ft }, P ) is called a martingale if E[|Xt |] < ∞,

E[Xt |Fs ] = Xs ,

0 ≤ s ≤ t ≤ ∞.

The systematic study of connections between martingales and Feller processes was initiated by D.Stroock and S.Varadhan in the 1960s [181]. Here, we give the following theorem: Theorem 2.1 ([57, Chapter 4, Proposition 1.7 and Theorem 4.1]) Let {Xt }t ≥0 be a càdlàg Feller process on a locally compact separable metric space E with initial distribution μ = P [X0 ∈ ·] and let (G(0) , D(G(0) )) be its generator. Let D be a core of D(G(0)), i.e. a subset D ⊂ D(G(0) ) such that (G(0) , D(G(0) )) is the closure of the operator (G(0) , D). For each f ∈ D, the process 

t

f (Xt ) − f (X0 ) −

(G(0)f )(Xs ) ds,

t ≥0

(2.2)

0

is a martingale with respect to the same filtration for which {Xt } is a Markov process. Moreover, {Xt } is the unique (in distribution) E-valued stochastic process with initial distribution μ such that the process defined by (2.2) is a martingale for any f ∈ D. A stopping time on the filtered probability space ( , F, {Ft }, P ) is a random variable τ : → [0, ∞] such that {τ ≤ t} ∈ Ft for all t ≥ 0. A stochastic process {Xt }t ≥0 on the space ( , F, {Ft }, P ) is said to be a local martingale if there exists an increasing sequence {τn }n∈N of stopping times with limn τn = +∞

2.1 Continuous-Time Markov Processes

13

a.s. and such that for every n ∈ N the process {Xt ∧τn }t ≥0 is a martingale, where Xt ∧τn = Xt 1{τn ≥t } + Xτn 1{τn 0 for all x, y ∈ I , where τy = inf{t ≥ 0|Xt = y}. An irreducible diffusion process {Xt } on an open interval I ⊂ R is not in general a Feller process. However, it is Cb -Feller in the sense that its transition semigroup  (Tt f )(x) = E[f (Xt )] is such that Tt Cb (I ) ⊂ Cb (I ). We can therefore define the η-resolvent operator of {Xt } as  Rη : Cb (I ) −→ Cb (I ),

∞

Rη f :=

e−ηt Tt f dt

(η > 0)

(2.3)

0

  and define the Cb -generator G(b) , D(G(b) ) of {Xt } as the operator with domain   D(G(b) ) = Rη Cb (I ) , given by (G(b) u)(x) = ηu(x) − g(x)

for u = Rη g, g ∈ Cb (I ), x ∈ I.

(2.4)

of η, (cf. It is possible to show that G(b) is independent   [65, p. 295]). Further, if {Tt } is a Feller semigroup with generator G(0) , D(G(0) ) , then we have D(G(0)) =   Rη C0 (I ) and G(0)f = G(b) f for all f ∈ D(G(0) ). A canonical scale s is a strictly increasing continuous function s : I −→ R. A speed measure m is a positive Radon measure on I with support supp(m) = I . We say that (s, m, k) is a canonical triplet if s is a canonical scale, m is a speed measure and k is a positive Radon measure on I (called the killing measure).

14

2 Preliminaries

The following theorem provides a correspondence between (generators of) onedimensional diffusions and canonical triplets: Theorem 2.3 ([65, Section 2.2]) If {Xt } is an irreducible diffusion process on the open interval I , then there exists a canonical triplet (s, m, k) on I such that (G(b) f )(x) =

dDs f − f dk (x), dm

f ∈ D(G(b) ), x ∈ I

(2.5)

(x+ε)−f (x) (where Ds f (x) := limε↓0 fs(x+ε)−s(x) ) in the sense that the measure dDs f − f dk is absolutely continuous with respect to dm and the corresponding Radon-Nikodym derivative has a representative which belongs to Cb (I ) and is equal to G(b) f . Conversely, if (s, m, k) is an arbitrary canonical triplet on I , then there exists an irreducible diffusion process {Xt } on I whose Cb -generator is given by (2.5) for all f ∈ D(G(b) ).

Let {Xt } be a diffusion on I = (a, b), where −∞ ≤ a < b ≤ +∞, and let (s, m, k) be its canonical triple. Write j = m + k, and for c ∈ I consider the integrals 

c

Ia =





y

ds(x) j (dy), 

a b

Ib =



a b

Jb =

y



c

ds(x) j (dy), 

ds(x) j (dy), c

Ja =

a b

c



y y

ds(x) j (dy). c

c

The endpoint e ∈ {a, b} is said to be: regular if Ie < ∞, Je < ∞; exit if Ie < ∞, Je = ∞;

entrance if Ie = ∞, Je < ∞; natural if Ie = ∞, Je = ∞,

(2.6)

this classification being independent of the choice of c. This is the so-called Feller boundary classification of the diffusion process {Xt }, and it determines the behaviour of {Xt } near the endpoints of (a, b) (see [89] for details). In particular, if the endpoint e is entrance or natural then the process {Xt } cannot reach e in finite time. Moreover, if neither a nor b is a regular endpoint then there exists a unique diffusion process {Xt } on (a, b) with canonical triple (s, m, k). (Uniqueness here means that the domain D(G(b) ) of the Cb -generator is uniquely determined by (s, m, k), see [65].) Theorem 2.3 ensures, in particular, that each second-order differential operator of the form a(x)

d d2 − c(x) + b(x) 2 dx dx

(x ∈ I ),

2.1 Continuous-Time Markov Processes

15

where a, b, c ∈ C(I ) with a > 0 and c ≥ 0 on I , is the generator of an irreducible diffusion process {Xt }t ≥0. The associated canonical triplet is 

x

s(x) =

e−B(y)dy,

m(dx) =

x0

eB(x) dx, a(x)

k(dx) = c(x)

eB(x) dx, a(x)

x ) where B(x) := x0 ba(ξ (ξ ) dξ and x0 ∈ I is arbitrary. Consider the stochastic differential equation (SDE) dXt = b(Xt ) dt + σ (Xt ) dWt ,

(2.7)

√ where σ (x) = 2a(x) and {Wt }t ≥0 is a standard Brownian motion. An I -valued stochastic process {Xt }t ≥0 is said  t to be a solution  t of the SDE (2.7) if it satisfies the integral equation Xt = X0 + 0 b(Xs )ds + 0 σ (Xs )dWs , where the latter term is a stochastic integral with respect to the standard Brownian motion. By [99, Chapter 5, Theorem 5.15], the SDE (2.7) has a unique weak solution {Xt } up to a possibly finite lifetime ζ := inf{t ≥ 0 | Xt ∈ / I }. If both endpoints a and b are entrance or natural, then it follows from [99, Chapter 5, Theorem 5.29] that {Xt } is a diffusion process on I whose lifetime is infinite a.s. and whose generator is the differential d2 d operator a(x) dx 2 + b(x) dx . Even though diffusions are not always Feller processes on the open interval I , they become Feller processes after a suitable extension to the boundaries of the interval: Proposition 2.7 ([65, Sections 4 and 6]) Let {Xt } be an irreducible diffusion on I = (a, b), and let I be the interval obtained by attaching the regular or entrance endpoints of {Xt } to the interval I . Then there exists a Feller process {Xt } with state space I satisfying the following conditions: • The process {Xt } is an extension of {Xt }, in the sense that Xt (ω) = X t (ω) for 0 ≤ t ≤ τI (ω) := inf{t ≥ 0 | Xt (ω) ∈ / I }; • If a ∈ I (respectively b ∈ I ), then {Xt } is instantaneously reflecting at the endpoint a (resp. b), in the sense that we have Px [Xt = a for a.e. t ≥ 0] = 1 (resp. Px [X t = b for a.e. t ≥ 0] = 1); • The transition semigroup of {X t } is a Feller semigroup whose infinitesimal generator is given by  dDs f −f dk ∈ C0 (I )   dm   D(G ) = u ∈ C0 (I )  Ds u(a) = 0 if the endpoint a is regular or entrance , (0)

Ds u(b) = 0 if the endpoint b is regular or entrance

(G(0)f )(x) =

dDs f − f dk (x) dm

  f ∈ D(G(0) ), x ∈ I .

16

2 Preliminaries

If {Xt } has no regular endpoints, then {Xt } is the unique extension of {Xt } to a strong Markov process with continuous paths on the interval I . Lévy Processes, Infinitely Divisible Distributions and Convolution Semigroups A stochastic process {Xt }t ≥0 on Rd with X0 = 0 is said to be a Lévy process if it is stochastically continuous, has independent increments (i.e. Xt − Xs is independent of {Xu : u ≤ s} for all s < t) and has stationary increments (i.e. Xt +s − Xs has the same distribution as Xt − X0 for all t, s ≥ 0). It is clear from this definition that any drifted Brownian motion on Rd started at zero (i.e. any process of the form Bt = αt + A Wt with α ∈ Rd , A a symmetric nonnegative definite d × d-matrix and {Wt } a d-dimensional standard Brownian motion) is a Lévy process. In the definition of Lévy process, some authors also require that {Xt } is càdlàg. This is unimportant because of the following proposition: Proposition 2.8 ([166, Theorem 11.5]) If {Xt }t ≥0 is a Lévy process on Rd , then: (a) {Xt } has a càdlàg modification; (b) The transition semigroup (Tt f )(x) := Ex [f (Xt )] ≡ E[f (Xt + x)] is a Feller semigroup on Rd (and, therefore, {Xt } is a Feller process). There is a one-to-one correspondence between Lévy processes, convolution semigroups and infinitely divisible distributions. Before stating this result, we recall some notions. The (ordinary) convolution of two measures μ, ν ∈ MC (Rd ) is  defined by (μ ∗ ν)(B) := Rd Rd δx+y (B)μ(dx)ν(dy) for each Borel subset B ⊂ Rd . A probability measure μ ∈ P(Rd ) is said to be infinitely divisible if for each integer n ∈ N, there exists a measure νn ∈ P(Rd ) such that μ = νn∗n , where νn∗n denotes the n-fold convolution of νn with itself. A family {μt }t ≥0 ⊂ P(Rd ) is called a convolution semigroup if we have μs ∗ μt = μs+t (s, t ≥ 0), μ0 = δ0 and w μt −→ δ0 as t ↓ 0. Proposition 2.9 ([166, Theorem 7.10]) Let {Xt }t ≥0 be a Feller process on Rd . The following assertions are equivalent: (i) {Xt } is a Lévy process; (ii) There exists a convolution semigroup {μt }t ≥0 ⊂ P(Rd ) such that E[f (Xt )] =  d Rd f (y)μt (dy) for each f ∈ Bb (R ). If these conditions hold then μt is, for all t ≥ 0, an infinitely divisible measure. Conversely, if μ ∈ P(Rd ) is an infinitely divisible measure then there exists a Lévy  process {Xt } such that E[f (X1 )] = Rd f (y)μ(dy) for f ∈ Bb (Rd ). Given a Rd -valued random variable X with law μ = P[X ∈ · ], the Fourier transform of μ (also  called the characteristic function of X) is defined as (Fμ)(z) := E[eiz·X ] ≡ Rd eiz·x μ(dx) (z ∈ Rd ). The celebrated Lévy-Khintchine formula provides an explicit characterization of the characteristic function (or Fourier transform of the law) of a Lévy process:

2.2 Sturm-Liouville Theory

17

Theorem 2.4 (Lévy-Khintchine Representation [166, Theorem 8.1]) Let {Xt }t ≥0 be a Lévy process and {μt } ⊂ P(Rd ) the associated convolution semigroup. We have E[eiz·Xt ] ≡ (Fμt )(z) = e−t φ(z)

(t ≥ 0, z ∈ Rd )

(2.8)

iz ·y ν(dy), 1 + |y|2

(2.9)

for some function φ(·) of the form  φ(z) = z ·Qz + iα ·z +

Rd \{0}

 1 − eiz·y +

where Q is a symmetric nonnegative definite d × d-matrix, α ∈ Rd and ν is a Lévy  |y|2 measure on Rd , i.e. a positive measure on Rd \{0} such that Rd \{0} 1+|y| 2 ν(dy) < ∞. Conversely, for any function φ(·) of the form (2.9) there exists a convolution semigroup {μt } ⊂ P(Rd ) with (Fμt )(z) = e−t φ(z). The function φ(·) in (2.9) is called the Lévy symbol of the process {Xt }. One can show that the integral term in the expression for the Lévy symbol is, for every Lévy measure ν, the characteristic function of a discontinuous Lévy process, and therefore the following result holds: Proposition 2.10 Let {Xt } be a càdlàg Lévy process on Rd with Lévy-Khintchine representation (2.8)–(2.9). The following are equivalent: (i) {Xt } has a.s. continuous paths; (ii) ν = 0; √ (iii) Xt = αt + QWt , where {Wt } is a standard Brownian motion on Rd .

2.2 Sturm-Liouville Theory Sturm-Liouville operators with positive coefficients are infinitesimal generators of one-dimensional diffusions. In Chaps. 3 and 4 we will address the problem of constructing convolution-like operators associated with diffusions whose generators are (reducible to) Sturm-Liouville operators. This section collects the necessary background material from Sturm-Liouville theory. We will consider the Sturm-Liouville expression

(u)(x) :=

 1  −(pu ) (x) + q(x)u(x) , r(x)

x ∈ (a, b) ⊂ R,

(2.10)

18

2 Preliminaries

where we assume that the coefficients are such that p, r > 0 and q ≥ 0 on (a, b), p, r and q are locally integrable on (a, b) and  a

c



c y

 dx  r(y) + q(y) dy < ∞, p(x)

(2.11)

where c ∈ (a, b) is arbitrary. −f dk with s(x) = The Sturm-Liouville expression (2.10) is of the form − dDs fdm  x dy , m(dx) = r(x)dx and k(dx) = q(x)dx. As in Sect. 2.1, an endpoint e∈ x0 p(y) {a, b} is called regular, entrance, exit or natural according to the classification (2.6),    c  y dx   c  c dx  where Ia = a a p(x) r(y) + q(y) dy, Ja = a y p(x) r(y) + q(y) dy, Ib =    b  y dx   b  b dx  c y p(x) r(y) + q(y) dy and Jb = c c p(x) r(y) + q(y) dy. The standing assumption that the coefficients {p, q, r} satisfy (2.11) means that the endpoint a is regular or entrance.

2.2.1 Solutions of the Sturm-Liouville Equation It is a basic fact from the theory of ordinary differential equations that the vector space of solutions of the Sturm-Liouville equation (u) = λu is two-dimensional, and that a basis is formed by the (unique) solutions u1,λ (x), u2,λ (x) which satisfy the initial conditions u1,λ (c) = sin α,

u 1,λ (c) = cos α,

u2,λ (c) = − cos α,

u 2,λ (c) = sin α,

where α ∈ [0, π) and c is any (interior) point of the interval (a, b). When the initial conditions are instead given at an endpoint of the interval, the possibility of solving the Sturm-Liouville problem depends on the boundary classification for the set of coefficients {p, q, r}. Our starting lemma asserts that under the assumption (2.11) we have existence and uniqueness of solution for the Sturm-Liouville problem with Neumann-type condition at the left endpoint. Let us recall that an entire function h : C −→ C is said to be of exponential type if there exist constants c, M > 0 such that |h(z)| ≤ Mec|z| for all z ∈ C. Lemma 2.1 The initial value problem

(w) = λw

(a < x < b, λ ∈ C),

w(a) = 1,

(pw )(a) = 0

(2.12)

has a unique solution wλ (·). Moreover, λ → wλ (x) is, for fixed x, an entire function of exponential type. We emphasize that the boundary assumption (2.11) for this lemma includes Sturm-Liouville equations where the left endpoint can be either limit point or limit circle, while the usual existence and uniqueness theorems for Sturm-Liouville

2.2 Sturm-Liouville Theory

19

problems with initial condition at an endpoint rely on the assumption that the endpoint is limit circle (cf. e.g. [11, Section 5]). Here we recall the well-known Weyl limit point/limit circle endpoint classification: the endpoint a (respectively b) is c b called limit point if a |uλ (x)|2 r(x)dx = ∞ (respectively c |uλ (x)|2 r(x)dx = ∞) c for some solution of (u) = λu and limit circle if a |uλ (x)|2 r(x)dx < ∞ b (respectively c |uλ (x)|2 r(x)dx < ∞) for all solutions of (u) = λu. A regular end point is always a limit circle endpoint, but an entrance endpoint can be either a limit point or a limit circle endpoint [54, Theorem 2.1]. Lemma 2.1 is not new—a special case is established in [96, Lemma 3]—but seems to be little known. We give a self-contained proof based on [96].  x dξ Proof of Lemma 2.1 Pick an arbitrary β ∈ (a, b). Define s(x) := c p(ξ ) and    x S(x) = a s(β) − s(ξ ) q(ξ ) + r(ξ ) dξ . From the boundary assumption (2.11) it follows that 0 ≤ S(x) ≤ S(β) < ∞ for x ∈ (a, β]. Let η0 (x; λ) = 1,  x     ηj (x; λ) = s(x) − s(ξ ) ηj −1 (ξ ; λ) q(ξ ) − λr(ξ ) dξ

(j = 1, 2, . . .). (2.13)

a

One can check (using induction) that |ηj (x; λ)| ≤ Therefore, the function wλ (x) =

∞ 

ηj (x; λ)

1 j!



(1 + |λ|)S(x)

(a < x ≤ β, λ ∈ C)

j

for all j .

(2.14)

j =0

is well-defined as an absolutely convergent series. The entireness of λ → wλ (x) follows at once from the Weierstrass theorem for compactly convergent series of holomorphic functions, and the estimate |wλ (x)| ≤

∞  j 1 (1 + |λ|)S(x) = e(1+|λ|)S(x) ≤ e(1+|λ|)S(β) j!

(a < x ≤ β)

j =0

(2.15) shows that λ → wλ (x) is of exponential type. For a < x ≤ β we have 

 y   1 wλ (ξ ) q(ξ ) − λr(ξ ) dξ dy a p(y) a  x   (s(x) − s(ξ ))wλ (ξ ) q(ξ ) − λr(ξ ) dξ =1+

1+

x

a

20

2 Preliminaries



x

=1+

(s(x) − s(ξ ))

a

=1+

 ∞

ηj (ξ ; λ)

  q(ξ ) − λr(ξ ) dξ

j =0

∞  

x

  (s(x) − s(ξ ))ηj (ξ ; λ) q(ξ ) − λr(ξ ) dξ

j =0 a

=1+

∞ 

ηj +1 (x; λ) = wλ (x),

j =0

i.e., wλ (x) satisfies 

x

wλ (x) = 1 + a

1 p(y)



y

  wλ (ξ ) q(ξ ) − λr(ξ ) dξ dy.

(2.16)

a

This integral equation is equivalent to (2.12), so the proof is complete. Corollary 2.1 If λ < 0, then the solution of (2.12) is strictly increasing. If λ < 0 and the endpoint b is exit or natural, then the solution of (2.12) is unbounded. Proof Rewriting the functions ηj (x; λ) from the proof of Lemma 2.1 as  ηj (x; λ) = a

x

1 p(y)

 a

y

  ηj −1 (ξ ; λ) q(ξ ) − λr(ξ ) dξ dy,

we see at once (using induction onj ) that each ηj (·; λ) is positive and strictly increasing, and therefore wλ (·) = ∞ j =0 ηj (·; λ) is strictly increasing. Moreover,  b 1 y  limx↑b η1 (x; λ) = a p(y) a q(ξ ) − λr(ξ ) dξ dy ≥ min{1, |λ|}Jb = ∞, hence wλ is unbounded.  c It  c isdxalso worth noting that the following converse of Lemma 2.1 holds: if a y p(x) r(y)dy = ∞ (so that (2.11) fails to hold) then for λ < 0 there exists no solution of (w) = λw satisfying the boundary conditions w(a) = 1 and c  c dx (pw )(a) = 0. Indeed, if the integral a y p(x) r(y)dy diverges, then it follows from [88, Sections 5.13–5.14] that any solution w of (w) = λw (λ < 0) either satisfies w(a) = 0 or (pw )(a) = +∞, so in particular (2.12) cannot hold. In the sequel, {am }m∈N will denote a sequence b > a1 > a2 > . . . with lim am = a. Next we verify that the solution wλ (·) for the Sturm-Liouville equation on the interval (a, b) is approximated by the corresponding solutions on the intervals (am , b). We will use this fact on Chap. 4. Lemma 2.2 For m ∈ N and λ ∈ C, let wλ,m (x) be the unique solution of the boundary value problem

(w) = λw

(am < x < b),

w(am ) = 1,

(pw )(am ) = 0.

(2.17)

2.2 Sturm-Liouville Theory

21

Then lim wλ,m (x) = wλ (x)

lim (pwλ,m )(x) = (pwλ )(x)

and

m→∞

m→∞

(2.18)

pointwise for each a < x < b and λ ∈ C. Proof In the same way as in the proof of Lemma 2.1 we can check that the solution of (2.17) is given by wλ,m (x) =

∞ 

(am < x < b, λ ∈ C),

ηj,m (x; λ)

j =0

x    where η0,m (x; λ) = 1 and ηj,m (x; λ) = am s(x) − s(ξ ) ηj −1,m (ξ ; λ) q(ξ ) −   j λr(ξ ) dξ . As before we have |ηj,m (x; λ)| ≤ j1! (1 + |λ|)S(x) for am < x ≤ β (where S is the function from the proof of Lemma 2.1). Using this estimate and induction on j , it is easy to see that ηj,m (x; λ) → ηj (x; λ) as m → ∞ (a < x ≤ β, λ ∈ C, j = 0, 1, . . .). Noting that the estimate on |ηj,m (x; λ)| allows us to take the limit under the summation sign, we conclude that wλ,m (x) → wλ (x) as m → ∞ (a < x ≤ β). Finally, by (2.16) we have for a < x ≤ β

lim (pwλ,m )(x) m→∞

 = −λ lim

x

m→∞ a m



wλ,m (ξ ) r(ξ )dξ

x

= −λ

wλ (ξ ) r(ξ )dξ a

= (pwλ )(x), using dominated convergence and the estimates |wλ,m (x)| |wλ (x)| ≤ e(1+|λ|)S(β) .

≤

e(1+|λ|)S(β) ,

The following lemma provides a sufficient condition for the solution wλ (·) to be uniformly bounded in the variables x ∈ (a, b) and λ ≥ 0: Lemma 2.3 If q ≡ 0, λ ≥ 0 and x → p(x)r(x) is an increasing function, then the solution of (2.12) is bounded: |wλ (x)| ≤ 1

for all a < x < b, λ ≥ 0.

(2.19)

Proof (Adapted from [210, Proposition 4.3]) Let us start by assuming that p(a)r(a) > 0. For λ = 0 the result is trivial because w0 (x) ≡ 1. Fix λ > 0. Multiplying both sides of the differential equation (wλ ) = λwλ by 2pwλ , we

22

2 Preliminaries

1 obtain − pr [(pwλ )2 ] = λ(wλ2 ) . Integrating the differential equation and then using integration by parts, we get

  λ 1 − wλ (x)2 =





 1 (pwλ )(ξ )2 dξ p(ξ )r(ξ ) a   x   (pwλ )(ξ ) 2 (pwλ )(x)2 + p(ξ )r(ξ ) dξ, = p(x)r(x) p(ξ )r(ξ ) a x

a < x < b,

where we also used the fact that (pwλ )(a) = 0 and the assumption that p(a)r(a) > 0. The right hand side is nonnegative, because x → p(x)r(x) is increasing and therefore (p(ξ )r(ξ )) ≥ 0. Given that λ > 0, it follows that 1 − wλ (x)2 ≥ 0, so that |wλ (x)| ≤ 1. If p(a)r(a) = 0, the above proof can be used to show that the solution of (2.17) is such that |wλ,m (x)| ≤ 1 for all a < x < b, λ ≥ 0 and m ∈ N; then Lemma 2.2 yields the desired result.

2.2.2 Eigenfunction Expansions Eigenfunction expansion theorems for ordinary and partial differential operators are a key tool for the construction of generalized convolutions. Under the running assumption that the left endpoint is regular or entrance, the Sturm-Liouville operator (2.10) has a self-adjoint realization with Neumann-type boundary conditions, and the corresponding spectral expansion gives rise to an invertible integral transform whose kernel is the solution wλ (·): (For brevity we write Lp (r) :=   p L (a, b); r(x)dx .) Theorem 2.5 The operator L(2) : D(L(2)) ⊂ L2 (r) −→ L2 (r),

L(2) u = (u),

where ⎧  

2  ⎪ ⎪ 2 (r)  u, u ∈ ACloc (a, b), (u) ∈ L (r), , if b is limit point, ⎪ u ∈ L ⎪  ⎨ (pu )(a) = 0 D(L(2) ) :=   

2 ⎪  ⎪ 2 (r)  u, u ∈ ACloc (a, b), (u) ∈ L (r), , if b is limit circle ⎪ ⎪ u ∈ L ⎩  (pu )(a) = (pu )(b) = 0 (2.20)

2.2 Sturm-Liouville Theory

23

is positive and self-adjoint. There exists a unique locally finite positive Borel measure ρL on R such that the map h → Fh, where 

b

(Fh)(λ) :=

h(x) wλ (x) r(x)dx



 h ∈ Cc [a, b), λ ≥ 0 ,

(2.21)

a

induces an isometric isomorphism F : L2 (r) −→ L2 (R; ρL ) whose inverse is given by  −1 ϕ(λ) wλ (x) ρ L (dλ), (2.22) (F ϕ)(x) = R

the convergence of the latter integral being understood with respect to the norm of L2 (r). The spectral measure ρ L is supported on R+ 0 . Moreover, the operator F is a spectral representation of L(2) , i.e. we have     D(L(2)) = u ∈ L2 (r)  λ·(Fu)(λ) ∈ L2 R+ , ρ L , 0   F(L(2) h) (λ) = λ·(Fh)(λ), h ∈ D(L(2)).

(2.23) (2.24)

Proof The fact that (L(2) , D(L(2) )) is a positive self-adjoint operator is a known result [185, Chapter 9]. The existence of a spectral transformation F associated with the operator L is a consequence of the standard Weyl-Titchmarsh-Kodaira theory of eigenfunction expansions of Sturm-Liouville operators (cf. [201, Chapter 8]). In the general case the eigenfunction expansion is written in terms of two linearly independent solutions of (u) = λu and a 2 × 2 matrix measure. However, by Lemma 2.1, the function wλ (x) is square-integrable near x = 0 with respect to the measure r(x)dx and, for fixed x, it is an entire function of λ. Therefore, the possibility of writing the expansion in terms only of the eigenfunction wλ (x) follows from the results of [53, Sections 9 and 10]. The isometric integral transform F will be called the L-transform. Remark 2.1 Assume that the coefficients of the Sturm-Liouville expression (2.10) are such that p , r are locally absolutely continuous on (a, b). Let u be a solution of

(u) = λu, and consider a transformation of independent and dependent variables of the form  x H (ξ )dξ, u = Z(x)v, y= c

24

2 Preliminaries

where the functions Z, H are positive and sufficiently smooth. A straightforward computation yields that the function v(y) is a solution of the Sturm-Liouville 1 −( pv ) (y) +  equation 

(v)(y) := r(y) q (y)v(y) = λv(y), where  r=

rZ 2 , H

p  = pH Z 2 ,

 q=

d  dZ  qZ 2 +Z pH . H dy dy

  We can write 

(v) = U −1 (Uv), where U : L2 (γ −1 (a), γ −1 (b)), r −→ L2 (r) is the isometry defined by (U −1 u)(y) =

(Uv)(x) = Z(x) v(γ (x)), with γ (x) := defined by

x c

u(γ −1 (y)) , Z(γ −1 (y))

H (ξ )dξ and γ −1 its inverse function. Therefore, the operator  F

  ( Fh)(λ) := F(Uh) (λ) =



γ −1 (b) γ −1 (a)

h(y)

wλ (γ −1 (y))  r(y)dy, Z(γ −1 (y))

  −1 ( F ϕ)(y) := U −1 (F−1 ϕ) (y)  := U −1 LU of the is a spectral representation of the self-adjoint realization L operator 

. Under suitable additional assumptions (for instance, if Z(a) = 1, (pZ )(a) = 0 and is limit point at b), one can check that this spectral representation coincides with that obtained by applying Theorem 2.5 to the transformed operator 

; in particular, the spectral measure given by Theorem 2.5 is invariant under such transformations of variable. Aspecial case is the so-called Liouville transformation u = [p(x)r(x)]1/4v, √ y= r(x)/p(x) dx. This choice yields a simplified operator 

without first-order 1 1 2 q

 term, namely (v)(y) = v (y)+ q (y)v(y), where  q = +(pr)− 4 d 2 [(pr) 4 ]. This r

dy

is called the Liouville normal form of the operator . Theorem 2.5 establishes the existence of a spectral measure ρ L such that the L-transform maps the space L2 (r) isometrically onto L2 (R; ρL ), but it provides no information on how to compute the measure ρL . When the Sturm-Liouville operator has no natural endpoints, the spectral measure is discrete and can be obtained by determining the eigenvalues and the norms of the eigenfunctions: Proposition 2.11 Suppose that the endpoint b is regular, entrance or exit. Let uλ (·) be a nontrivial solution of (u) = λu (λ ∈ C) such that uλ (b) = 1, (pu λ )(b) = 0,

if b is regular or entrance, (2.25)

uλ ∈ L2 ((c, b), r(x)dx) for some c ∈ (a, b),

if b is exit

(2.26)

2.2 Sturm-Liouville Theory

25

and let Wr(wλ , uλ ) := p(wλ u λ − uλ wλ ) be the modified Wronskian of the solutions wλ and uλ . Then Wr(wλ , uλ ) is independent of x and its positive zeros 0 ≤ λ1 < λ2 < λ3 < . . . ↑ ∞ are eigenvalues of the self-adjoint operator L. The spectrum of L is {λk }k∈N , and its (purely discrete) spectral measure is given by ρL =

∞ 

wλk −2 δ . L2 (r) λk

k=1

Proof See [122, Section 5.1]. If the endpoint b is natural, the spectrum of L has, in general, a more complicated structure. We refer to the work of Linetsky [122] for a complete characterization of the structure of the spectrum of a large class of Sturm-Liouville operators with natural endpoints. The following results describe two approaches for computing the spectral measure of operators whose endpoint b is natural—the so-called real variable approach, where ρL is obtained as a limit of discrete measures which correspond to eigenvalue problems on approximating intervals, and an alternative approach which relies on complex analysis and the so-called Weyl-Titchmarsh mfunction: Proposition 2.12 Suppose that the endpoint b is natural. For β ∈ (a, b), let 0 ≤ β λ1,β < λ2,β < . . . ↑ ∞ be the zeros of the function λ → wλ (β) and let ρ be the L measure β

ρL =

∞ 

wλk,β −2 2,β δλk,β ,

k=1



β

where f 2,β =

|f (x)|2 r(x)dx.

a

There exists a right continuous, monotone increasing function (·) on R such that β limβ↑b ρ L (−∞, λ] = (λ) at all points of continuity of . Moreover, the spectral measure of L is the Lebesgue-Stieltjes measure with distribution function (λ), i.e. we have ρ L (λ1 , λ2 ] = (λ2 ) − (λ1 ) for all λ1 , λ2 ∈ R with λ1 < λ2 . Proof See [122, Section 5.2]. Notice that Lemma 2.2 guarantees existence of a solution θ of (θ ) = λθ which is real entire in λ (i.e. for fixed x ∈ (a, b) the function λ → θλ (x) is entire, and θλ (x) ∈ R for λ ∈ R), and such that Wr(wλ , θλ ) = 1. After this remark, we can formulate the following characterization of the spectral measure. Proposition 2.13 Suppose that the endpoint b is natural, and let θλ (·) be a solution of (θ ) = λθ which is real entire in λ. There exists a function m : C \ R −→ C, called the Weyl-Titchmarsh m-function, which is uniquely defined by the

26

2 Preliminaries

requirement that ψλ (x) := θλ (x) + m(λ)wλ (x) belongs to L2 ((c, b), r(x)dx) for some c ∈ (a, b). The spectral measure of the operator L is given by ρL (λ1 , λ2 ] = lim lim δ↓0 ε↓0



1 π

λ2 +δ λ1 +δ

Im(m(λ + iε))dλ

(λ1 , λ2 ∈ R, λ1 < λ2 ). (2.27)

Proof See [53, Sections 9 and 10]. It is often important to know whether the inversion integral for the L-transform is absolutely convergent. A sufficient condition, which is valid for any Sturm-Liouville operator satisfying the left boundary assumption (2.11), is given in the next lemma:   c  y dx  q(y) + r(y) dy < ∞, and J = (a, b) Lemma 2.4 Set J = [a, b) if a a p(x) otherwise. Then: (a) For each μ ∈ C \ R, the integrals  R+ 0



wλ (x) wλ (y) ρ L (dλ) |λ − μ|2

and

R+ 0

(pwλ )(x) (pwλ )(y) ρL (dλ) |λ − μ|2 (2.28)

converge uniformly on compact squares in J × J . (b) If h ∈ D(L(2) ), then  h(x) =





(ph )(x) =

(Fh)(λ) wλ (x) ρL (dλ),

(2.29)

(Fh)(λ) (pwλ )(x) ρ L (dλ),

(2.30)

R+ 0

R+ 0

where the right-hand side integrals converge absolutely and uniformly on compact subsets in J . Proof (a) It is known that the resolvent of the Sturm-Liouville operator (L(2) , D(L(2) )) is given by (L(2) − μ)−1 g(x) =



b

g(y)G(x, y, μ) r(y)dy,

g ∈ L2 (r), μ ∈ C \ R,

a

where G(x, y, μ) =

⎧ ⎨

1 Wr(wμ ,uμ ) wμ (x)uμ (y), 1 ⎩ Wr(wμ ,uμ ) wμ (y)uμ (x),

x < y, x≥y

2.2 Sturm-Liouville Theory

27

and uλ (·) (λ ∈ C \ R) is a nontrivial solution of (u) = λu satisfying (2.25) if b is regular or entrance and (2.26) if b is exit or natural; moreover, the Ltransform of the resolvent kernel is   wλ (x) . F G(x, ·, μ) (λ) = λ−μ (These facts follow from general results in Sturm-Liouville spectral theory, see [53, Lemma 10.6].) We have 

wλ (x)wλ (y) ρL (dλ) = |λ − μ|2

R+ 0



b

a

G(x, ξ, μ)G(y, ξ, μ) r(ξ )dξ =

  1 Im G(x, y, μ) , Im(μ)

where the first equality follows from the isometric property of F and the second equality is a consequence of the resolvent formula (L(2) − μ1 )−1 − (L(2) − ∂ μ2 )−1 = (μ1 − μ2 )(L(2) − μ1 )−1 (L(2) − μ2 )−1 . Letting ∂ξ[1] := p(ξ ) ∂ξ ,    [1]  it is easy to check that the functions Im G(x, y, μ) , ∂x Im G(x, y, μ) and   ∂x[1] ∂y[1] Im G(x, y, μ) are continuous in a < x, y < b. From this we can conclude, after a careful estimation of the differentiated integrals (see the proof of [144, §21.2, Corollary 3]), that  R+ 0

  (pwλ )(x) (pwλ )(y) 1 ∂x[1] ∂y[1] Im G(x, y, μ) ρ L (dλ) = 2 |λ − μ| Im(μ)

and that the integrals (2.28) converge uniformly for x, y in compact subsets of J. (b) By Theorem 2.5 and the classical theorem on differentiation under the integral sign for Riemann-Stieltjes integrals, to prove (2.29)–(2.30) it only remains to justify the absolute and uniform convergence of the integrals in the right-hand sides. (2) from Recall  +  Theorem 2.5 that the condition h ∈ D(L ) implies that Fh ∈ + L2 R0 , ρL and also λ (Fh)(λ) ∈ L2 R0 , ρ L . As a consequence, we obtain 

  (Fh)(λ)wλ (x) ρ (dλ) L +

R0

      wλ (x)   wλ (x)          ρ (dλ)   ρ (dλ) + (Fh)(λ)  λ (Fh)(λ)  ≤ λ+i  L λ+i  L R+ R+ 0 0     wλ (x)    ≤ λ (Fh)(λ)ρ + (Fh)(λ)ρ  λ + i ρ 

< ∞,

28

2 Preliminaries

  where  · ρ denotes the norm of the space L2 R; ρL , and similarly 

 

     (Fh)(λ) (pw )(x) ρ (dλ) ≤ λ (Fh)(λ)ρ + (Fh)(λ)ρ  (pwλ )(x)  < ∞. λ L  λ+i  +

R0

ρ

  (x)  

λ  and  (pwλ )(x)  We know from part (a) that the integrals which define  wλ+i λ+i ρ ρ converge uniformly, hence the integrals in (2.29)–(2.30) converge absolutely and uniformly on compact subsets of J .

2.2.3 Diffusion Semigroups Generated by Sturm-Liouville Operators Being a positive self-adjoint operator, the Neumann realization (L(2) , D(L(2))) of the Sturm-Liouville expression (2.10) is the (negative of the) infinitesimal generator (2) on L2 (r). Since T (2) = e−t L (the of a strongly continuous semigroup {T (2)} t ≥0

t

t

latter being defined via the spectral calculus), the eigenfunction expansion of this semigroup is a by-product of Theorem 2.5: Proposition 2.14 The semigroup {Tt(2) }t ≥0 generated by (L(2) , D(L(2) )) is sub(2) Markovian, i.e. such that Tt hL∞ (r) ≤ hL∞ (r) for all h ∈ L2 (r) ∩ L∞ (r) and (2) Tt h ≥ 0 whenever h ≥ 0. Moreover, for t > 0 and h ∈ L2 (r) this semigroup admits the representations (Tt h)(x) = F−1 [e−t ·(Fh)(·)](x) = (2)



e−t λwλ (x) (Fh)(λ) ρL (dλ)

(2.31)

h(y) p(t, x, y) r(y)dy,

(2.32)

e−t λ wλ (x)wλ (y) ρL (dλ)

  x, y ∈ J

R+ 0



b

= a

where p(t, x, y) := F

−1

[e

−t ·

 w(·) (x)](y) =

R+ 0

(2.33) (here J is defined as in Lemma 2.4) and the latter integral converges absolutely and uniformly on compact squares in J × J for each fixed t > 0.

2.2 Sturm-Liouville Theory

29

Proof One can check (see [33, Section 1.1]) that L(2) is the positive self-adjoint operator associated with the unbounded sesquilinear form EL : D(EL ) × D(EL ) −→ C defined by    D(EL ) = u ∈ L2 (r) ∩ L2 (q)  u ∈ ACloc (R+ ), u ∈ L2 (p) ,  ∞  ∞ EL (u, v) = u (x)v (x)p(x)dx + u(x)v(x)q(x)dx, 0

0

    2 2 where L2 (p) = L2 (a, b);  p(x)dx and  L (q) = L (a, b); q(x)dx . According to [33, Section 2.3], EL , D(EL ) is closed and Markovian. (The closedness means that D(EL ) is a Hilbert space with respect to the inner product EL (u, v) + u, vL2 (r) , while the Markovianity means that if u ∈ D(EL ) then v := max(min(u, 1), 0) ∈ D(EL ) and EL (v, v) ≤ EL (u, u).) Using the well-known Beurling-Deny criterion (e.g. [33, Theorem 1.1.3]), it follows that the semigroup (2) {Tt } is sub-Markovian. The representation (2.31) is a direct consequence of the spectral theorem for unbounded self-adjoint operators. It follows from Lemma 2.4(a) that the right hand side of (2.33) converges absolutely and uniformly in compact subsets of J ×J , hence   for x ∈ J the function λ → e−t λ wλ (x) belongs to L2 R; ρL . The representation (2.32) is therefore obtained by combining (2.31) with the isometric property of F. As noted in the beginning of this section, the (negative of the) Sturm-Liouville expression considered here is of the form (2.5). Assume that the endpoint b is not exit, and let I = [a, b) if b is natural and I = [a, b] if b is regular or entrance. By Theorem 2.3 and Proposition 2.7, there exists a diffusion process {Xt }t ≥0 on I which is a Feller process whose infinitesimal generator is the operator (−L(0), D(L(0) )), where L(0) u = (u),     u, u ∈ ACloc (a, b), (u) ∈ C0 (I ) (0)  . D(L ) = u ∈ C0 (I )  (pu )(a) = 0, (pu )(b) = 0 if b is regular or entrance (2.34) The Feller semigroup generated by (L(0), D(L(0) )) is the restriction to C0 (I ) of the L∞ (r)-extension of the semigroup {Tt(2)}, cf. [33, Equation (1.1.9)]. The next corollary gives some consequences of the preceding remarks. Corollary 2.2 Assume that the endpoint b is not exit. Let {Tt }t ≥0  and {Xt }t ≥0 be,  respectively, the Feller semigroup and diffusion generated by −L(0) , D(L(0)) . Then {Tt }t ≥0 is consistent with the strongly continuous contraction semigroup {Tt(2) } generated by (L(2) , D(L(2) )), in the sense that Tt h = Tt(2) h if h ∈ C0 (I ) ∩ L2 (r). The function p(t, x, ·) defined in (2.33) is the density (with respect to r(y)dy) of the

30

2 Preliminaries

transition kernel of the Feller semigroup {Tt }t ≥0, i.e. we have 

b

(Tt h)(x) = Ex [h(Xt )] =

h(y) p(t, x, y) r(y)dy



 t > 0, x ∈ J, h ∈ C0 (I ) .

a

If q ≡ 0, then {Tt }t ≥0 is a conservative Feller semigroup and therefore p(t, x, ·) r(·) is, for each t > 0 and x ∈ J , the density of a probability measure on I .

2.2.4 Remarkable Particular Cases The general family of Sturm-Liouville operators studied above includes many differential operators which are of hypergeometric type in the sense that the solutions of (u) = λu can be written in terms of hypergeometric functions. In such cases, it is often possible to determine, using Propositions 2.11–2.13 and known identities from the theory of special functions, a closed-form expression for the spectral measure. As the examples below demonstrate, one can recover, in particular, the inversion theorem for many common integral transforms, as well as an explicit (spectral) representation for the transition probabilities of important diffusion processes. The following examples will be further reworked in the subsequent chapters of this book. We start with an example which is nearly trivial, but quite instructive: Example 2.1 The Sturm-Liouville operator

=−

d2 , dx 2

0 −1. The Jacobi differential operator

= −(1 − x 2 )

d2 d − (β − α − (α + β + 1)x) , dx dx 2

−1 < x < 1

is of the form (2.10) with q ≡ 0, r(x) = (1−x)α (1+x)β and p(x) = (1−x)1+α (1+ x)1+β . The endpoint 1 is regular if −1 < α < 0 and entrance if α ≥ 0; similarly, the endpoint −1 is regular if −1 < β < 0 and entrance if β ≥ 0. In all cases, the Neumann self-adjoint realization (L, D(L)) has a purely discrete spectrum. The function  1−x wλ (x) = 2 F1 η − τ, η + τ ; α + 1; (η = 12 (α + β + 1), λ = τ 2 − η2 ) 2 is a solution of (u) = λu such that wλ (1) = 1 and (pwλ )(1) = 0. Here 2 F1 denotes the hypergeometric function [145, Chapter 15]. One can verify that this is an eigenfunction of (L, D(L)) if and only if λ = n(2η + n) (n ∈ N) [126]. Therefore, the eigenfunctions are the Jacobi polynomials wk(2η+k) (x) ≡ (α,β) (−1)k 1 dk [(1−x)k+α (1+x)k+β ]. Since Rk 2L2 (r) = 2k (α+1)k (1−x)α (1+x)β dx k 22η−1 k! (k+α+1)(k+β+1) [44, Section 15.2], the integral transform pair (2.21)–(2.22) [(1+α)k ]2 (k+η) (k+2η) (α,β)

Rk

(x) :=

is the Jacobi series expansion h(x) =

∞  k=0

[(α + 1)k ]2 (k + η) (k + 2η) (α,β) (Fh)(k)Rk (x), 22η−1 k! (k + α + 1)(k + β + 1)

1 (α,β) where (Fh)(k) = −1 h(x) Rk (x) (1−x)α (1+x)β dx. Accordingly, the transition probability density of the diffusion process on [−1, 1] generated by is p(t, x, y) =

∞  k=0

e−t k(2η+k)

[(α + 1)k ]2 (k 2η−1 2 k! (k + α

+ η) (k + 2η) (α,β) (α,β) R (x)Rk (y). + 1)(k + β + 1) k

In the literature, this stochastic process is known as the Jacobi diffusion [100, 122]. Next we present in some detail an example of how one can determine the spectral measure of a Sturm-Liouville operator whose spectrum is not discrete and whose fundamental solutions are nontrivial special functions of hypergeometric type.

32

2 Preliminaries

Example 2.3 The Bessel process with drift μ > 0 and index α−1 (α > 0) is, 2 according to [123], the diffusion generated by the differential operator

=−

d α d2 + 2μ , − dx 2 x dx

0 < x < ∞.

This Sturm-Liouville operator is obtained by choosing q ≡ 0 and p(x) = r(x) = x α e2μx . One can check that the endpoint 0 is regular if 0 < α < 1 and entrance if α ≥ 1, while the endpoint +∞ is natural. The solution of the initial value problem (2.12) is α

α

wλ (x) = (2iτ )− 2 e−μx x − 2 M− αμ , α−1 (2iτ x), 2iτ

(2.36)

2

z  ( 12 +ν−κ)n 1 +ν+n 2 is the Whittaker where λ = τ 2 + μ2 and Mκ,ν (z) := e− 2 ∞ n=0 (1+2ν)nn! z function of the first kind [145, §13.14]. (The fact that (2.36) is a solution of (u) = λu follows from [153, Equation 2.1.2.108], and we can use the results of [145, §13.14(iii) and §13.15(ii)] to check that wλ (0) = 1 and (pwλ )(0) = 0.) If α ∈ / N, a suitable linearly independent solution of (u) = λu is α

α

θλ (x) = (1 − α)−1 (2iτ ) 2 −1 e−μx x − 2 M− αμ , 1−α (2iτ x). 2iτ

2

(Using [145, §13.2(i)] and [123, Remark 1], one verifies that θλ (x) is real entire; [145, Equation 13.2.33] yields that Wr(wλ , θλ ) = 1. The case α ∈ N can be treated using [145, §13.2(v)].) Now, it follows from [145, Equation 13.14.21] that a solution of the Sturm-Liouville equation which is square-integrable with respect to r(x)dx near infinity is ψλ (x) =

μ ( α2 (1 + iτ )) α α (2iτ ) 2 −1 e−μx x − 2 W− αμ , α−1 (2iτ x) 2iτ 2 (α)

(λ ∈ C \ R),

where Wα,ν (x) is the Whittaker function of the second kind [145, §13.14]. By [145, Equation 13.14.33] this solution can be written as θλ (x) + m(λ)wλ (x), where m(λ) = −(α)−2 (2τ )α−1

μ sin( π2α (1 + iτ ))  α  2 (1 + sin(π α)

μ  α μ  iτ )  2 (1 − iτ )

(λ ∈ C \ R).

Taking the limit we obtain  α−2    2 α−1 exp − παμ  α (1 + 1 2τ 2τ 2 π (α) lim m(λ + iε) = ε↓0 π 0,

μ 2 , iτ )

λ > μ2 , λ < μ2

which, by Proposition 2.13, is the density of the (absolutely continuous) spectral measure ρ L .

2.2 Sturm-Liouville Theory

Letting σ (τ ) := of index transforms

33

2α−1 τα π (α)2

α

(Fh)(τ ) = (2iτ )− 2



  α   (1 + exp − παμ 2τ 2

∞

−1

(F

ϕ)(x) = (2ix)

− α2 −μx

it follows that the pair

α

h(x) M− αμ , α−1 (2iτ x) eμx x 2 dx, 2iτ

0

μ 2 , iτ )



∞

e

0

2

ϕ(τ ) M− αμ , α−1 (2iτ x) τ − 2 σ (τ )dτ α

2iτ

2

(2.37) (2.38)

defines an isometry between the spaces L2 (R+ , x α e2μx dx) and L2 (R+ , σ (τ )dτ ). In the limit μ → 0, using [145, Equations 10.27.6 and 13.18.8] we recover the Hankel transform (2.47) whose kernel is the Bessel function of the first kind. From the above it also follows that the transition density of the Bessel process with drift is given by p(t, x, y) = (−4xy)− 2 e−μ(x+y) ×  ∞ 2 2 × e−t (τ +μ ) M− αμ , α−1 (2iτ x) M− αμ , α−1 (2iτy) τ −α σ (τ )dτ. α

2iτ

0

2

2iτ

2

This spectral representation for the law of the Bessel process was established by Linetsky in [123], based on the related results of Titchmarsh in [187, §4.17]. However, the pair of confluent hypergeometric type integral transforms (2.37)– (2.38) is apparently little known; in particular, it is not reported in reference monographs on integral transforms such as [158, 204, 207]. Example 2.4 Another integral transform related to the Whittaker functions is obtained by considering the operator

= −x 2

d2 d − (1 + 2(1 − α)x) , 2 dx dx

0 < x < ∞,

(2.39)

which is of the form (2.10) with q ≡ 0, r(x) = x −2α e−1/x and p(x) = x 2(1−α)e−1/x . The operator (2.39) is the generator of the Shiryaev process, which will be studied in detail in Chap. 3. Here the solution of the initial value problem (2.12) is a normalized Whittaker W function (Proposition 3.1). For α ≤ 12 , the   corresponding spectral measure has density σ (τ ) = π −2 τ sinh(2πτ ) 12 − α + 2 iτ  , where we write λ = τ 2 + ( 1 − α)2 . The L-transform specializes into 2



∞

(Fh)(τ ) = 0

(F

−1

h(x) Wα,iτ ( x1 ) x −α e− 2x dx,

1 1 ϕ)(x) = 2 x α e 2x π

1

 0

∞

(2.40)

  2 ϕ(τ ) Wα,iτ ( x1 ) τ sinh(2πτ ) 12 − α + iτ  dτ. (2.41)

34

2 Preliminaries

Accordingly, (2.33) specializes into an explicit spectral representation for the transition density of the Shiryaev process. The integral transform (2.40)–(2.41) is a modified form (cf. Remark 2.1) of the so-called index Whittaker transform, which was first introduced by Wimp [202] as a particular case of an integral transform having the Meijer-G function in the kernel. Its Lp theory was studied in [180]. The index Whittaker transform includes as a particular case the Kontorovich-Lebedev transform, which is one of the most wellknown index transforms [204, 207] and has a wide range of applications in physics. The spectral measure σ (τ )dτ can be deduced using either the approach based on the Weyl-Titchmarsh m-function or the real variable approach (we refer to [177, Example 2] and [124] respectively). Example 2.5 The coefficients q ≡ 0, p(x) = r(x) = (sinh x)2α+1 (cosh x)2β+1 (with β ∈ R, α > −1, α ± β + 1 ≥ 0) give rise to the Jacobi operator

=−

d d − [(2α + 1) coth x + (2β + 1) tanh x] , dx dx 2

0 < x < ∞.

(2.42)

The so-called Jacobi function wλ (x) = φτ(α,β)(x) := 2 F1



1 2 (η

− iτ ), 12 (η + iτ ); α + 1; −(sinh x)2



(where η = α + β + 1, λ = τ 2 + η2 )) can be shown to be the unique solution of the Sturm-Liouville initial value problem (2.12). Using Proposition 2.13, one can show (cf. [105] and references therein, see also [177, Example 3]) that the spectral    ( 1 (η+iτ ))( 1 (η+iτ )−β) 2  , so measure is absolutely continuous with density σ (τ ) =  2 1+iτ iτ2 ( 2 )( 2 )(α+1)  that the L-transform becomes  ∞ h(x) φτ(α,β)(x) (sinh x)2α+1(cosh x)2β+1 dx, (Fh)(τ ) = 0

(F

−1



∞

ϕ)(x) = 0





 ( 1 (η + iτ ))( 1 (η + iτ ) − β) 2   (α,β) 2 ϕ(τ ) φτ (x)  2 1+iτ  dτ.   ( 2 )( iτ2 )(α + 1) (2.43)

This is the so-called (Fourier-)Jacobi transform, which is closely related (via a suitable change of variables) to the Olevskii transform, the index hypergeometric transform or, in the case α = β, the generalized Mehler-Fock transform [206]. If β = − 12 , the Feller process generated by the Neumann self-adjoint realization of (2.42) is known as the hyperbolic Bessel process; more generally, it is called a hypergeometric diffusion [21]. Like in the previous examples, the transition probabilities admit the explicit integral representation p(t, x, y) =  ∞ −t (τ 2 +η2 ) (α,β) (α,β) φτ (x)φτ (y)σ (τ )dτ . 0 e

2.3 Generalized Convolutions and Hypergroups

35

For ease of presentation, in Examples 2.3–2.5 the range of the parameters α, β and μ was chosen so that the spectral measure is purely absolutely continuous. In general, the measure decomposes into a discrete and an absolutely continuous part, both of which can be determined using the results of Sect. 2.2.2 (for details, see [122]). For instance, if we let α > 12 in the Sturm-Liouville operator of Example 2.4 and let Nα be the integer part of α − 12 , then the spectral measure becomes [115, 123] ρL (dλ) =

Nα  2α − 1 − 2n δn(2α−1−n) (dλ) n!(2α − n) n=0

  2 + 1[(α− 1 )2 ,∞)(λ) τ sinh(2πτ ) 12 − α + iτ  dτ, 2

so that a finite sum must be added to the inversion formula (2.41) for the index Whittaker transform. The examples presented above do not exhaust the class of Sturm-Liouville operators whose spectral measure is known in closed form. Additional examples can be found e.g. in [48, 67, 116, 179].

2.3 Generalized Convolutions and Hypergroups Motivated by various considerations in analysis and in probability, various authors have introduced axiomatic notions of convolution-like structures and the closely related translation-like operators. These notions and the corresponding theories are broadly similar, but differ on significant aspects. Here we review some axiomatic definitions which have played an influential role in past studies on generalized harmonic analysis and give context to the materials in later chapters. Generalized Translation Operators The early historical development of the theory of abstract harmonic analysis is mostly due to the work of Levitan in the 1940s. Embracing the ideas laid out by Delsarte in [45], Levitan embarked on a comprehensive study of the so-called generalized translation operators, defined in [119] as follows (see also [121]): Definition 2.1 Let E be a locally compact space endowed with a positive measure m. The linear operators Ty : Lp (E, m) −→ Lp (E, m) (y ∈ E, 1 ≤ p ≤ ∞) are said to constitute a family of generalized translation operators if the following conditions hold: T1. Ty Tx = Tx Ty , where we write (Ty f )(x) := (Tx f )(y); T2. There exists an element e ∈ E such that Te = Id; T3. Ty f Lp (E,m) ≤ cp (y) f Lp (E,m) , where the functions cp (·) are positive and bounded on compact subsets of E; T4. If f ∈ Lp (E, m) and ε > 0, then for each y ∈ E there exists Uy ⊂ E, a

neighbourhood of y such that Ty f − Ty f Lp (E,m) < ε for all y ∈ Uy .

36

2 Preliminaries

Generalized translations are naturally related to generalized convolutions by analogues of equality (2.44), below. We refer to Levitan’s monograph [121] for a comprehensive presentation of his results on Lie-type theorems with respect to generalized translation operators and other analogues of classical constructions in harmonic analysis. Hypercomplex Systems Influenced by Levitan’s work, Berezansky and S. Krein initiated, in the early 1950s, a study of the so-called hypercomplex systems, whose definition reads as follows [13–15]: Definition 2.2 Let E be a complete separable locally compact space. A measure c(A, B, y) (A, B Borel subsets of E, y ∈ E) is called a structure measure if it is a positive regular Borel measure with respect to A (respectively B) for fixed (B, y) (resp. (A, y)), and the following properties hold: S1. If A, B are relatively compact then c(A, B,   ·) ∈ Cc (E); S2. The identity E c(A, B, x) c(dx, C, y) = E c(B, C, x) c(A, dx, y) holds for all y ∈ E and A, B, C Borel subsets with A, B relatively compact; S3. There exists a positive regular Borel measure m such that m(A)m(B) = E c(A, B, y)m(dy) for all relatively compact Borel sets A, B. If c(A, B, y) is a structure measure on E, then the space L1 (E, m) with the convolution   (f ∗ g)(x) = f (y)g(ξ ) c(dy, dξ, x) E

E

is said to be a hypercomplex system. A complete presentation of the extensive theory of harmonic analysis on hypercomplex systems developed by Berezansky and coauthors can be found on the monograph [15], which also provides an extensive discussion of examples related to orthogonal polynomials or Sturm-Liouville operators. Urbanik Convolutions The definition of Urbanik (generalized) convolutions on R+ 0 was introduced by Urbanik in the 1960s and was thoroughly studied in a series of papers by the same author [190–194]. + Definition 2.3 For a > 0, let a : R+ 0 −→ R0 be the multiplication map x → a (x) := ax. A bilinear operator  on MC (R+ 0 ) is said to be an Urbanik convolution if the following axioms hold:

U1. U2. U3. U4.

+ If μ, ν ∈ P(R+ 0 ), then μ  ν ∈ P(R0 ); μ  ν = ν  μ and μ  (ν  π) = (μ  ν)  π for all μ, ν, π ∈ MC (R+ 0 );

If μn −→ μ, then μn  ν −→ μ  ν for all ν ∈ MC (R+ 0 ); δ0  μ = μ for all μ ∈ MC (R+ ); 0 w

w

2.3 Generalized Convolutions and Hypergroups

U5.

37

a (μ  ν) = (a μ)  (a ν) for all μ, ν ∈ P(R+ 0 ) and a > 0; w

U6. There exists a sequence {cn }n∈N ⊂ R+ such that cn δ1n −→ μ for some measure μ = δ0 . The topics studied in Urbanik’s papers include infinite divisibility of probability measures with respect to any convolution satisfying axioms U1–U6. The following result, gives an analogous to the Fourier transform and a Lévy-Khintchine-type representation: Proposition 2.15 Let  be an Urbanik convolution, and assume there exists a mapping ℘ : P(R+ 0 ) −→ R such that the identities ℘ (cμ1 + (1 − c)μ2 ) = c℘ (μ1 ) + (1 − c)℘ (μ2 ),

℘ (μ1  μ2 ) = ℘ (μ1 )℘ (μ2 )

hold for all 0 ≤ c ≤ 1, μ1 , μ2 ∈ P(R+ 0 ). Then: (a) There exists a one-to-one correspondence μ ↔ Hμ between measures μ ∈ + + P(R+ 0 ) and functions Hμ on R0 such that for μ, μ1 , μ2 ∈ P(R0 ), a > 0, λ ≥ 0 we have   H(μ1  μ2 ) (λ) = (Hμ1 )(λ)·(Hμ2 )(λ),



 H(a μ) (λ) = (Hμ)(aλ).

The map μ → Hμ can be written as an integral transform Hμ(λ) =  + + (λx)μ(dx) with kernel : R 0 −→ R. R 0

(b) A function h : R+ 0 −→ R can be written as h = Hμ for some -infinitely divisible measure μ (that is, a measure μ ∈ P(R+ 0 ) such that for each n ∈ N n ) if and only if it has the there exists νn ∈ P(R+ ) for which μ = ν n 0 representation   h(λ) = exp −kλ() +

R+ 0

(λx) − 1 ν(dx) , τ (x)

λ ≥ 0,

where k > 0, () is the so-called characteristic exponent of the convolution  (see [23, 190]), ν is a finite Borel measure on R+ 0 , τ (x) := 1 − (x) for 0 ≤ x ≤ x0 , τ (x) := 1 − (x0 ) for x ≥ x0 , and x0 > 0 is chosen so that (·) < 1 on (0, x0 ]. The notions of -convolution-like semigroups and -Lévy-like processes with respect to an Urbanik convolution  can be defined exactly like the corresponding notions for the Kingman convolution, as explained in Sect. 2.4 below. The properties of such Lévy-like processes and their infinitesimal generators have been studied in [23, 195]. For instance, it has been shown in [23] that it is possible to define a stochastic integral with respect to a given -Lévy-like process and, moreover, the transition probabilities of this stochastic integral can be characterized using the integral transform H described above.

38

2 Preliminaries

The so-called Kendall convolution is an example of an Urbanik convolution which has been the subject of numerous recent papers, due to its connection with the Williamson transform and the theory of Archimedean copulas [91–93]. We refer to [139–141] for further recent work on Urbanik convolutions and its particular cases. Hypergroups The following notion of hypergroup was introduced by Jewett in the mid-1970s [94] and (modulo slight changes in the axioms) independently also by Dunkl [49] and Spector [178]. The definition below, which singles out the more general concept of weak hypergroup, is taken from [82]. Definition 2.4 Let E be a locally compact space and ∗ a bilinear operator on MC (E). The pair (E, ∗) is said to be a weak hypergroup if the following axioms are satisfied: H1. If μ, ν ∈ P(E), then μ ∗ ν ∈ P(E); H2. μ ∗ (ν ∗ π) = (μ ∗ ν) ∗ π for all μ, ν, π ∈ MC (E); H3. The map (μ, ν) → μ∗ν is continuous (in the weak topology) from MC (E)× MC (E) to MC (E); H4. There exists an element e ∈ E such that δe ∗ μ = μ ∗ δe = μ for all μ ∈ MC (E); H5. There exists a homeomorphism (called involution) x → xˇ of E onto itself such that (x) ˇ = x and (δx ∗ δy ) = δyˇ ∗ δxˇ , where (δx ∗ δy ) is defined via   f (ξ ) (δx ∗ δy ) (dξ ) = f (ξˇ )(δx ∗ δy )(dξ ); H6. supp(δx ∗ δy ) is compact for all x, y ∈ E.

ˇ

ˇ

ˇ

ˇ

A weak hypergroup (E, ∗) is called a hypergroup if, in addition, ˇ H7. e ∈ supp(δx ∗ δy ) if and only if y = x; H8. (x, y) → supp(δx ∗ δy ) is continuous from E × E into the space of compact subsets of E (endowed with the Michael topology, see [94]). One should realize that the definitions of hypergroup and weak hypergroup include a compactness axiom on the support (axiom H6) which is comparable to axiom S1 in the definition of hypercomplex system (Definition 2.2). It is also important to note that other meanings for the word ‘hypergroup’ can be found on the literature on abstract harmonic analysis, particularly on older papers; we refer to the survey paper of Litvinov [127] for a thorough historical overview. However, the axioms of Definition 2.4 have been widely recognized as being appropriate for studying harmonic analysis, and are now well-established as the standard definition of hypergroup. In what follows, the word ‘hypergroup’ always refers to a structure satisfying axioms H1–H8. The reference monograph on the theory of hypergroups is the book of Bloom and Heyer [19], which contains a complete bibliography of the theory developed up to the mid-1990s. A comprehensive treatment of wavelet theory on hypergroups is given in the book of Trimèche [188]. Recent work on hypergroup theory can be found on the book of Székelyhidi [184] and references therein.

2.3 Generalized Convolutions and Hypergroups

39

The correspondence between the notions of (weak) hypergroup, generalized translation operator and hypercomplex system is described in the following proposition. As a preparation, we notethat a positive Borel measure m on E is said to be left  invariant if E f d(δx ∗ m) = E f dm for all x ∈ E and f ∈ Cc (E). Right invariant measures are defined similarly. It should be kept in mind that if the hypergroup (E, ∗) is commutative (i.e. μ ∗ ν = ν ∗ μ for all measures μ, ν ∈ MC (E)), then there exists a left (and right) invariant measure on E [19, Theorem 1.3.15]. Proposition 2.16 (a) If (E, ∗) is a weak hypergroup endowed with a left invariant measure m, then the family of operators {Tx }x∈E defined by  (Tx f )(y) :=

f (ξ )(δx ∗ δy )(dξ ),

f Borel measurable

E

satisfies axioms T1 and T2 of Definition 2.1, as well as the following properties (which hold for x, y ∈ E): (i) Tx 1 = 1, and if f ≥ 0 then Tx f ≥ 0; (ii) Tx f L2 (E,m) ≤ f L2 (E,m) ; (iii) If f ∈ Cc (E), then the function (x, y) → (Tx f )(y) is continuous in each variable; (iv) (Tx f )(y) = (Tyˇ fˇ)(x) ˇ ˇ for all f ∈ L2 (E, m), where we set fˇ(x) = f (x); (v) For every pair of relatively compact subsets B1 , B2 ⊂ E there exists a compact set E0 ⊂ E such that if suppf ∩ E0 = ∅ then (Tx f )(y) = 0 for m-a.e. x ∈ B1 , y ∈ B2 . Conversely, if {Tx }x∈E is a family of linear operators satisfying the conditions above, then the convolution ∗ on MC (E) defined by   (μ ∗ ν)(B) :=

(Tx 1B )(y) μ(dx)ν(dy) E

(B a Borel subset of E)

E

(2.44) endows E with a weak hypergroup structure. (b) If (E, ∗) is a weak hypergroup endowed with a left invariant measure m, then the space L1 (E, m) with the convolution  (f ∗ g)(x) :=

(Tx f )(y) g(y) ˇ m(dy) E

is a hypercomplex system with structure measure



E (T

x

1A )(y) 1B (y)dy. ˇ

40

2 Preliminaries

Proof See [15, pp. 60–62]. We note that a converse to the correspondence between weak hypergroups and hypercomplex systems stated in part (b) can be deduced as a corollary of part (a): if (L1 (E, m), ∗) is a hypercomplex system and the induced translation operators defined via Tx f, g := (f ∗ g)(x) ˇ (where x → xˇ is an involution on E) satisfy the conditions listed in part (a) of the proposition, then the convolution defined in (2.44) endows E with a weak hypergroup structure. Concerning the relation between hypergroups and Urbanik convolutions, one should be aware that, in general, hypergroups on R+ 0 do not satisfy the homogeneity axiom U5 of Urbanik convolutions, and Urbanik convolutions do not satisfy the compactness axiom H6 of hypergroups. We refer to [19, 190] for examples of both types. However, the Kingman convolution described in the next section (Definition 2.6) is an example both of a hypergroup structure on R+ 0 and of an Urbanik convolution. A hypergroup homomorphism between (E1 , ∗1 ) and (E2 , ∗2 ) is a map τ : MC (E1 ) −→ MC (E2 ) such that τ (μ ∗1 ν) = τ (μ) ∗2 τ (ν) for all μ, ν ∈ MC (E1 ) and τ (δx ) is a Dirac measure for all x ∈ E1 . If τ is bijective, then it is said to be a hypergroup isomorphism. Given a hypergroup (E1 , ∗1 ) and a continuous bijection τ : E1 −→ E2 , one can define a convolution ∗2 on E2 by letting   δx ∗2 δy = τ (δτ −1 (x) ∗1 δτ −1 (y)) and (μ ∗2 ν)(·) = E2 E2 (δx ∗2 δy )(·) μ(dx)ν(dy). (Here τ (δτ −1 (x) ∗1 δτ −1 (y) ) stands for the pushforward of the measure δτ −1 (x) ∗1 δτ −1 (y) under the map ξ → τ (ξ ).) It is then straightforward to check that the hypergroup axioms hold for (E2 , ∗2 ), so that (E1 , ∗1 ) and (E2 , ∗2 ) are isomorphic hypergroups. Let (E1 , ∗1 ), . . . , (En , ∗n ) be a finite family of hypergroups and write E = E1 × . . . × En . Define the convolution operator ∗ : MC (E) × MC (E) −→ MC (E) by   (μ ∗ ν)(·) = E

E

  (δx1 ∗1 δy1 ) ⊗ . . . ⊗ (δxn ∗n δyn ) (·) μ(dx)ν(dy).

One can easily verify that this operator satisfies all the hypergroup axioms. The hypergroup (E, ∗) is called the product of the hypergroups (E1 , ∗1 ), . . . , (En , ∗n ). As noted above, the structural properties captured in the hypergroup axioms allow for the development of several analogues of standard theorems in classical harmonic analysis [19, 184, 188]. Here we only highlight the Lévy-Khintchine type theorem stated below. (We say that a measure μ ∈ MC (E) is symmetric if ˇ for all Borel subsets B ⊂ E. The notion of infinitely divisible μ(B) = μ(B) distribution is defined as in Proposition 2.15(b).) Proposition 2.17 Let (E, ∗) be a commutative hypergroup. Any symmetric infinitely divisible measure μ ∈ P(E) can be represented as μ = γ ∗ e(θ ),

2.3 Generalized Convolutions and Hypergroups

41

where: • θ = lim (n·μn )|E\{e} is a σ -finite positive measure; n→∞ • e(θ ) isthe so-called ∗-Poisson measure associated with θ , defined as e(θ ) := θ ∗n e−θ ∞ n=0 n! ; • γ is a ∗-Gaussian measure, i.e. an infinitely divisible measure such that limn→∞ n·γn (E \ V ) = 0 for all open sets V containing e. ∗n (Here μn and γn are the measures such that μ = μ∗n n and γ = γn .) The  representation is unique, i.e. if μ =  γ ∗ e(θ ) for a σ -finite positive measure  θ and a ∗-Gaussian measure  γ , then θ =  θ and γ =  γ.

Proof See [162, Theorems 4.4 and 4.7]. Stochastic Convolutions Unlike the definitions of convolution-like structures presented above, the so-called stochastic convolutions, which were introduced and studied by Volkovich in [196, 198], include the existence of a compatible (generalized) characteristic function in their defining axioms: Definition 2.5 Let E be a locally compact space. A bilinear operator ◦ on MC (E) is said to be a stochastic convolution (in the sense of Volkovich) if it has the following properties: V1. If μ, ν ∈ P(E), then μ ◦ ν ∈ P(E); V2. There exists a separable complete metric space S and a bounded real continuous function ω(x, σ ) on E × S such that the ◦-characteristic function  μ (σ ) :=

ω(x, σ )μ(dx)

(μ ∈ P(E), σ ∈ S)

E

determines uniquely the probability measure μ, and no proper closed subset of S has the same property; V3. There exists e ∈ E such that ω(e, σ ) = 1 for all σ ∈ S; w V4. μn −→ μ if and only if μn (σ ) → μ (σ ) for all σ ∈ S; V5. μ3 = μ1 ◦ μ2 if and only if μ3 (σ ) = μ1 (σ )μ2 (σ ) for all σ ∈ S; V6. Let P ⊂ P(E). The set D(P) of all divisors (with respect to the convolution ◦) of measures ν ∈ P is relatively compact if and only if P is relatively compact. The following Lévy-Khintchine type theorem of Volkovich provides us with a characterization of the family of infinitely divisible distributions with respect to a given stochastic convolution. (The reader should compare it with the corresponding results for Urbanik convolutions and hypergroups stated in Propositions 2.15(b) and 2.17 respectively.)

42

2 Preliminaries

Proposition 2.18 Let ◦ be a stochastic convolution on E. A function  : E −→ R is a ◦-characteristic function of an infinitely divisible measure μ ∈ P(E) if and only it can be represented in the form  (σ ) = γ (σ ) exp

  ω(x, σ ) − 1 ν(dx) ,

(2.45)

E\{e}

where: • ν is a σ -finite measure on E \ {e} which is finite on the complement of any neighbourhood of e and such that 



 1 − ω(x, σ ) ν(dx) < ∞;

E

• γ (·) is the ◦-characteristic function of an infinitely divisible measure γ such that γ = e(aν) ◦ η

  a > 0, ν, η ∈ P(E), η infinitely divisible

⇒

ν = δe .

Proof See [196, pp. 465–466]. Any Urbanik convolution is a stochastic convolution [198], as well as all known examples of hypergroups on R+ 0 (cf. Sect. 4.1 below). Additional examples are listed in [198]. We can therefore interpret the defining properties V1–V6 of stochastic convolutions as a less restrictive set of axioms which still enable one to study infinite divisibility of measures on the convolution algebra and establish an analogue of the usual Lévy-Khintchine representation. The additional structure provided by the stronger axioms of hypergroups or of Urbanik convolutions gives rise to many other analogues of fundamental results of harmonic analysis, such as laws of large numbers and characterizations of Lévy processes (these results can be found in the literature cited above). Generalized Integral Convolutions This notion, which is complementary to that of generalized translation operator and of hypercomplex system, is not based on probabilistic motivations; rather, the motivation is to construct analogues of the ordinary convolution associated with prescribed integral transforms. Let K : C −→  C be an integral transformation between function spaces C and  C. An operator ∗ such that K(f ∗ g) = Kf · Kg is said to be a generalized integral convolution associated with the integral transform K [73, 97]. (Note the analogy with the corresponding property between the Fourier transform and the ordinary convolution.) This notion has been generalized by allowing for the presence of a weight function β ∈  C on the right-hand side, so that K(f ∗ g) = β · Kf · Kg [98], or by considering the more general property K3 (f ∗ g) = K1 f · K2 g, where K1 , K2 , K3 are different integral transforms [186, 204, 207].

2.4 Harmonic Analysis with Respect to the Kingman Convolution

43

It is noted in [46] that if ∗ is a generalized integral convolution associated with an integral transform K and L is a linear operator on C such that KL = MK, where M is a multiplication operator on  C, then the linear operator ∗ commutes with the convolution, i.e. the identity L(f ∗ g) = (Lf ) ∗ g holds. As mentioned in the Introduction, this gives rise to applications in differential equations, because many integral transforms determine isomorphisms between differential operators and multiplication operators. Many generalized integral convolutions have been constructed for onedimensional integral transforms of the form (Kf )(x) = I k(x, y)f (y)dy, where I is an interval of the real line and k : I × I −→ R. We refer to [73] for an extensive list of examples where the convolution has been constructed in closed form, and to [186, 207] for an approach based on double Mellin-Barnes integrals which is applicable to a general family of integral transforms with Meijer-G and Fox-H functions in the kernel.

2.4 Harmonic Analysis with Respect to the Kingman Convolution We saw in Sect. 2.1 that the drifted Brownian motion {Bt } has the convolution semigroup property, namely it satisfies Px [Bt ∈ ·] = μt ∗ δx for some convolution semigroup {μt }. In Sect. 2.3, we reviewed various notions of generalized convolutions. Now, we conclude the chapter with an example of a generalized convolution and corresponding convolution semigroups. The Kingman convolution is the seminal example of a binary operator ◦ on the space of probability measures which allows us to obtain the following analogue of the convolution semigroup property of drifted Brownian motion: for a diffusion process {Xt } other than the Brownian motion (in the case of the Kingman convolution, the Bessel process), we have Px [Xt ∈ ·] = μt ◦ δx , where {μt } is a family of measures satisfying μt +s = μt ◦ μs . In this section we briefly present the construction of the Kingman convolution and some properties which mirror well-known facts in classical harmonic analysis. This construction should be kept in mind throughout the subsequent chapters, as it serves as a benchmark for our later work in developing structures of generalized harmonic analysis associated with other diffusion processes. Let {Xt }t ≥0 be the Bessel process with index η > − 12 (started at x0 ≥ 0), defined √ as Xt = Zt where {Zt }t ≥0 is the unique strong solution of the SDE  dZt = 2(η + 1)dt + 2 Zt dWt ,

Z0 = x02

(cf. [164, §XI.1]). The process {Xt } is a one-dimensional diffusion with infinitesimal 2

η+ 1

d 2 d generator G = 12 dx 2 + x dx . In the case 0 < η < 1, the boundary x = 0 is instantaneously reflecting, while in the case η ≥ 1 the endpoint x = 0 is never

44

2 Preliminaries

reached by {Xt }. The transition probabilities of the Bessel process are given by the closed-form expression pt,x (dy) := P [Xt ∈ dy|X0 = x] ⎧  2 2   √xy  −1 −η η+1 exp − x +y I ⎪ dy, if x, t > 0, ⎪ η t ⎨t x y  y2t2  −η t −η−1 2 2η+1 = if x = 0, t > 0, exp − 2t dy, (η+1) y ⎪ ⎪ ⎩δ (dy), if t = 0, x

(2.46)

 (z/2)η+2k where (·) is the Gamma function and Iη (z) := ∞ k=0 k!(η+k+1) is the modified Bessel function of the first kind with index η [145, §10.25]. η+ 1

2

d 2 d The infinitesimal generator 12 dx 2 + x dx is associated with the invertible integral transform H : L2 (R+ ; x 2η+1dx) −→ L2 (R+ ; τ 2η+1dτ ) defined by



∞

(Hf )(τ ) =

f (x)J η (τ x) x 2η+1 dx,

0

(H

−1

2−2η ϕ)(x) = (η + 1)2



(2.47)

∞

ϕ(τ )J η (τ x) τ

2η+1

dτ,

0

 (−1)k (z/2)η+2k where J η (z) := 2η (η+1)z−η Jη (z) and Jη (z) := ∞ k=0 k!(η+k+1) is the Bessel function of the first kind [145, §10.2]. The operator H, which is known as the Hankel transform [84], is a particular case of the general Sturm-Liouville integral transform (2.21)–(2.22), introduced in Sect. 2.2. To see this, notice that the function x → 2

η+ 1

d 2 d J η (τ x) is the unique solution of the boundary value problem − 12 dx 2 − x dx u = 2 2η+1

u (x) = 0 (see [145]), which is a problem of type τ u, u(0) = 1, limx↓0 x 2η+1 2η+1 (2.1) with r(x) = x , p(x) = x 2 and q = 0.

Proposition 2.19 Define the extension of the Hankel transform to finite complex measures by  (Hμ)(τ ) :=

R+ 0

J η (τ x)μ(dx),

(μ ∈ MC (R+ 0 ), τ ≥ 0).

(2.48)

Then (Hμ)(τ ) is, for each μ ∈ MC (R+ 0 ), a continuous function of τ ≥ 0 which determines uniquely the measure μ. Moreover, the Hankel transform of the transition probabilities (2.46) equals (Hpt,x )(τ ) = e−t τ J η (τ x) 2

(t > 0, x ≥ 0).

Proof Since |J η (y)| ≤ 1 for y ≥ 0 [84, Theorem 2a], dominated convergence yields that τ → (Hμ)(τ ) is continuous. By [101, Lemma 2], (Hμ)(τ ) determines

2.4 Harmonic Analysis with Respect to the Kingman Convolution

45

uniquely the measure μ. The fact that (Hpt,x )(τ ) = e−t τ J η (τ x) can be verified using [156, Equation 2.12.39.3]. 2

+ + We will say that ◦ : MC (R+ 0 ) × MC (R0 ) −→ MC (R0 ) is a generalized convolution for the Bessel process if the transition probabilities are such that

pt,x = μt ◦ δx ,

(2.49)

where {μt }t ≥0 ⊂ P(R+ 0 ) is such that μt +s = μt ◦ μs for every t, s ≥ 0. It follows from Proposition 2.19 that if ◦ is such that H(μ ◦ ν) ≡ (Hμ)·(Hν) for all μ, ν ∈ P(R+ 0 ), then ◦ is a generalized convolution for the Bessel process. This suggests that a crucial requirement for the   generalized convolution ◦ is that it should satisfy the product formula H(δx ◦ δy ) (τ ) ≡ (Hδx )(τ )·(Hδy )(τ ) or, equivalently, J η (τ x) J η (τy) = R+ J η (τ ξ )(δx ◦ δy )(dξ ), where the measure δx ◦ δy should not 0 depend on τ . It turns out that such a product formula indeed exists, and we will see below that it gives rise to a convolution for which the desired (generalized) convolution semigroup property (2.49) holds. Theorem 2.6 (Product Formula for the Bessel Function of the First Kind) The following identity holds for all x, y > 0, τ ≥ 0 and η > 0: 21−2η (η + 1) (xy)−2η J η (τ x) J η (τy) = √ π (η + 12 )  x+y

η−1/2 J η (τ ξ ) (ξ 2 − (x − y)2 )((x + y)2 − ξ 2 ) ξ dξ. |x−y|

(2.50) Proof This follows from a classical integration formula for the Bessel function [200, p. 411], [84]. + + Definition 2.6 The operator ◦ : MC (R+ 0 ) × MC (R0 ) −→ MC (R0 ) defined by

 (μ ◦ ν)(B) :=

 + R+ 0 R0

γx,y (B) μ(dx)ν(dy)

(μ, ν ∈ MC (R+ 0 )),

where B is an arbitrary Borel subset of R+ 0 , γx,0 = γ0,x = δx and γx,y (dξ ) = 2η+1 k(x, y, ξ )ξ dξ , with 21−2η (η + 1) k(x, y, ξ ) = √ (xyξ )−2η π (η + 12 )

2 η−1/2 (ξ − (x − y)2 )((x + y)2 − ξ 2 ) 1[|x−y|,x+y](ξ ), is called the Kingman convolution (of order η) [101, 194].

x, y, ξ > 0

46

2 Preliminaries

One can easily verify that the Kingman convolution preserves the space P(R+ 0) (i.e. the Kingman convolution of two probability measures is indeed a probability measure) and, moreover, that it is trivialized by the Hankel transform of measures: Proposition 2.20 Let π, μ, ν ∈ MC (R+ 0 ). We have π = μ ◦ ν if and only if (Hπ)(τ ) = (Hμ)(τ )·(Hν)(τ ) for all τ ≥ 0. We observe that the theorem, definition and proposition above are counterparts of iz·x the following facts from classical harmonic analysis: the kernel  e iz·ξ of the Fourier iz·x iz·y transform satisfies the trivial product formula e e  =  Rd e δx+y (dξ ); the ordinary convolution is computed as (μ ∗ ν)(B) = Rd Rd γx,y (B) μ(dx)ν(dy), where γx,y = δx+y is the measure of the product formula; we have π = μ ∗ ν if and only if (Fπ)(z) = (Fμ)(z) · (Fν)(z) for all z ∈ Rd , where F is the Fourier transform. Proposition 2.20 should also be compared with statement (a) of Proposition 2.15. Indeed, it can be checked that the Kingman convolution is also a Urbanik convolution, whose ◦-characteristic function μ → Hμ has the kernel (x) ≡ J η (τ x). Using Proposition 2.20, one can verify (see Chap. 4 below or [19]) that the Kingman convolution satisfies axioms H1–H8 of Definition 2.4 and, therefore, induces a hypergroup structure in R+ 0 , which is known as the Bessel-Kingman hypergroup. When η = n/2 − 1 (n ∈ N), this hypergroup can be constructed as n the projection on R+ 0 of the space of all radial measures on R , cf. [163, Section 7]. From Definition 2.6, the notions of Kingman convolution semigroups and Kingman Lévy processes can be constructed as follows. Definition 2.7 A family {μt }t ≥0 ⊂ P(R+ 0 ) is said to be a Kingman convolution semigroup if μs ◦ μt = μs+t for all s, t ≥ 0,

μ0 = δ 0

w

and μt −→ δ0 as t ↓ 0.

Convolution semigroups are similarly defined for a general Urbanik convolution or a general hypergroup, see [23, Section 4] or [19, Section 5.2] respectively. Corollary 2.3 Let μt = pt,0 , where {pt,x }t,x≥0 are the transition probabilities (2.46) of the Bessel process started at zero. Then {μt }t ≥0 is a Kingman convolution semigroup. Moreover, we have pt,x = μt ◦ δx for all t, x ≥ 0 (i.e. ◦ is a generalized convolution for the Bessel process). Proof See the comments before Theorem 2.6 and observe that the weak continuity w μt −→ δ0 as t ↓ 0 follows from the fact that the Bessel process is a Feller process (Proposition 2.7). The next two results show that (an analogue of) two important properties of ordinary convolution semigroups—the fact that a convolution semigroup determines a Feller semigroup on R (cf. Proposition 2.8), and the Lévy-Khintchine representation (cf. Theorem 2.4)—can also be established for Kingman convolution semigroups.

2.4 Harmonic Analysis with Respect to the Kingman Convolution

47

Proposition 2.21 Let {μt }t ≥0 ⊂ P(R+ 0 ) be a Kingman convolution semigroup. Then the family {Tt }t ≥0 defined by + Tt : Cb (R+ 0 ) −→ Cb (R0 ),  Tt f = Tμt f, where (Tμtf )(x) :=

R+ 0

f d(δx ◦ μt )

is a conservative Feller semigroup. Proof See [163, Proposition 2.1]. Theorem 2.7 (Lévy-Khintchine Type Representation) If {μt }t ≥0 ⊂ P(R+ 0 ) is a Kingman convolution semigroup, then (Hμt )(τ ) = e−t ψ(τ )

(2.51)

for some function ψ(·) of the form  ψ(τ ) = cτ 2 +

R+

  1 − J η (τ x) ν(dx),

(2.52)

where c ≥ 0 and ν is a measure on R+ which is finite on the complement of any neighbourhood of 0 and such that for τ ≥ 0 we have 

 R+

 1 − J η (τ x) ν(dx) < ∞.

Conversely, for each function of the form (2.52) there exists a Kingman convolution semigroup {μt } such that (Hμt )(τ ) = e−t ψ(τ ) for all τ ≥ 0. In particular, the functions ψβ (τ ) := τ β (0 < β < 2) belong to the set of admissible functions of the form (2.52). Proof See [190, Theorem 13] and [194, Theorem 2]. Unlike the Lévy symbol (2.9) in the Lévy-Khintchine representation for ordinary convolution semigroups, the symbol ψ(·) in the Lévy-Khintchine type formula (2.51)–(2.52) for the Kingman convolution has no imaginary terms. This is unsurprising, because the Hankel transform of a probability measure on R+ 0 is real-valued, while the Fourier transform of probability measures on Rd is complex-valued. The resemblance between the two formulas becomes yet more evident when {μt } ⊂ P(Rd ) is an ordinary convolution semigroup of symmetric measures: an ordinary convolution semigroup is symmetric if and  only if α = 0 and the measure ν is symmetric, so that φ(z) = z · Qz + Rd \{0} 1 − cos(z · y) ν(dy). (Since J − 1 (ξ ) = 2 cos ξ [145, §10.16], this right-hand side is, for d = 1, the limiting form of the representation (2.52) when η ↓ − 12 .)

48

2 Preliminaries

Theorem 2.7 can also be compared with the Lévy-Khintchine representation for Urbanik convolutions in statement (b) of Proposition 2.15. An analogous representation in the context of Sturm-Liouville hypergroups is given in [30, Theorem 7]. A Kingman Lévy process is a Feller process associated with a Kingman convolution semigroup. By the above Lévy-Khintchine type representation, the class of Kingman Lévy processes generalizes the Bessel processes in an analogous way as the class of (ordinary) Lévy processes generalizes the Brownian motion. We note, in particular, that the class of Kingman Lévy processes includes many processes which do not admit continuous versions (cf. [163, Theorem 2.2]). For further properties of the Kingman convolution and the associated Lévy processes, we refer to [19, 23, 101, 163, 190]. The results stated thus far refer to the probabilistic properties of the Kingman convolution (seen as a binary operator on the space of probability measures). There is also an extensive literature on the Hankel convolution of functions, which is defined by 

∞ ∞

(f ◦ g)(x) := 0

f (y)k(x, y, ξ )y 2η+1 dy g(ξ ) ξ 2η+1 dξ.

(2.53)

0

In other words, the Hankel convolution f ◦g is defined as the density of the Kingman convolution of the measures μf (dx) = f (x)x 2η+1dx and μg (dx) = g(x)x 2η+1 dx. It is thus clear that the Kingman convolution is the generalized integral convolution associated with the Hankel transform (2.47). It is clear that H(f ◦ g) = (Hf ) · (Hg) for f, g ∈ L1 (R+ ; x 2η+1 dx); this result is the Hankel counterpart  of the usual convolution theorem F(f ∗ g) = (Ff )·(Fg), where (f ∗ g)(x) = Rd f (x − y)g(y)dy and F is the Fourier transform on Rd . The Hankel convolution has many other properties which are parallel to those of the ordinary convolution of functions, such as a Young-type inequality. Let us recall that the classical Young convolution inequality [62, Proposition 8.9] states that if f ∈ Lp1 (Rd ), g ∈ Lp2 (Rd ) (p1 , p2 ∈ [1, ∞]) and s ∈ [1, ∞] is defined by 1 1 1 s = p1 + p2 − 1, then the integral defining (f ∗ g)(x) converges for a.e. x and f ∗ gLs (Rd ) ≤ f Lp1 (Rd ) · gLp2 (Rd ) . The following analogue holds for the p Hankel convolution. (We write Lη := Lp (R+ ; x 2η+1dx).) p

p

Proposition 2.22 Let f ∈ Lη 1 , g ∈ Lη 2 (p1 , p2 ∈ [1, ∞]) and let s ∈ [1, ∞] be defined by 1s = p11 + p12 − 1. Then (f ◦ g)(x) converges for a.e. x > 0 and f ◦ gLsη ≤ f Lpη 1 ·gLpη 2 . p

q

If f ∈ Lη and g ∈ Lη with

1 p

+

1 q

= 1, then f ◦ g ∈ Cb (R+ ).

2.4 Harmonic Analysis with Respect to the Kingman Convolution

49

Proof See [84, Theorem 2b]. Additional examples of analogues of classical properties which have been established for the Hankel convolution include: an analogue of the Marcinkiewicz multiplier theorem [78], a characterization of variation diminishing convolution kernels similar to that for the ordinary convolution [84], a parallel theory for Hankel convolution equations [35], among others.

Chapter 3

The Whittaker Convolution

The goal of this chapter is to construct a generalized convolution for the onedimensional diffusion process known as the Shiryaev process. The properties of this convolution-like operator will allow us to interpret the Shiryaev process as a Lévylike process, thereby providing a positive answer to the general question formulated in the Introduction. The Shiryaev process (started at y0 ≥ 0) is defined in [149] as the unique strong solution {Yt }t ≥0 of the SDE dYt = (1 + μYt )dt + σ Yt dWt ,

Y0 = y0

(3.1)

where μ ∈ R, σ > 0 and {Wt }t ≥0 is a standard Brownian motion. The infinitesimal generator of the Shiryaev process is the differential operator A defined as Au(y) = σ 2 2



2 y u (y) + (1 + μy)u (y). The Shiryaev process, whose defining SDE (3.1) was first derived by Shiryaev in the context of quickest detection problems [168], has various applications in mathematical finance; in particular, it plays a fundamental role in the problem of Asian option pricing under the famous Black-Scholes model [47, 124]. See [152] for a survey of other applications in physics and finance. We will√restrict our attention to the standardized Shiryaev process with parameters σ = 2 and μ = 2(1 − α) ∈ R, i.e. the one-dimensional diffusion generated by the operator Aα u(y) = y 2 u

(y) + (1 + 2(1 − α)y)u (y) =



1  pα u (y), rα (y)

(3.2)

where rα (ξ ) := ξ −2α e−1/ξ and pα (ξ ) := ξ 2(1−α) e−1/ξ . This restriction does not introduce any loss of generality, because we know (cf. [21, Section II.8]) that if d2 {Xt }t ≥0 is a one-dimensional diffusion generated by Aα,γ ,c := γ x 2 dx 2 + γ (c + © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Sousa et al., Convolution-like Structures, Differential Operators and Diffusion Processes, Lecture Notes in Mathematics 2315, https://doi.org/10.1007/978-3-031-05296-5_3

51

52

3 The Whittaker Convolution

d 2(1 − α)x) dx (c, γ > 0), then the process {Yt = 1c Xγ t }t ≥0 is a one-dimensional diffusion generated by (3.2). In fact, once we have constructed the convolution structure for the standardized Shiryaev process, the convolution structure for Aα,γ ,c is simply a by-product which is obtained via elementary changes of variable (cf. Remark 3.11). From Sect. 3.3 onwards we will mostly assume that α ≤ 12 ; this is a necessary and sufficient condition for the underlying product formula to have the positivity and conservativeness property which is required for the induced convolution to be a binary operator on the space of probability measures.

3.1 A Special Case: The Kontorovich–Lebedev Convolution Before advancing to our general discussion of convolutions associated with the standardized Shiryaev process (3.2), we take a look into a motivating particular case: the so-called Kontorovich–Lebedev convolution, which was introduced by Kakichev in 1967 [97] and whose properties and applications have been studied in numerous works [85, 86, 154, 204, 205, 207]. Definition 3.1 (a) The Kontorovich–Lebedev transform (FKL f )(·) of a function f ∈ L2 (R+ , dx) is defined by 

∞

(FKL f )(τ ) =

(3.3)

f (x)Kiτ (x) dx, 0

where Kν (x) is the modified Bessel function of the second kind [145, §10.25]. (b) Let f, g : R+ −→ C. If the double integral (f ∗ g)(x) := KL

1 2x





∞ ∞

exp 0

0

−

xy xξ yξ − − f (y)g(ξ )dydξ 2ξ 2y 2x

(3.4)

exists for almost every x ∈ R+ , then we call it the Kontorovich–Lebedev convolution of f and g. The connection between the Kontorovich–Lebedev transform and convolution is analogous to that between the Hankel transform and convolution, discussed in Sect. 2.4. Indeed, like in (2.50), (2.53), the definition of the Kontorovich–Lebedev convolution is based on the product formula for the kernel of the Kontorovich– Lebedev transform, which is given by 1 Kν (x)Kν (y) = 2





∞

Kν (ξ ) exp 0

xy xξ yξ dξ − − − 2ξ 2y 2x ξ

(x, y > 0, ν ∈ C). (3.5)

3.2 The Product Formula for the Whittaker Function

53

This identity is a classical result known as the Macdonald formula, which can be found in standard texts on special functions, cf. [56, §7.7.6] or [204, Equation (1.103)]. Since the modified Bessel function of the second kind and the Whittaker W func1 1 tion are related by Kν (x) = π 2 (2x)− 2 W0,ν (2x) (cf. Remark 3.1 below), the def1 ∞ 1 inition (3.3) can be rewritten as (FKL f )(τ ) = π 2 0 f (x) W0,iτ (2x) (2x)− 2 dx. The Kontorovich–Lebedev transform is therefore (up to a change of variables) the particular case α = 0 of the index Whittaker transform (2.40). Accordingly, a straightforward corollary of (3.5) is that the product formula  W 0,ν (x) W 0,ν (y) =

∞

W 0,ν (ξ ) k0 (x, y, ξ ) e−1/ξ dξ,

(3.6)

0

where  1 1 x y ξ 1 1 + + − − − exp k0 (x, y, ξ ) := 2(πxyξ )1/2 2x 2y 2ξ 4yξ 4xξ 4xy 1

holds for the functions W 0,ν (y) := e 2y W0,ν ( y1 ), i.e., for the generalized eigenfunctions of the generator A0 of the standardized Shiryaev process with parameter α = 0 (cf. Proposition 3.1 below). One can check (using the identity (3.17) below with ν = − 12 ) that the measures k0 (x, y, ξ ) e−1/ξ dξ constitute a family of probability measures: {k0 (x, y, ξ ) e−1/ξ dξ }x,y>0 ⊂ P(R+ 0 ). As pointed out in the Introduction, this probabilistic property of the product formula means that the normalized Kontorovich–Lebedev convolution  ∞ ∞ k0 (x, y, ξ )f (y)g(ξ )e−1/y dye−1/ξ dξ, (3.7) (f  g)(x) := KL

0

0

together with its extension to probability measures, are especially suitable for developing generalized harmonic analysis. We note that, despite this, the normalized form (3.7) has rarely been adopted in the literature on the Kontorovich–Lebedev convolution.

3.2 The Product Formula for the Whittaker Function For our purposes, the Kontorovich–Lebedev transform is merely a particular case of the index Whittaker transform; this makes it natural to conjecture that a probabilistic product formula similar to (3.6) might hold for the generalized eigenfunctions of Aα with α = 0. The goal of this section is to prove that this conjecture is true. It should be noted that no counterpart of the product formula (3.6) for the operator Aα (α = 0) can be found on standard texts on special functions; in order to establish

54

3 The Whittaker Convolution

such counterpart, we will rely on integration formulas and other known results from the theory of special functions. To prepare our work, we begin by summarizing some relevant facts on the Whittaker W functions and their role as eigenfunctions of the Sturm–Liouville operator Aα . Proposition 3.1 The unique solution of the boundary value problem − Aα u = λu

(y ∈ R+ , λ ∈ C),

u(0) = 1,

(pα u )(0) = 0.

(3.8)

is given by 1

Wα, λ(y) := y α e 2y Wα, λ( y1 ), where kind.

λ

=



(3.9)

( 12 − α)2 − λ and Wα,ν (x) is the Whittaker function of the second 1

Throughout this chapter, the function Wα,ν (y) = y α e 2y Wα,ν ( y1 ) will be called the normalized Whittaker W function. To prove Proposition 3.1, one just needs to check, using the basic properties of the Whittaker function stated below, that (3.9) is a solution of −Aα u = λu which satisfies the given boundary conditions. Remark 3.1 (Some Basics on the Whittaker Function of the Second Kind) Let α, ν ∈ C. The Whittaker function Wα,ν (x) is, by definition, the solution of  2 2 u = 0 which is determined Whittaker’s differential equation ddxu2 + − 41 + αx + 1/4−ν x2 uniquely by the property x

Wα,ν (x) ∼ x α e− 2 ,

|x| → ∞, Re x > 0.

(3.10)

The Whittaker W function is an analytic function of x on the half-plane Re x > 0, and for fixed x it is an entire function of the first and the second parameter [145, §13.14(ii)]. For Re x > 0 and Re α < 12 + Re ν, it admits the integral representation (cf. [155], integral 2.3.6.9) x

Wα,ν (x) =

e− 2 x α ( 12 − α + ν)



∞ 0

 1 s − 12 +α+ν e−s s − 2 −α+ν 1 + ds. x

(3.11)

The Whittaker W function is an even function of the parameter ν [145, Equation 13.14.31]. For α = 12 ± ν, 32 ± ν, . . ., its asymptotic behaviour near the origin is, cf. [145, §13.14(iii)]   1 Wα,ν (x) = O x 2 −Re ν (Re ν ≥ 0, ν = 0),   1 Wα,0 (x) = O − x 2 log x ,

x → 0.

(3.12)

3.2 The Product Formula for the Whittaker Function

55

The asymptotic expansion for Wα,ν (x) as |x| → ∞ is given by [145, Equation 13.19.3] Wα,ν (x) ∼ e− 2 x α x

∞ 1  ( 2 − α + ν)k ( 12 − α − ν)k (−x)−k , k!

|x| → ∞, Re x > 0.

k=0

(3.13) The Whittaker function satisfies the recurrence relation and the differentiation formula [145, Equations 13.15.13 and 13.15.23] 1

1

x 2 Wα+ 1 ,ν+ 1 (x) = (x + 2ν)Wα,ν (x) + ( 12 − α − ν) x 2 Wα− 1 ,ν− 1 (x), 2

2

2

2

(3.14)

 d n      x x ex/2 x −α−1 Wα,ν (x) = 12 + ν − α n 12 − ν − α n ex/2 x n−α−1 Wα−n,ν (x), dx

(3.15)  where n ∈ N and (a)n = n−1 j =0 (a + j ) is the Pochhammer symbol. When the parameter α is equal to zero (resp., equal to 12 + ν), the Whittaker function reduces to the modified Bessel function of the second kind (resp., to an elementary function) [145, §13.18(i), (iii)], 1

1

W0,ν (2x) = π − 2 (2x) 2 Kν (x), W 1 +ν,ν (x) = x 2

1 2 +ν

e−x/2.

(3.16) (3.17)

By [204, Theorem 1.11], for α ∈ R the asymptotic expansion of the Whittaker function with imaginary parameter ν = iτ as τ → ∞ is 1 2

Wα,iτ (x) = (2x) τ

α− 21 −πτ/2

e

   x  π 1

 + − α + τ 1 + O(τ −1 ) , cos τ log 4τ 2 2 (3.18)

the expansion being uniform in 0 < x ≤ M (M > 0). The desired generalization of the product formula (3.5) to the Whittaker function Wα,ν (x) with general parameter α ∈ C is stated next: Theorem 3.1 The product Wα,ν (x)Wα,ν (y) of two Whittaker functions of the second kind with different arguments admits the integral representation 

∞

Wα,ν (x)Wα,ν (y) =

Wα,ν (ξ ) κα (x, y, ξ ) 0

dξ ξ2

(x, y > 0, α, ν ∈ C), (3.19)

56

3 The Whittaker Convolution

where 1

1

κα (x, y, ξ ) := 2−1−α π − 2 (xyξ ) 2 exp



 x y ξ (xy + xξ + yξ )2 xy + xξ + yξ + + − D2α 2 2 2 8xyξ (2xyξ )1/2

being Dμ (z) the parabolic cylinder function [56, Section 8.2]. Remark 3.2 Before the proof, let us collect some facts on the parabolic cylinder function Dμ (z) which will be needed in the sequel. The parabolic cylinder function is given in terms of the Whittaker function by μ

Dμ (z) = 2 2 + 4 z− 2 W μ + 1 , 1 1

1

2

4 4

 z2  2 .

 2 2 This function is a solution of the differential equation ddzu2 + μ + 12 − z4 u = 0, and it is an entire function of the parameter μ. An integral representation for the parabolic cylinder function is [145, Equation 12.5.3] z2

zμ e − 4 Dμ (z) =  1   2 (1 − μ)



∞

e 0

−s − 12 (1+μ)

s

 μ 2s 2 1 + 2 ds z

(Re z > 0, Re μ < 1). (3.20)

The asymptotic form of Dμ (z) for large z is [56, Equation 8.4(1)] z2

Dμ (z) ∼ zμ e− 4 ,

z → ∞.

(3.21)

The recurrence relation and differentiation formula for Dμ (z) are [56, Equations 8.2(14) and 8.2(16)] Dμ+1 (z) = zDμ (z) − μDμ−1 (z),  z2 d n − z2 e 4 Dμ (z) = (−1)n e− 4 Dμ+n (z) n dz

(3.22) (n ∈ N),

(3.23)

and the parabolic cylinder function reduces to an exponential function when its parameter equals zero [56, Equation 8.2(9)], z2

D0 (z) = e− 4 .

(3.24)

We will prove Theorem 3.1 through a sequence of lemmas, where we shall assume that α is a negative real number and ν is purely imaginary. In the final step of the proof, an analytic continuation argument will be used to remove this restriction. Our first lemma gives an alternative product formula which is less useful than (3.19) because its kernel also depends on the second parameter of the Whittaker function.

3.2 The Product Formula for the Whittaker Function

57

Lemma 3.1 If α ∈ (−∞, 0) and τ ∈ R, then the integral representation x

p

(xp)α e− 2 − 2

Wα,iτ (x)Wα,iτ (p) =

× |( 12 − α + iτ )|2   ∞  ∞ 1 ξ 1 w + + wξ dw dξ ξ −1−α e− 2 Wα,iτ (ξ ) w−2α exp − w − × x p xp 0 0 (3.25)

is valid for x, p > 0. Proof From relation 2.21.2.17 in [157] it follows that Wα,iτ (x)Wα,iτ (p) p

= (xp) 2 −iτ e − 2 − 2 1

x

1

x

p

(xp) 2 −iτ e − 2 − 2 = (1 − 2α)

 1  − α − iτ, 1 − 2iτ ; x − α − iτ, 1 − 2iτ ; p 2 2

1

 0

∞

− 1 +α+iτ e −w w−2α (w + x)(w + p) 2 ×



1 xp 1 − α − iτ, − α − iτ ; 1 − 2α; 1 − dw 2 2 (w + x)(w + p)

× 2 F1 

p

(xp)α e − 2 − 2 (1 − 2α) x

=

∞

e −w w−2α ×

0

× 2 F1



(3.26)

1 1 1 w2 1 − α − iτ, − α + iτ ; 1 − 2α; − + w− dw. 2 2 x p px

Here (a, b; x) := ex/2x −b/2 W b −a, b − 1 (x) is the confluent hypergeometric 2 2 2 function of the second kind [55, Chapter VI] (also known as the Tricomi function or the Kummer function of the second kind); in the last step we used the  z transformation formula 2 F1 (a, b; c; z) = (1 − z)−a 2 F1 a, c − b; c; z−1 for the Gauss hypergeometric function, cf. [145, Equation 15.8.1]. Next, according to integral 2.19.3.5 in [157], the Gauss hypergeometric function in (3.26) admits the integral representation  2 F1

1 1 1 1 w2 − α − iτ, − α + iτ ; 1 − 2α; − + w− 2 2 x p px   ∞   (1 − 2α) −1−α exp − ξ − 1 + 1 + w wξ W = ξ α,iτ (ξ )dξ, 2 x p xp |( 1 − α + iτ )|2 0 2

58

3 The Whittaker Convolution

and thus we have x

p

(xp)α e− 2 − 2

Wα,iτ (x)Wα,iτ (p) =

|( 12 − α + iτ )|2   ∞  ∞ 1 w ξ 1 −w −2α −1−α + + e w ξ exp − − wξ Wα,iτ (ξ ) dξ dw. 2 x p xp 0 0 (3.27)

Using the assumption Re α < 0 and the limiting forms (3.10), (3.12) ∞ of the Whittaker function, we see that the integrals 0 e−w w−2α dw and  ∞ −1−α − ξ e 2 Wα,iτ (ξ )dξ converge absolutely. Therefore, we can use Fubini’s 0 ξ theorem to reverse the order of integration in (3.27); doing so, we obtain (3.25). The previous lemma gives an integral representation for |( 12 − α + iτ )|2 Wα,iτ (x)Wα,iτ (p) whose kernel does not depend on τ . Integral representations for |( 12 −α +iτ )|2 Wα,iτ (x) which share the same property are also known. In the next two lemmas we take advantage of these integral representations and of the uniqueness theorem for Laplace transforms in order to deduce that the product formula (3.19) holds when α is a negative real number and ν = iτ ∈ iR. Lemma 3.2 The identity  22α x −α Wα,iτ (x)

∞

y

e− 2y − 2 y α−2 Wα,iτ (y)dy s

0

 = 0

∞

2α  ! x 2s − 12  2s 1/2 ξ  2s 1/2 1+ + 1 exp − 1+ Wα,iτ (ξ ) ξ α−2 dξ + 1+ xξ xξ 2 2 xξ

(3.28) holds for α ∈ (−∞, 0), τ ∈ R and x, s > 0. Proof Using the change of variable s = 2wξ(1 +

w x ),

we rewrite (3.25) as

|( 12 − α + iτ )|2 Wα,iτ (x)Wα,iτ (p)  ∞  ∞  ξ x p 2s − 12 1 − s e− 2 ξ α−2 Wα,iτ (ξ ) e 2p s −2α 1 + × = (xp)α e− 2 − 2 2 xξ 0 0 2α !   x  ξ 2s 12 2s 12 +1 exp ds dξ + 1− 1+ × 1+ xξ 2 2 xξ  2α  ∞  ∞ p 2s 12 2s − 12  1 − s 1+ e 2p s −2α +1 × 1+ = (xp)α e− 2 2 xξ xξ 0 0 ! x ξ  2s 12 × exp − Wα,iτ (ξ ) ξ α−2 dξ ds, + 1+ 2 2 xξ

(3.29)

3.2 The Product Formula for the Whittaker Function

59

where the absolute convergence of the iterated integral (see the proof of the previous lemma) justifies the change of order of integration. On the other hand, by relation 2.19.5.18 in [157] we have |( 12 − α + iτ )|2 Wα,iτ (p)



p

= 22α−1 (1 − 2α)pα e− 2 p

= 22α−1 pα e− 2 = 22α−1 pα e

− p2

 ∞

0 ∞

e 

∞

1 1 −1+2α − y α−2 + e 2y Wα,iτ (y) dy 2y 2p

s s − 2y − 2p −2α

s

y

ds e− 2 y α−2 Wα,iτ (y) dy

0 0  ∞ ∞ s s − 2p − 2y − y2 α−2 −2α

e

0

s

e

y

Wα,iτ (y) dy ds.

0

(3.30) Comparing (3.29) and (3.30), and recalling the injectivity of Laplace transform, we deduce that (3.28) holds. Lemma 3.3 The product formula (3.19) holds for α < 0, τ ∈ R and x, y > 0. Proof We begin by deriving the following representation for the function of s appearing in the right-hand side of (3.28): 2α  ! x ξ  2s 12 2s − 12  2s 12 +1 exp − + 1+ 1+ xξ xξ 2 2 xξ  1! ∞     x 2s 12 ξ 2s 2 1 exp − u 1 + exp − + 1+ γ (−2α, u) du = (−2α) 2 2 xξ xξ 0 1  (πxξ )− 2 ∞  ξ x = γ (−2α, u)× u+ + (−2α) 0 2 2 !  ∞   1 x ξ 2 y − 12 y exp − 2s + xξ − u+ + dy du × 4y 2 2 xξ 0 1   xξ  1 (πxξ )− 2 ∞ − 2ys e exp − = y− 2 × (−2α) 0 4y    ∞ ξ x ξ 2 y x exp − u + + γ (−2α, u)du dy, u+ + × 2 2 2 2 xξ 0



1+

60

3 The Whittaker Convolution

where γ (·, ·) is the incomplete Gamma function [56, Chapter IX]. In the first two equalities we have used integral 8.14.1 in [145] and integral 2.3.16.3 in [155], respectively, and the positivity of the integrand allows us to change the order of integration. Substituting in (3.28), we find that  1 1 1 (−2α)2 2 +2α π − 2 x 2 −α Wα,iτ (x) 

 5 ξ − 2 +α Wα,iτ (ξ )

∞

y

e− 2y − 2 y α−2 Wα,iτ (y)dy s

0

 xξ  1 y− 2 exp − 4y 0 0    ∞ x x ξ exp − u + + u+ + × 2 2 2 0  ∞  ∞   s 1 5 xξ − Wα,iτ (ξ ) e 2y y − 2 ξ − 2 +α exp − = 4y 0 0    ∞ ξ x x × exp − u + + u+ + 2 2 2 0

=

∞

∞

e

s − 2y

ξ 2 y 2 xξ



ξ 2 y 2 xξ



γ (−2α, u)du dy dξ

γ (−2α, u)du dξ dy,

(3.31) where the order of integration can be interchanged because of the absolute convergence of the triple integral, which follows from the inequality γ (−2α, u) ≤ (−2α) and the equalities 

   5 ξ − 2 +α Wα,iτ (ξ )

 xξ  1 × y − 2 exp − 2y 0 0    ∞ ξ x ξ 2 y x exp − u + + du dy dξ × u+ + 2 2 2 2 xξ 0     ∞ xξ y xy ξy − 3 s x ∞ − 5 +α  − − − − y 2 dy dξ ξ 2 exp − = Wα,iτ (ξ ) 2 0 2y 4y 2 4ξ 4x 0    ∞   1 1 ξ  2s 12  x 2s − 21 + 1+ ξ −3+α 1 + exp − = 2− 2 (πx) 2 Wα,iτ (ξ ) dξ xξ 2 2 xξ 0 ∞

∞

e

s − 2y

0 and suppose 1 that M ≤ 12 − Re α ≤ M and 0 ≤ Re ν ≤ M. Then for t > 0 we have   ∞   s − 12 +α+ν  −s − 12 −α+ν  e s ds  1+ t |( 12 − α + ν)|  0 t  ∞  e− 2 s M −s −1 1/M 2M ≤ e s (s + s ) 1 + ds t |( 12 − α + ν)| 0 # "   1 1 v  = 1 1 1 1 (t)+(2M)v 1 3M (t) , (M− +1), (M+ ) (1−M), M 2 M 2 M 2 2 |( 12 − α + ν)|

    v α,ν (t) = v α,−ν (t) =

e− 2 t

62

3 The Whittaker Convolution

where we have used the integral representation (3.11). Moreover, letting n ∈ N, a repeated application of the recurrence relation (3.14) shows that v α+ n2 ,ν+ n2 (t) = Q(1) n,α,ν

1 1 v α,ν (t) + Q(2) v 1 1 (t), n,α,ν t t α− 2 ,ν− 2

where the Q(i) n,α,ν (·) are polynomials of degree at most n whose coefficients depend on α and ν. Therefore, for M1 ≤ 12 − Re α ≤ M − 12 and −M + 12 ≤ Re ν ≤ M we have           v α+ n ,ν+ n (t) ≤ Q(1) 1 v α,ν (t) + Q(2) 1 v 1 1 (t) 2

n,α,ν t

2

n,α,ν t

α− 2 ,ν− 2

  1   (2)  1   + Q  G(α, ν)× ≤ Q(1) n,α,ν t n,α,ν t "   # 1 ×  M v 1 (M− 1 +1), 1 (M+ 1 ) (t) + (2M)v 1 (1−M), 3M (t) , 2

2

M

2

M

2

(3.34) where

G(α, ν) =

Similarly, for   D2α (t) =

⎧ ⎪ ⎪2|( 12 − α + ν)|−1 , ⎨ |( 12 ⎪ ⎪ ⎩|( 1 2 1 M

≤

1 2

−α

+ ν)|−1

− α − ν)|−1 +

  

t2

e− 4

|( 12 − α)|

=

e− 4 t |( 21

− α)| t

|( 12 − α)|

t

∞

2 Re α 

(t −2M + t − M ) 2

− ν)|−1 ,

≤ Re ν ≤ M,

0 ≤ Re ν < 12 ,

− α − ν)|−1 ,

e

−s − 12 −α

s

0



∞ 0

(t −2M

−α

1 2

−M +

1 2

≤ Re ν < 0.

− Re α ≤ M the integral representation (3.20) gives

t2

≤

+

|( 32 |( 32

  2s α  1 + 2 ds  t

 1 2s 12 − M1 e−s s −1 (s M + s M ) 1 + 2 ds t

"    2  2 # 2 1 + t− M)  M v 3 − 1 , 1 t2 + (M)v 3 − 1 (M+ 1 ), 1 + 1 (M+ 1 ) t2 4

M 4

4

2

M

4

2

M

and, by (3.22), for each n ∈ N we have D2α+n (t) = Q(3) n,α (t)D2α (t) + (j ) (t)D (t), being Q (·) polynomials of degree at most n with coefficients Q(4) 2α−1 n,α n,α depending on α, hence      D2α+n (t) ≤ |( 1 − α)|−1 + |(1 − α)|−1 |Q(3) (t)| + |Q(4) (t)| t (t −2M + t − M2 ) n,α,ν n,α 2 "    2  2 # 1 v 3 − 1 , 1 t2 + (M)v 3 − 1 (M+ 1 ), 1 + 1 (M+ 1 ) t2 . ×  M 4

M 4

4

2

M

4

2

M

(3.35)

3.2 The Product Formula for the Whittaker Function

63

Using the inequalities (3.34), (3.35) and the limiting forms (3.10), (3.12) for the Whittaker function, one can verify without difficulty that  sup (α,ν)∈RM

 dξ   Wα+ n2 ,ν+ n2 (ξ ) κα+ n2 (x, y, ξ ) 2 < ∞, ξ

∞

0

 1   where RM = (α, ν)  M ≤ 12 − Re α ≤ M − 12 , −M + 12 ≤ Re ν ≤ M . Since M and n are arbitrary, the known results on the analyticity of parameter-dependent ∞ integrals (e.g. [135]) yield that 0 Wα,ν (ξ ) κα (x, y, ξ ) dξ is an entire function of ξ2 the parameter α and the parameter ν. As the left-hand side of (3.19) is also an entire function of α and ν, by analytic continuation we conclude that the product formula (3.19) extends to all α, ν ∈ C, as we wanted to show. Remark 3.3 (a) The product formula (3.19) can be equivalently written in terms of the normalized Whittaker W function as  ∞ Wα,ν (ξ ) kα (x, y, ξ ) ξ −2α e−1/ξ dξ, (3.36) Wα,ν (x) Wα,ν (y) = 0

where 1

kα (x, y, ξ ) := (xyξ )α e 2x 1

1 1 + 2y + 2ξ

κα ( x1 , y1 , ξ1 )

1

= 2−1−α π − 2 (xyξ )− 2 +α exp



 1 1 (x + y + ξ )2 x+y+ξ 1 D2α . + + − x y ξ 8xyξ (2xyξ )1/2

(3.37) (b) It follows from (3.16) and (3.24) that in the particular case α = 0, (3.19) specializes into 1 Kν (x)Kν (y) = 2





∞

Kν (ξ ) exp 0

xξ yξ dξ xy − − , − 2ξ 2y 2x ξ

which is the product formula for the modified Bessel function stated in (3.5). (c) Since the parabolic cylinder function Dν (t) is a positive function of t > 0 whenever ν ∈ (−∞, 1] (as can be seen e.g. from the representation (3.20)), we have kα (x, y, ξ ) > 0

for all α ≤

1 2

and x, y, ξ > 0.

(3.38)

This positivity property means that the convolution operator induced by the product formula (3.36) (cf. Sect. 3.5) is positivity-preserving.

64

3 The Whittaker Convolution

(d) A useful upper bound for the kernel of the product formula (3.36), valid for α ∈ R, is the following:  2   kα (x, y, ξ ) ≤ A(y) (xyξ )− 12 (x + y + ξ )2α exp 1 − (x + y − ξ ) ξ 4xyξ

(x, y, ξ > 0),

(3.39) where 1

A(y) = 2−1−α π − 2 ·



 t2 max t −2α e 4 Dα (t) < ∞

This upper bound follows from the inequality by (3.21), the function

(y > 0).

t ≥y −1/2

t −2α e

t 2/4

(x+y+ξ )2 2xyξ

≥

1 y

and the fact that,

D2α (t) is bounded on the interval [y −1/2, ∞).

The normalized Whittaker W function includes, as a particular case, the (generalized) Bessel polynomial Bn (x; α) (α ∈ R\{1, 32 , 2, . . .}, n ∈ N0 ) introduced in [109] as the polynomial of degree n, with constant term equal to 1, which is a solution of the Sturm–Liouville equation y 2 u

(y)+(1 +2(1 −α)y)u (y) = n(n+1 −2α)u(y). By [145, Equation 13.14.9] and [109, §15], these polynomials are given by 1

Bn (x; α) = W α, 1 −α+n (x) = x 2α e x 2

d n 2(n−α) − 1 [x e x] dx n

(n ∈ N0 , α = 1, 32 , 2, . . .).

The Bessel polynomials are one of the four canonical families of classical orthogonal polynomials (see [132, 133]). We refer to the book [77] for a detailed exposition on the properties and applications of the Bessel polynomials. As an immediate corollary of (3.36), the following product formula holds for the Bessel polynomials: Corollary 3.1 The product Bn (x; α)Bn (y; α) of Bessel polynomials admits the integral representation 

∞

Bn (x; α)Bn (y; α) =

Bn (ξ ; α) kα (x, y, ξ ) ξ −2α e−1/ξ dξ,

0

where kα (x, y, ξ ) is defined by (3.37).

3.3 Whittaker Translation We now define the generalized translation operator induced by the product formula (3.36) for the normalized Whittaker function:

3.3 Whittaker Translation

65

Definition 3.2 Let 1 ≤ p ≤ ∞ and α ≤ 12 . The linear operator  (Tyα f )(x) =

∞

f (ξ )kα (x, y, ξ ) rα (ξ )dξ

    f ∈ Lp R+ ; rα (x)dx , x, y > 0 ,

0

(3.40) where kα (x, y, ξ ) is defined by (3.37) and rα is as in (3.2), will be called the Whittaker translation operator (of order α). y

The operator Tα is called a translation operator because it is obtained from the ordinary translation operator (T y f )(x) = f (x + y) ≡ f (ξ )dδx+y (dξ ) by replacing the measure δx+y of the product formula for the Fourier kernel by the measure kα (x, y, ξ ) of the product formula for the Whittaker function. Accordingly, many of the properties given in Proposition 3.2 below resemble the properties of the ordinary translation operator. We first establish the following lemma which gives the closed-form expression for the Whittaker translation of the power function ϑ(x) = x β . Lemma 3.4 For α, β ∈ C, we have 

∞ 0

ξ β kα (x, y, ξ ) rα (ξ )dξ = (x + y)β W α,α− 1 −β 2

 xy  x+y

(x, y > 0). (3.41)

In particular,

∞ 0

kα (x, y, ξ ) rα (ξ )dξ = 1 for α ∈ C and x, y > 0.

Proof Fix x, y > 0, and suppose that α 0, β ∈ [−M, M]) one can verify, as in the previous proof, that  sup ¯ (α,β)∈R M

∞ 0

  ξ β kα+ n (x, y, ξ ) ξ −2α−n  e−1/ξ dξ < ∞, 2

   ¯ M = (α, β)  1 ≤ 1 − Re α ≤ M − 1 , −M ≤ Re β ≤ M , being where R M 2 2 M > 0 and n ∈ N arbitrary. Both sides of (3.41) are therefore entire functions of the parameter a and the parameter β; consequently, the principle of analytic continuation gives (3.41) in the general case. By (3.17), the right-hand side of (3.41) equals 1 when β = 0. The next proposition gives the basic continuity and Lp properties of the Whittaker translation operator. We consider the weighted Lp spaces   Lp (rα ) := Lp R+ ; rα (x)dx

(1 ≤ p ≤ ∞, −∞ < α ≤ 12 )

with the usual norms 

∞

f p,α =

1/p |f (x)|p rα (x)dx

0

f ∞ ≡ f ∞,α = ess sup |f (x)|. 0 0; then, kα (x, y, ξ ) ≤ A1 (y) ξ

− 12

 (1 + ξ ) exp

ξ , − 4My

1 M

≤ x ≤ M, ξ > 0, (3.43)

3

1

1 where A1 (y) = 2A(y)y 2α− 2 (1 + y)M 2 exp( M 2 + 2y ); this estimate is obtained using (3.39), together with the inequalities x +y +ξ ≤(1+x)(1+y)(1+ξ ) and ∞ (x + y + ξ )2α−1 ≤ y 2α−1. Clearly, (3.43) implies that 0 |kα (x, y, ξ )|q rα (ξ )dξ y 1 converges absolutely and uniformly in x ∈ [ M , M], and it follows that Tα f ∈ C(R+ ). To prove the Lp -continuity of the translation, let f ∈ Cc (R+ ) and 1 < p < 1 ∞. Fix M > 0 such that the support of f is contained in [ M , M]. Interchanging the role of x and ξ in the estimate (3.43), we easily see that

 |Ty+h α f (x)| ≤ f ∞

M 1 M

kα (x, y + h, ξ ) rα (ξ )dξ

x , ≤ f ∞ A2 (y + h) x (1 + x) exp − 4M(y + h) (3.44) M where A2 (y) = A1 (y) 1/M rα (ξ )dξ . It is easy to check that the function A2 (y) is locally bounded on R+ , so it follows from (3.44) that there exists g ∈ Lp (rα ) y+h such that |Tα f (x)| ≤ g(x) for all 0 < x < ∞ and all |h| < δ (where y δ > 0 is sufficiently small). We have already proved that (Tα f )(x) ≡ (Txα f )(y) is continuous in y, hence by Lp -dominated convergence we conclude that y+h y Tα f − Tα f p,α → 0 as h → 0. As in the proof of the Lp -continuity of the ordinary translation, for general f ∈ Lp (rα ) the result is proved by taking a sequence of functions fn ∈ Cc (R+ ) which tend to f in the norm  · p,α . (d) We start by studying the behaviour of the integral Eδ kα (x, y, ξ ) rα (ξ )dξ as    x → 0, where Eδ = ξ ∈ R+  |y − ξ | > δ and δ ∈ (0, y) is some fixed constant. We have  (x + y − ξ )2 x+y+ξ −1/ξ kα (x, y, ξ )e exp − ≤C , x, ξ > 0 |x + y − ξ | 8xyξ (3.45) − 21



3.3 Whittaker Translation

69

(where C < ∞ is independent of x and ξ ). This follows by combining (3.39) 2 with the boundedness of the function |t|e−t and the inequality (x + y + δ 2α−1 2α−1 ξ) ≤y . Furthermore, if x ≤ 2 and ξ ∈ Eδ , the inequalities     2ξ x+y+ξ   ≤ 1 + 4ξ = 1 + |x + y − ξ | x+y−ξ δ   (x + y − ξ )2 1 1 (y − ξ )2 exp − ≤ exp − − 8xyξ 4y 4ξ 4δyξ lead us to kα (x, y, ξ )ξ −2α e−1/ξ ≤ C ξ −2α (1 + ξ ) exp

 −

1 (y − ξ )2 , − 4ξ 4δyξ

x ≤ 2δ , ξ ∈ Eδ .

(3.46) Since the right-hand side of (3.46) clearly belongs to L1 (Eδ ), the dominated convergence theorem is applicable, and letting x → 0 in (3.45) we find that  lim

x↓0 Eδ

   lim kα (x, y, ξ ) rα (ξ )dξ = 0. kα (x, y, ξ ) rα (ξ )dξ = Eδ x↓0

(3.47)

Let us now fix ε > 0, and write Vδ = R+ \ Eδ . Since f is continuous, we can choose δ > 0 such that |f (ξ ) − f (y)| < ε for all ξ ∈ Vδ . By this choice of δ and the positivity of kα (x, y, ξ ), we find     ∞   y   (T f )(x) − f (y) =  kα (x, y, ξ ) f (ξ ) − f (y) rα (ξ )dξ  α  0          ≤ kα (x, y, ξ ) f (ξ ) − f (y) rα (ξ )dξ  + ε kα (x, y, ξ )rα (ξ )dξ Eδ

≤ 2f ∞

Vδ

 kα (x, y, ξ ) rα (ξ )dξ + ε. Eδ

  y By (3.47), it follows that lim supx↓0 (Tα f )(x)−f (y) ≤ ε. Since ε is arbitrary, the proof of part (d) is finished. M (e) We begin by claiming that for each M > 0 we have 0 kα (x, y, ξ ) rα (ξ )dξ → 0 as x → ∞. Indeed, if x > 2M and ξ ≤ M, combining (3.45) with the inequalities 2ξ 2M x+y+ξ =1+ ≤1+ , |x + y − ξ | x+y−ξ y+M   (x + y − ξ )2 1 M 1 exp − − − ≤ exp , 8xyξ 4y 4ξ 4yξ

(3.48) (3.49)

70

3 The Whittaker Convolution

we see that kα (x, y, ξ )ξ

−2α −1/ξ

e

≤Cξ

−2α

M 1 − , − 4ξ 4yξ

 exp

x ≥ 2M, ξ ≤ M,

where the right-hand side is integrable on the interval (0, M]; hence, if we let x → ∞ in (3.45), by dominated convergence we obtain  lim

x→∞ 0

M



M

kα (x, y, ξ ) rα (ξ )dξ = 0

 lim kα (x, y, ξ ) rα (ξ )dξ = 0.

x→∞

(3.50) Let f ∈ Bb (R+ ) be such that limx→∞ f (x) = 0, and let ε > 0. Choose M such that |f (x)| < ε for all x ≥ M. Then  |(Tyα f )(x)| ≤ f ∞  ≤ f ∞

M

 kα (x, y, ξ ) rα (ξ )dξ + ε

0

∞

kα (x, y, ξ ) rα (ξ )dξ M

M

kα (x, y, ξ ) rα (ξ )dξ + ε,

0 y

so that (3.50) yields lim supx→∞ |(Tα f )(x)| ≤ ε, where ε is arbitrary. We observe that, as a consequence of Proposition 3.2, the Whittaker translation (3.40) with the convention that (Txα f )(0) = (T0α f )(x) = f (x) for all x, satisfies the properties   + Tyα Cb (R+ 0 ) ⊂ Cb (R0 )

  + Tyα C0 (R+ 0 ) ⊂ C0 (R0 )

and

(y ≥ 0), (3.51)

as well as the obvious symmetry property (Tyα f )(x) = (Txα f )(y)

(x, y ≥ 0).

It is also easy to check that the Whittaker translation is symmetric with respect to 1 the measure rα (x)dx, in the sense that for f, g ∈ Cb (R+ 0 ) ∩ L (rα ) we have 

∞ 0

 (Tyα f )(x)g(x) rα (x)dx

∞

= 0

f (x)(Tyα g)(x) rα (x)dx.

(3.52)

3.4 Index Whittaker Transforms

71

3.4 Index Whittaker Transforms The integral transform determined by the generator (3.2) of the Shiryaev process, which we will call the index Whittaker transform (of order α), is defined by 

∞

(Wα f )(τ ) :=

f (y) Wα,iτ (y) rα (y)dy,

τ ≥ 0.

(3.53)

0

(This is a modified form of the index Whittaker transform defined in [180].) This integral transform is a fundamental tool for studying the Whittaker convolution (defined in the next section), since it is the object which will play a role similar to that of the Hankel transform in the construction of the Kingman convolution. As noted in Example 2.4, the spectral expansion of the differential operator Aα yields the following theorem: Proposition 3.3 For α < isometric isomorphism

1 2,

the index Whittaker transform (3.53) defines an

  Wα : L2 (rα ) −→ L2 R+ ; ρα (τ )dτ ,   2 where ρα (τ ) := π −2 τ sinh(2πτ ) 12 − α + iτ  , whose inverse is given by (Wα−1 ϕ)(x)



∞

=

ϕ(τ ) Wα,iτ (x) ρα (τ )dτ,

(3.54)

0

the convergence of the integrals (3.53) and   (3.54) being understood with respect to the norm of the spaces L2 R+ ; ρα (τ )dτ and L2 (rα ), respectively. Moreover, the differential operator (3.2) is connected with the index Whittaker transform via the identity

     Wα (−Aα f ) (τ ) = τ 2 + ( 12 − α)2 · Wα f (τ ),

f ∈ D(2) α ,

where 

 2

+ 2

D(2) α := u ∈ L (rα )  u, u ∈ ACloc (R ), Aα u ∈ L (rα ), (pα u )(0) = 0       = u ∈ L2 (rα )  τ 2 + ( 12 − α)2 · Wα f (τ ) ∈ L2 R+ ; ρα (τ )dτ . We note that, for α
0, α < 12 , τ ≥ 0).

72

3 The Whittaker Convolution

Applying the inverse Whittaker transform (3.54), we find that for x, y, ξ > 0 and α < 12 we have 

∞

kα (x, y, ξ ) =

Wα,iτ (x) Wα,iτ (y) Wα,iτ (ξ ) ρα (τ )dτ,

(3.55)

0

where the integral on the right-hand side converges absolutely, as can be verified   1 using the asymptotic forms (3.18) and ( 12 − α + iτ ) ∼ (2π) 2 τ −α exp(− πτ 2 ), τ → +∞ (cf. [145, Equation 5.11.9]). The product formula (3.36) ensures that for each fixed α < 12 the normalized Whittaker functions Wα,ν (·) (ν ∈ C) are solutions of the functional equation 

∞

ω(x)ω(y) =

ω(ξ ) kα (x, y, ξ ) rα (ξ )dξ

(x, y > 0).

(3.56)

0

Using the representation (3.55) for the kernel of the product formula, one can prove a lemma which rules out the existence of other nontrivial solutions for this functional equation: Lemma 3.5 Let α < 12 and ν ≥ 0. Suppose that the function ω(x) is such that there exists C > 0 for which   ω(x) ≤ C W α,ν (x)

for a.e. x > 0,

(3.57)

and that ω(x) is a nontrivial solution of the functional equation (3.56). Then ω(x) = W α,ρ (x) for some ρ ∈ C with |Re ρ| ≤ ν. Proof We begin by noting that Aα,x kα (x, y, ξ ) = Aα,y kα (x, y, ξ )  ∞   Wα,iτ (x) W α,iτ (y) W α,iτ (ξ ) τ 2 + ( 12 − α)2 ρα (τ )dτ, =− 0

(3.58) where Aα,x and Aα,y denote the differential operator (3.2) acting on the variable x and y respectively. The identity (3.58) is obtained via differentiation of (3.55) under the integral sign, which is admissible because the differentiated integrals converge absolutely and locally uniformly, as can be verified in a straightforward way using the identity   d W α,ν (y) = ν 2 − ( 12 − α)2 W α−1,ν (y) dy

(3.59)

3.4 Index Whittaker Transforms

73

(which follows from (3.15)) and the asymptotic expansion (3.18). (Recall also that, by 3.1, the function W α,ν (·) satisfies the differential equation Aα u =  2Proposition ν − ( 12 − α)2 u.) Now, assuming that the right-hand side of the functional equation (3.56) can also be differentiated under the integral sign, it follows from (3.58) that     Aα,x ω(x) ω(y) = Aα,y ω(y) ω(x)

(x, y > 0).

(3.60)

Here the possibility of interchanging derivative and integral follows again from the locally uniform convergence of the differentiated integrals, which can be straightforwardly checked using (3.57), the identity   y+ξ −x ∂kα (x, y, ξ ) = kα+ 1 (x, y, ξ ) − x −2 + (1 − 2α)x −1 kα (x, y, ξ ) 2 2 ∂x 2x yξ (3.61) (which is a consequence of (3.23)) and the upper bound (3.39) for the function kα (x, y, ξ ). Notice that (3.60) holds for arbitrary values of x and y. Therefore, we must have Aα,y ω(y) Aα,x ω(x) = =λ ω(x) ω(y) for some λ ∈ C, meaning that ω(x) is a solution of the Sturm–Liouville equation   Aα ω(x) = ρ 2 − ( 12 − α)2 ω(x), where ρ is the principal square root of λ + ( 12 − α)2 . Consequently, the function ω(x) is a linear combination of the functions W α,ρ (x) and ∞  ( 12 − α + ρ)k −( 1 −α+ρ+k) 1 1 x 2 x α e 2x Mα,ρ ( x1 ), ≡ M α,ρ (x) := (1 + 2ρ + k)k! (1 + 2ρ) k=0

where Mα,ρ (x) is the Whittaker function of the first kind [145, §13.14]. (Here we are 1 Mα,ρ (z) using the well-known fact that the Whittaker functions Wα,ρ (z) and (1+2ρ)  1 d 2u are, for 2 − α + ρ = 0, −1, −2, . . ., two linearly independent solutions of dz2 + − 1/4−ρ 2  1 α u = 0. Recall also that the vector space of solutions of a Sturm– 4 + z + z2 Liouville equation is two-dimensional.) However, it follows from the limiting forms for the Whittaker M function [145, Equation 13.14.20] that M α,ρ (x) is, for all ρ ∈ C, unbounded as x goes to zero, and this violates (3.57). In addition,  the limiting forms (3.10), (3.12) for the Whittaker function show that W α,ρ (x) ≤ C W α,ν (x) holds if and only if |Re ρ| ≤ ν. Therefore, we must have ω(x) = W α,ρ (x) for ρ belonging to the strip |Re ρ| ≤ ν.

74

3 The Whittaker Convolution

We proceed with the definition of the index Whittaker transform of finite complex measures, which will allow us to interpret (3.53) as the index Whittaker transform of an absolutely continuous measure with density f (·)rα (·): Definition 3.3 Let μ ∈ MC (R+ 0 ). The index Whittaker transform of the measure μ is the function defined by the integral  $ μ(λ) ≡ $ μ(λ; α) =

R+ 0

Wα, λ(y) μ(dy),

λ ≥ 0.

(3.62)

For convenience this transformation is regarded as a function of λ = τ 2 +( 12 −α)2  (recall that λ = ( 12 − α)2 − λ). Before we state the basic properties of the index Whittaker transform of finite measures, we prove an auxiliary result of independent interest: an integral representation for the normalized Whittaker function W α,ν (y) which we call the Laplace-type representation for Wα,ν (y) because it is of the same form as the Laplace representation for the characters of Sturm–Liouville hypergroups, cf. [210, (4.7)–(4.8)]. The integral representation below cannot be found in standard references on integration formulas for special functions such as [155–157]. Theorem 3.2 The normalized Whittaker W function admits the integral representation  ∞  ∞ νs Wα,ν (y) = e ηα,y (s)ds = 2 cosh(νs)ηα,y (s)ds (α, ν ∈ C, y > 0), −∞

0

(3.63) where ηα,y is the function defined by 1

1

ηα,y (s) := 2−1−α π − 2 y − 2 +α exp



  s   s  1 1 1 1 − cosh2 D2α 2 2 y − 2 cosh y 2y 2 2

and Dμ (z) is the parabolic cylinder function. Proof Only the first equality in (3.63) needs proof. Let us temporarily assume that ν ≥ 0 and −∞ < α < 12 , and let ξ > 0. We begin by noting the identity  ξ



∞

1−2α

exp 0

1 −2α ξ 2y − y − W α,ν (y) dy 4 y

= 22−2α K2ν (ξ )  ∞   s  ds, eνs exp −ξ cosh = 2−2α 2 −∞

(3.64)

3.4 Index Whittaker Transforms

75

which is a consequence of integrals 2.4.18.12 in [155] and 2.19.4.7 in [157]. To deduce the theorem from this identity, we will use the injectivity property of the Laplace transform, after rewriting the right-hand side as an iterated integral. To that end, we point out that, according to integral 2.11.4.4 in [156], for s, ξ > 0 we have   s  ξ 2α−1 exp −ξ cosh 2    ∞   1  s  1 1 1 ξ 2y α−1 − 12 2 s − 2 −α − − cosh =2 π exp − D2α 2 2 y 2 cosh y dy. 4 2y 2 2 0 Substituting in (3.64) and interchanging the order of integration (which is valid because, as noted in Remark 3.3(c), we have Dμ (y) > 0 for y > 0 and μ < 1, and therefore the iterated integral has positive integrand), we find that 



ξ 2y 1 −2α exp − W α,ν (y) dy = − y 4 y 0   ∞ 1 ξ 2 y − 1 −α exp − = 2−1−α π − 2 y 2 × 4 0    ∞  s   s  1 1 1 cosh2 D2α 2 2 y − 2 cosh ds dy. exp νs − × 2y 2 2 −∞ ∞

Given that the Laplace transform is one-to-one, this identity yields e

− y1

1

1

W α,ν (y) = 2−1−α π − 2 y − 2 +α

   s   s  1 1 1 cosh2 D2α 2 2 y − 2 cosh ds, exp νs − 2y 2 2 −∞



∞

finishing the proof for the case −∞ < α < 12 , ν ∈ R. ∞ To extend (3.63) to all α, ν ∈ C, it is enough to show that −∞ eνs ηα,x (s)ds is an entire function of the parameter α and the parameter ν (so that the usual analytic continuation argument can be applied). For t > 0 and α ∈ C with Re α ≤ 0, the integral representation (3.20) gives 2   − t4 2 Re α   ∞   t 2s α  −s − 12 −α  D2α (t) = e e s 1 + 2 ds  t |( 12 − α)|  0 t2

≤

=

e− 4 t 2 Re α |( 12



− α)|

( 12 − Re α) |( 12

− α)|

∞

1

e−s s − 2 −Re α ds

0 t2

e− 4 t 2 Re α .

76

3 The Whittaker Convolution (3)

Furthermore, for each n ∈ N0 we have D2α+n (t) = Qn,α (t)D2α (t) + (j ) Q(4) n,α (t)D2α−1 (t) (cf. proof of Theorem 3.1), being Qn,α (·) polynomials of degree at most n whose coefficients are continuous functions of α. It is easy to see that (j ) |Qn,α (t)| ≤ Cn (α) (1 + t n ) for some function Cn (α) that depends continuously on α ∈ C and, consequently,  

 D2α+n (t) ≤ Cn (α) (1 + t n ) D2α (t) + D2α−1 (t) ≤ Cn (α) e

2

− t4

t

2 Re α

(1 + t ) n

! (1 − Re α) −1 t + , |(1 − α)| |( 12 − α)|

( 12 − Re α)

  t2 sup D2α+n (t) ≤ CM,n e− 4 (t −2M−1 + t n ),

(3.65)

|α|≤M Re α≤0

where M > 0 and n ∈ N0 are arbitrary and the constant CM,n depends on M and n. Using (3.65), we see that  sup (α,ν)∈RM

         exp νs − 1 cosh2 s D2α+n 2 12 y − 21 cosh s  ds < ∞,  2y 2 2  −∞ ∞

   where RM = (α, ν)  |α| ≤ M, Re α ≤ 0, |ν| ≤ M . Applying the standard results on the analyticity of parameter-dependent integrals (e.g. [135]), we obtain ∞ the entireness in α and in ν of −∞ eνs ηα,x (s)ds, completing the proof. It is worth observing that ηα,y (s) ≥ 0

for all α ≤ 12 , y > 0

and so it follows from Theorem 3.2 that |Wα,ν (y)| ≤ Wα,ν0 (y)

whenever α ≤ 12 , |Re ν| ≤ ν0 (ν0 ≥ 0).

(3.66)

Together with the identity Wα, 1 −α (y) = 1 (see (3.17)), this implies that 2

|Wα,ν (y)| ≤ 1

for all y > 0, α ≤ 12 , ν in the strip |Re ν| ≤

1 2 −α.

(3.67)

3.4 Index Whittaker Transforms

77

We are now ready to establish some important facts on the index Whittaker transform (3.62): Proposition 3.4 For α < 12 , the index Whittaker transform $ μ ≡ $ μ(·; α) of μ ∈ + MC (R0 ) has the following properties: {μj } ⊆ (i) $ μ is uniformly continuous on R+ 0 . Moreover, if a family of measures  + MC (R0 ) is such that the family of restricted measures μj |R+ is tight and uniformly bounded, then {% μj } is uniformly equicontinuous on R+ 0.

1 ) is uniquely determined by $ μ (ii) Each measure μ ∈ MC (R+ | ( − α)2 , ∞ . 0 2

+ (iii) If {μn } is a sequence of measures belonging to M+ (R+ 0 ), μ ∈ M+ (R0 ), and w

μn −→ μ, then μ %n −−−−→ $ μ

uniformly on compact sets.

n→∞

(iv) If {μn } is a sequence of measures belonging to M+ (R+ 0 ) whose index Whittaker transforms are such that μ %n (λ) −−−−→ f (λ) n→∞

pointwise in λ ≥ 0

(3.68)

for some real-valued function f which is continuous at a neighborhood of zero, w then μn −→ μ for some measure μ ∈ M+ (R+ μ ≡ f. 0 ) such that $ Proof (i) Let us prove the second statement, which implies the first. ε > 0. By the  Fix 1 tightness assumption, we can choose M > 0 such that μj (0, M )∪(M, ∞) < ε. Moreover, noting that |Re λ | ≤ 12 − α, it is easily seen that | exp( λ1 s) − 1

exp( λ2 s)| ≤ | λ1 − λ2 |s e( 2 −α)s for all s, λ1 , λ2 ≥ 0 and, consequently, from Theorem 3.2 we get  Wα,

λ1 (y) − Wα,

  λ2 (y) ≤ |

 λ1

−

λ2 |

∞

−∞

s e( 2 −α)s ηα,y (s) ds, 1

(3.69)

where the integral on the right-hand side converges uniformly with respect to y in compact subsets of R+ and is therefore a continuous function of y > 0. By continuity of λ → λ , we can choose δ > 0 such that |

λ1

−

λ2 |


0.

In fact, by continuity this equality holds for all t ≥ 0, because the integrals converge uniformly with respect to t ≥ 0. Consequently, 

∞ 0

e−ys μ1 (dy) =



∞

e−ys μ2 (dy)

for all s ≥ 0.

0

Since the measures μj are uniquely determined by their Laplace transforms [102, Theorem 15.6], it follows that μ1 = μ2 . (iii) Since Wα, λ(·) is continuous and bounded, the pointwise convergence μ %n (λ) → $ μ(λ) follows from the definition of weak convergence. For the

3.4 Index Whittaker Transforms

79 w

restricted measures, we clearly have μn |R+ −→ μ|R+. Using the well-known Prokhorov’s theorem which states that a family of measures on a complete separable metric space is relatively compact in the weak topology if and only if it is tight [102, Theorem 13.29], we see that {μn |R+} is tight and therefore (by part (i)) {% μn } is uniformly equicontinuous. Invoking a general result which asserts that if {gn } is a uniformly equicontinuous sequence of functions then gn → g pointwise implies that gn → g uniformly on compact sets [102, Lemma 15.22], we conclude that the convergence μ %n → $ μ is uniform on compact sets. (iv) We only need to show that the sequence {μn } is tight. Indeed, if {μn } is tight, then Prokhorov’s theorem yields that for any subsequence {μnk } there exists a w further subsequence {μnkj } and a measure μ ∈ M+ (R+ 0 ) such that μnkj −→ μ. Then, due to part (iii) and to (3.68), we have $ μ(λ) = f (λ) for all λ ≥ 0, which implies (by part (ii)) that all such subsequences have the same weak limit; consequently, the sequence μn itself converges weakly to μ. To prove the tightness, take ε > 0. Since f is continuous at a neighborhood  2δ   of zero, we have 1δ 0 f (0) − f (λ) dλ −→ 0 as δ ↓ 0; therefore, we can choose δ > 0 such that 1 δ



2δ



 f (0) − f (λ) dλ < ε.

0

Next we observe that, as a consequence of (3.12) and the dominated conver  2δ  gence theorem, we have 0 1 − Wα, λ(y) dλ −→ 2δ as y → ∞, meaning that we can pick M > 0 such that  0

2δ

  1 − Wα, λ(y) dλ ≥ δ

for all y > M.

By our choice of M and Fubini’s theorem,   1 μn [M, ∞) = δ ≤ ≤ =

1 δ 1 δ 1 δ



∞

δ μn (dy)  

M

∞  2δ

M

0

∞  2δ

0



0

0 2δ

  1 − Wα, λ(y) dλ μn (dy)   1 − Wα, λ(y) dλ μn (dy)

  μ %n (0) − μ %n (λ) dλ.

80

3 The Whittaker Convolution

Hence, using the dominated convergence theorem,  2δ   1   μ %n (0) − μ lim sup μn [M, ∞) ≤ lim sup %n (λ) dλ δ n→∞ 0 n→∞    1 2δ %n (0) − μ = lim μ %n (λ) dλ δ 0 n→∞   1 2δ  f (0) − f (λ) dλ < ε = δ 0

due to the choice of δ. Since ε is arbitrary, we conclude that {μn } is tight, as desired. Remark 3.4 Parts (iii) and (iv) of the proposition above show that the following analogue of the Lévy continuity theorem holds for the index Whittaker transform: the index Whittaker transform is a topological homeomorphism between P(R+ 0) with the weak topology and the set $ P of index Whittaker transforms of probability measures with the topology of uniform convergence in compact sets.

3.5 Whittaker Convolution of Measures The parameter α < 12 will be fixed throughout the rest of this chapter. Motivated by the connection between the Bessel product formula and the Kingman convolution, we now define the Whittaker convolution in order that the convolution of Dirac measures is the kernel of the product formula (3.36) for the normalized Whittaker W function: Definition 3.4 Let μ, ν ∈ MC (R+ 0 ). The measure μ  ν defined by α



 R+ 0

f (x) (μ  ν)(dx) = α

 (Tyα f )(x) μ(dx)ν(dy),

+ R+ 0 R0

f ∈ Cb (R+ 0)

is called the Whittaker convolution (of order α) of the measures μ and ν. It clearly follows from this definition (and Definition 3.2) that for x, y > 0 the Whittaker convolution of Dirac measures δx  δy is the absolutely continuous α

measure defined by (δx  δy )(dξ ) = kα (x, y, ξ ) rα (ξ )dξ. α

Due to (3.41) and (3.38), the Whittaker convolution of two probability measures μ, ν ∈ P(R+ 0 ) is also a probability measure. Furthermore, Proposition 3.6 below shows that the Whittaker convolution is commutative, associative and such that (c1 μ1 + c2 μ2 )  ν = c1 (μ1  ν) + c2 (μ2  ν) for μ1 , μ2 , ν ∈ P(R+ 0 ) and c1 , c2 ∈ C. α

Consequently:

α

α

3.5 Whittaker Convolution of Measures

81

Proposition 3.5 The vector space MC (R+ 0 ) (with usual addition and scalar multiplication), endowed with the convolution multiplication , is a commutative algebra α

over C whose multiplicative identity is the Dirac measure δ0 .  y Since R+ f (ξ )(δx  δy )(dξ ) = (Tα f )(x), the fact that kα (x, y, ξ ) is strictly α

0

positive for x, y, ξ > 0 yields that supp(δx  δy ) = R+ 0 for all x, y > 0, in α

sharp contrast with the compactness axiom H6 which is part of the definition of a hypergroup (Definition 2.4). It is worth mentioning that positive product formulas which lead to convolution operators not satisfying the hypergroup requirements on supp(δx  δy ) have also been found for certain families of orthogonal polynomials α

[39]. We now state the fundamental connection between the Whittaker convolution and the index Whittaker transform (3.62). (The analogous trivialization property for the Kingman convolution was stated in Proposition 2.20.) Proposition 3.6 Let μ, μ1 , μ2 ∈ MC (R+ 0 ). We have μ = μ1  μ2 if and only if α

$ μ(λ) = μ %1 (λ) μ %2 (λ)  y Proof In view of (3.36), we have Tα Wα,

for all λ ≥ 0.  λ

(x) = Wα, λ(x) Wα, λ(y), hence

 μ 1  μ2 (λ) = α

 =  =

R+ 0

R+ 0

Wα, λ(x) (μ1  μ2 )(dx) α

 R+ 0



Tyα Wα,

 λ

(x) μ1 (dx)μ2 (dy)



+ R+ 0 R0

Wα, λ(x) Wα, λ(y) μ1 (dx)μ2 (dy) = μ %1 (λ)% μ2 (λ),

λ ≥ 0.

This proves the “only if” part, and the converse follows from the uniqueness property in Proposition 3.4(ii). Remark 3.5 It was noted above that the Whittaker convolution cannot be interpreted as a particular case of the axiomatic framework of hypergroups. One can show that, in addition, the Whittaker convolution also does not constitute an example of an Urbanik convolution algebra (cf. Definition 2.3), because the homogeneity axiom U5 fails to hold for the Whittaker convolution. Indeed, let a ∈ R+ \ {1} and assume that the identity a (δx  δy ) = a (δx )  α

a (δy ) holds for all x, y > 0. Then   Wα, λ(ax) Wα, λ(ay) =  a (δx )(λ)·  a (δy )(λ) = [a (δx )  a (δy )]% (λ) α

α

82

3 The Whittaker Convolution

= [a (δx  δy )]%(λ) α  = Wα, λ(aξ ) (δx  δy )(dξ ) α

 =

Wα, λ(aξ ) kα (x, y, ξ ) rα (ξ )dξ.

Therefore, x → Wα, λ(ax) is, for each λ ≥ 0, a solution of the functional equation (3.56). However, it follows from Lemma 3.5 that any bounded solution of this functional equation is of the form Wα,σ (x) for some σ ∈ C with |Re σ | ≤ 12 − α. One can check (using e.g. the asymptotic expansion (3.13), see also (3.74) below) that the identity Wα, λ(ax) ≡ Wα,σ (x) does not hold for any pair (λ, σ ) ∈ R+ 0 × C, so we obtain a contradiction. As advertised in the introduction to this chapter, the Whittaker convolution  can α

be extended to a more general family of convolutions associated with the differential d d2 operators γ x 2 dx 2 + γ (c + 2(1 − α)x) dx (γ , c > 0). We will see in Remark 3.11 that this is achieved via the rescaled convolutions     μ  ν := c (1/c μ)  (1/c ν) = μ  ν if c = 1 . α,c

α

α

3.5.1 Infinitely Divisible Distributions The set of -infinitely divisible distributions is defined as α

   + α n  Pα,id = μ ∈ P(R+ , 0 ) for all n ∈ N there exists νn ∈ P(R0 ) such that μ = (νn ) (3.72)

where (νn )α n denotes the n-fold Whittaker convolution of νn with itself. Lemma 3.6 Let μ ∈ Pα,id . Then 0 < $ μ(λ) ≤ 1 for all λ ≥ 0. Moreover, μ has no nontrivial idempotent divisors, i.e., if μ = ϑ  ν (with ϑ, ν ∈ P(R+ 0 )) where ϑ is α

idempotent with respect to the Whittaker convolution (that is, it satisfies ϑ = ϑ  ϑ), α

then ϑ = δ0 .

Proof The inequality $ μ(λ) ≤ 1 is obvious from (3.67). The positivity can be proved as follows (the argument is similar to that of [166, Lemma 7.5]): for every n ∈ N μ(λ) = ν$n (λ)n . We can define there exists νn such that $  φ(λ) := lim ν$n (λ) = n→∞

1, if $ μ(λ) = 0, 0, if $ μ(λ) = 0

3.5 Whittaker Convolution of Measures

83

and then (by the continuity of $ μ, Proposition 3.4(i)) we have φ(λ) = 1 in a neighbourhood of 0. Therefore (Proposition 3.4(iv)) φ is the index Whittaker transform of a probability measure; in particular, it is continuous on R+ 0 , so we conclude that φ ≡ 1. Thus $ μ has no zeros, and by continuity it follows that $ μ(λ) > 0 for all λ.  2 Assume that μ = ϑ  ν with ϑ idempotent. Then $ ϑ (λ) = $ ϑ (λ) for all λ, and α consequently $ ϑ(λ) only takes the values 0 and 1. However, $ μ(λ) = $ ϑ(λ)$ ν(λ) = 0; hence $ ϑ(λ) = 1 for all λ, i.e., ϑ = δ0 . The first part of the lemma shows that the index Whittaker transform of any measure μ ∈ Pα,id is of the form μ(λ) = e−ψμ (λ) , $ where ψμ (λ) (λ ≥ 0) is a positive continuous function such that ψμ (0) = 0, which we shall call the log-Whittaker transform of μ. The next result shows that the logWhittaker transform of an infinitely divisible distribution grows at most linearly: Proposition 3.7 Let μ ∈ Pα,id . Then ψμ (λ) ≤ Cμ (1 + λ)

for all λ ≥ 0

for some constant Cμ > 0 which is independent of λ. The proof relies on the following lemma: Lemma 3.7 The normalized Whittaker W function satisfies the inequality   1 − Wα,ν (y) ≤ ( 12 − α)2 − ν 2 y

for each y ≥ 0 and ν ∈ [0, 12 − α] ∪ iR.

  Proof By Proposition 3.1, Wα,ν (·) solves the equation −Aα u = ( 12 − α)2 − ν 2 u, and thus we have !   d d − ξ 2−2α e−1/ξ Wα,ν (ξ ) = ( 12 − α)2 − ν 2 ξ −2α e−1/ξ Wα,ν (ξ ). dξ dξ Recalling that W α,ν (x) satisfies the boundary conditions given in (3.8), after integrating both sides between 0 and y and then between 0 and x we obtain    1 2 2 1 − Wα,ν (x) = ( 2 − α) − ν

x 0



y

y 2α−2 e1/y

ξ −2α e−1/ξ Wα,ν (ξ ) dξ dy.

0

(3.73)

84

3 The Whittaker Convolution

Using (3.67) and the inequality ( yξ )2−2α ≤ 1 (which holds for 0 < ξ ≤ y due to the assumption α < 12 ), we thus find that   1 − Wα,ν (x) ≤ ( 12 − α)2 − ν 2   ≤ ( 12 − α)2 − ν 2

 

  = ( 12 − α)2 − ν 2 x

x

 y 2α−2e1/y

0 x

ξ 2α e−1/ξ dξ dy

0



y

e1/y

0

y

ξ −2 e−1/ξ dξ dy

0

as required. Proof of Proposition 3.7 Let νn ∈ P(R+ 0 ) be defined as in (3.72), so that 1 ν$n (λ) ≡ exp(− n ψμ (λ)). Due to the inequality 1 − e−τ ≤ τ (τ ≥ 0) and the fact that limn n(1 − e−k/n ) = k for each k ∈ R, we have   n 1 − ν$n (λ) ≤ ψμ (λ) for all n ∈ N,

  lim n 1 − ν$n (λ) = ψμ (λ).

n→∞

Pick λ1 > 0. It follows from the asymptotic expansion (3.13) (which can be differentiated term by term, cf. [145, §2.1(iii)]) that    dn W α,ν (y) −−−→ (−1)n 12 −α+ν n 12 −α−ν n n y→0 dy In particular, limy→0 d dy Wα,

(y) ≤

λ1

1 1 − Wα, λ1

− λ21

d dy W α,

λ

(n = 0, 1, 2, . . .).

(3.74)

(y) = −λ, hence there exists ε > 0 such that

for all 0 < y ≤ ε, and then we have 

1 λ1(y) = − λ1



y 0

d Wα, dx

(x) dy ≥

λ1

y 2

for all 0 ≤ y ≤ ε. (3.75)

Using also Lemma 3.7, we get  n

[0,ε)

  1 − Wα, λ(y) νn (dy) ≤ λn

 [0,ε)

y νn (dy)





 (y) νn (dy)

≤

2λn λ1

≤

 2λ 2λn  1 − ν$n (λ1 ) ≤ ψμ (λ1 ). λ1 λ1

[0,ε)

1 − Wα,

λ1

(3.76)

3.5 Whittaker Convolution of Measures

85

Next, from the asymptotic expansion given in (3.18) we easily see that there exists λ2 > 0 such that  Wα,

 1 (y) ≤ 2

for all y ≥ ε

λ2

and using (3.67), we obtain  n

[ε,∞)

  1 − Wα, λ(y) νn (dy) ≤ 2n

 [ε,∞)

 ≤ 4n

νn (dy) 

[ε,∞)

1 − Wα,

 (y) νn (dy)

(3.77)

λ2

  ≤ 4n 1 − ν$n (λ2 ) ≤ 4ψμ (λ2 ). Combining (3.76) and (3.77), one sees that  n(1 − ν$n (λ)) = n

where Cμ = max (3.78) yields

R+ 0



 1 − Wα,

 2λ (y) νn (dy) ≤ ψμ (λ1 )+4ψμ (λ2 ) ≤ Cμ (1 + λ), λ1 (3.78)

λ2

2 λ1 ψμ (λ1 ), 4ψμ (λ2 )



. Taking the limit n → ∞ in the inequality

ψμ (λ) ≤ Cμ (1 + λ) which completes the proof.

3.5.2 Lévy–Khintchine Type Representation One can prove that an analogue of the classical Lévy–Khintchine formula holds for the log-Whittaker transforms of -infinitely divisible distributions. To establish this α

result, one needs to adapt the notions of compound Poisson and Gaussian measures to the context of the Whittaker convolution algebra: Definition 3.5 Let μ ∈ P(R+ 0 ) and a > 0. The measure eα (aμ) defined by eα (aμ) = e−a

∞  an n=0

n!

μα n

(the infinite sum converging in the weak topology) is said to be the -compound α

Poisson measure associated with aμ.

86

3 The Whittaker Convolution

This definition is completely analogous to that of the classical compound Poisson measure. From the definition it immediately follows that eα (aμ) ∈ P(R+ 0 ). Moreover, its index Whittaker transform can be easily deduced using Proposition 3.6: −a e α (aμ)(λ) = e

∞  an n=0

n!

α n μ& (λ) = e−a

∞    n an  μ(λ) − 1) . $ μ(λ) = exp a($ n! n=0

Since eα ((a+b)μ) = eα (aμ)eα (bμ), every -compound Poisson measure belongs α

to Pα,id .

α

Definition 3.6 A measure μ ∈ P(R+ 0 ) is called a -Gaussian measure if μ ∈ Pα,id α

and μ = eα (aν)  ϑ α



a > 0, ν ∈ P(R+ 0 ), ϑ ∈ Pα,id



⇒

ν = δ0 .

Remark 3.6 This definition is similar to the definition of Gaussian measures on locally compact abelian groups as in [146, Chapter IV]. It is analogous with the classical notion of a Gaussian measure on R by the following result:  a k ∗k Let e(aν) := e−a ∞ k=0 k! μ , where ∗ is the ordinary convolution. For a measure μ ∈ P(R), the following conditions are equivalent:  2 (i) μ(dx) = √ 1 exp − (x−c) dx for some c ∈ R and σ > 0; 2σ 2 2π σ (ii) μ is infinitely divisible, and if μ = e(aν) ∗ ϑ (with a > 0, ν ∈ P(R) and ϑ ∈ P(R) infinitely divisible), then ν = δ0 . (The implication (i) ⇒ (ii) is a consequence of the Lévy-Cramer theorem [125, §III.1] which asserts that if μ ∈ P(R) is a Gaussian measure (in the sense of (i)) and μ = μ1 ∗ μ2 with μ1 , μ2 ∈ P(R), then μ1 and μ2 are also Gaussian measures. The converse implication follows from the fact that if an infinitely divisible μ ∈ P(R) is such that μ = e(aν)∗ϑ implies ν = δ0 , then the Lévy measure in the classical Lévy– Khintchine formula must be the zero measure, which means that μ is a Gaussian measure; see the discussion after Equation (16.8) in [102].) The Lévy–Khintchine type representation for measures μ ∈ Pα,id reads as follows: Theorem 3.3 The log-Whittaker transform of a measure μ ∈ Pα,id can be represented in the form  ψμ (λ) = ψγ (λ) +

R+

  1 − Wα, λ(x) ν(dx),

(3.79)

3.5 Whittaker Convolution of Measures

87

where ν is a σ -finite measure on R+ which is finite on (ε, ∞) for all ε > 0 and such that    1 − Wα, λ(x) ν(dx) < ∞ R+

and γ is a -Gaussian measure with log-Whittaker transform ψγ (λ). Conversely, α

each function of the form (3.79) is a log-Whittaker transform of some μ ∈ Pα,id . This theorem is a particular case of the Lévy–Khintchine type theorem for stochastic convolutions stated in Proposition 2.18. To keep this chapter more self-contained, a sketch of the proof is presented below; it relies on an algebraictopological technique which has been earlier used to establish a canonical representation for infinitely divisible distributions on locally compact abelian groups [146, 147]. Proof Let μ ∈ Pα,id , let ∞ > a1 > a2 > . . . with lim an = 0, and let In = [0, an ), Jn = [an , ∞). Consider the set Qμ of all proper divisors of μ of the form eα (π) such that π(I1 ) = 0. (A measure ν ∈ P(R+ 0 ) is said to be a proper divisor of μ ∈ Pα,id if ν ∈ Pα,id and μ = ν  θ for some θ ∈ Pα,id .) Using α

Proposition 3.4 and the properties of the normalized Whittaker W function, one can prove (see [198, Corollary 1]) that the set D(P) of all divisors (with respect to the Whittaker convolution) of measures ν ∈ P is relatively compact whenever   1− P ⊂ P(R+ 0 ) is relatively compact. This, in turn, implies that supeα (π)∈Qμ R+ 0   Wα, λ(x) π(dx) < ∞ and therefore, by compactness of the set of proper divisors of μ, there exists a divisor μ1 = eα (π1 ) ∈ Qμ such that π1 (J1 ) is maximal among all elements of Qμ . Write μ = μ1  β 1

(β1 ∈ Pα,id ).

α

Applying the same reasoning to β1 with I1 replaced by I2 , we get β1 = μ2  β2 = α

eα (π2 )  β2 . If we perform this successively, we get α

μ = βn  θ n , α

where θn = μ1  μ2  . . . μn , α

α

μk = eα (πk )

with πk (Ik ) = 0 and πk (Jk ) having the specified maximality property. The sequences {βn } and {θn } are relatively compact; letting β and θ be limit points, we have (β, θ ∈ Pα,id ).

μ=βθ α

Suppose, by contradiction, that β is not -Gaussian, and let eα (η), with η = δ0 , be α

a divisor of β. Clearly η(Jk ) > 0 for some k; given that each βn divides βn−1 , we

88

3 The Whittaker Convolution

have βk = eα (η)  ν (ν ∈ Pα,id ). If we let  η be the restriction of η to the interval Jk , α

then η)  eα (η −  η)  ν, βk−1 = eα (πk +  α

α

which is absurd (because (πk +  η)(Jk ) > πk (Jk ), contradicting the maximality property which defines πk ). To determine the log-Whittaker transform of θ  , note that θn = eα (!n ) is the -compound Poisson measure associated with !n := nk=1 πk ,  α  thus ψθn (λ) = R+ 1 − Wα, λ(x) !n (dx). Since {!n } is an increasing sequence 0 of measures and each eα (!n ) dividing μ, there exists a σ -finite measure ν such that  ψθ (λ) = lim n

 =

R+ 0

 R+ 0

 1 − Wα, λ(x) !n (dx)

  1 − Wα, λ(x) ν(dx) < ∞

(μ ∈ Pα,id ensures the finiteness of the integral); from the relative compactness of D({μ}) it is possible to conclude that ν(Jk ) < ∞ for all k. For the converse, let νn be the restriction of ν to the interval Jn defined as above. It is verified without difficulty that the right-hand side of (3.79) is continuous at zero, w hence by Proposition 3.4(d) β  eα (νn ) −→ μ ∈ P(R+ 0 ), and μ ∈ Pα,id because α

Pα,id is closed under weak convergence. Remark 3.7 Proposition 3.11 below ensures that the function ψc (λ) = −cλ is, for each c > 0, the log-Whittaker transform of a -Gaussian measure. However, the above Lévy–Khintchine representation provides no information on whether the log-Whittaker transform of the -Gaussian measure γ must be of the form ψγ (λ) = −cλ for some c > 0. Such a characterization of Gaussian measures has been established for Urbanik convolution algebras [190, 191] and for Sturm– Liouville hypergroups on R+ 0 [30, 163]; however, the proofs of these results depend on assumptions which are not satisfied by the Whittaker convolution. (The proof for Urbanik convolutions relies on the homogeneity axiom U5 of Definition 2.3, while the proof for Sturm–Liouville hypergroups depends on a regularity property of the associated Sturm–Liouville type integral transform which cannot be easily extended to the index Whittaker transform.) A related characterization of Gaussian measures has also been established on spaces endowed with a generalized characteristic function ωλ (x)μ(dx) having properties similar to those of Proposition 3.4, and in which there exists not only a convolution with respect to the variable x but also a positivity-preserving convolution with respect to the dual variable λ (see [198]). We leave open the problem of extending these characterizations to the Whittaker convolution.

3.6 Lévy Processes with Respect to the Whittaker Convolution

89

3.6 Lévy Processes with Respect to the Whittaker Convolution 3.6.1 Convolution Semigroups Having in mind the study of Lévy-like processes on the Whittaker convolution algebra, we first introduce the notion of a Whittaker convolution semigroup. Definition 3.7 A family {μt }t ≥0 ⊂ P(R+ 0 ) is called a -convolution semigroup if it α

satisfies the conditions • μs  μt = μs+t for all s, t ≥ 0; α

• μ0 = δ 0 ; w • μt −→ δ0 as t ↓ 0. Remark 3.8 Similarly to the classical case (cf. [166, Section 7]), the -infinitely α

divisible distributions are in one-to-one correspondence with the -convolution α

semigroups: (i) If {μt } is a -convolution semigroup, then μt is (for each t ≥ 0) a -infinitely α

α

divisible distribution. α n (Indeed, for each n ∈ N the measure μt /n ∈ P(R+ = 0 ) is such that (μt /n ) μt .) (ii) If μ is a -infinitely divisible distribution with log-Whittaker transform ψμ (λ), α

then the semigroup {μt } defined by μ $t (λ) = exp(−t ψμ (λ)) is the unique α

convolution semigroup such that μ1 = μ. (To prove this, it suffices to justify that exp(−t ψμ (λ)) is, for each t > 0, the index Whittaker transform of a probability measure. If t = pq ∈ Q, this is  p  +  p  true because (ν q ) α (λ) = exp − q ψμ (λ) , where νq ∈ P(R0 ) is defined  pn  as in (3.72). If t > 0 is irrational, let qn ⊂ Q be a sequence converging α pn to t and define μt ∈ P(R+ ; the 0 ) as the weak limit of the measures (νqn ) existence of the weak limit follows from Proposition 3.4(iv), and it is clear that μ $t (λ) = exp(−t ψμ (λ)).) From this it follows, in particular, that -convolution semigroups admit a Lévy– α

Khintchine type representation (Theorem 3.3). Unsurprisingly, each -convolution semigroup is associated with a conservative α

Feller semigroup of operators which commute with the Whittaker translation: Proposition 3.8 Let {μt }t ≥0 be a -convolution semigroup. Then the family {Tt }t ≥0 α

of convolution operators defined by + Tt : Cb (R+ 0 ) −→ Cb (R0 ),

t Tt f := Tμ α f,

(3.80)

90

3 The Whittaker Convolution

where Tνα (ν ∈ MC (R+ 0 )) is the operator defined by  (Tνα f )(x)

:=

R+ 0

(Tyα f )(x) ν(dy),

is a conservative Feller semigroup. Furthermore, we have Tt Tνα f = Tνα Tt f for all x x t ≥ 0 and ν ∈ MC (R+ 0 ) (in particular, Tt Tα f = Tα Tt f for x ≥ 0). One should note that this result is similar to the Feller property for Kingman convolution semigroups, stated in Proposition 2.21.  μ  y + Proof For μ,ν ∈ P(R0 ) we have (Tα f )(x)ν(dx) = (Tα f )(x)μ(dx)ν(dy) = f (x) μ  ν (dx). Therefore, by associativity and commutativity of the Whittaker α

convolution,  ν (Tμ α (Tα f ))(x) =

 =  =

R+ 0

R+ 0

R+ 0

ν Tμ α (Tα f ) dδx

  Tνα f d μ  δx = α

 R+ 0

   f d ν  μ  δx α

  μν    f d μ  ν  δx = Tα α f (x) α

α

α



 μ, ν ∈ P(R+ 0) , (3.81)

and the convolution semigroup property yields that T0 = Id and Tt Ts = Tt +s for   + t, s ≥ 0. The property Tt C0 (R+ 0 ) ⊂ C0 (R0 ) follows at once from (3.51) and the dominated convergence theorem. The positivity and conservativeness of Tt follows from the corresponding property of the Whittaker translation (Proposition 3.2(a)). To prove the strong continuity of the semigroup, let f ∈ C0 (R+ 0 ) and x ≥ 0. From the definition of weak convergence and the fact that (T0α f )(x) = f (x) we deduce that         y (Tt f )(x) − f (x) =  (Tα f )(x) − f (x) μt (dy)    −−→  t ↓0

R+ 0

R+ 0

   y  (Tα f )(x) − f (x) δ0 (dy) = 0.

and therefore Tt f − f ∞ −→ 0 as t ↓ 0 (cf. Proposition 2.2). The concluding statement is a consequence of (3.81).

3.6 Lévy Processes with Respect to the Whittaker Convolution

91

Proposition 3.9 Let {Tt } be a Feller semigroup determined by the -convolution α   (p) semigroup {μt }t ≥0. Then, for each 1 ≤ p < ∞, Tt |C (R+ ) has an extension {Tt } c

0

which is a strongly continuous contraction semigroup on Lp (rα ). Moreover, the (p) operators Tt are given by 

(p)  t Tt f (x) = (Tμ α f )(x) :=



  f ∈ Lp (rα ) .

(Tyα f )(x) μt (dy)

R+ 0

(3.82)

Proof By Proposition 3.2(b) and Minkowski’s integral inequality, we have

 ≤

∞ 



 (p)  Tt f  ≤ p,α

R+ 0

0

R+ 0

p |(Tyα f )(x)| μt (dy)

!1

p

rα (x)dx (3.83)

Tyα f p,α μt (dy)

≤ f p,α , (p)

showing that the operators Tt defined by (3.82) are contractions on Lp (rα ). To + prove the strong continuity, let f ∈ Lp (rα ), ε > 0 and choose g ∈ C∞ c (R ) such that f − gp,α ≤ ε. Then it follows from (3.83) and the strong continuity of the Feller semigroup {Tt } that   (p) lim sup Tt f − f p,α t ↓0

  (p) (p)  ≤ lim sup Tt f − Tt g p,α + f − gp,α + Tt g − gp,α t ↓0

≤ 2ε + C ·lim sup Tt g − g∞ = 2ε, t ↓0



(C < ∞ because the support supp(g) ⊂ R+ is   (p) compact). Since ε is arbitrary, we find that limt ↓0 Tt f − f p,α = 0 for each f ∈ Lp (rα ).

where C = [

supp(g) rα (x)dx]

1/p

It is worth pointing out that, taking advantage of the correspondence between functions f ∈ L2 (rα ) and their index Whittaker transforms (Proposition 3.3), the (2) action of the L2 -Markov semigroup {Tt } can be explicitly written as Wα (Tt f )(τ ) = e−t ψ(τ (2)

2 +( 1 −α)2 ) 2

·(Wα f )(τ ),

f ∈ L2 (rα ),

(3.84)

92

3 The Whittaker Convolution

where ψ is the log-Whittaker transform of the -convolution semigroup {μt } (i.e., α

+ of the measure μ1 ). Indeed, for f ∈ Cc (R+ 0 ) and μ ∈ MC (R0 ) we have



Wα (Tμ αf)





∞

(τ ) =  =  =  =

 Wα,iτ (x)

0

 R+ 0

R+ 0

R+ 0

R+ 0

(Tyα f )(x)μ(dy) rα (x)dx

 Wα (Tyα f ) (τ ) μ(dy)

 

 R+ 0

R+ 0

R+ 0

Wα,iτ (x)kα (x, y, ξ ) rα (x)dx f (ξ )rα (ξ )dξ μ(dy)

Wα,iτ (y)Wα,iτ (ξ )f (ξ )rα (ξ )dξ μ(dy)

=$ μ(τ 2 + ( 12 − α)2 )·(Wα f )(τ ) (the second, third and fourth equalities being by changing the order of  obtained μ  integration and using (3.36)). The identity Wα (Tα f ) (τ ) = $ μ(τ 2 + ( 12 − α)2 ) · 2 (Wα f )(τ ) extends, by continuity, to all f ∈ L (rα ), and then Remark 3.8 yields (3.84). The index Whittaker transform also allows us to give the following characterization of the generator of the semigroup {Tt(2) }: Proposition 3.10 Let {μt } be a -convolution semigroup with log-Whittaker transα

(2) {Tt }

be the associated Markovian semigroup on L2 (rα ). Then the form ψ and let (2) infinitesimal generator (G(2) , D(G(2) )) of the semigroup {Tt } is the self-adjoint operator given by 

 Wα (G(2) f ) (τ ) = −ψ(τ 2 + ( 12 − α)2 ) · (Wα f )(τ ),

f ∈ D(G(2) ),

where   D(G ) = f ∈ L (rα )  

(2)

2

0

    ψ(τ 2 + ( 1 − α)2 )2 (Wα f )(τ )2 ρα (τ )dτ < ∞ . 2

∞

Proof The proof is very similar to that of the corresponding result for the Fourier transform and the generator of an ordinary convolution semigroup (see [16, Theorem 12.16]), so we only give a sketch. It follows from (3.52) that Tt(2)f, g = f, Tt(2)g for f, g ∈ L2 (rα ), i.e. {Tt(2)} is a semigroup of symmetric operators in L2 (rα ). It is well-known from the theory of semigroups on Hilbert spaces that the generators of self-adjoint contraction semigroups are the operators −A where A is a positive self-adjoint operator [42, Theorem 4.6]. Hence, in particular, the generator (G(2) , D(G(2) )) is self-adjoint.

3.6 Lévy Processes with Respect to the Whittaker Convolution

93

(2)

Letting f ∈ D(G(2)), so that L2 -limt ↓0 1t (Tt f − f ) = G(2) f ∈ L2 (rα ), from (3.84) we get L2 -lim t ↓0

 1  −t ψ e − 1 · Wα f = Wα (G(2) f ) t

 ) := ψ(τ 2 + ( 1 − α)2 )). The convergence holds almost (here we write ψ(τ 2 everywhere along a sequence {t n }n∈N such that  tn → 0, so we conclude that  · Wα f ∈ L2 R+ ; ρα (τ )dτ . Wα (G(2) f ) = −ψ    · Wα f ∈ L2 R+ ; ρα (τ )dτ , then we Conversely, if we let f ∈ L2 (rα ) with −ψ have L2 -lim t ↓0

   1 (2)  · Wα f ∈ L2 R+ ; ρα (τ )dτ Wα (Tt f ) − Wα f = −ψ t

and the isometry gives that L2 -limt ↓0 D(G(2)).

1 t

 (2)  Tt f − f ∈ L2 (rα ), meaning that f ∈

3.6.2 Lévy and Gaussian Processes Definition 3.8 Let {μt }t ≥0 be a -convolution semigroup. An R+ 0 -valued Markov α

process X = {Xt }t ≥0 is said to be a -Lévy process associated with {μt }t ≥0 if its α

transition probabilities are given by

 P Xt ∈ B|Xs = x = (μt−s  δx )(B), α

0 ≤ s ≤ t, x ≥ 0, B a Borel subset of R+ 0.

In other words, a -Lévy process is a Feller process associated with the Feller α

semigroup defined in (3.80). Consequently, the general connection between Feller semigroups and Feller processes (Sect. 2.1) ensures that for each (initial) distribution ν ∈ P(R+ 0 ) and -convolution semigroup {μt }t ≥0 there exists a -Lévy process X α

α

associated with {μt }t ≥0 and such that P [X0 ∈ ·] = ν. Being a Feller process, any -Lévy process is stochastically continuous and has a càdlàg modification α

(Proposition 2.3). As expected (cf. Corollary 2.3 for the Kingman convolution), the -Lévy α

processes are a subclass of Feller processes which includes the Shiryaev process generated by the Sturm–Liouville operator (3.2): Proposition 3.11 The Shiryaev process {Yt }t ≥0 is a -Lévy process. α

94

3 The Whittaker Convolution

Proof For t, x ≥ 0 let us write pt,x (dy) ≡ Px [Yt ∈ dy]. According to Corollary 2.2, we have  pt,x (dy) =

R+ 0

e−t (τ

2 +( 1 −α)2 ) 2

Wα,iτ (x) Wα,iτ (y) ρα (τ )dτ rα (y)dy,

t, x > 0, (3.85)

where the integral converges absolutely. Consequently, by Proposition 3.3, −t λ p & Wα, λ(x), t,x (λ) = e

t, x ≥ 0

(3.86)

(the weak continuity of pt,x justifies that the equality also holds for t = 0 and for −t λ . It is clear from x = 0). This shows that pt,x = pt,0  δx where p & t,0 (λ) = e α

the properties of the index Whittaker transform of measures that {pt,0 }t ≥0 is a α

convolution semigroup; therefore, Y is a -Lévy process. α

Remark 3.9 The proposition above ensures that there exists a -convolution semiα

group {μt } such that μ $t (λ) = e−t λ. An interesting problem, which we do not address in this work, is to prove or disprove the existence of convolution semigroups β %β (λ) = e−t λβ , where 0 < β < 1. A positive answer has been {μt }t ≥0 such that μ t given for Urbanik convolution algebras (see [190]), but the proof depends on the homogeneity axiom U5 which is not satisfied by the Whittaker convolution. In the context of Urbanik convolution algebras, if ν is a measure with generalized β characteristic function e−t λ , then ν satisfies the property given a, b > 0, there exists c > 0 such that a ν  b ν = c ν. Such measures are therefore called stable measures with respect to the generalized convolution. (This is analogous to the classical notion of a strictly stable probability distribution on R as a measure ν ∈ P(R) such that given a, b ∈ R there exists d

c ∈ R for which we have aX1 + bX2 = cX, where X, X1 , X2 are mutually d

independent random variables with common distribution ν and = denotes equality in distribution; see e.g. [60, Chapter VI] for the basics on stable distributions on R.) For a discussion of the notion of stable measures in the context of generalized (hypergroup) convolutions not satisfying the homogeneity axiom, we refer to [211]. Since -Lévy processes are Feller processes, they can be characterized as the α

solution of the corresponding martingale problem, as stated in Theorem 2.1. The next proposition provides some additional equivalent martingale characterizations of -Lévy processes. For an R+ 0 -valued càdlàg process X = {Xt }t ≥0 , a linear α

3.6 Lévy Processes with Respect to the Whittaker Convolution

95

operator A : D −→ C(R+ D ⊂ C(R+ 0 ) with domain 0 ) and a function f ∈ D,  A,f A,f we introduce the notation ZX = ZX,t t ≥0, where 

A,f

ZX,t := f (Xt ) − f (X0 ) −

t

(Af )(Xs ) ds.

(3.87)

0

Proposition 3.12 Let {μt }t ≥0 be a -convolution semigroup with log-Whittaker α

transform ψ and let (A, DA ) be the infinitesimal generator of the Feller semigroup determined by {μt } (cf. Proposition 3.8). Let X be an R+ 0 -valued càdlàg Markov process. The following assertions are equivalent: (i) X is a -Lévy process associated with {μt }; α

t ψ(λ)W (ii) {e α, λ(Xt )}t ≥0 is a martingale   t for each λ ≥  0; (iii) Wα, λ(Xt )−Wα, λ(X0 )+ψ(λ) 0 Wα, λ(Xs ) ds t ≥0 is a martingale for each λ ≥ 0; A,Wα, (·) (iv) ZX is a martingale for each λ ≥ 0; A,f (v) ZX is a martingale for each f ∈ DA . λ

Proof The proof is identical to that of the corresponding martingale characterization for Lévy processes on commutative hypergroups [163, Theorem 3.4]. A -convolution semigroup {μt }t ≥0 such that μ1 is a -Gaussian measure will α

α

be called a -Gaussian convolution semigroup, and a -Lévy process associated α

α

with a -Gaussian convolution semigroup is said to be a -Gaussian process. α

α

An alternative characterization of -Gaussian convolution semigroups (which in α

particular implies that any -Gaussian convolution semigroup is fully composed α

of -Gaussian measures) is given in the next lemma. α

Lemma 3.8 Let μ ∈ Pα,id and let {μt } be the -convolution semigroup {μt } such α

that μ1 = μ. Then, the following conditions are equivalent: (i) μ is a -Gaussian measure; α

(ii) limt ↓0 1t μt [ε, ∞) = 0 for every ε > 0;   (iii) limt ↓0 1t (μt  δx ) R+ 0 \ (x − ε, x + ε) = 0 for every x ≥ 0 and ε > 0. α

Proof (i)  ⇒ (ii): Let {tn }n∈N be a sequence such that tn → 0 as n → ∞, and let νn = eα t1n μtn . We have lim ν$n (λ) = lim exp

n→∞

n→∞

!  1 μ %1 (λ)tn − 1 = μ %1 (λ), tn w

λ>0

(3.88)

and therefore, by Proposition 3.4(iv), νn −→ μ1 as n → ∞. From this it follows, cf. [196], that if πn denotes the restriction of t1n μtn to [a, b) \ Va , then {πn } is relatively

96

3 The Whittaker Convolution

compact; if π is a limit point, then eα (π) is a divisor of μ1 . Since μ1 is Gaussian, eα (π) = δa , hence π must be the zero measure, showing that (ii) holds. (ii) ⇒ (i): As in (3.88), for λ > 0 we have 1 μ %1 (λ) = lim exp n→∞ tn



 [a,b)

 λ (x) − 1 μtn(dx)

Wα,

  1 = lim exp Wα, n→∞ tn Va

!

!  λ (x) − 1 μtn(dx) ,

 where the second equality is due to (ii), noting that t1n [a,b)\Va (Wα, λ (x) −     w 1)μtn(dx) ≤ 2t μtn [a, b) \ Va . Given that νn = eα t1n μtn −→ μ1 , we have (again, see [196]) 

 Wα,

$ μ1 (λ) = exp (a,b)

!  (x) − 1 η(dx) , λ

λ>0

for some σ -finite measure η on (a, b) which, by the above, vanishes on the complement of any neighbourhood of the point a. Therefore, μ1 is Gaussian. (ii) ⇐⇒ (iii): To prove the nontrivial direction, assume that (ii) holds, and fix x, ε > 0 with 0 < x < ε. Write Eε = R+ 0 \ (x − ε, x + ε), and let 1ε denote its indicator function. We start the proof by establishing an upper bound for the function (Txα 1ε )(y), with y > 0 small. Using the estimate (3.39), together with the inequalities  1+

x ξ

|x−ξ |5 5

(8xyξ ) 2

2α

≤ (1 + ξ −1 )(1 + x + δ),  |x−ξ |2 exp − 8xyξ ≤ 1 +

y ξ

(x, ξ > 0, y < δ),

it is easily seen that y (x − ξ )2 1 − − q(x, y, ξ )rα (ξ ) ≤ C1 y |x − ξ | ξ(1 + ξ ) exp − 2ξ 4xξ 8xyξ  1 (x − ξ )2 − (y ≤ δ, ξ ∈ Eε ), ≤ C2 y 2 ξ(1 + ξ ) exp − 2ξ 8δxξ 2

−5



3.6 Lévy Processes with Respect to the Whittaker Convolution

97

where the constants C1 , C2 > 0 depends only on x, δ and ε. Consequently, for y < δ we have  (Txα 1ε )(y) = kα (x, y, ξ )rα (ξ )dξ Eε



≤ C2 y

∞

2

 ξ(1 + ξ ) exp

0

1 (x − ξ )2 − − dξ 2ξ 8δxξ

≤ C3 y 2 ,

(3.89)

the convergence of the integral justifying that the last inequality holds for a possibly larger constant C2 . Let λ > 0 be arbitrary. If δ > 0 is sufficiently small, then from (3.75) and (3.89) it follows that (Txα 1ε )(y) ≤ 2C3 λ−1 y, 1 − Wα, λ(y)

y ≤ δ,

and therefore, there exists δ > 0 (which depends on λ) such that (Txα 1ε )(y) ≤ 1 − Wα, λ(y) for all y ∈ [0, δ ). We then estimate 1 1 (μt  δx )(Eε ) = α t t ≤

1 t

1 ≤ t =

 (Txα 1ε )(y)μt (dy)

 

R+ 0

  1 1 − Wα, λ(y) μt (dy) + μt [δ , ∞) t [0,δ ) R+ 0

  1 1 − Wα, λ(y) μt (dy) + μt [δ , ∞) t

 1 1 1−μ $t (λ) + μt [δ , ∞). t t

  Since we are assuming that (ii) holds and we know that limt ↓0 1t 1 − μ $t (λ) = − log $ μ(λ) (cf. proof of Proposition 3.7), the above inequality gives 1 lim sup (μt  δx )(Eε ) ≤ − log $ μ(λ). α t t ↓0 By the properties of the index Whittaker transform, the right-hand side is continuous and vanishes for λ = 0, so from the arbitrariness of λ we see that limt ↓0 1t (μt  α

δx )(Eε ) = 0, as desired.

Denoting, as in (3.85), the law of the Shiryaev process started at x by   pt,x , it follows from Propositions 2.4 and 2.7 that limt ↓0 1t pt,x R+ \ (x − ε, x + ε) = 0 for 0 any x ≥ 0 and ε > 0, meaning that the Shiryaev process is a -Gaussian process. α

98

3 The Whittaker Convolution

It turns out that, as a consequence of the previous lemma, any other -Gaussian α

process is also a one-dimensional diffusion: Corollary 3.2 Let X = {Xt }t ≥0 be a -Gaussian process associated with the α

α

Gaussian convolution semigroup {μt }t ≥0. Then:

(i) X has a modification whose paths are a.s. continuous; (ii) Let (G, D(G)) be the infinitesimal generator of the Feller semigroup determined by {μt }. Then G is a local operator, i.e., (Gf )(x) = (Gg)(x) whenever f, g ∈ D(G) and f = g on some neighborhood of x ≥ 0. Proof We know from Lemma 3.8 that the associated -Gaussian convolution α   \ (x − ε, x + ε) = 0 for every semigroup is such that limt ↓0 1t (μt  δx ) R+ 0 α

x ≥ 0 and ε > 0. Using Proposition 2.4, we conclude that the càdlàg modification of X has a.s. continuous sample paths. The locality of the generator is then proved  = {X t }t ≥0 which is the by applying Proposition 2.5 to the R-valued process X  extension of X obtained by setting Xt (ω) = x whenever the initial distribution is ν = δx , x < 0.

3.6.3 Some Auxiliary Results on the Whittaker Translation In this subsection we return to the Whittaker translation operator (3.40), which we will now interpret as an operator on the space      1 −β + x β )  +  |f (x)| ≤ b1 exp x + b2 (x X := f ∈ L0 (R )  , for some b1 , b2 ≥ 0 and 0 ≤ β < 1

(3.90)

being L0 (R+ ) the space of Lebesgue measurable functions f : R+ −→ C. The goal of this digression is to determine some properties which will be useful for introducing (in the next subsection) the notion of moment functions with respect to the Whittaker convolution. We first note that the condition f ∈ X ensures that for each x, y > 0 y ∞ the Whittaker translation (Tα f )(x) = 0 f (ξ )kα (x, y, ξ ) rα (ξ )dξ exists as an absolutely convergent integral, as can be verified using (3.39). Lemma 3.9 Fix y, M > 0. Let f ∈ X and p ∈ N0 . Then, for each ε > 0 there exists δ, M0 > 0 such that  EM

 ∂p    |f (ξ )|  p kα (x, y, ξ )rα (ξ ) dξ < ε ∂x

1 where EM = (0, M ] ∪ [M, ∞).

for all x ∈ (0, δ] and M ≥ M0 ,

3.6 Lévy Processes with Respect to the Whittaker Convolution

99

Proof Fix k ≥ − 12 + max{α, 0}. Note that if σ ∈ C then (after a new choice of b2 and β) the function ξ → ξ σ f (ξ ) also belongs to X. Let δ < y4 . If |ξ − y| ≥ 2δ and 1

x ≤ δ, using (3.39), the boundedness of the function |t|k+ 2 e−|t | and the inequalities  2y 2α  δ 2α−2k−1 (x + y + ξ )2α ≤ 1 + , δ 4 |x + ξ − y|2k+1

α ≥ 0,

(x + y + ξ )2α y 2α ≤ , |x + ξ − y|2k+1 δ 2k+1

α ≤ 0,

we find that x −k |f (ξ )| kα (x, y, ξ ) rα (ξ ) ≤

 (x + ξ − y)2 2α −β β (x + y + ξ ) exp b (ξ + ξ ) − 2 1 4xyξ (xyξ )k+ 2 C

 (x + ξ − y)2 , ≤ C exp b2 (ξ −β + ξ β ) − 8xyξ

|ξ − y| ≥ 2δ, x ≤ δ, (3.91)

∞ where C depends only on y and δ. Since 0 ≤ β < 2, the integral 0 exp b2 (ξ −β +

−y)2 dξ converges uniformly in x ∈ [0, δ]. Combining this with the ξ β ) − (x+ξ 8xyξ inequality (3.91), we conclude that M0 > 0 can be chosen so large that 

x −k |f (ξ )| kα (x, y, ξ ) rα (ξ ) dξ < δ

for all 0 < x
0. Then: y

(i) lim (Tα f )(x) = f (y); x→0

y

∂ (ii) lim pα (x) ∂x (Tα f )(x) = 0. x→0

100

3 The Whittaker Convolution

Proof (i) We will first show that it is enough to prove the result for f ∈ C2c (R+ ). Suppose that part (i) of the lemma holds for f ∈ C2c (R+ ). Let g ∈ X ∩ C2 (R+ ) and ε, M > 0; then, choose δ > 0 and gc ∈ C2c (R+ ) such that g(ξ ) = gc (ξ ) for all 1 ξ ∈ [M , M] and |(Tyα g)(x) − (Tyα gc )(x)| < ε

for all x ∈ (0, δ]

1 (to see that this is possible, apply the case p = 0 of Lemma 3.9). If y ∈ [ M , M], we obtain

lim sup |(Tyα g)(x) − g(y)| ≤ ε + lim |(Tyα gc )(x) − gc (y)| = ε. x→0

x→0

y

As M and ε are arbitrary, we conclude that limx→0 (Tα g)(x) = g(y) for all y > 0 and g ∈ X ∩ C2 (R+ ). y Let us now prove that limx→0 (Tα f )(x) = f (y) holds for f ∈ C2c (R+ ). Using the integral representation for kα (x, y, ξ ) given in (3.55), we write  (Tyα f )(x)

=



∞

∞

f (ξ ) 0

 =

Wα,iτ (x) Wα,iτ (y) Wα,iτ (ξ ) ρα (τ )dτ rα (ξ )dξ

0 ∞

(Wα f )(τ ) Wα,iτ (x) Wα,iτ (y) ρα (τ )dτ,

0

(3.93) where the second equality is obtained by changing the order of integration, which is valid because f has compact support. It was noted above that the index Whittaker transform Wα is a particular case of the Sturm–Liouville integral transform (2.21)–(2.22), thus it follows from Lemma 2.4(b) that  (Wα f )(τ ) Wα,iτ (y) ∈ L1 R+ ; ρα (τ )dτ . Recalling also that |Wα,iτ (x)| ≤ 1 (x, τ ≥ 0) and |W α,iτ (0)| = 1, cf. (3.67) and (3.74), by dominated convergence we obtain that  ∞   lim (Tyα f )(x) = (Wα f )(τ ) lim Wα,iτ (x) Wα,iτ (y) ρα (τ )dτ = f (y), x→0

x→0

0

concluding the proof. (ii) Identical reasoning as in part (i) shows that it is enough to prove the result for f ∈ C2c (R+ ). Taking f ∈ C2c (R+ ), differentiation of (3.93) under the integral sign gives ∂ pα (x) (Tyα f )(x) = ∂x

 0

∞

 

(Wα f )(τ ) pα Wα,iτ (x) Wα,iτ (y) ρα (τ )dτ.

3.6 Lévy Processes with Respect to the Whittaker Convolution

101

If we now apply (3.74), by dominated convergence we conclude that lim pα (x) y ∂ ∂x (Tα f )(x)

x→0

= 0.

3.6.4 Moment Functions Moment functions for generalized convolutions are functions having the same additivity property which is satisfied by the monomials under the classical convolution. Such functions have been applied to the study of limit theorems for hypergroup convolution structures (see the discussion in [19, pp. 530–531]). Let us introduce, in a similar way, the notion of moment functions with respect to the Whittaker convolution: Definition 3.9 The sequence of functions {ϕk }k=1,...,n is said to be a -moment sequence (of length n) if ϕk ∈ X for k = 1, . . . , n (cf. (3.90)) and (Tyα ϕk )(x) =

k   k ϕj (x)ϕk−j (y) j

(k = 1, . . . , n; x, y ≥ 0),

α

(3.94)

j =0

where ϕ0 (x) := 1 (x ≥ 0). It is worth  ∞recalling that for x, y > 0 the left-hand side of (3.94) is given by the integral 0 ϕk (ξ )kα (x, y, ξ ) rα (ξ )dξ , which converges absolutely. This actually implies that -moment functions are necessarily smooth: α

Lemma 3.11 If {ϕk }k=1,...,n is a -moment sequence, then ϕk ∈ C∞ (R+ ) for all k. α

Proof Let M > 0 and 1 ≤ k ≤ n. Let f ∈ C∞ c (2M, 3M) be such that  3M / (2M, 3M). Then 2M f (x) rα (x)dx = 1, and set f (x) = 0 for x ∈  ∞  ∞ k   k ϕk−j (x)f (x) rα (x)dx = (Tyα ϕk )(x) f (x) rα (x)dx ϕj (y) j 0 0 j =0



∞

= 0

ϕk (x) (Tyα f )(x) rα (x)dx,

where the second equality follows from the identity (3.52), which is easily seen to + hold also for f ∈ C∞ c (R ) and g ∈ X. Hence if we prove that the right-hand side is an infinitely differentiable function of 0 < y < M, then by induction it follows that each ϕk ∈ C∞ (0, M) and, by arbitrariness of M, ϕk ∈ C∞ (R+ ).

102

3 The Whittaker Convolution

By (3.39), we have (Tyα f )(x)

1 x (y − ξ )2 1 − − − dξ 2y 2ξ 4yξ 4xyξ 2M  1 x 1 1 − , − ≤ C2 f ∞ (xy)− 2 (1 + x 2α ) exp 2y 12M 2 12x 1

≤ C1 f ∞ (xy)− 2



3M

1

ξ − 2 −2α (x + y + ξ )2α exp



where C1 and C2 are constants depending only on M. Since ϕk ∈ X, we find that 1

ϕk (x) (Tyα f )(x) rα (x) ≤ C (xy)− 2 (1 + x −2α ) exp



x 1 1 + b2 (x β + x −β ) − , − 2y 12M 2 12x

(3.95) where C > 0, b2 ≥ 0 and 0 ≤ β < 1 do not depend on y.Denoting the right-hand ∞ side of (3.95) by J (x, y), it is easily seen that the integral 0 J (x, y)dx converges ∞ y locally uniformly and, therefore, 0 ϕk (x) (Tα f )(x) m(x)dx is a continuous function of 0 < y < M. Using the identity (3.61) (with x and y interchanged) and y ∂n similar arguments, one can derive an upper bound for the derivatives ∂y n (Tα f )(x) ∞ y (n = 1, 2, . . .) and then deduce that 0 ϕk (x) (Tα f )(x) rα (x)dx is n times continuously differentiable. Proposition 3.13 {ϕk }k=1,...,n is a -moment sequence if and only if there exist α λ1 , . . . , λn ∈ R such that Aα ϕk (x) =

k   k λj ϕk−j (x), j

ϕk (0) = 0,

(pα ϕk )(0) = 0,

(3.96)

j =1

for k = 1, . . . , n, where ϕ0 ≡ 1 and Aα is the differential operator (3.2). Proof Let {ϕk }k=1,...,n be a -moment sequence. First we will show that ϕk ∈ α

C∞ (R+ ) ∩ C1 (R+ 0 ) with ϕk (0) = (pα ϕk )(0) = 0. By Lemma 3.11, ϕk ∈ ∞ + X ∩ C (R ). It thus follows from Lemma 3.10 that for fixed y > 0 we have y y ∂ limx→0 (Tα ϕk )(x) = ϕk (y) and limx→0 pα (x) ∂x (Tα ϕk )(x) = 0. If we rewrite (3.94) as

ϕk (x) = (Tyα ϕk )(x) −

k−1   k ϕj (x)ϕk−j (y) j

(3.97)

j =0

and let x → 0 on the right-hand side, we deduce (by induction on k) that limx→0 ϕk (x) = 0 for all k. After differentiating both sides of (3.97), we similarly find that limx→0 (pα ϕk )(x) = 0 for each k.

3.6 Lévy Processes with Respect to the Whittaker Convolution

103

   We now prove that ϕk satisfies Aα ϕk = kj =1 jk λj ϕk−j , omitting the details which are similar to the proof of [184, Theorem 4.5]. We know from (3.58) that Aα,x kα (x, y, ξ ) = Aα,y kα (x, y, ξ ). Moreover, from the identity (3.61) it follows y that the integral defining (Tα ϕk )(x) can be differentiated under the integral sign. Therefore, the right-hand side of (3.94) is, for each k, a solution of Aα,x u = Aα,y u, i.e. k   k j =0

j

(Aα ϕj )(x) ϕk−j (y) =

k   k ϕj (x) (Aα ϕk−j )(y). j j =0

   Assume by induction that Aα ϕ (x) = j =1 j λj ϕ −j (x) for = 1, . . . , k − 1. Using the induction hypothesis and rearranging the terms in a suitable way, we find that (Aα ϕk )(x) −

k−1 

λj ϕk−j (x) = (Aα ϕk )(y) −

j =1

k−1 

for all x, y > 0

λj ϕk−j (y)

j =1

and, consequently, (Aα ϕk )(x) −

k−1 

λj ϕk−j (x) = λk

j =1

for some λk ∈ R. For the converse, suppose that {ϕk }k=1,...,n are solutions of (3.96). Integrating, we y x

k  k   obtain ϕk (x) = − 0 pα1(y) 0 rα (ξ ) j =1 j λj ϕk−j (ξ ) dξ dy. Straightforward bounds on this integral yield that ϕk ∈ X (see the proof of Proposition 3.15). We can assume by induction that (Tyα ϕr )(x)

r   r ϕ (x)ϕr− (y) =

for r = 1, . . . , k − 1

=0

y

and the goal is to prove that k,y (x) := (Tα ϕk )(x) − vanishes identically. We compute

k  j =0 j ϕj (x)ϕk−j (y)

k

k  

 k (Aα ϕj )(x)ϕk−j (y) Aα,x k,y (x) = Tyα (Aα ϕk )(x) − j j =0

=

k  k  k     k k j λj (Tyα ϕk−j )(x) − ϕk−j (y) λ ϕj − (x) j j

j =1

j =0

=1

104

3 The Whittaker Convolution

k−j  k  k  k      k k−j k j λj ϕ (x)ϕk−j − (y) − ϕk−j (y) λ ϕj − (x) = j

j

j =1

j =0

=0

=1

= 0. y

y

Here, the first equality follows from the identity Tα (Aα ϕ)(x) ≡ Aα,x (Tα ϕ)(x), which can be verified using (3.58) and integration by parts; the second equality applies (3.96); the induction hypothesis gives the third equality; and the last step is obtained by rearranging the sums. Furthermore, limx→0 k,y (x) = ∂ limx→0 pα (x) ∂x k,y (x) = 0 (due to Lemma 3.10); by uniqueness of solution, k,y (x) ≡ 0, showing that (3.94) holds. The functions  ϕα,k (k ∈ N) defined as the unique solution of ϕα,k (x) = −k(1 − 2α) ϕα,k−1 (x) − k(k − 1) ϕα,k−2(x), Aα  ϕα,k (0) = 0, 

ϕα,k )(0) (pα 

(3.98)

=0

(where  ϕα,−1 (x) := 0 and  ϕα,0 (x) := 1) are said to be the canonical -moment α

functions. By integration of the differential equation, we find the explicit recursive expression 

x

ϕα,k (x) = k  0

1 pα (y)



y

 rα (ξ ) (1 − 2α) ϕα,k−1 (ξ ) + (k − 1) ϕα,k−2(ξ ) dξ dy.

0

(3.99) Moreover, as a consequence of the uniqueness of solution for (3.98) and the Laplace representation (3.63), the canonical moment functions can also be represented as ∂ k  Wα,σ (x) ∂σ k σ = 12 −α !  ∞ k σs  ∂ (e )  = ηα,x (s)ds ∂σ k σ = 12 −α −∞  ∞ 1 s k e( 2 −α)s ηα,x (s)ds. =

 ϕα,k (x) =

−∞

The first (canonical) moment function can be written in closed form: Proposition 3.14 We have  ϕα,1 (x) =

  1 1  0, 1 G31  (1 − 2α) 23 x 0, 0, 1 − 2α

(3.100)

3.6 Lévy Processes with Respect to the Whittaker Convolution

105

 a1 ,...,ap  where Gmn pq z | b1 ,...,bq denotes the Meijer-G function [157, Section 8.2]. In the 1

particular case α = 0, we have  ϕα,1 (x) = e x (0, x1 ), where (a, z) is the incomplete Gamma function [56, Chapter IX]. y x Proof We know from (3.99) that  ϕα,1 (x) = (1 − 2α) 0 pα1(y) 0 rα (ξ )dξ dy. Consequently,  ϕα,1 (x) = (1 − 2α)   = (1 − 2α)

∞

v

1 x



∞

e

w2α−2 e−w dw dv

v ∞

1 x

−2α v

v −2α ev (−1 + 2α, v)dv

−1 dv 1 −2α, −1 x   1 0, 1 31 1  = G ,  (1 − 2α) 23 x 0, 0, 1 − 2α

=

1 (1 − 2α)



∞

   G21 12 v 

where the first equality is obtained via a change of variables, the second equality follows from the definition of the incomplete Gamma function, the third step is due to [157, Relations 8.2.2.15 and 8.4.16.13] and the final step applies [128, Equation  1 0,1  1 1 x 5.6.4(6)]. The result for α = 0 follows from the identity G31 23 x | 0,0,1 = e (0, x ), cf. [157, Relations 8.2.2.9 and 8.4.16.13]. Actually, the right-hand side of (3.100) can be written (for α < 12 ) as a sum of simpler special functions. Such representation can be obtained by applying [7, Equation (A13)]. Returning to moment functions of general order, it is clear from the explicit representation (3.99) that  ϕα,k (x) > 0 for all x > 0 and k ∈ N. We note that ∞ 1 2  ϕα,2 ≥  ϕα,1 (by Jensen’s inequality applied to  ϕα,k (x) = −∞ s k e( 2 −α)s ηα,x (s)ds) and that the Taylor expansions of the first two moment functions as x → 0 are  ϕα,1 (x) = (1 − 2α)x − (1 − 2α)(1 − α)x 2 + o(x 2),

(3.101)

 ϕα,2 (x) = 2x − (1 + 2α − 4α 2 )x 2 + o(x 2)

(these relations can be deduced from the asymptotic expansion (3.13), taking into ∂ k  account that  ϕα,k (x) = ∂σ k σ = 1 −α Wα,σ (x)). Concerning the growth of the moment functions as x → ∞, we have:

2

Proposition 3.15 Let ε > 0. For each k ∈ N,  ϕα,k (x) = O(x ε ) as x → ∞.   Proof Due to (3.12), it suffices to prove that  ϕα,k (x) = O W α, 1 −α+ε (x) as 2 x → ∞ for each k ∈ N. This is trivial for k = 0 since  ϕα,0 ≡ 1 = O(x ε) = O W α, 1 −α+ε (x) . By induction, suppose that  ϕα,j (x) = O W α, 1 −α+ε (x) for 2

2

106

3 The Whittaker Convolution

j = 0, . . . , k − 1. This implies that  ϕα,j (x) ≤ C · W α, 1 −α+ε (x) for all x ≥ 0 2 and j = 0, . . . , k − 1 (where C > 0 does not depend on x). Recalling (3.73) and (3.99), we find 

x

ϕα,k (x) ≤ C  0

= C·

1 pα (y)



y 0

rα (ξ ) W α, 1 −α+ε (ξ )dξ dy 2

  1 W α, 1 −α+ε (x) − 1 2 ε(1 − 2α + ε)

  and therefore  ϕα,k (x) = O W α, 1 −α+ε (x) , proving the proposition. 2

The previous proposition shows that the modified moments E[ ϕα,k (X)] will only diverge if the tails of the random variable X are very heavy. The next result shows that the modified moments can be computed via the index Whittaker transform: Proposition 3.16 Let μ ∈ P(R+ 0 ) and k ∈ N. The following assertions are equivalent:  ϕα,k (x)μ(dx) < ∞; (i) R+  0  (ii) σ → R+ Wα,σ (x)μ(dx) is k times differentiable on [0, 12 − α]. 0

If (i) and (ii) hold, then



R+ 0

∂ k Wα,σ (x) ∂σ k

[0, − α] and, in particular, 1 2



R+ 0

μ(dx) =

∂k ∂σ k

 ϕα,k (x)μ(dx) =



Wα,σ (x)μ(dx) R+ 0



for all σ ∈  R+ Wα,σ (x)μ(dx) .



∂ k  ∂σ k σ = 12 −α

0

Proof The following proof is similar to that of the corresponding result for Sturm–Liouville hypergroups (see [208, Theorem 4.11] for further details). Write {k}

Wα,σ (x) :=

∂ k Wα,σ (x) . ∂σ k

By the Laplace representation (3.63),

{k} Wα,σ (x) =



∞

s k (eσ s + (−1)k e−σ s ) ηα,x (s)ds.

0

Since sinh and cosh are both increasing and convex functions on R+ , we have {k} {k} 0 ≤ Wα,σ (x) ≤ Wα,σ (x) ≤  ϕα,k (x) 1 2

for 0 ≤ σ1 ≤ σ2 ≤

1 2

− α, (3.102)

{k}

ϕα,k (x) − Wα,σ1 (x)  1 2

− α − σ1

{k}

≤

ϕα,k (x) − Wα,σ2 (x)  1 2

− α − σ2

for 0 ≤ σ1 ≤ σ2
0, then the process ϕ1 (Xt ) − E[ϕ1 (Xt )] t ≥0 is a martingale; (b) If, in addition, E[ϕ2 (Xt )] exists for all t > 0, then the process   ϕ2 (Xt ) − 2ϕ1 (Xt )E[ϕ1 (Xt )] − E[ϕ2 (Xt )] + 2E[ϕ1 (Xt )]2 t ≥0 is a martingale.

108

3 The Whittaker Convolution

In particular, if we let Y be the Shiryaev process started at Y0 = 0 and let λ1 , λ2 be as in Proposition 3.13, then the processes {ϕ1 (Yt ) + λ1 t}t ≥0 and {ϕ2 (Yt ) + 2λ1 tϕ1 (Yt ) + λ2 t + λ21 t 2 }t ≥0 are martingales. Proof To prove (a), we let 0 ≤ s < t and compute  E[ϕ1 (Xt ) | Xs ] =

R+ 0

  μ ϕ1 d μt−s  δXs = (Tα t−s ϕ1 )(Xs ) = α

Taking the expectation of both sides yields consequently,



R+ 0

 R+ 0

ϕ1 dμt−s +ϕ1 (Xs ).

ϕ1 dμt −s = E[ϕ1 (Xt )]−E[ϕ1 (Xs )];

 " #  

 E ϕ1 (Xt ) − E[ϕ1 (Xt )]  ϕ1 (Xs ) − E[ϕ1 (Xs )] = E ϕ1 (Xt ) − E[ϕ1 (Xt )]  Xs = ϕ1 (Xs ) − E[ϕ1 (Xs )], which shows that ϕ1 (Xt ) − E[ϕ1 (Xt )] is a martingale. Part (b) can be proved by similar arguments. Let Y be the Shiryaev process started at zero and {p  t,0 }t ≥0 be the associated -convolution semigroup. Then by (3.86) we have R+ Wα,σ (x) pt,0 (dx) = α

0

1

et (σ −( 2 −α) ) for σ ∈ [0, 12 − α], and it therefore follows from Proposition 3.16  ϕα,1 dpt,0 = (1 − 2α)t and E[ ϕα,2 (Xt )] = 2t + (1 − 2α)2 t 2 . that E[ ϕα,1 (Xt )] =  2

2

λ2

λ1 1 ϕα,1 and ϕ2 = (1−2α)  ϕα,2 − It follows from Proposition 3.13 that ϕ1 = − 1−2α 2   2λ21 ϕ . Consequently, E[ϕ1 (Yt )] = −λ1 t and E[ϕ2 (Yt )] = λ21 t 2 −λ2 t, + λ2  (1−2α)3 1−2α α,1 so that the final statement holds.

3.6.5 Lévy-Type Characterization of the Shiryaev Process In this subsection we will show that the martingale property given in the last statement of the previous proposition is in fact a (Lévy-type) characterization of the Shiryaev process. For this purpose, it is convenient to focus on the moment functions φ1 and φ2 that correspond to the choice λ1 = −1 and λ2 = 0, i.e. 

x

φ1 (x) ≡ φα,1 (x) = 0



1 pα (y)

 0

y

rα (ξ ) dξ dy =

1  ϕα,1 (x), 1 − 2α

 y 1 φ2 (x) ≡ φα,2 (x) = 2 rα (ξ ) φ1 (ξ ) dξ dy 0 pα (y) 0 " # 1 2 ϕα,2 (x) −   ϕα,1 (x) . = 2 (1 − 2α) 1 − 2α x

3.6 Lévy Processes with Respect to the Whittaker Convolution

109

In the following results, we write    k +

 Ck, (R+ 0 ) := f ∈ C (R0 ) : f [0,ε) ∈ C [0, ε) for some ε > 0 . Lemma 3.12



2 + (a) If f ∈ C2,4 (R+ 0 ) with f (0) = f (0) = 0, then there exists h ∈ C (R0 ) with 2 f (x) = h(φ1 (x )) for x ≥ 0. (b) There exists a unique function h0 ∈ C2 (R+ 0 ) such that h0 (φ1 (x)) = φ2 (x) for x ≥ 0, and it satisfies h

0 (x) > 0 for all x ≥ 0.

Proof From (3.101) we find that the Taylor expansions of the functions φ1 (x 2 ) and φ2 (x 2 ) as x → 0 are of the form φ1 (x 2 ) = c1 x 2 + c2 x 4 + o(x 4 ) and φ2 (x 2 ) = c3 x 4 + o(x 4), with c1 , c3 > 0. Consequently, part (a) can be proved using the same arguments as in [163, Lemma 5.7]. Letting f (x) = φ2 (x 2 ), we deduce that in particular there exists h0 ∈ C2 (R+ 0 ) such that h0 (φ1 (x)) = φ2 (x) for all x ≥ 0. A straightforward adaptation of the proof of [163, Lemma 5.8] yields that h

0 (x) > 0 for x ≥ 0. Lemma 3.13 Let X = {Xt }t ≥0 be an R+ 0 -valued process with a.s. continuous paths Aα ,φj defined by (3.87) are local martingales for and such that the processes ZX j = 1, 2. Then

Aα ,φ1  =2 ZX t

 0

t

Xs2 (φ1 (Xs ))2 ds

almost surely.

A ,g Moreover, ZX α is a local martingale whenever g ∈ C2,4 (R+ 0 ). Proof This proof is analogous to that of [163, Lemma 6.2], to which we refer for further details. Let h ∈ C2 (R+ 0 ). Given that Aα φ1 = 1, an application of the chain rule A ,φ shows that Aα (h(φ1 ))(x) = x 2 h

(φ1 (x)) (φ1 (x))2 + h (φ1 (x)). Since ZX α 1 is a local martingale, we can apply Itô’s formula for continuous semimartingales [138, Theorem 6.2] to the process h(φ1 (Xt )) and deduce that d(h(φ1 (Xt ))) = h (φ1 (Xt )) dφ1 (Xt ) + 12 h

(φ1 (Xt )) d[φ1 (X)]t . Consequently, d(h(φ1 (Xt ))) − Aα (h(φ1 (Xt ))) dt   = h

(φ1 (Xt )) 12 d[φ1 (X)]t − Xt2 (φ1 (Xt ))2 dt + dVth ,

(3.105)

 t  where Vth := 0 h (φ1 (Xs ))(dφ1 (Xs ) − ds) is a local martingale (cf. [138, A ,φ Proposition 2.63]; note that dφ1 (Xt ) − dt = dZX,tα 1 is the differential of a local martingale). If, in particular, h is the function h0 from Lemma 3.12(b), then

110

3 The Whittaker Convolution

t

A ,φ d(h0 (φ1 (Xs ))) − Aα (h0 (φ1 (Xs ))) ds = ZX,tα 2 is also a local martingale, and from (3.105) we find that 0



t 0

h

0 (φ1 (Xs ))



1 2 d[φ1 (X)]s

− Xs2 (φ1 (Xs ))2 ds

 is a local martingale.

t   But 0 h

0 (φ1 (Xs )) 12 d[φ1 (X)]s − Xs2 (φ1 (Xs ))2 ds is also a process of locally finite variation (cf. [138, Proposition 2.73]; note that 12 d[φ1 (X)]s − Xs2 (φ1 (Xs ))2 ds is the differential of a process of locally finite variation), hence it is a.s. equal to zero (see [138, Theorem 2.11]). Consequently, taking into account that h

0 > A ,φ 0 (Lemma 3.12(b)), we have d[ZX α 1 ]t − 2Xt2 (φ1 (Xt ))2 dt = d[φ1 (X)]t − 2Xt2 (φ1 (Xt ))2 dt = 0 a.s., proving the first assertion.  The result just proved, combined with (3.105), implies that h(φ1 (Xt )) −  t 2 + 0 Aα (h(φ1 ))(Xs )ds t ≥0 is, for each h ∈ C (R0 ), a local martingale. Applying Lemma 3.12(a) with f (x) := g(x 2 ) ∈ C2,4 (R+ 0 ), we find that g(x) ≡ h(φ1 (x)) for some h ∈ C2 (R+ ), and this proves the second assertion. 0 We are finally ready to establish the martingale characterization of the Shiryaev process. (We call it a Lévy-type characterization because it resembles the Lévy characterization of Brownian motion stated in Theorem 2.2. A parallel result for hypergroup structures is given in [163, Theorem 6.3].) Theorem 3.4 (Lévy-Type Characterization for the Shiryaev Process) Let Y = {Yt }t ≥0 be an R+ 0 -valued Markov process with a.s. continuous paths. The following assertions are equivalent: (i) Y is the Shiryaev process; (ii) {φ1 (Yt ) − t}t ≥0 and {φ2 (Yt ) − 2tφ1 (Yt ) + t 2 }t ≥0 are martingales (or local martingales); t A ,φ A ,φ (iii) ZY α 1 is a local martingale with [ZY α 1 ]t = 2 0 Ys2 (φ1 (Ys ))2 ds. Proof (i) ⇒ (ii): This follows from Proposition 3.17. A ,φ (ii) ⇒ (iii): Assume that (ii) is true. Since dZY,tα 1 = dφ1 (Yt ) − dt, the A ,φ process ZY α 1 is a local martingale. Furthermore,     A ,φ dZY,tα 2 = dφ2 (Yt ) − 2φ1 (Yt )dt = d φ2 (Yt ) − 2tφ1 (Yt ) + t 2 + 2t dφ1 (Yt ) − dt (where integration by parts [138, Proposition 2.28] gives the second equality) −A ,φ and therefore the process ZY α 2 is also a local martingale. By Lemma 3.13, t −A ,φ [ZY α 1 ]t = 2 0 Ys2 (φ1 (Ys ))2 ds. (iii) ⇒ (i): Assuming that (iii) holds, Eq. (3.105) and the proof of Lemma 3.13 W (·) show that, for each λ ≥ 0, ZYAα , α, is a local martingale and (by boundedness λ

3.7 Whittaker Convolution of Functions

111

on compact time intervals, cf. [138, Corollary 1.145]) a true martingale. Proposition 3.12 now yields that Y is the Shiryaev process. Remark 3.10 In this section we have focused on continuous-time stochastic processes which are additive with respect to the Whittaker convolution. In a similar way, one can introduce the discrete-time counterparts of the processes studied above. An R+ 0 -valued Markov chain {Sn }n∈N0 with S0 = 0 is said to be -additive if α

there exist measures μn ∈ P(R+ 0 ) such that

n ∈ N, x ≥ 0, B a Borel subset of R+ 0.

P [Sn ∈ B|Sn−1 = x] = (μn  δx )(B), α

If μn = μ for all n, then {Sn } is said to be a -random walk. One can give an explicit α

construction for -additive Markov chains, cf. [19, Section 7.1] (see also Sect. 4.5.3 α

below). In the context of hypergroups, moment functions have been successfully applied to the study of the limiting behaviour of additive Markov chains (cf. [19, Chapter 7]). Parallel results hold for the Whittaker convolution. For instance, letting {Sn } be a -additive Markov chain constructed as above, the following strong laws of large α

numbers are established as in [19, Theorems 7.3.21 and 7.3.24]: (a) If {rn }n∈N  is a sequence of positive2 numbers such that limn rn = ∞ and ∞ 1 ϕα,1 (Xn )] < ∞, then ϕα,2 (Xn )] − E[ n=1 rn E[  1  ϕα,1 (Sn )] = 0 lim √  ϕα,1 (Sn ) − E[ n rn

P -a.s.

(b) If {Sn } is a -random walk such that E[ ϕα,2(X1 )θ/2 ] < ∞ for some 1 ≤ θ < 2, α

then E[ ϕα,1 (X1 )] < ∞ and lim n

 1  ϕα,1 (Sn ) − nE[  ϕα,1 (X1 )] = 0

n1/θ

P -a.s.

3.7 Whittaker Convolution of Functions After having studied the probabilistic properties of the Whittaker convolution, we return to the study of the basic properties of this convolution, which we shall now regard as a binary operator on weighted Lp spaces. Definition 3.10 Let f, g : R+ → C be complex-valued functions. If the double integral  (f  g)(x) := α

0

∞

(Txα f )(ξ ) g(ξ ) rα (ξ )dξ

112

3 The Whittaker Convolution



∞ ∞

=

kα (x, y, ξ ) f (y) g(ξ ) rα (y)dy rα (ξ )dξ 0

0

exists for almost every 0 < x < ∞, then we call it the Whittaker convolution (of order α) of the functions f and g. Note that this definition is obtained from Definition 3.4 by letting μ and ν be the absolutely continuous measures defined by μ(dx) = f (x) rα (x)dx and ν(dx) = g(x) rα (x)dx. The Whittaker convolution of functions is positivity-preserving (i.e., f  g ≥ 0 whenever f, g ≥ 0) and commutative (i.e., f  g = g  f ). Moreover, it α

α

α

is a generalization of the Kontorovich–Lebedev convolution: indeed, for α = 0 we have (f  g)(x) = (f  g)(x) = 0

KL

1 −1 −3 1 1 π 2 x 2 e 2x (f ∗ g)( 2x ), KL 4

where  is the normalized Kontorovich–Lebedev convolution (3.7), ∗ is the KL

KL

classical Kontorovich–Lebedev convolution (3.4), f(x) = 3

1 g(x) = x − 2 e−x g( 2x ).

3 1 x − 2 e−x f ( 2x )

and

3.7.1 Mapping Properties in the Spaces Lp (rα ) The well-known Young convolution inequality has a natural analogue for the Whittaker convolution of functions belonging to the family of Lp spaces defined in (3.42). (The following result should also be compared with the Young inequality for the Hankel convolution stated in Proposition 2.22.) Proposition 3.18 (Young Inequality for the Whittaker Convolution) Let p1 , p2 ∈ [1, ∞] such that p11 + p12 ≥ 1. For f ∈ Lp1 (rα ) and g ∈ Lp2 (rα ), the

L-convolution f  g is well-defined and, for s ∈ [1, ∞] defined by α

1 s

=

1 p1

+ p12 − 1,

it satisfies f  gs,α ≤ f p1 ,α gp2 ,α α

(in particular, f  g ∈ Ls (rα )). Consequently, the Whittaker convolution is a α

continuous bilinear operator from Lp1 (rα ) × Lp2 (rα ) into Ls (rα ). Proof The proof is analogous to that of the Young inequality for the ordinary convolution. Define t11 = p11 − 1s and t12 = p12 − 1s . Observe that

1/s . |(Txα f )(y)| |g(y)| ≤ |(Txα f )(y)|p1 /t1 |g(y)|p2 /t2 |(Txα f )(y)|p1 |g(y)|p2

3.7 Whittaker Convolution of Functions

Since

1 s

+

1 t1

+



∞ 0

1 t2

113

= 1, we have by Hölder’s inequality and Proposition 3.2(b)

|(Txα f )(y)| |g(y)| rα (y)dy

 ≤

∞

0

1   |(Txα f )(y)|p1 rα (y)dy 

∞

× 0

∞

t1

1

t2

|g(y)| rα (y)dy p2

0

1 |(Txα f )(y)|p1 |g(y)|p2 rα (y)dy 

≤ f pp11 /t1 gpp22 /t2

∞ 0

s

1/s |(Txα f )(y)|p1 |g(y)|p2 rα (y)dy

.

Using again Proposition 3.2(b) we conclude that p /t

p /t

p /s

p /s

f  gs,α ≤ f p11 ,α1 gp22 ,α2 f p11 ,α gp22 ,α = f p1 ,α gp2 ,α . α

Another analogue of a well-known property of the ordinary convolution is the fact that the Whittaker convolution of functions belonging to Lp spaces with conjugate exponents defines a continuous function: Proposition 3.19 Let p, q ∈ [1, ∞] with Lq (rα ), then f  g ∈ Cb (R+ ).

1 p

+

= 1. If f ∈ Lp (rα ) and g ∈

1 q

α

Proof The previous proposition ensures the boundedness of f  g. For the α

continuity, let x0 > 0; then for 1 < p < ∞ we have     (f  g)(x) − (f  g)(x0 ) =   α

α

∞ 0

(Txα f )(ξ )

− (Txα0 f )(ξ )



≤ Txα f − Txα0f p,α gq,α → 0

  g(ξ ) rα (ξ )dξ  as x → x0

by Hölder’s inequality and Proposition 3.2(c). In the case p = ∞ (and by symmetry p = 1), the continuity of f  g follows by dominated convergence, using parts (a) α

and (c) of Proposition 3.2. Some fundamental connections between the Whittaker convolution (and translation), the index Whittaker transform and the differential operator Aα are given in the following proposition. Proposition 3.20 Let y > 0 and τ ≥ 0. Then:   y (a) If f ∈ L2 (rα ), then Wα (Tα f ) (τ ) = Wα,iτ (y) (Wα f )(τ );   (b) If f ∈ L2 (rα ) and g ∈ L1 (rα ), then Wα (f  g) (τ ) = (Wα f )(τ ) (Wα g)(τ ); α

114

3 The Whittaker Convolution y

y

(c) If f ∈ L2 (rα ) and g ∈ L1 (rα ), then Tα (f  g) = (Tα f )  g; (d) If f ∈ (e) If f ∈

α α y y y (2) D(2) α , then Tα f ∈ Dα and Aα (Tα f ) = Tα (Aα f ); (2) 1 D(2) α and g ∈ L (rα ), then f  g ∈ Dα and Aα (f  g) α α

= (Aα f )  g. α

Proof (a) Let f ∈ L1 (rα ) ∩ L2 (rα ). Using Fubini’s theorem and the product formula (3.36), we compute 

 Wα (Tyα f ) (τ ) =



∞ ∞ 0

f (ξ )kα (x, y, ξ ) rα (ξ )dξ Wα,iτ (x) rα (x)dx

0



= Wα,iτ (y)

∞

f (ξ ) Wα,iτ (ξ ) rα (ξ )dξ

0

= Wα,iτ (y) (Wα f )(τ ). By denseness and continuity, the equality extends to all f ∈ L2 (rα ), as required. (b) For f ∈ L1 (rα ) ∩ L2 (rα ) and g ∈ L1 (rα ) we have 

 Wα (f  g) (τ ) = α



∞ ∞

0

 =

0

0 ∞

(Txα f )(ξ ) g(ξ ) rα (ξ )dξ Wα,iτ (x) rα (x)dx

  g(ξ ) Wα (Tξα f ) (τ ) rα (ξ )dξ 

= (Wα f )(τ )

∞

g(ξ ) Wα,iτ (ξ ) rα (ξ )dξ = (Wα f )(τ ) (Wα g)(τ ),

0

where we have used Fubini’s theorem and part (a). Again, denseness yields the result. (c) By the previous properties,  



Wα Tyα (f  g) (τ ) = Wα (Tyα f )  g (τ ) = Wα,iτ (y) (Wα f )(τ ) (Wα g)(τ ). α

α

y

y

Since both Tα (f  g) and (Tα f )  g are elements of the space L2 (rα ) (see α

α

y

y

Proposition 3.21 below), this implies that Tα (f  g) = (Tα f )  g. α α y  (2)  (2) (d) Recalling the inequality (3.67), it is evident that Tα Dα ∈ Dα . Since the  y y index Whittaker transforms of Aα (Tα f ) and Tα (Aα f ) are both equal to τ 2 +  ( 12 − α)2 Wα,iτ (y)(Wα f )(τ ), the result follows. (e) The proof is similar to that of (d).

3.7 Whittaker Convolution of Functions

115

We have seen in Proposition 3.18 that if f ∈ L2 (rα ) and g ∈ Lp (rα ) (1 ≤ p < 2p

2) then the Whittaker convolution f  g exists and belongs to L 2−p (rα ). Using the α

index Whittaker transform, this result can be strengthened as follows: Proposition 3.21 Let f ∈ L2 (rα ) and g ∈ Lp (rα ) (1 ≤ p < 2). Then f  g ∈ α

L2 (rα ), and we have f  g2,α ≤ Cp f 2,α gp,α , α

  where Cp = Wα,0 q,α < ∞ (being

1 p

+

1 q

= 1).

Proof The fact that Wα,0 q,α is finite for each 2 < q ≤ ∞ is easily verified using the limiting forms (3.10), (3.12). Now, for f, g ∈ Cc (R+ ) we have   f  g2,α = (Wα f )·(Wα g)L2 (ρα ) α     ≤ sup (Wα g)(τ ) ·  Wα f L2 (ρα ) τ ≥0

  ≤ Wα,0 

q,α

gp,α f 2,α ,

  where we denoted L2 (ρα ) = L2 R+ ; ρα (τ )dτ ; we have used the isometric property of the index Whittaker transform, and the final step relies on the inequality |Wα,iτ (x)| ≤ Wα,0 (x) (proved in (3.66)) and on Hölder’s inequality. As usual, the result for f ∈ L2 (rα ) and g ∈ Lp (rα ) follows from the denseness of Cc (R+ ) in these Lp spaces. Corollary 3.3 (a) If f, g ∈ L2 (rα ), then f  g ∈ Lq (rα ) for all 2 < q ≤ ∞, with α

f  gq,α ≤ Cq f 2,α g2,α α

  being Cq = Wα,0 q,α. (b) Let 1 ≤ p1 < 2 and 1 ≤ p2 ≤ 2 such that 1 t

=

1 p1

+

s ∈ [2, t].

1 p2

− 1. If f ∈

Lp1 (rα )

and g ∈

1 1 3 p1 + p2 ≤ 2 . Let t Lp2 (rα ), then f  g ∈ α

be defined by Ls (rα ) for all

116

3 The Whittaker Convolution

Proof The following proof is adapted from [61, Section 5]. (a) Let h ∈ Lp (rα ) ( p1 + q1 = 1) and f, g ∈ Cc (R+ ). Using Proposition 3.21 and Fubini’s theorem, we obtain  ∞   ∞    ≤ (f  g)(x)h(x)r (x)dx (|f |  |h|)(x)|g(x)|rα (x)dx α   α

0

0

α

  ≤ g2,α |f |  |h|2,α ≤ Cq g2,α f 2,α hp,α . α

Therefore f  gq,α = α

sup h∈Lp (rα ) hp,α ≤1

   

  (f  g)(x)h(x)rα (x)dx  ≤ Cq f 2,α g2,α α

∞ 0

and the usual continuity argument yields the result. (b) By the Young inequality (Proposition 3.18) f  g ∈ Lt (rα ), and we know that α

L2 (rα ) ∩ Lt (rα ) ⊂ Ls (rα ), thus we just need to show that f  g ∈ L2 (rα ). α

Observe that g1 := g · 1{|g|≥1} ∈ L1 (rα ) ∩ Lp2 (rα ) and g2 := g · 1{|g| 0, f ∈ Lα,0

    1 1 if and only if f ∈ L1 (0, 1], x −2α e− x dx ∩ L1 [1, ∞), x − 2 −α log x dx ,

(3.106) and therefore the spaces Lα,ν are ordered: Lα,ν1 ⊂ Lα,ν2 whenever ν1 > ν2 .

3.7 Whittaker Convolution of Functions

117

It is also interesting to note that the family {Lα,ν }ν≥0 contains the space L1 (rα ) ≡ Lα, 1 −α . (Recall that by (3.17) we have Wα, 1 −α (y) = 1.) 2 2 The following lemma collects some properties of the index Whittaker transform in the spaces Lα,ν (α < 12 , ν ≥ 0): Lemma 3.14 If f ∈ Lα,ν , then its index Whittaker transform (Wα f )(τ ) = ∞ f (y) Wα,iτ (y) rα (y)dy is, for every τ belonging to the complex strip |Im τ | ≤ ν, 0 well-defined as an absolutely convergent integral, and it satisfies (Wα f )(τ ) −−−−→ 0

uniformly in the strip |Im τ | ≤ ν.

τ →∞

(3.107)

Moreover, if (Wα f )(τ ) = 0 for all τ ≥ 0, then f (x) = 0 for almost every x > 0. Proof The absolute convergence of the integral defining Wα f is clear from the inequality (3.66). It follows from (3.63) and the Riemann-Lebesgue lemma that for each y > 0 we have W α,iτ (y) −→ 0 as τ → ∞ uniformly in the strip |Im τ | ≤ ν, hence dominated convergence gives (3.107). Letting μ be the (possibly unbounded) measure μ(dx) = f (x) rα (x)dx, the same proof of Proposition 3.4(ii) shows that if $ μ(τ 2 + ( 12 − α)2 ) ≡ (Wα f )(τ ) = 0 for all τ ≥ 0, then μ is the zero measure, so that f (x) = 0 a.e. Proposition 3.22 For f, g ∈ Lα,ν , the Whittaker convolution f  g is well-defined α

and satisfies f  gLα,ν ≤ f Lα,ν gLα,ν α

(in particular, f g ∈ Lα,ν ). Moreover, properties (a) and (b) in Proposition 3.20 are α

valid when f and g belong to Lα,ν and τ is a complex number such that |Im τ | ≤ ν. Proof We compute f  gLα,ν α



∞ ∞ ∞

≤ 0

 = 

0

0

0

∞ ∞ ∞

0 ∞

=

0

|f (y)|kα (x, y, ξ )rα (y)dy |g(ξ )|rα (ξ )dξ W α,ν (x)rα (x)dx W α,ν (x)kα (x, y, ξ )rα (x)dx |f (y)|rα (y)dy |g(ξ )|rα (ξ )dξ 

∞

|f (y)| W α,ν (y)rα (y)dy

0

|g(ξ )| W α,ν (ξ )rα (ξ )dξ

0

= f Lα,ν gLα,ν , where the positivity of the integrand justifies the change of order of integration, and the second equality follows from the product formula (3.36). The final statement is proved using the same calculations as before.

118

3 The Whittaker Convolution

Corollary 3.4 The Banach space Lα,ν , equipped with the convolution multiplication f · g ≡ f  g, is a commutative Banach algebra without identity element. α

Proof Proposition 3.22 shows that the Whittaker convolution defines a binary operation on Lα,ν for which the norm is submultiplicative. The commutativity and associativity of the   Whittaker convolution in the space Lα,ν follows from the property Wα (f  g) (τ ) = (Wα f )(τ ) (Wα g)(τ ) and the injectivity property of α

Lemma 3.14. Suppose now that there exists e ∈ Lα,ν such that f  e = f for all f ∈ Lα,ν . α

This means that (Wα f )(τ ) (Wα e)(τ ) = (Wα f )(τ )

for all f ∈ Lα,ν and τ ≥ 0.

Clearly, this implies that (Wα e)(τ ) = 1 for all τ ≥ 0, which contradicts Lemma 3.14. This shows that there exists no identity element for the Whittaker convolution on the space Lα,ν . We will see that the index Whittaker transform on the Banach algebra Lα,ν admits a Wiener-Lévy type theorem which resembles the classical Wiener-Lévy theorem on integral equations with difference kernel (cf. [71, §17, Corollary 1], [111, p. 164]). For this, we will need the following lemma: Lemma 3.15 Let J : Lα,ν −→ C be a linear functional satisfying J (f  g) = J (f )·J (g)

for all f, g ∈ Lα,ν .

α

(3.108)

∞ Then J (f ) = 0 f (ξ ) W α,iτ (ξ ) rα (ξ )dξ for some τ belonging to the complex strip |Im τ | ≤ ν, including infinity. Notice that τ = ∞ corresponds, by (3.107), to the zero functional on Lα,ν . Proof By the standard theorem on duality of Lp spaces, 

∞

J (f ) =

f (ξ ) ω(ξ ) rα (ξ )dξ, 0

where

ω W α,ν

∈ L∞ (rα ), i.e., (3.57) holds. Since J (f  g) = J (f ) · J (g), for f, g ∈ α

Lα,ν we have 

∞



∞

f (ξ ) ω(ξ ) rα (ξ )dξ ·

0



∞ ∞

= 0

0

g(ξ ) ω(ξ ) rα (ξ )dξ 0

(Tξα f )(y) g(y)rα (y)dy ω(ξ )rα (ξ )dξ

3.7 Whittaker Convolution of Functions



∞ ∞

= 

0

=

0

∞ ∞

0

0

119

(Tyα f )(ξ ) ω(ξ )rα (ξ )dξ g(y)rα (y)dy (Tyα ω)(ξ ) f (ξ )rα (ξ )dξ g(y)rα (y)dy,

where the last equality follows from the commutativity of the Whittaker convolution, cf. Corollary 3.4. (The commutativity easily extends to f ∈ Lα,ν and ω ∞ W α,ν ∈ L (rα ) via a continuity argument.) Since f and g are arbitrary,  ω(x) ω(y) = (Tyα ω)(x) ≡

∞

ω(ξ ) kα (x, y, ξ ) rα (ξ )dξ

for a.e. x, y > 0

0

and the conclusion follows from Lemma 3.5. Theorem 3.5 (Wiener-Lévy Type Theorem) Let f ∈ Lα,ν (α < 12 , ν ≥ 0) and " ∈ C. The following assertions are equivalent: (i) " + (Wα f )(τ ) = 0 for all τ belonging to the complex strip |Im τ | ≤ ν, including infinity; (ii) There exists a unique function g ∈ Lα,ν such that 1 = "−1 + (Wα g)(τ ) " + (Wα f )(τ )

(|Im τ | ≤ ν).

(3.109)

Before the proof, we need to introduce some relevant notions from Gelfand’s theory of maximal ideals in commutative Banach algebras (cf. e.g. [170, Chapter 6]). Let V be a commutative Banach algebra with identity element. A proper ideal on V is a nonempty linear subspace I  V such that v · x = x · v ∈ I whenever v ∈ V and x ∈ I. A proper ideal I is said to be maximal in V if I = J whenever J is a proper ideal such that I ⊂ J. A linear functional F : V −→ C is said to be a multiplicative linear functional if F ≡ 0 and F (x ·y) = F (x)F (y) for all x, y ∈ V . (This implies that F (e) = 1, where e is the identity element in V .) We will make use of the following basic results [170, Proposition 6.1.12 and Theorem 6.2.2]: • If v ∈ V is not invertible, then v is contained in a maximal ideal of V ; • If F is a multiplicative linear functional on V , then I = Ker(F ) ≡ {v ∈ V | F (v) = 0} is a maximal ideal on V , and conversely if I is a maximal ideal on V then there exists a unique multiplicative linear functional F : V −→ C such that I = Ker(F ). Proof of Theorem 3.5 (i) ⇒ (ii): Let Vα,ν be the Banach algebra obtained from Lα,ν by formally adjoining an identity element e, that is, Vα,ν := {"e + f (·) | " ∈ C, f ∈ Lα,ν } endowed with the norm "e + f  =  |"| + f Lα,ν  . The index Whittaker transform is naturally extended to Vα,ν as Wα ("e + f ) (τ ) := " + (Wα f )(τ ) (|Im τ | ≤ ν). It follows from Proposition 3.22 that Jα,τ : Vα,ν −→ C,

  Jα,τ ("e + f ) := Wα ("e + f ) (τ )

(3.110)

120

3 The Whittaker Convolution

is, for each τ in the strip |Im τ | ≤ ν (including infinity), a multiplicative linear functional on Vα,ν . We claim that there are no multiplicative linear functionals in Vα,ν other than the functionals Jα,τ defined in (3.110). Indeed, if J : Vα,ν −→ C is a multiplicative linear functional, then restricting to Lα,ν we obtain a functional f → J (f ) on Lα,ν such that (3.108) holds. By Lemma 3.15, J (f ) = (Wα f )(τ ) for some τ in the strip |Im τ | ≤ ν (including infinity), and thus by linearity J ("e+f ) = " + (Wα f )(τ ). Hence J = Jα,τ , as we had claimed. Assume that " + (Wα f )(τ ) = 0 for all τ with |Im τ | ≤ ν. By the above, we have "e + f ∈ / Ker(J ) for all multiplicative linear functionals J : Vα,ν −→ C, and using the results stated before the proof we deduce that "e + f is invertible on Vα,ν . Denoting the inverse by " e + g (" ∈ C, g ∈ Lα,ν ), we obtain     " + (Wα f )(τ ) · " + (Wα g)(τ ) = 1

(|Im τ | ≤ ν).

We know that limτ →∞ W α,iτ (y) = 0 for y > 0, hence as in Lemma 3.14 it follows that the left hand side equals "" when τ = ∞. We thus have " = "−1 , so that (3.109) holds. (ii) ⇒ (i): This implication is straightforward: given that g ∈ Lα,ν , Lemma 3.14 ensures that |(Wα g)(τ )| ≤ (Wα |g|)(iν) < ∞ for all τ in the strip |Im τ | ≤ ν. Since (3.109) holds, it follows that " +(Wα f )(τ ) = −1 1 = 0 " +( Wα g)(τ ) for all τ with |Im τ | ≤ ν.

3.8 Convolution-Type Integral Equations In this final section of the chapter we demonstrate that the Whittaker convolution, and especially the analogue of the Wiener-Lévy theorem proved above, can be used to study the existence of solution for integral equations of the second kind which can be represented as Whittaker convolution equations, in the sense defined as follows: Definition 3.11 The integral equation of the second kind 

∞

f (x) +

J (x, y)f (y) dy = h(x),

(3.111)

0

where h is a known function and f is to be determined, is said to be a Whittaker convolution equation if there exists α < 12 and θ ∈ Lα,0 such that J (x, y) = (Txα θ )(y) rα (y) ≡ (Txα θ )(y) y −2α e−1/y . In other words, (3.111) is a Whittaker convolution equation if it can be written in the form f (x) + (f  θ )(x) = h(x) α

for some α
0, α < 12 ),

where r(ξ ) := 1c rα ( ξc ) and p(ξ ) := cγ ξ 2(1−α)e−1/ξ . It should be noted that this extension includes the infinitesimal generator of a general (nonstandardized) 2 Shiryaev process (as defined in (3.1)) with drift μ > σ2 . The crucial observation here is that the solutions of the more general Sturm–Liouville problem −Au = λu (with Neumann boundary conditions) can also be expressed in terms of the Whittaker W function and, therefore, the corresponding product formula can be obtained by applying elementary changes of variables to the product formula determined in Sect. 3.2. The kernels of the more general product formula also constitute a family of probability densities, so the induced notions of generalized translation, convolution, Lévy processes and moment functions (defined in analogy with those of the previous sections) have essentially the same properties as before.

3.8 Convolution-Type Integral Equations

125

Table 3.1 Basic results and definitions for the extended Whittaker convolution  

Solution of the Sturm–Liouville W α,# ( xc ) # := ( 12 − α)2 − γλ boundary value problem [−Au = λu, u(0) = 1, (pu )(0) = 0] ∞ Product formula for the W α,ν ( xc ) W α,ν ( yc ) = 0 W α,ν ( ξc ) k(x, y, ξ ) r (ξ )dξ

 Sturm–Liouville solutions k(x, y, ξ ) := kα ( xc , yc , ξc ) ∞ Extended Whittaker translation (Ty f )(x) := 0 f (ξ ) q(x, y, ξ ) r (ξ )dξ   y/c operator ≡ Tα f (c ·) ( xc ) ∞ Extended index Whittaker transform (Wf )(τ ) := 0 f (y) Wα,iτ ( yc ) r(y)dy   of functions and measures ≡ Wα f (c ·) (τ )   (Wμ)(λ) := R+ Wα,# ( yc ) μ(dy) ≡  1/c μ(γ λ) ∞0 x Extended Whittaker convolution of (fg)(x) := 0 (T f )(ξ ) g(ξ ) r(ξ )dξ   functions and measures = f (c ·)  g(c ·) ( xc ) α   (μν)(dξ ) := R+ R+ q(x, y, ξ )r (ξ )dξ μ(dx) ν(dy) 0

-infinitely divisible measures, -convolution semigroups and -Lévy processes (Canonical) -moment functions

0

Replace  by  in the previous definitions α

ϕk (x) = ϕk ( xc ), where ϕk (·) are (canonical) -moment functions α

Extended Whittaker convolution equation

An equation of the form (3.111), with J (x, y) = (Tx θ)(y)r (y)

Table 3.1 collects the product formula and the definitions of the fundamental objects which underlie the construction of this extension of the Whittaker convolution structure. Using these definitions and the same proofs as before, it is straightforward to check that the main results of the previous sections—such as the Lévy-type martingale characterization for the nonstandardized Shiryaev process or the Wiener-Lévy type theorem for the index Whittaker transform—are also valid for the extended Whittaker convolution structure.

Chapter 4

Generalized Convolutions for Sturm-Liouville Operators

This chapter is dedicated to the problem of constructing Sturm-Liouville convolutions, i.e. generalized convolution operators associated with Sturm-Liouville differential expressions. The convolutions constructed here will, in particular, allow us to interpret the diffusion process generated by the Neumann realization (L(2) , D(L(2))) of the Sturm-Liouville operator as a Lévy-like process. We consider Sturm-Liouville operators of the form (2.10) but without zero order term, that is,

=−

1 d  d  p , r dx dx

x ∈ (a, b)

(4.1)

(−∞ ≤ a < b ≤ ∞). Throughout the chapter we always assume that the coefficients are such that p(x), r(x) > 0 for all x ∈ (a, b), p, p , r, r ∈ c  c dx ACloc (a, b) and a y p(x) r(y)dy < ∞. Remark 4.1 We shall make extensive use of the fact that the differential expression (4.1) can be transformed into the standard form d2 A d 1 d  d   A =− 2 − .

=− A dξ dξ dξ A dξ This is achieved by setting A(ξ ) :=



p(γ −1 (ξ )) r(γ −1 (ξ )),

(4.2)

where γ −1 is the inverse of the increasing function  x' r(y) γ (x) = dy, p(y) c © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Sousa et al., Convolution-like Structures, Differential Operators and Diffusion Processes, Lecture Notes in Mathematics 2315, https://doi.org/10.1007/978-3-031-05296-5_4

127

128

4 Generalized Convolutions for Sturm-Liouville Operators

 r(y) c ∈ (a, b) being a fixed point (if p(y) is integrable near a, we may also take c = a). Indeed, we know from Remark 2.1 that a given function ωλ : (a, b) → C satisfies (ωλ ) = λωλ if and only if  ωλ (ξ ) := ωλ (γ −1 (ξ )) satisfies 

( ωλ ) = λ ωλ .

4.1 Known Results and Motivation As noted in Sect. 2.3, the roots of the problem of constructing Sturm-Liouville convolutions originate in the work of Delsarte and Levitan on generalized translation operators determined by ordinary differential operators [45, 117, 118, 120, 121]. This theory was later developed by Chébli [30], who constructed convolutions satisfying the hypergroup axioms for Sturm-Liouville operators belonging to a 2

η+ 1

d 1 2 d family which includes the Bessel operator 12 dx 2 + x dx (η > − 2 ) and the Jacobi 1 operator (2.42) with α ≥ β ≥ 2 . In turn, Zeuner [209, 210] introduced the general notion of a Sturm-Liouville hypergroup and extended the results of Levitan and Chébli to a larger class of differential operators. (Further remarks on the historical development of the topic can be found in [19, pp. 256–257].) The definition of Sturm-Liouville hypergroup proposed by Zeuner reads as follows:

Definition 4.1 ([209]) A hypergroup (R+ 0 , ∗) is said to be a Sturm-Liouville hypergroup if there exists a function A on R+ 0 satisfying the condition SL0

1 + A ∈ C(R+ 0 ) ∩ C (R ) and A(x) > 0 for x > 0

such that, for every function f ∈ C∞ c,even , the convolution  vf (x, y) =

R+ 0

f (ξ )(δx ∗ δy )(dξ )

(4.3)

  2 belongs to C2 (R+ 0 ) and satisfies ( x vf )(x, y) = ( y vf )(x, y), (∂y vf )(x, 0) = 0 1 ∂ ∂ (x > 0), where x = − A(x) ∂x (A(x) ∂x ). The following fundamental existence theorem for Sturm-Liouville hypergroups was established in [210]: Theorem 4.1 Suppose that A satisfies SL0 and is such that SL1

One of the following assertions holds:

(x) = αx0 + α1 (x) for x in a neighbourhood of 0, where SL1.1 A(0) = 0 and AA(x) ∞ α0 > 0 and α1 ∈ C (R) is an odd function; SL1.2 A(0) > 0 and A ∈ C1 (R+ 0 ).

A SL2 There exists η ∈ C1 (R+ 0 ) such that η ≥ 0, the functions φη := A − η,

A ψ η := 12 η − 14 η2 + 2A ·η are both decreasing on R+ and limx→∞ φη (x) = 0.

4.1 Known Results and Motivation

129

Define the convolution ∗ via (4.3) where, for f ∈ C∞ c,even , vf denotes the unique solution of x vf = y vf , vf (x, 0) = vf (0, x) = f (x), (∂y vf )(x, 0) = (∂x vf )(0, y) = 0. Then (R+ 0 , ∗) is a Sturm-Liouville hypergroup. Remark 4.2 In this chapter we focus on the construction of convolutions defined on noncompact intervals of R. However, it is worth noting that Sturm-Liouville hypergroups have also been studied on compact intervals: a hypergroup ([b1 , b2 ], ∗) is said to be a Sturm-Liouville hypergroup of compact type [19, Definition 3.5.76] if there exists a function A ∈ C[b1 , b2 ] ∩ C1 (b1 , b2 ) such that A(x) > 0 for b1 < b x < b2 and b12 A(x)dx = 1 which satisfies the following requirement: for each  f ∈ C∞ c (b1 , b2 ) the convolution [b1 ,b2 ] f (ξ )(δx ∗ δy )(dξ ) is twice differentiable and satisfies ( x vf )(x, y) = ( y vf )(x, y) and (∂y vf )(x, b1 ) = (∂y vf )(x, b2 ) = 0 1 ∂ ∂ (b1 < x < b2 ), where x = − A(x) ∂x (A(x) ∂x ) is the associated Sturm-Liouville operator. The simplest example (where A is constant) is the two-point support hypergroup ([0, β], ) defined as δx  δy = 12 (δ|x−y| + δβ−|β−x−y| ). Another β

β

important example (cf. [19, p. 242]) is that of the compact Jacobi hypergroup ([−1, 1],  ): for (α, β) such that −1 < β ≤ α and either β ≥ − 12 or α + β ≥ 0, α, β

(α,β)

(α,β)

this hypergroup is defined as δx  δy := νx,y , where νx,y is the unique measure α, β such that the product formula  Rn(α,β) (x)Rn(α,β) (y) =

[−1,1]

Rn(α,β) (ξ ) ν(α,β) x,y (dξ )

(n ∈ N0 )

(4.4)

(α,β)

defined in Example 2.2. (It is proved in holds for the Jacobi polynomials Rn (α,β) [70] that for each x, y ∈ [−1, 1] there exists a unique measure νx,y ∈ P[−1, 1] such that (4.4) holds.) Further existence theorems for Sturm-Liouville hypergroups of compact type associated with suitable families of differential operators have been established in [19, 210]. Theorem 4.1 is the existence theorem which underlies the general theory of one-dimensional Sturm-Liouville convolution-like operators developed in [19, 163, 184, 210]; it includes, as particular cases, all the concrete examples of hypergroup structures on R+ 0 which have been reported in earlier literature (cf. [19, 68]). In the previous chapter we studied the problem of existence of a Sturm-Liouville convolution for the particular case of the generator (3.2) of the Shiryaev process, and we established the following result: y

Proposition 4.1 Let Tα and  be the Whittaker translation and convolution α

(Definitions 3.2 and 3.4 respectively), and let f ∈ C∞ c,even . Then the function  vf (x, y) := (Tyα f )(x) ≡

R+ 0

f (ξ )(δx  δy )(dξ ) α

130

4 Generalized Convolutions for Sturm-Liouville Operators

is a solution of Aα,x vf = Aα,y vf , vf (x, 0) = vf (0, x) = f (x), (∂y[1] vf )(x, 0) = (∂x[1] vf )(0, y) = 0. (Here Aα,x and Aα,y denote the differential operator (3.2) ∂ .) acting on the variable x and y respectively, and ∂ξ[1] := pα (ξ ) ∂ξ Proof The fact that vf is a solution of Aα,x vf = Aα,y vf follows from the proof of Lemma 3.5, and the claimed boundary conditions at the axes x = 0 and y = 0 were proved in Lemma 3.10. This result is not a particular case of Theorem 4.1 because the Sturm-Liouville operator Aα does not belong to the family of operators satisfying assumptions SL1– SL2. (Note that Aα is transformed, via the change of variables z = log x, into the d2 −z ) d defined on the interval (a, b) = (−∞, ∞).) operator dz 2 + (1 − 2α + e dz Moreover, it was observed in Sect. 3.5 that, unlike the convolutions of Theorem 4.1, the Whittaker convolution does not satisfy the compactness axiom H6 of hypergroups. However, many of the properties of the Whittaker convolution established in Chap. 3 are remarkably similar to those of Sturm-Liouville hypergroups. This leads to natural questions, namely whether one can construct other Sturm-Liouville convolutions which do not satisfy the compactness axiom and, more specifically, whether it is possible to achieve this by extending the PDE approach of [210] to the Sturm-Liouville operator Aα and other operators of a similar sort. A positive answer to these questions is given within this chapter.

4.2 Laplace-Type Representation The possibility of constructing a generalized convolution associated with the SturmLiouville expression (4.1) is strongly connected with the positivity-preservingness of solutions of the hyperbolic Cauchy problem x v = y v, v(x, 0) = v(0, x) = f (x), (∂y v)(x, 0) = (∂x v)(0, y) = 0. We now introduce an assumption which will be seen to be sufficient for the Cauchy problem to be positivity preserving. Recall that the function A, defined in (4.2), is the coefficient associated with the transformation of into the standard form (Remark 4.1).  b  r(y) Assumption MP We have γ (b) = c p(y) dy = ∞, and there exists η ∈



A C1 (γ (a), ∞) such that η ≥ 0, the functions φη := AA − η, ψ η := 12 η − 14 η2 + 2A ·η are both decreasing on (γ (a), ∞) and φη satisfies limξ →∞ φη (ξ ) = 0.

The reader will notice that this assumption is similar to condition SL2 in the existence theorem for Sturm-Liouville hypergroups stated above (Theorem 4.1), but it is more general as it does not require the function η to be C1 at the left endpoint of the interval. As in Sect. 2.2, in the sequel we denote by wλ (·) the unique solution of (2.12), and {am }m∈N will denote a sequence b > a1 > a2 > . . . with lim am = a. Having in mind the product formula that we shall establish for Sturm-Liouville expressions

4.2 Laplace-Type Representation

131

(4.1) which satisfy Assumption MP, in this section we prove the related fact that the solution wλ (x) of the initial value problem (2.12) admits a representation as the Fourier transform of a subprobability measure. To this end, we need a few lemmas. We start by stating some important properties which hold for all Sturm-Liouville operators of the form (4.1) which satisfy Assumption MP. Lemma 4.1 If Assumption MP holds, then (a) The function

A

A

:=

is nonnegative, and there exists a finite limit σ

A (ξ ) + limξ →∞ 2A(ξ ) ∈ R0 . If λ ≤ σ 2 , then wλ (x) > 0 for all x ∈ [a, b). If λ > σ 2 , then wλ (·) has infinitely many zeros

(b) (c) on [a, b). (d) b is a natural endpoint for the Sturm-Liouville operator . Proof The proofs of (a) and (b) are rather technical and rely on a careful study of (the coefficients of) the differential operator 

; see, respectively, Section 2 and Proposition 4.2 of [210]. Concerning part (c), we first √ apply the Liouville transformation (Remark 2.1) to deduce that that the function A(ξ ) wλ (γ −1 (ξ )) is a solution of −v

+(q−λ)v = 0, where q(ξ ) =

 A (ξ ) 2 2A(ξ )

+

 A (ξ ) 

2A(ξ )

=

1 2 1 φ (ξ ) + ψ η (ξ ) + φ η (ξ ), 4 η 2

ξ ∈ (γ (a), ∞).

(4.5) We know from Assumption MP and [210, Lemma 2.9] that limξ →∞ φη (ξ ) = 0 and limξ →∞ η (ξ ) = 0. In turn, the fact that φη is positive and decreasing clearly implies that φ η ∈ L1 ([c, ∞), dξ ) for c > γ (a) and, therefore, limξ →∞ φ η (ξ ) = 0. We thus have limξ →∞ q(ξ ) = σ 2 . Using a basic oscillation criterion √ for second order ordinary differential equations [48, XIII.7.37], we conclude that A(ξ ) wλ (γ −1 (ξ )) has infinitely many zeros on [γ (a), ∞) whenever λ > σ 2 , so that (c) holds. Part (d) follows from the general fact that the existence of oscillatory solutions for the Sturm-Liouville equation (that is, solutions with infinitely many zeros) implies that the essential spectrum of any self-adjoint realization of the SturmLiouville expression is nonempty [48, XIII.7.39], which in turn implies that, in the Feller boundary classification, at least one endpoint must be natural [137, Theorem 3.1]. Our second lemma states that the family of Sturm-Liouville expressions satisfying Assumption MP is closed under changes of variable determined by the multiplication of the coefficients by (squared) strictly positive solutions of the Sturm-Liouville problem. It is based on a known result on changes of spectral functions for Sturm-Liouville operators and Krein strings ([110], see also [51, Section 6.9]).

132

4 Generalized Convolutions for Sturm-Liouville Operators

 d d p dx be a Sturm-Liouville expression satisfying Lemma 4.2 Let = − 1r dx Assumption MP. For −∞ < κ ≤ σ 2 , consider the modified differential expression

κ = −

1 d  κ d  p , r κ dx dx

x ∈ (a, b),

where pκ = wκ2 ·p and r κ = wκ2 ·r. Then Assumption MP also holds for κ, and the function κ

wλ (x) :=

wκ+λ (x) wκ (x)

(4.6)

is, for each λ ∈ C, the unique solution of κ(w) = λw, w(a) = 1 and (pκw )(a) = 0. Moreover, the spectral measure associated with κ (Theorem 2.5) is given by κ

ρL (λ1 , λ2 ] = ρ L (λ1 + κ, λ2 + κ]

(−∞ < λ1 ≤ λ2 < ∞).

Proof Fix −∞ < κ ≤ σ 2 . The functions A and Aκ associated to the operators

and κ respectively (defined as in (4.2)) are connected by Aκ = w κ2 · A, where −1 w κ (ξ ) = wκ (γ (ξ )). In order to show that Assumption MP holds for κ , write a˜ m = γ (am ) and 2 (ξ ) · A(ξ ), where a˜ consider the function Aκ,m(ξ ) := w κ,m m ≤ ξ < ∞ and w 

κ,m w λ,m (ξ ) = wλ,m (γ −1 (ξ )). Let ηκ,m := η + 2 w κ,m , where η satisfies the conditions

of Assumption MP. By Lemma 4.1(b) we have ηκ,m ∈ C1 [a˜ m , ∞), and it is easily seen (cf. [210, Example 4.6]) that φηκ,m :=

(Aκ,m)

− ηκ,m = φη , Aκ,m

ψ ηκ,m = ψ η − κ,

ηκ,m(a˜ m ) = η(a˜ m ) ≥ 0,

and then one can show that ηκ,m ≥ 0 (see [210, Remark 2.12]), hence Assumpw κ (ξ ) tion MP holds for the function Aκ,m. If we now let ηκ(ξ ) := η(ξ ) + 2 w κ (ξ ) = κ,m (ξ ) (where γ (a) < ξ < ∞; the second equality is due to (2.18)), limm→∞ η then it is clear that the limit function ηκ satisfies Assumption MP for the function Aκ associated with the operator κ. A simple computation gives −

1 r κ

p

 !

κ+λ

w

κ

wκ

=−



1

p wκ+λ wκ − p wκ+λ wκ

wκ2 ·r

=−

 1

wκ+λ (x)

(wκ+λ ) wκ − wκ+λ (wκ ) = λ , 2 wκ (x) wκ

4.2 Laplace-Type Representation κ

133

κ

so that κ(wλ ) = λwλ . The boundary conditions at a are also straightforwardly checked. To prove the last assertion, notice that the eigenfunction expansions associated with and κ are related through the identity  f  (λ) = (Ff )(κ + λ), Fκ wκ

f ∈ L2 (r)

  (where, as in Sect. 2.2, we write Lp (r) := Lp (a, b); r(x)dx ), and therefore Ff L2 (R,ρ

L

)

= f L2 (r)

 f = w

κ

   

  = (Ff )(κ + ·)L2 (R,ρκ) . L L2 (r κ)

Recalling the uniqueness of the spectral measure for which the isometric property κ in Theorem 2.5 holds, we deduce that ρ (λ1 , λ2 ] = ρL (λ1 + κ, λ2 + κ]. L Corollary 4.1 If 0 < λ ≤ σ 2 , then wλ (·) is strictly decreasing and such that limx↑b wλ (x) = 0.

λ −1 λ Proof By the previous lemma, wλ (x) = w−λ (x) . By Corollary 2.1, w−λ (x) is strictly increasing and unbounded, yielding the result. The remaining ingredient for the proof of the Laplace representation is the weak maximum principle for the hyperbolic PDE ∂x2 u = ∂y2 u + φη (y) ∂y u − ψ η (y) u. (This equation is equivalent, up to a change of variables, to the PDE ∂x2 u = − y u.) In the following lemma and corollary we state and prove this maximum principle in a general form which also serves as a preparation for our study of the hyperbolic PDE x u = y u (Sect. 4.3). Lemma 4.3 Let the functions φ1 , φ2 , ψ 1 , ψ 2 : (γ (a), ∞) −→ R be such that φ2 , ψ 2 are decreasing,

0 ≤ φ1 ≤ φ2 ,

0 ≤ ψ1 ≤ ψ2,

lim φ2 (ξ ) = 0.

ξ →∞

(4.7)

Denote by ℘ j (j = 1, 2) the differential expression ℘ j (v) := −v

− φj v + ψ j v = −



1  A φj v + ψ j v, A φj

x where A φj (x) = exp( β φj (ξ )dξ ) (with β > γ (a) arbitrary). For γ (a) < c ≤ y ≤ x, consider the triangle c,x,y := {(ξ, ζ ) ∈ R2 | ζ ≥ c, ξ + ζ ≤ x + y, ξ − ζ ≥ x − y}, and let v ∈ C2 ( c,x,y ). Then the following integral equation holds: A φ1 (x)A φ2 (y) v(x, y) = H + I0 + I1 + I2 + I3 − I4 ,

(4.8)

134

4 Generalized Convolutions for Sturm-Liouville Operators

where

H := 12 A φ2 (c) A φ1 (x − y + c) v(x − y + c, c) + A φ1 (x + y − c) v(x + y − c, c)],

(4.9)  I0 := 12 A φ2 (c)  I1 :=

1 2

y

c

x+y−c

A φ1 (s)(∂y v)(s, c) ds,

x−y+c

(4.10)

 A φ1 (x − y + s)A φ2 (s) φ2 (s) + φ1 (x − y + s) v(x − y + s, s) ds,

(4.11)  I2 :=

1 2

c

y

 A φ1 (x + y − s)A φ2 (s) φ2 (s) − φ1 (x + y − s) v(x + y − s, s) ds,

(4.12)

 I3 :=

1 2

 I4 :=

 A φ1 (ξ )A φ2 (ζ ) ψ 2 (ζ ) − ψ 1 (ξ ) v(ξ, ζ ) dξ dζ,

(4.13)

A φ1 (ξ )A φ2 (ζ ) (℘ 2,ζ v − ℘ 1,ξ v)(ξ, ζ ) dξ dζ,

(4.14)

c,x,y

1 2 c,x,y

and ℘ j,z denotes the differential expression ℘ j acting on the variable z. Proof Just compute 

 I4 − I3 =

1 2 c,x,y

 = I0 −

1 2

 −

1 2

y c

y c

 ∂

A φ1 (ξ )A φ2 (ζ ) (∂ξ v)(ξ, ζ ) ∂ξ  ∂

A φ1 (ξ )A φ2 (ζ ) (∂ζ v)(ξ, ζ ) dξ dζ − ∂ζ A φ1 (x − y + s)A φ2 (s) (∂ζ v + ∂ξ v)(x − y + s, s) ds

A φ1 (x + y − s)A φ2 (s) (∂ζ v − ∂ξ v)(x + y − s, s) ds 

y

= I0 + I1 −  + I2 − c

c y

 d

A φ1 (x − y + s)A φ2 (s) v(x − y + s, s) ds ds

 d

A φ1 (x + y − s)A φ2 (s) v(x + y − s, s) ds, ds

where in the second equality we used Green’s theorem, and the third equality follows easily from the fact that (A φj ) = φj A φj .

4.2 Laplace-Type Representation

135

Corollary 4.2 (Weak Maximum Principle) In the conditions of Lemma 4.3, let γ (a) < c ≤ y0 ≤ x0 . If u ∈ C2 ( c,x0 ,y0 ) satisfies (℘ 2,y u − ℘ 1,x u)(x, y) ≤ 0,

then u ≥ 0 in

(x, y) ∈

c,x0 ,y0 ,

u(x, c) ≥ 0,

x ∈ [x0 − y0 + c, x0 + y0 − c],

(∂y u)(x, c) ≥ 0,

x ∈ [x0 − y0 + c, x0 + y0 − c],

(4.15)

c,x0 ,y0 .

Proof Pick a function ω ∈ C2 [c, ∞) such that ℘ 2 ω < 0, ω(c) > 0 and ω (c) ≥ 0. Clearly, it is enough to show that for all ε > 0 we have v(x, y) := u(x, y) + ε ω(y) > 0 for (x, y) ∈ c,x0 ,y0 . By Lemma 4.3, the integral equation (4.8) holds for the function v. Assume by contradiction that there exist ε > 0, (x, y) ∈ c,x0 ,y0 for which we have v(x, y) = 0 and v(ξ, ζ ) ≥ 0 for all (ξ, ζ ) ∈ c,x,y ⊂ c,x0 ,y0 . It is clear from the choice of ω that v(·, c) > 0, thus we have H ≥ 0 in the right hand side of (4.8). Similarly, (∂y v)(·, c) = (∂y u)(·, c) + ε ω (c) ≥ 0, hence I0 ≥ 0. Since the functions φ1 , φ2 , ψ 1 , ψ 2 satisfy (4.7) and we are assuming that u ≥ 0 on c,x,y , we have I1 ≥ 0, I2 ≥ 0 and I3 ≥ 0. In addition, I4 < 0 because (℘ 2,ζ v − ℘ 1,ξ v)(ξ, ζ ) = (℘ 2,ζ u − ℘ 1,ξ u)(ξ, ζ ) + (℘ 2 ω)(ζ ) < 0. Consequently, (4.8) yields 0 = A φ1 (x)A φ2 (y)v(x, y) ≥ −I4 > 0. This contradiction shows that v(x, y) > 0 for all (x, y) ∈ c,x0 ,y0 . Finally, we state the announced Laplace-type representation for the solutions of the Sturm-Liouville initial value problem. Theorem 4.1 (Laplace-Type Representation) Let be a Sturm-Liouville expression of the form (4.1), and suppose that Assumption MP holds. Let wλ be the solution of the initial value problem (2.12). For each x ∈ [a, b) there exists a subprobability measure πx on R such that   iτ s e πx (ds) = cos(τ s) πx (ds) (τ ∈ C), (4.16) wτ 2 +σ 2 (x) = R

where σ = limξ →∞ to

R

A (ξ ) 2A(ξ ) .

|wτ 2 +σ 2 (x)| ≤ 1

In particular, the boundedness property (2.19) extends

on the strip |Im(τ )| ≤ σ

(a ≤ x < b).

(4.17)

We first show that a similar representation holds for the solutions wλ,m of the initial value problem on the approximating intervals (am , b) (Lemma 2.2); the result of Theorem 4.1 will then be deduced by a limiting argument. Proposition 4.2 Let be a Sturm-Liouville expression of the form (4.1), and suppose that Assumption MP holds. Let wλ,m be defined as in Lemma 2.2. For each

136

4 Generalized Convolutions for Sturm-Liouville Operators

m ∈ N and x ∈ [am , b) there exists a subprobability measure πx,m on R such that  wτ 2 +σ 2 ,m (x) =

 R

eiτ s πx,m (ds) =

R

(τ ∈ C).

cos(τ s) πx,m (ds)

(4.18)

Proof Throughout the proof we assume, without loss of generality, that we have chosen c = am in the definition of the function γ introduced in Remark 4.1, so that γ (am ) = 0. We begin by proving that the result holds when σ = 0. Let η, φη , ψ η be defined y as in Assumption MP. The function ϑ λ,m (y) := exp( 12 0 η(ξ )dξ ) wλ,m (γ −1 (y)) is the solution of ℘(u) = λu

(0 < ξ < ∞),

u (0) = 0,

u(0) = 1,

where ℘(v) := −v

−φη v +ψ η v. From this it follows that the function uτ (x, y) := cos(τ x) ϑ τ 2 ,m (y) (x, y ∈ R+ 0 ) is, for each τ ∈ C, a solution of the hyperbolic PDE ∂x2 u = −℘ y u. It follows from Corollary 4.2 that the Cauchy problem (∂x2 + ℘ y )u = 0,

u(x, 0) = f (x),

(∂y u)(x, 0) = 0

has the property that if f ∈ C∞ c (R), f ≥ 0 then the solution uf is such that uf (x, y) ≥ 0 for all x ≥ y ≥ 0. Thus f → uf (x, y) is a positive linear functional ∞ on C∞ c (R) and, consequently , uf (x, y) = R f dμx,y,m for all f ∈ Cc (R), where μx,y,m is, for each x ≥ y ≥ 0, a finite positive Borel measure; moreover, it follows from the domain of dependence for the Cauchy problem that μx,y,m has compact support. In particular we can write  cos(τ x) ϑ τ 2 ,m (y) =

R

cos(τ s) μx,y,m (ds),

x ≥ y ≥ 0.

(4.19)

Assume that each measure μx,y,m is symmetric (if not, replace it by its symmetrization), and let y,m = μy,y,m ∗ 12 (δy + δ−y ) − μ2y,y,m. We then have R



 R

cos(τ s) y,m (ds) =

 R

cos(τ s) μy,y,m (ds)  −

R

R

cos(τ s)

cos(τ s) μ2y,y,m(ds)

  = cos2 (τy) − 2 cos(τy) ϑ τ 2 ,m (y) = ϑ τ 2 ,m (y).

1

2 (δy

 + δ−y ) (ds)

4.2 Laplace-Type Representation

137

We claim that y,m is a positive measure. Indeed, we have    cos(τ x) ϑ τ 2 ,m (y) = cos(τ s) y,m ∗ 12 (δx + δ−x ) (ds), R

R

where the right-hand side is, by (4.19), a positive-definite function of τ ∈ R; therefore, the convolution y,m ∗ 12 (δx + δ−x ) is, for all x ≥ y ≥ 0, a positive R

Borel measure. Since the support of y,m is compact, the supports of y,m ∗ δx R and y,m ∗ δ−x are disjoint for x sufficiently large, and this implies that the R measures y,m ∗ δx and (consequently) y,m are both positive. Setting πx,m :=  1  γ (x) R  exp − 2 0 η(ξ )dξ γ (x),m , we conclude that (4.18) holds for all τ ∈ C. Since w0,m (x) ≡ 1, we have πx,m ∈ P(R) for all x ∈ [am , b). Suppose now that σ > 0. Then the result for the case σ = 0 can be applied 2 to the operator σ  defined in Lemma 4.2 and the corresponding eigenfunctions σ 2 

wλ,m (x) :=

wλ+σ 2 ,m (x) wσ 2 ,m (x) .

(Indeed, it follows from Lemma 4.1 that the function Aσ

associated to the operator σ  , defined as in (4.2), is such that limξ →∞ 2

2

(Aσ  ) (ξ ) 2 2Aσ  (ξ )

2

=

0.) Hence wτ 2 +σ 2 ,m (x) = wσ 2 ,m (x)

 R

cos(τ s) π x,m (ds),

where π x,m is, for each x ∈ [a, b), a symmetric probability measure. Setting πx,m := wσ 2 ,m (x) π x,m , we obtain (4.18). By Lemma 2.3 we have wσ 2 ,m (·) ≤ 1, hence each πx,m is a subprobability measure. Proof of Theorem 4.1 We proved in Proposition 4.2 that for each m ∈ N there exists a symmetric subprobability measure πx,m whose Fourier transform is the function τ → wτ 2 +σ 2 ,m (x) (τ ∈ R). We also know (from Lemmas 2.1–2.2) that wτ 2 +σ 2 ,m (x) −→ wτ 2 +σ 2 (x) pointwise as m → ∞, the limit function being continuous in τ . Applying the Lévy continuity theorem (e.g. [9, Theorem 23.8]), we conclude that wτ 2 +σ 2 (x) is the Fourier transform of a symmetric subprobability measure πx and, in addition, the measures πx,m converge weakly to πx as m → ∞. Therefore, for x > a we have  wτ 2 +σ 2 (x) = cos(τ s) πx (ds) (τ ∈ R). (4.20) R

In order to extend (4.20) to τ ∈ C, we let 0 ≤ φ1 ≤ φ2 ≤ . . . be functions with compact support such that φn ↑ 1 pointwise, and for fixed x > a, κ > 0 we

138

4 Generalized Convolutions for Sturm-Liouville Operators

compute  R

cosh(κs) πx (ds)  = lim

n→∞ R

φn (s) cosh(κs) πx (ds) 

= lim lim

n→∞ m→∞ R

φn (s) cosh(κs) πx,m (ds)



≤ lim

m→∞ R

cosh(κs) πx,m (ds) = lim wσ 2 −κ 2 ,m (x) = wσ 2 −κ 2 (x) < ∞. m→∞

From this estimate we easily see that the right-hand side of (4.20) is an entire function of τ ; therefore, by analytic continuation, (4.20) holds for all τ ∈ C. Finally, if |Im(τ )| ≤ σ then  |wτ 2 +σ 2 (x)| ≤

R

 | cos(τ s)|πx (ds) ≤

R

cosh(σ s) πx (ds) = w0 (x) = 1,

and therefore (4.17) is true. We finish this section by presenting a description of the spectrum of the Neumann realization of the Sturm-Liouville operator which will later be useful, and whose proof relies on the Laplace representation. Recall that the Neumann realization (L(2) , D(L(2))) was defined in Theorem 2.5 as the self-adjoint operator obtained by restricting the Sturm-Liouville operator to the domain which (considering that, by Lemma 4.1(d), the endpoint b is limit point) was defined in (2.20) as    D(L(2)) = u ∈ L2 (r)  u, u ∈ ACloc (a, b), (u) ∈ L2 (r), (pu )(a) = 0 . Proposition 4.3 Let be a Sturm-Liouville expression of the form (4.1), and suppose that Assumption MP holds. The spectral measure ρ L of Proposition 2.5 is such that supp(ρL ) = [σ 2 , ∞). In addition, L has purely absolutely continuous spectrum in (σ 2 , ∞). Proof It follows from the proof of Lemma 4.1 that the operator L is unitarily d2 equivalent to a self-adjoint realization of the differential expression − dξ 2 + q (γ (a) < ξ < ∞), where q is defined by (4.5) and satisfies q = q1 + q2 , with limξ →∞ q1 (ξ ) = σ 2 and q2 ∈ L1 ([c, ∞), dξ ) for c > γ (a). Using a general result on the spectral properties of Sturm-Liouville operators stated in [201, Theorem 15.3], we conclude that the spectrum of L is purely absolutely continuous on (σ 2 , ∞) and the essential spectrum equals [σ 2 , ∞). (The result of [201] is stated for Sturm-Liouville operators whose left endpoint is regular, but we can apply it here because a well-known result [185, Theorem 9.11] ensures that the essential spectrum of L is the union of the essential spectrums of self-adjoint realizations of

restricted to the intervals (a, c) and (c, b), a < c < b. Recall also that, as noted in

4.3 The Existence Theorem for Sturm-Liouville Product Formulas

139

the proof of Lemma 4.1, Sturm-Liouville operators with no natural endpoints have a purely discrete spectrum.) It remains to show that L has no eigenvalues on [0, σ 2 ]. Indeed, if we assume that 0 ≤ λ0 ≤ σ 2 is an eigenvalue of L, then wλ0 belongs to D(L(2) ) and therefore, by the Laplace representation (4.16), wλ belongs to D(L(2)) for all λ ≥ σ 2 ; since the eigenvalues are discrete, this is a contradiction.

4.3 The Existence Theorem for Sturm-Liouville Product Formulas As in the particular cases of the Kingman and the Whittaker convolutions, the product formula for the solutions wλ of the Sturm-Liouville problem (2.12) is the tool which will allow us to introduce a generalized convolution associated with the operator . The probabilistic property of the product formula (i.e. the property that the kernel of the product formula is composed of probability measures) is the requirement which will ensure that the convolution preserves the space of probability measures. The aim of this section is to show that Assumption MP is a sufficient condition for the existence of such a probabilistic product formula. Namely, we will prove the following result: Theorem 4.2 (Product Formula for wλ ) Let be a Sturm-Liouville expression of the form (4.1), and suppose that Assumption MP holds. For each x, y ∈ [a, b) there exists a measure νx,y ∈ P[a, b) such that the product wλ (x) wλ (y) admits the integral representation  wλ (x) wλ (y) =

[a,b)

wλ (ξ ) νx,y (dξ ),

x, y ∈ [a, b), λ ∈ C.

(4.21)

4.3.1 The Associated Hyperbolic Cauchy Problem The proof of Theorem 4.2 relies crucially on the basic properties (existence, uniqueness and positivity-preservingness of solution) of the hyperbolic Cauchy problem associated with , i.e., of the boundary value problem defined by ( x h)(x, y) = ( y h)(x, y) h(x, a) = f (x), (∂y[1] h)(x, a) = 0, ∂ . where ∂y[1] = p(y) ∂y

(x, y ∈ (a, b)), (4.22)

140

4 Generalized Convolutions for Sturm-Liouville Operators

Since y − x =

p(x) ∂ 2 r(x) ∂x 2

2

∂ − p(y) r(y) ∂y 2 + lower order terms, the equation x h = y h

is hyperbolic at the line y = a if p(a) r(a) > 0; otherwise, the initial conditions of the Cauchy problem are given at a line of parabolic degeneracy. If γ (a) =  c r(y) − a p(y) dy > −∞, then we can remove the degeneracy via the change of variables x = γ (ξ ), y = γ (ζ ) (cf. Remark 4.1), through which the partial

ζ u, with initial differential equation is transformed to the standard form 

ξ u =  condition at the line ζ = γ (a). In the case γ (a) = −∞, the standard form of the equation is also parabolically degenerate in the sense that its initial line is ζ = −∞. Theorem 4.3 (Existence of Solution) Let be a Sturm-Liouville expression of the form (4.1), and suppose that x → p(x)r(x) is an increasing function. If f ∈ D(L(2) ) and (f ) ∈ D(L(2) ), then the function  hf (x, y) :=

[σ 2 ,∞)

wλ (x) wλ (y) (Ff )(λ) ρL (dλ)

(4.23)

solves the Cauchy problem (4.22). For ease of notation, unless necessary we drop the dependence in h and denote (4.23) by h(x, y). Proof Let us begin by justifying that x h can be computed via differentiation under the integral sign. Since  x wλ is a solution of the initial value problem (2.12), we have (pwλ )(x) = −λ a wλ (ξ ) r(ξ )dξ and therefore (by Lemma 2.3) |(pwλ )(x)| ≤ x λ a r(ξ )dξ . Hence 

  (Ff )(λ) (pw )(x) wλ (y) ρ (dλ) λ L [σ 2 ,∞)  x    ≤ r(ξ )dξ · λ (Ff )(λ) wλ (y) ρ L (dλ) < ∞, a

(4.24)

[σ 2 ,∞)

where the convergence (which is uniform in compacts) follows from (2.24) and Lemma 2.4. The convergence of the differentiated integral yields that  (∂x[1]h)(x, y) = [σ 2 ,∞)(Ff )(λ) (pwλ )(x) wλ (y) ρ L (dλ). Since ( wλ )(x) =  λwλ (x), in the same way we check that [σ 2 ,∞) (Ff )(λ) ( wλ )(x) wλ (y) ρL (dλ) converges absolutely and uniformly on compacts and is therefore equal to ( x h)(x, y). Consequently,  ( x h)(x, y) = ( y h)(x, y) =

[σ 2 ,∞)

λ (Ff )(λ) wλ (x) wλ (y) ρ L (dλ).

(4.25)

Concerning the boundary conditions, Lemma 2.4(b) together with the fact that wλ (a) = 1 imply that h(x, a) = f (x), and from (4.24) we easily see that

4.3 The Existence Theorem for Sturm-Liouville Product Formulas

141

limy↓a (∂y[1] h)(x, y) = 0. This shows that h is a solution of the Cauchy problem (4.22). Under the assumptions of the theorem, the solution (4.23) of the hyperbolic Cauchy problem satisfies the following conditions: (α) h(·, y) ∈ D(L(2) ) for all a < y < b; (β) There exists a zero ρL -measure set #0 ⊂ [σ 2 , ∞) such that for each λ ∈ [σ 2 , ∞) \ #0 we have F[ y h(·, y)](λ) = y [Fh(·, y)](λ)

for all a < y < b,

(4.26)

lim ∂y[1]F[h(·, y)](λ) = 0.

lim[Fh(·, y)](λ) = (Ff )(λ), y↓a

(4.27)

y↓a

Indeed, by Theorem 2.5 we have [Fh(·, y)](λ) = (Ff )(λ) wλ (y) for all λ ∈ supp(ρL ) and a < y < b. Since f ∈ D(L(2) ) and |wλ (·)| ≤ 1 (Lemma 2.3), it is clear from (2.23) that h(x, y) satisfies (α). Moreover, it follows from (4.25) that F[ y h(·, y)](λ) = λ (Ff )(λ) wλ (y) = y [Fh(·, y)](λ), hence (4.26) holds. The properties (4.27) follow immediately from Lemma 2.1. Next we show that the solution from the above existence theorem is the unique solution satisfying conditions (α)–(β): Theorem 4.4 (Uniqueness) Let be a Sturm-Liouville expression of the form (4.1), and supposethat x → p(x)r(x) is an increasing function. Let f ∈ D(L(2) ) and let  h1 , h2 ∈ C2 (a, b)2 be two solutions of ( x h)(x, y) = ( y h)(x, y). Suppose that both h1 and h2 satisfy conditions (α)–(β). Then h1 (x, y) ≡ h2 (x, y)

for all x, y ∈ (a, b).

(4.28)

Proof Fix λ ∈ R+ 0 \ #0 and let j (y, λ) := [Fhj (·, y)](λ). We have

y j (y, λ) = F[ y hj (·, y)](λ) = F[ x hj (·, y)](λ) = λ j (y, λ),

a am . x Proof Let a˜ m := γ (am ) and B(x) := exp( 12 a˜ m η(ξ )dξ ). It follows from Proposition 4.4 that the function um (x, y) := B(x)B(y)hm (γ −1(x), γ −1(y)) is a solution of the Cauchy problem (℘ x um )(x, y) = (℘ y um )(x, y),

x, y > a˜ m ,

(4.36)

um (x, a˜ m ) = B(x) f (γ −1 (x)),

x > a˜ m ,

(4.37)

(∂y um )(x, a˜ m ) = 12 η(a˜ m ) B(x) f (γ −1 (x)),

x > a˜ m ,

(4.38)

2

∂ ∂ where ℘ x := − ∂x 2 − φ η (x) ∂x + ψ η (x). Clearly, um satisfies the inequalities (4.15) for arbitrary x0 ≥ y0 ≥ a˜ m (here ℘ 1 = ℘ 2 and c = a˜ m ). By Corollary 4.2, um (x0 , y0 ) ≥ 0 for all x0 ≥ y0 > a˜ m ; consequently, (4.35) holds. The proof that f ≤ C implies hm ≤ C is straightforward: if we have f ≤ C,   then  um (x, y) = B(x)B(y) C − hm (γ −1(x), γ −1(y)) is a solution of (4.36) with initial conditions

   um (x, a˜ m ) = B(x) C − f (γ −1 (x)) ≥ 0,   (∂y um )(x, a˜ m ) = 12 η(a˜ m ) B(x) C − f (γ −1 (x)) ≥ 0, thus the reasoning of the previous paragraph yields that C−hm ≥ 0 for x ≥ y > a˜ m . Corollary 4.3 (Positivity of Solution for the Cauchy Problem (4.22)) Let be a Sturm-Liouville expression of the form (4.1), and suppose that Assumption MP holds. If f ∈ D(L(2) ), (f ) ∈ D(L(2) ) and f ≥ 0, then the function h given by (4.23) is such that h(x, y) ≥ 0

for x, y ∈ (a, b).

If, in addition, f ≤ C, then h(x, y) ≤ C for x, y ∈ (a, b).

4.3 The Existence Theorem for Sturm-Liouville Product Formulas

145

Proof This is an immediate consequence of Proposition 4.5 together with the pointwise convergence property (4.31). (By (4.23) we have f (x, y) = f (y, x), thus the conclusion holds for all x, y ∈ (a, b).)

4.3.2 The Time-Shifted Product Formula Before proving that there exists a product formula of the form (4.21) for the SturmLiouville solutions {wλ (·)}λ∈C , we will show that a similar product formula holds for the family of functions {e−t λwλ (·)}λ∈C . This auxiliary result will be called the time-shifted product formula because the latter family is obtained by applying the diffusion semigroup generated by to the solutions wλ (·). Indeed, we saw in Sect. 2.2.3 that e−t λ wλ (x) = (Tt wλ )(x) = [Fp(t, x, ·)](λ), where {Tt }t ≥0 denotes the Feller semigroup generated by the Neumann realization of and p(t, x, y) denotes the Feller transition density (2.33). By the inversion formula (2.22) for the L-transform, a natural candidate for the measure of the product formula for {wλ (·)}λ∈C is  νx,y (dξ ) =

[σ 2 ,∞)

wλ (x) wλ (y) wλ (ξ ) ρ L (dλ) r(ξ )dξ.

This is only a formal solution, because in general the integral does not converge. But it suffices to include the regularization term e−t λ in order to obtain an integral which (under the assumptions of the existence and uniqueness theorems above) always converges absolutely: Lemma 4.4 Let be a Sturm-Liouville expression of the form (4.1), and suppose that x → p(x)r(x) is an increasing function. Let t0 > 0 and K1 , K2 compact subsets of (a, b). The integral  [σ 2 ,∞)

e−t λ wλ (x) wλ (y) wλ (ξ ) ρL (dλ)

converges absolutely and uniformly on (t, x, y, ξ ) ∈ [t0 , ∞) × K1 × K2 × [a, b). Proof This follows from Lemma 2.3 and the uniform convergence property of the integral representation of the transition density of the Feller semigroup {Tt }t ≥0 (Proposition 2.14). In what follows we write  qt (x, y, ξ ) :=

[σ 2 ,∞)

e−t λ wλ (x) wλ (y) wλ (ξ ) ρL (dλ).

(4.39)

146

4 Generalized Convolutions for Sturm-Liouville Operators

This function, which is (at least formally) the density of the measure of the timeshifted product formula, is for fixed t, x, y the density (with respect to r(ξ )dξ ) of a subprobability measure: Lemma 4.5 Let be a Sturm-Liouville expression of the form (4.1), and suppose that Assumption MP holds. The function qt (x, y, ξ ) is nonnegative and such that b + a qt (x, y, ξ ) r(ξ )dξ ≤ 1 for all (t, x, y) ∈ R × (a, b) × (a, b). Throughout the proof (and in the sequel) we write D(2,0) := D(L(2)) ∩ D(L(0) ), where    D(L(0)) = u ∈ C0 [a, b)  u, u ∈ ACloc (a, b), (u) ∈ C0 [a, b), (pu )(a) = 0 is the domain of the Feller semigroup {Tt }t ≥0 (cf. Sect. 2.2.3). Note that if g ∈ C2c [a, b) with g ∈ Cc (a, b), then g ∈ D(2,0); consequently, any indicator function of an interval I ⊂ [a, b) is the pointwise limit of functions gn ∈ D(2,0) . Proof Since qt (x, y, ·) ∈ Cb [a, b), it suffices to show that for all g ∈ D(2,0) with 0 ≤ g ≤ 1 we have 0 ≤ Qt,g (x, y) ≤ 1

  t > 0, x, y ∈ (a, b) ,

b where Qt,g (x, y) := a g(ξ ) qt (x, y, ξ ) r(ξ )dξ . Fix t > 0 and g ∈ D(2,0) with 0 ≤ g ≤ 1. Since [Fqt (x, y, ·)](λ) = −t e λ wλ (x) wλ (y), it follows from the isometric property of the L-transform (Theorem 2.5) that  Qt,g (x, y) = e−t λ wλ (x) wλ (y) (Fg)(λ) ρ L (dλ). [σ 2 ,∞)

Differentiating under the integral sign we easily check (by dominated convergence and using Lemma 2.4(b)) that x Qt,g = y Qt,g , (∂y[1] Qt,g )(x, a) = 0 and  Qt,g (x, a) =

[σ 2 ,∞)

e−t λ wλ (x) (Fg)(λ) ρ L (dλ) = (Tt g)(x),

where the last equality follows from (2.31). The fact that 0 ≤ g ≤ 1 clearly implies that 0 ≤ (Tt g)(x) ≤ 1 for x ∈ (a, b). One can verify via (2.23) that the function f (x) = (Tt g)(x) is such that f ∈ D(L(2) ) and (f ) ∈ D(L(2) ). It then follows from the positivity property of the hyperbolic Cauchy problem (Corollary 4.3) that 0 ≤ Qt,g (x, y) ≤ 1 for all x, y ∈ (a, b), as claimed. Proposition 4.6 (Time-Shifted Product Formula) Let be a Sturm-Liouville expression of the form (4.1), and suppose that Assumption MP holds. The product

4.3 The Existence Theorem for Sturm-Liouville Product Formulas

147

e−t λ wλ (x) wλ (y) admits the integral representation e−t λ wλ (x) wλ (y) =

 b a

wλ (ξ ) qt (x, y, ξ ) r(ξ )dξ,

t > 0, x, y ∈ (a, b), λ ≥ 0,

(4.40) where the integral in the right hand side is absolutely convergent. b In particular, a qt (x, y, ξ ) r(ξ )dξ = 1 for all t > 0, x, y ∈ (a, b). Proof The absolute convergence of the integral in the right hand side is immediate from Lemmas 2.3 and 4.5. By Theorem 2.5, the equality in (4.40) holds ρ L -almost everywhere. Since supp(ρL ) = [σ 2 , ∞) (Lemma 4.3), the fact that both sides of (4.40) are continuous functions of λ ≥ 0 allows us to extend by continuity the equality (4.40) to all λ ≥ σ 2 . If σ = 0, we are done. Suppose that σ > 0. By (4.17) and Lemma 4.5, together with standard results on the analyticity of parameter-dependent integrals, the function τ → b a wτ 2 +σ 2 (ξ ) qt (x, y, ξ ) r(ξ )dξ is an analytic function of τ in the strip |Im(τ )| < 2 2 σ . It is also clear that τ → e−t (τ +σ ) wτ 2 +σ 2 (x) wτ 2 +σ 2 (y) is an entire function. By analytic continuation we see that these two functions are equal for all τ in the strip |Im(τ )| < σ ; consequently, (4.40) holds. The last statement is obtained by setting λ = 0.

4.3.3 The Product Formula for wλ as the Limit Case Unsurprisingly, the product formula (4.21) is deduced by taking the limit as t ↓ 0 in the time-shifted product formula (4.40). If the functions wλ (·) belong to C0 [a, b), the limit can be straightforwardly taken in the vague topology of measures. As shown below, the class of modified Sturm-Liouville operators described in Lemma 4.2 can then be used to extend the product formula to the case where the functions wλ (·) do not belong to C0 [a, b) Theorem 4.5 (Product Formula for wλ ) Let be a Sturm-Liouville expression of the form (4.1), and suppose that Assumption MP holds. For x, y ∈ (a, b) and t > 0, let νt,x,y ∈ P[a, b) be the measure defined by νt,x,y (dξ ) = qt (x, y, ξ ) r(ξ )dξ . Then w for each x, y ∈ (a, b) there exists a measure νx,y ∈ P[a, b) such that νt,x,y −→ νx,y as t ↓ 0. Moreover, the product wλ (x) wλ (y) admits the integral representation  wλ (x) wλ (y) =

[a,b)

wλ (ξ ) νx,y (dξ ),

In particular, Theorem 4.2 holds.

x, y ∈ (a, b), λ ∈ C.

(4.41)

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4 Generalized Convolutions for Sturm-Liouville Operators

Proof Let {tn }n∈N be an arbitrary decreasing sequence with tn ↓ 0. It is a basic fact that any sequence of probability measures contains a vaguely convergent subsequence (e.g. [10, p. 213]), thus there exists a subsequence {tnk } and a measure v νx,y ∈ M+ [a, b) such that νtnk ,x,y −→ νx,y as k → ∞. Let us show that all such subsequences {νtnk ,x,y } have the same vague limit. Suppose that tk1 , tk2 are two v

j

j

different sequences with tk ↓ 0 and that νt j ,x,y −→ νx,y as k → ∞ (j = 1, 2). For g ∈ D(2,0) we have   j g(ξ ) νx,y (dξ ) = lim

k

k→∞ [a,b)

[a,b)

g(ξ ) νt j ,x,y (dξ )



= lim

k

j

k→∞ [σ 2 ,∞)

e−tk λ wλ (x) wλ (y) (Fg)(λ) ρ L (dλ)



=

[σ 2 ,∞)

wλ (x) wλ (y) (Fg)(λ) ρL (dλ)

(the second equality was justified in the proof of Lemma  4.5, and dominated convergence yields the last equality). In particular, [a,b) g(ξ ) ν1x,y (dξ ) =  2 (2,0) , and this implies that ν 1 2 x,y = ν x,y . Since [a,b) g(ξ ) ν x,y (dξ ) for all g ∈ D v

all subsequences have the same vague limit, we conclude that νt,x,y −→ νx,y as t ↓ 0. A (ξ ) Suppose first that σ := limξ →∞ 2A(ξ ) > 0. Then Corollary 4.1 ensures that 2 limx↑b wλ (x) = 0 for 0 < λ ≤ σ , and by the Laplace-type representation (4.16) we have wλ (·) ≤ wσ 2 (·) for λ > σ 2 , hence wλ ∈ C0 [a, b) for all λ > 0. Accordingly, by taking the limit as t ↓ 0 of both sides of (4.40) we deduce that the product formula (4.41) holds for all λ > 0. To prove that (4.41) is valid in the general case, let κ < 0 be arbitrary. We know that the operator κ defined in Lemma 4.2 satisfies Assumption MP; by Lemma 4.1 κ ) (ξ ) we have limξ →∞ (A > 0 and consequently (by the reasoning in the previous 2Aκ (ξ ) paragraph) the corresponding Sturm-Liouville solutions (4.6) belong to C0 [a, b) for all λ > 0. From the previous part of the proof, κ κ wλ (x) wλ (y)

 =

b

a

κ

wλ (ξ ) νκ x,y (dξ ),

x, y ∈ (a, b), λ > 0,

κ

κ

(4.42)

with νx,y constructed as before. We easily verify that qt (x, y, ξ )r κ(ξ ) = κ etκ wκ (ξ ) wκ (ξ ) wκ (x)wκ (y) qt (x, y, ξ )r(ξ ) and, consequently, ν x,y (dξ ) = wκ (x)wκ (y) νx,y (dξ ). It thus follows from (4.42) that 

b

wκ+λ (x) wκ+λ (y) = a

wκ+λ (ξ ) νx,y (dξ ),

x, y ∈ (a, b), λ > 0,

4.3 The Existence Theorem for Sturm-Liouville Product Formulas

149

where κ < 0 is arbitrary; hence (4.41) holds for all λ ∈ R. If we then set λ = τ 2 +σ 2 in (4.41), we straightforwardly verify that both sides are entire functions of τ (for the right hand side, this follows from the Laplace-type representation (4.16) and the fact that the integral converges for all λ < 0), so by analytic continuation the product formula holds for all λ ∈ C. Given that w0 (x) ≡ 1, setting λ = 0 in (4.41) shows that νx,y ∈ P[a, b); consequently, the measures νt,x,y converge to νx,y in the weak topology (cf. [10, Theorem 30.8]). Clearly, the product formula (4.41) can be extended to x, y ∈ [a, b) by setting νx,a := δx and νa,y := δy , hence Theorem 4.2 holds. It is worth commenting that the reasoning used in this proof also allows us to justify that the time-shifted product formula (4.40) is valid for all λ ∈ C. As shown in the proof above, the measure νx,y of the product formula (4.41) is characterized by the identity 

 [a,b)

f (ξ ) νx,y (dξ ) =

[σ 2 ,∞)

wλ (x) wλ (y) (Ff )(λ) ρ L (dλ),

f ∈ D(2,0). (4.43)

Furthermore, the relation between this measure and the measure νt,x,y (dξ ) = qt (x, y, ξ ) r(ξ )dξ of the time-shifted product formula (4.40) can be written explicitly: Corollary 4.4 The measure νt,x,y can be written in terms of the measure νx,y and the transition kernel p(t, x, y) of the Feller semigroup generated by the SturmLiouville operator as 

b

νt,x,y (dξ ) =

νz,y (dξ ) p(t, x, z) r(z)dz

  t > 0, x, y ∈ (a, b) .

a

Proof Recalling (2.31) and the proof of the previous proposition, we find that for g ∈ D(2,0) we have 

b a

 

[a,b) b

=  = 

g(ξ )ν z,y (dξ ) p(t, x, z) r(z)dz



[σ 2 ,∞)

a

[σ 2 ,∞)

wλ (z) wλ (y) (Fg)(λ) ρ L (dλ) p(t, x, z) r(z)dz

e−t λ wλ (x) wλ (y) (Fg)(λ) ρ L (dλ)

b

=

g(ξ ) qt (x, y, ξ ) r(ξ )dξ, a

hence the measures νt,x,y (dξ ) and

b a

νz,y (dξ ) p(t, x, z) r(z)dz are the same.

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4 Generalized Convolutions for Sturm-Liouville Operators

4.4 Sturm-Liouville Transform of Measures In analogy with the definition of the index Whittaker transform of measures (Definition 3.3), it is natural to define the L-transform of finite complex measures so that (2.21) is the L-transform of an absolutely continuous measure with density f (·)r(·): Definition 4.3 Let μ ∈ MC [a, b). The L-transform of the measure μ is the function defined by the integral  μ(λ) = $

[a,b)

wλ (x) μ(dx),

λ ≥ 0.

It is immediate from Lemma 2.3 that |$ μ(λ)| ≤ $ μ(0) = μ for all μ ∈ M+ [a, b). In addition, this definition leads to various properties which, as in the case of the Whittaker transform (cf. Proposition 3.4), resemble those of the Fourier transform of complex measures: Proposition 4.7 Let be a Sturm-Liouville expression of the form (4.1), and suppose that Assumption MP holds. Let $ μ be the L-transform of μ ∈ MC [a, b). The following properties hold: (i) $ μ is continuous on R+ 0 . Moreover, if a family of measures {μj } ⊂ MC [a, b) is tight and uniformly bounded, then {% μj } is equicontinuous on R+ 0. (ii) Each measure μ ∈ MC [a, b) is uniquely determined by $ μ|[σ 2 ,∞) . (iii) If {μn } is a sequence of measures belonging to M+ [a, b), μ ∈ M+ [a, b), and w μn −→ μ, then μ %n −−−−→ $ μ n→∞

uniformly for λ in compact sets.

(iv) Suppose that limx↑b wλ (x) = 0 for all λ > 0. If {μn } is a sequence of measures belonging to M+ [a, b) whose L-transforms are such that μ %n (λ) −−−−→ f (λ) n→∞

pointwise in λ ≥ 0

for some real-valued function f which is continuous at a neighborhood of zero, w then μn −→ μ for some measure μ ∈ M+ [a, b) such that $ μ ≡ f. Proof (i) It suffices to prove the second statement. Set C = supj μj . Fix λ0 ≥ 0 and ε > 0. By the tightness assumption, we can choose β ∈ (a, b) such that |μj |(β, b) < ε for all j . Since the family of derivatives {∂λ w(·) (x)}x∈(a,β] is locally bounded on R+ 0 (to verify this, differentiate the series (2.14) term by

4.4 Sturm-Liouville Transform of Measures

151

term and then compute an upper bound as in (2.15)), we can choose δ > 0 such that |λ − λ0 | < δ

⇒

|wλ (x) − wλ0 (x)| < ε for all a < x ≤ β.

Consequently,     μ %j (λ) − μ %j (λ0 ) =   ≤

(a,b)

    wλ (x) − wλ0 (x) μj (dx)

  wλ (x) − wλ (x)|μj |(dx) + 0

(β,b)



  wλ (x) − wλ (x)|μj |(dx) ≤ (2 + C)ε 0

(a,β]

for all j , provided that |λ − λ0 | < δ, which means that {% μj } is equicontinuous at λ0 . (ii) Let μ ∈ MC [a, b) be such that $ μ(λ) = 0 for all λ ≥ σ 2 . We need to show that μ is the zero measure. For each g ∈ D(2,0) we have for a < x < b  (Fg)(λ) wλ (x) $ μ(λ) ρ L (dλ) 0 = [σ 2 ,∞)



 =

[a,b) [σ 2 ,∞)

(Fg)(λ)wλ (x)wλ (y) ρL (dλ) μ(dy),

where the change of order of integration is valid because, by Lemmas 2.3 and 2.4(b), the double integral converges absolutely; therefore 

 [a,b)

g(y) μ(dy) =



[a,b) [σ 2 ,∞)

 =

(Fg)(λ) wλ (y) ρL (dλ)μ(dy)



lim

[a,b) x↓a [σ 2 ,∞)



= lim

(Fg)(λ) wλ (x) wλ (y) ρ L (dλ)μ(dy)



x↓a [a,b) [σ 2 ,∞)

(Fg)(λ) wλ (x) wλ (y) ρ L (dλ)μ(dy)

= 0, using Lemma 2.4, the identity (4.43) and dominated convergence. This shows that [a,b) g(y) μ(dy) = 0 for all g ∈ D(2,0) and, consequently, μ is the zero measure. (iii) Since wλ (·) is continuous and bounded, the pointwise convergence μ %n (λ) → $ μ(λ) follows from the definition of weak convergence of measures. By Prokhorov’s theorem {μn } is tight and uniformly bounded, thus (by part (i)) {% μn } is equicontinuous on R+ 0 . The same argument from the proof of Proposition 3.4(c) yields that the convergence μ %n → $ μ is uniform on compact sets.

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4 Generalized Convolutions for Sturm-Liouville Operators

(iv) The proof follows the same argument as that of Proposition 3.4(iv), replacing the interval R+ 0 and the function W α, λ (·) by [a, b) and wλ (·) respectively. Remark 4.3 I. If limx↑b wλ (x) = 0 for all λ > 0, then as in Remark 3.4 we obtain the following analogue of the Lévy continuity theorem: the L-transform is a topological homeomorphism between P[a, b) with the weak topology and the set $ P of Ltransforms of probability measures with the topology of uniform convergence in compact sets. II. Much like weak convergence, vague convergence of measures can be formulated via the L-transform, provided that limx↑b wλ (x) = 0 for all λ > 0. Indeed, we can state: v

II.1 If {μn } ⊂ M+ [a, b), μ ∈ M+ [a, b), and μn −→ μ, then lim μ %n (λ) = $ μ(λ) pointwise for each λ > 0; II.2 If {μn } ⊂ M+ [a, b), {μn } is uniformly bounded and lim μ %n (λ) = f (λ) v + pointwise in λ > 0 for some function f ∈ Bb (R ), then μn −→ μ for some measure μ ∈ M+ [a, b) such that $ μ ≡ f. (The first part is trivial, and the second part is proved as follows: since any uniformly bounded sequence of positive measures contains a vaguely convergent subsequence, for any subsequence {μnk } there exists a further subsequence {μnkj } and a measure v

μ such that μnkj −→ μ; then II.1 implies that $ μ(λ) = f (λ) for λ > 0, so the vague v

limit of such a subsequence is unique and, consequently, μn−→ μ.) Consider the following stronger version of Assumption MP:  d d p dx satisfies Assumption MP and Assumption MP∞ The operator = − 1r dx its coefficients satisfy limx↑b p(x)r(x) = ∞. This assumption will play an important role in the subsequent sections, mostly because it ensures that the properties stated in the Remark 4.3 hold. Indeed, one can state:  d d Lemma 4.6 Let = − 1r dx p dx be a Sturm-Liouville operator satisfying Assumption MP. Then Assumption MP∞ holds if and only if limx↑b wλ (x) = 0 for all λ > 0. Proof This follows from known results on the asymptotic behaviour of solutions of

the Sturm-Liouville equation −u

− AA u = λu, see [64, proof of Lemma 3.7]. We note that, in particular, the lemma states that the condition limx↑b wλ (x) = 0 (λ > 0) holds whenever σ > 0. This particular case had already been pointed out in the proof of Theorem 4.5.

4.5 Sturm-Liouville Convolution of Measures

153

4.5 Sturm-Liouville Convolution of Measures In what follows we always assume that the Sturm-Liouville expression satisfies Assumption MP. (In general we allow for operators such that limx↑b p(x)r(x) < ∞; whenever this is not the case, we will explicitly state that Assumption MP∞ is required to hold.) As usual (cf. Definitions 2.6 and 3.4, Proposition 2.16), we define the convolution ∗ : MC [a, b) × MC [a, b) −→ MC [a, b) as the natural extension of the mapping (x, y) → δx ∗ δy := νx,y (where νx,y is the measure of the product formula (4.41)), and we define the translation of functions as the integral with respect to the convolution of Dirac measures: Definition 4.5 Let μ, ν ∈ MC [a, b). The complex measure  (μ ∗ ν)(dξ ) =

 [a,b) [a,b)

νx,y (dξ ) μ(dx) ν(dy)

is called the L-convolution of the measures μ and ν. The L-translation of a Borel measurable function f : [a, b) −→ C is defined as   f (ξ ) νx,y (dξ ) ≡ f (ξ ) (δx ∗ δy )(dξ ), x, y ∈ [a, b). (Ty f )(x) := [a,b)

[a,b)

More generally, the L-translation by μ ∈ M+ [a, b) is defined as (Tμ f )(x) :=  [a,b) f (ξ ) (δx ∗ μ)(dξ ). We will see that, in the same spirit of Sects. 3.5–3.8, analogues of many basic notions of (generalized) probabilistic harmonic analysis can be developed on the measure algebra determined by the L-convolution. Our first proposition states the unsurprising fact that the L-convolution is trivialized by the Sturm-Liouville transform of measures: Proposition 4.8 Let μ, ν, π ∈ MC [a, b). We have π = μ ∗ ν if and only if $ π (λ) = $ μ(λ)$ ν(λ)

for all λ ≥ 0.

Proof Identical to that of Proposition 3.6 (replacing Wα,

λ

(·) by wλ (·), etc.).

The following result collects some basic properties of the measure algebra determined by the L-convolution. Proposition 4.9 The space (MC [a, b), ∗), equipped with the total variation norm, is a commutative Banach algebra over C whose identity element is the Dirac measure δa . The subset P[a, b) is closed under the L-convolution. Moreover, the map (μ, ν) → μ∗ν is continuous (in the weak topology) from MC [a, b)×MC [a, b) to MC [a, b).

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4 Generalized Convolutions for Sturm-Liouville Operators

Proof Since μ  ∗ν = $ μ ·$ ν (Proposition 4.8), the commutativity, associativity and bilinearity of the L-convolution follow at once from the uniqueness property of the L-transform (Proposition 4.7(ii)). One can verify directly from the definition of the L-convolution that the submultiplicativity property μ ∗ ν ≤ μ · ν holds, and that equality holds whenever μ, ν ∈ M+ [a, b); it is also clear that the convolution of positive measures is a positive measure. We conclude that the Banach algebra property holds and that P[a, b) is closed under convolution. If limx↑b wλ (x) = 0 for all λ > 0, the identity ν& x,y (λ) = wλ (x)wλ (y) implies (by Proposition 4.7(iv)) that (x, y) → νx,y is continuous in the weak topology. If the functions wλ (x) do not vanish at the limit x ↑ b, let κ < 0 be arbitrary and let h ∈ Cb [a, b). Since wκ is increasing and unbounded (Corollary 2.1), whκ ∈ κ

C0 [a, b). If we let νx,y be the measure defined in the proof of Theorem 4.5, then by κ Remark 4.3.III the map (x, y) → νx,y is continuous, and thus  (x, y) −→

1 f (ξ ) κ νx,y (dξ ) = wκ (x)wκ (y) [a,b) wκ (ξ )

 [a,b)

f (ξ ) νx,y (dξ )

 is continuous. This shows that (x, y) → [a,b) f (ξ ) νx,y (dξ ) is continuous for all f ∈ Cb [a, b) and therefore (x, y) → νx,y is continuous in the weak topology. w w Finally, for f ∈ Cb [a, b) and μn , νn ∈ MC [a, b) with μn −→ μ and νn −→ ν we have     lim f (ξ )(μn ∗ νn )(dξ ) = lim f d νx,y μn (dx)νn (dy) n

n

[a,b)

 =  =

[a,b) [a,b)





[a,b) [a,b)

[a,b)

[a,b)

[a,b)

f d νx,y μ(dx)ν(dy)

f (ξ )(μ ∗ ν)(dξ ),

due to the continuity of the function in parenthesis; this proves that (μ, ν) → μ ∗ ν is continuous. Next we summarize some useful facts about the generalized translation introduced in Definition 4.5. For simplicity we write  · p ≡  · Lp (r) (1 ≤ p ≤ ∞). Proposition 4.10 Let μ ∈ M+ [a, b). The L-translation operator Tμ has the following properties: (i) Let 1 ≤ p ≤ ∞. If f ∈ Lp (r), then (Tμ f )(x) is a Borel measurable function of x ∈ [a, b) and satisfies Tμ f p ≤ μ·f p . (ii) If f ∈ L2 (r), then F(Tμ f )(λ) = $ μ(λ) (Ff )(λ) for ρ L -a.e. λ. (iii) If f ∈ Cb [a, b), then Tμ f ∈ Cb [a, b). (iv) Suppose that Assumption MP∞ holds. If f ∈ C0 [a, b), then Tμ f ∈ C0 [a, b).

4.5 Sturm-Liouville Convolution of Measures

155

Proof (i) It suffices to prove the result for nonnegative f . The map ν → μ ∗ ν is weakly continuous (Proposition 4.9) and takes M+ [a, b) into itself. By a technical result proved in [94, Section 2.3], this implies that, for each Borel measurable h ≥ 0, the function x → (Tμ f )(x) is Borel measurable. It follows that  b μ [a,b) g(x)(μ ∗ r)(dx) := a (T g)(x)r(x)dx (g ∈ Cc [a, b)) defines a positive Borel measure. For a ≤ c1 < c2 < b, let 1[c1 ,c2 ) be the indicator function of [c1 , c2 ), let fn ∈ D(2,0) be a sequence of nonnegative functions such that fn → 1[c1 ,c2 ) pointwise, and write C = {g ∈ C∞ c (a, b) | 0 ≤ g ≤ 1}. We compute  (μ ∗ r)[c1 , c2 ) = lim n

fn (x)(μ ∗ r)(dx)

[a,b)



= lim sup n g∈C

b



= lim sup n g∈C

(Ffn )(λ) (Fg)(λ) $ μ(λ) ρL (dλ)

[σ 2 ,∞)

 = lim sup n g∈C

b

b

≤ μ·lim n



fn (x) (Tμ g)(x) r(x)dx

a



= μ·

(Tμ fn )(x) g(x) r(x)dx

a

fn (x) r(x)dx a

c2

r(x)dx, c1

where the third and fourth equalities follow from (4.43) and the isometric property of the L-transform (Theorem 2.5), and the inequality holds because Tμ g∞ ≤ μ · g∞ ≤ μ. Therefore, Tμ f 1 = f L1 ([a,b),μ∗r) ≤ μ · f 1 for each Borel measurable f ≥ 0. Since δx ∗ μ ∈ M+ [a, b), 1/p Hölder’s inequality yields that Tμ f p ≤ μ1/q ·Tμ |f |p 1 ≤ μ·f p for 1 < p < ∞. Finally, if f ∈ L∞ (r), f ≥ 0 then f = fb + f0 , where 0 ≤ fb ≤ f ∞ and f0 = 0 Lebesgue-almost everywhere. Since Tμ f0 1 ≤ μ · f0 1 = 0, we have Tμ f0 = 0 Lebesgue-a.e., and therefore Tμ f ∞ = Tμ fb ∞ ≤ μ·f ∞ . (ii) For f ∈ D(2,0), this identity follows at once from (4.43). The property extends to all f ∈ L2 (r) by the standard continuity argument. (iii) This follows immediately from the fact that (μ, ν) → μ ∗ ν is weakly continuous (Proposition 4.9). (iv) It remains to show that (Tμ h)(x) → 0 as x ↑ b. Since wλ (x)$ μ(λ) → 0 as v x ↑ b (λ > 0), it follows from Remark 4.3.II that δx ∗ μ −→ 0 as x ↑ b, where

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4 Generalized Convolutions for Sturm-Liouville Operators

0 denotes the zero measure; this means that for each f ∈ C0 [a, b) we have  (Tμ f )(x) =



[a,b)

f (ξ )(δx ∗ μ)(dξ ) −→

[a,b)

f (ξ ) 0(dξ ) = 0

as x ↑ b,

showing that Tμ f ∈ C0 [a, b).

4.5.1 Infinite Divisibility and Lévy-Khintchine Type Representation The set Pid of L-infinitely divisible distributions is defined in the usual way:    Pid = μ ∈ P[a, b)  for all n ∈ N there exists νn ∈ P[a, b) such that μ = νn∗n , where νn∗n denotes the n-fold L-convolution of νn with itself. Lemma 4.7 Suppose that Assumption MP∞ holds. If μ ∈ Pid , then μ(λ) = e−ψμ (λ) , $ where ψμ (λ) (λ ≥ 0) is a positive continuous function such that ψμ (0) = 0. Moreover, measures μ ∈ Pid have no nontrivial idempotent divisors, i.e., if μ = ϑ ∗ ν (with ϑ, ν ∈ P[a, b)) where ϑ is idempotent with respect to the L-convolution (that is, it satisfies ϑ = ϑ ∗ ϑ), then ϑ = δ0 . Proof Same as that of Lemma 3.6. The function ψμ (λ) described in the lemma will be called the log L-transform of μ. As in the case of the log-Whittaker transform (cf. Proposition 3.7), its growth is at most linear: Proposition 4.11 Suppose that Assumption MP∞ holds, and let μ ∈ Pid . Then ψμ (λ) ≤ Cμ (1 + λ)

for all λ ≥ 0

for some constant Cμ > 0 which is independent of λ. Proof Let νn ∈ P[a, b) be the measure such that ν$n (λ) ≡ exp(− n1 ψμ (λ)). The     inequality n 1 − ν$n (λ) ≤ ψμ (λ) (n ∈ N) and the limit limn→∞ n 1 − ν$n (λ) = ψμ (λ) are justified as in the proof of Proposition 3.7. Pick λ1 > 0. We know that limx↑b wλ1 (x) = 0 (Lemma 4.6), hence there exists β ∈ (a, b) such that |wλ1 (x)| ≤ 12 for all β ≤ x < b. Combining this with (2.19),

4.5 Sturm-Liouville Convolution of Measures

157

we deduce that for all λ ≥ 0 we have     n 1 − wλ (x) νn (dx) ≤ 2n [β,b)

[β,b)

 ≤ 4n

νn (dx)

  1 − wλ1 (x) νn (dx)

(4.44)

[β,b)

  ≤ 4n 1 − ν$n (λ1 ) ≤ 4ψμ (λ1 ). Next, it follows from the proof of Proposition 4.7(i) that we can choose λ2 > 0 such that 1 − wλ (x) < 12 for all 0 ≤ λ ≤ λ2 and all a < x ≤ β. Define η1 (x) := x 1 y a p(y) a r(ξ )dξ dy ≡ η1 (x; −1), where η1 (·, ·) is the function defined in (2.13). Recalling (2.16), we obtain  1 − wλ2 (x) = λ2

a

x

1 p(y)



y a

wλ2 (ξ )r(ξ )dξ dy ≥

On the other hand, by (2.19) we have 1−wλ(x) ≤ λ λη1 (x) for all x ∈ [a, b) and λ ≥ 0. Consequently,  n

 [a,β)

 1 − wλ (x) νn (dx) ≤ λn

λ2 η1 (x) 2

y 1 a p(y) a

x

for all a ≤ x ≤ β.

|wλ (ξ )|r(ξ )dξ dy ≤

 [a,β)

η1 (x) νn (dx)



  1 − wλ2 (x) νn (dx)

≤

2λn λ2

≤

 2λ 2λn  1 − ν$n (λ2 ) ≤ ψμ (λ2 ). λ2 λ2

[a,β)

(4.45)

Combining (4.44) and (4.45), one sees that for all n ∈ N and λ ≥ 0 we have n(1 − ν$n (λ)) ≤ Cμ (1 +λ), where Cμ = max 4ψμ (λ1 ), λ22 ψμ (λ2 ) . The conclusion follows by taking the limit as n → ∞. The log L-transforms of L-infinitely divisible distributions also admit an analogue of the classical Lévy-Khintchine representation. The relevant notions of compound Poisson and Gaussian measures are similar to those for the Whittaker convolution: Definition 4.6 Let μ ∈ P[a, b) and c > 0. The measure e(cμ) ∈ P[a, b) defined by e(cμ) = e−c

∞ n  c n=0

n!

μ∗n

(the infinite sum converging in the weak topology) is said to be the L-compound Poisson measure associated with cμ.

158

4 Generalized Convolutions for Sturm-Liouville Operators

It is immediate that e(cμ) ∈ Pid and that its log L-transform is ψe(cμ) (λ) = c(1 − $ μ(λ)). Definition 4.7 A measure μ ∈ P[a, b) is called an L-Gaussian measure if μ ∈ Pid and μ = e(cν) ∗ ϑ

  c > 0, ν ∈ P[a, b), ϑ ∈ Pid

⇒

ν = δa .

Theorem 4.5 (Lévy-Khintchine Type Formula) Suppose that Assumption MP∞ holds. The log L-transform of a measure μ ∈ Pid can be represented in the form  ψμ (λ) = ψα (λ) +



 1 − wλ (x) ν(dx),

(4.46)

(a,b)

where ν is a σ -finite measure on (a, b) which is finite on the complement of any neighbourhood of a and such that    1 − wλ (x) ν(dx) < ∞, (a,b)

and α is an L-Gaussian measure with log L-transform ψα (λ). Conversely, each function of the form (4.46) is a log L-transform of some μ ∈ Pid . This Lévy-Khintchine type representation, together with its counterpart for the Whittaker convolution (Theorem 3.3), are both particular cases of the general LévyKhintchine formula for stochastic convolutions stated in Proposition 2.18 and whose proof was sketched in Sect. 3.5.2. (The fact that the L-convolution satisfies axiom V6 of Definition 2.5 is argued in the same way as in Sect. 3.5.2.)

4.5.2 Convolution Semigroups Definition 4.8 A family {μt }t ≥0 ⊂ P[a, b) is called an L-convolution semigroup if it satisfies the conditions • μs ∗ μt = μs+t for all s, t ≥ 0; • μ0 = δ a ; w • μt −→ δa as t ↓ 0. If the Sturm-Liouville operator satisfies Assumption MP∞ , then there exists a one-to-one correspondence {μt }t ≥0 → μ1 ∈ Pid between the set of L-convolution semigroups and the set of L-infinitely divisible distributions. (This can be justified exactly as in Remark 3.8.) Consequently, any convolution semigroup {μt } has an   L-transform of the form μ $t (λ) = exp −t ψμ1 (λ) , where ψμ1 (·) is a function of the form (4.46).

4.5 Sturm-Liouville Convolution of Measures

159

The family of generalized translation operators determined by a given Lconvolution semigroup has the expected Feller-type properties: Proposition 4.12 Suppose that Assumption MP∞ holds, and let {μt }t ≥0 be an Lconvolution semigroup. Then the family {Tt }t ≥0 defined by + Tt : Cb (R+ 0 ) −→ Cb (R0 ),

Tt f := Tμt f

is a conservative Feller semigroup such that the identity Tt Tν f = Tν Tt f holds for all t ≥ 0 and ν ∈ MC [a, b). The restriction Tt |C [a,b) can be extended to a c

(p)

strongly continuous contraction semigroup {Tt } on the space Lp (r) (1 ≤ p < (p) (p) ∞). Moreover, the operators Tt are given by Tt f = Tμt f (f ∈ Lp (r)). Proof Similar to that of Propositions 3.8–3.9. Proposition 4.13 Suppose that Assumption MP∞ holds. Let {μt } be an L(2) convolution semigroup with log L-transform ψ and let {Tt } be the associated 2 Markovian semigroup on L (r). Then the infinitesimal generator (G(2) , D(G(2) )) (2) of the semigroup {Tt } is the self-adjoint operator given by   F(G(2) f ) (λ) = −ψ · (Ff ),

f ∈ D(G(2) ),

where    D(G(2)) = f ∈ L2 (r) 

    ψ(λ)2 (Ff )(λ)2 ρ (λ)dλ < ∞ . L

∞ 0

Proof Similar to that of Proposition 3.10.

4.5.3 Additive and Lévy Processes Definition 4.9 An [a, b)-valued Markov chain {Sn }n∈N0 is said to be L-additive if there exist measures μn ∈ P[a, b) such that P [Sn ∈ B|Sn−1 = x] = (μn ∗ δx )(B),

n ∈ N, x ∈ [a, b), B a Borel subset of [a, b).

(4.47) If μn = μ for all n, then {Sn } is said to be an L-random walk. An explicit construction can be given for L-additive Markov chains, based on the following lemma:

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4 Generalized Convolutions for Sturm-Liouville Operators

Lemma 4.8 There exists a Borel measurable  : [a, b) × [a, b) × [0, 1] −→ [a, b) such that (δx ∗ δy )(B) = m{(x, y, ·) ∈ B},

x, y ∈ [a, b), B a Borel subset of [a, b),

where m denotes Lebesgue measure on [0, 1].   Proof Let (x, y, ξ ) = max a, sup{z ∈ [a, b) : (δx ∗ δy )[a, z] < ξ } . Using the continuity of the L-convolution, one can show that  is Borel measurable, see [19, Theorem 7.1.3]. It is straightforward that m{(x, y, ·) ∈ [a, c]} = m{(δx ∗ δy )[a, c] ≥ ξ } = (δx ∗ δy )[a, c]. Let X1 , U1 , X2 , U2 , . . . be a sequence of independent random variables (on a given probability space ( , A, π)) where the Xn have distribution PXn = μn ∈ P[a, b) and each of the (auxiliary) random variables Un has the uniform distribution on [0, 1]. Set S0 = 0,

Sn = Sn−1 ⊕Un Xn ,

(4.48)

where X ⊕U Y := (X, Y, U ). Then the distributions PSn of the random variables Sn are such that PSn = PSn−1 ∗ μn (n ∈ N0 ) and, consequently, {Sn }n∈N0 is an L-additive Markov chain satisfying (4.47). The identity PSn = PSn−1 ∗ μn is easily checked:

 PSn (B) = P (Sn−1 , Xn , Un ) ∈ B   m{(x, y, ·) ∈ B}PSn−1 (dx)PXn (dy) =  =

[a,b) [a,b)



[a,b) [a,b)

(δx ∗ δy )(B)PSn−1 (dx)PXn (dy)

= (PSn−1 ∗ μn )(B). The continuous-time analogue of L-random walks are the L-Lévy processes, defined in analogy with Definition 3.8: Definition 4.10 An [a, b)-valued Markov process Y = {Yt }t ≥0 is said to be an L-Lévy process if there exists an L-convolution semigroup {μt }t ≥0 such that for 0 ≤ s ≤ t the transition probabilities of Y are given by P [Yt ∈ B|Ys = x] = (μt −s ∗ δx )(B),

x ∈ [a, b), B a Borel subset of [a, b). (4.49)

If we let ν ∈ P[a, b) be a given measure and {μt }t ≥0 a given L-convolution semigroup, then in the same manner as in the previous chapter one can construct a L-Lévy process satisfying (4.49) and such that P [X0 ∈ ·] = ν. Like in Corol-

4.5 Sturm-Liouville Convolution of Measures

161

lary 2.3 and Proposition 3.11, the class of Lévy processes includes the diffusion generated by the associated Sturm-Liouville operator (as defined in Sect. 2.2.3): Proposition 4.14 Suppose that Assumption MP∞ holds. The diffusion process X generated by the Neumann realization (L(2) , D(L(2) )) of is a L-Lévy process. Proof Similar to that of Proposition 3.11. An analogue of the well-known theorem on approximation of Lévy processes d

by triangular arrays holds for L-Lévy processes (below the notation −→ stands for convergence in distribution): Proposition 4.15 Suppose that Assumption MP∞ holds, and let X be an [a, b)valued random variable. The following assertions are equivalent: (i) X = Y1 for some L-Lévy process Y = {Yt }t ≥0; (ii) The distribution of X is L-infinitely divisible; d

j

n −→ X for some sequence of L-random walks S 1 , S 2 , . . . (with S = a) (iii) Sm 0 n and some integers mn → ∞.

Proof The equivalence between (i) and (ii) is a restatement of the one-to-one correspondence {μt }t ≥0 ←→ μ1 between L-infinitely divisible measures and Lconvolution semigroups. It is obvious that (i) implies (iii): simply let mn = n and S n the random walk whose step distribution is the law of Y1/n . Suppose that (iii) holds and let πn , μ be the distributions of Sjn , X respectively. Choose ε > 0 small enough so that $ μ(λ) > Cε > 0 for λ ∈ [0, ε], where Cε > 0 is a constant. By (iii) and Proposition 4.7(iii), π$n (λ)mn → $ μ(λ) uniformly on compacts, which implies that π$n (λ) → 1 for all λ ∈ [0, ε] and, therefore, by w w Proposition 4.7(iv) πn −→ δa . Now let k ∈ N be arbitrary. Since πn −→ δa , we can w ∗(m / k) assume that each mn is a multiple of k. Write νn = πn n , so that νn∗k −→ μ. By ∗mn relative compactness of D({πn }) (see [198, Corollary 1]), the sequence {νn }n∈N w has a weakly convergent subsequence, say νnj −→ μk as j → ∞, and from this it clearly follows that μ∗k k = μ. Consequently, (ii) holds. An L-convolution semigroup {μt }t ≥0 such that μ1 is an L-Gaussian measure is called an L-Gaussian convolution semigroup, and an L-Lévy process associated with an L-Gaussian convolution semigroup is called an L-Gaussian process. Proposition 4.16 (Alternative Characterizations of L-Gaussian Convolution Semigroups) Suppose that Assumption MP∞ holds and that A ∈ C3 (a, b) (where A is the function defined in (4.2)). Let Y = {Yt }t ≥0 be an L-Lévy process, let {μt }t ≥0 be the associated L-convolution semigroup and let (G(0) , D(G(0) )) be the infinitesimal generator of the Feller semigroup associated with Y . The following conditions are equivalent: measure; (i) μ1 is a Gaussian   (ii) limt ↓0 1t μt [a, b) \ Va = 0 for every neighbourhood Va of the point a;

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4 Generalized Convolutions for Sturm-Liouville Operators

  (iii) limt ↓0 1t (μt ∗ δx ) [a, b) \ Vx = 0 for every x ∈ [a, b) and every neighbourhood Vx of the point x; (iv) Y has a modification whose paths are a.s. continuous. If any of these conditions hold then the infinitesimal generator of Y is a local operator, i.e., (G(0) f )(x) = (G(0) g)(x) whenever f, g ∈ D(G(0) ) and h = g on some neighbourhood of x ∈ [a, b). Proof (i) ⇐⇒ (ii): This can be proved as in Lemma 3.8 (with the obvious adaptations). (ii) ⇐⇒ (iii): To prove the nontrivial direction, assume that (ii) holds, and fix x ∈ (a, b). Let Vx be a neighbourhood of the point x not containing a and write Ex = [a, b) \ Vx . Pick a function f ∈ C4 such that 0 ≤ f ≤ 1, f = 0 on Ex and f = 1 on some smaller neighbourhood Ux ⊂ Vx of the point x. (Using the assumption A ∈ C3 (a, b), it is easy to check that f, (f ) ∈ D(2,0).) We begin by showing that lim y↓a

1 − (Tx f )(y) =0 1 − wλ (y)

for each λ > 0.

(4.50)

Indeed, it follows from (4.43) that limy↓a (Tx f )(y) = 1, limy↓a ∂y[1] (Tx f )(y) = 0 and 

y (Tx f )(y) = λ (Ff )(λ) wλ (x) wλ (y) ρL (dλ) R+ 0

  = Tx (f ) (y) −−→ (f )(x) = 0, y↓a

hence using L’Hôpital’s rule twice we find that for λ > 0 we have x

y (T f )(y) Tx f )(y) limy↓a 1−( 1−wλ (y) = limy↓a λ wλ (y) = 0.  By(4.50), for each λ > 0 there exists aλ > a such that (Tx 1Ex )(y) ≤ Tx (1 − f ) (y) ≤ 1 − wλ (y) for all y ∈ [a, aλ). We then estimate 1 1 (μt ∗ δx )(Ex ) = t t ≤

1 t

1 ≤ t =

 (Tx 1Ex )(y)μt (dy)

 

[a,b)

  1 1 − wλ (y) μt (dy) + μt [aλ , b) t [a,aλ )   1 1 − wλ (y) μt (dy) + μt [aλ , b) t [a,b)

 1 1 1−μ $t (λ) + μt [aλ , b). t t

4.5 Sturm-Liouville Convolution of Measures

163

Given that  we are assuming that  (ii) holds  and, by the L-semigroup property, $t (λ) = limt ↓0 1t 1− μ %1 (λ)t = − log μ %1 (λ), the above inequality limt ↓0 1t 1− μ gives 1 lim sup (μt ∗ δx )(Ex ) ≤ − log μ %1 (λ). t t ↓0 This holds for arbitrary λ > 0. Since the right-hand side is continuous and vanishes for λ = 0, we conclude that limt ↓0 1t (μt ∗ δx )(Ex ) = 0, as desired. (iii) ⇒ (iv): This follows from Proposition 2.4. (iv) ⇒ (iii): This is a general fact which is known as Ray’s theorem on onedimensional diffusion processes. The proof can be found in [88, Theorem 5.2.1]. The final assertion follows from Proposition 2.5. Remark 4.4 As in the context of the Whittaker convolution, one can introduce the notion of a moment sequence associated with the L-convolution. The canonical L-moment functions  ϕk (k ∈ N) can be defined recursively as the solution of the initial value problem

( ϕk )(x) = −2σ k ϕk−1(x) − k(k − 1) ϕk−2(x),

ϕk (a) = 0, 

(p ϕk )(a) = 0

(where  ϕ−1 (x) := 0 and  ϕ0 (x) := 1). Equivalently, we can write 

x

ϕk (x) = k 

1 p(y)

0

=

∂k

 =

 

∂τ k  ∞ −∞

τ =σ



y

 r(ξ ) 2σ  ϕk−1 (ξ ) + (k − 1) ϕk−2 (ξ ) dξ dy

0

wσ 2 −τ 2 (x)

s k eσ s πx (ds).

Using the product formula (4.41), one can check that the canonical funcL-moment   tions are a solution of the functional equation (Ty ϕk )(x) = kj =0 jk ϕj (x)ϕk−j (y), meaning that the  ϕk play a role similar to that of the monomials under the classical convolution. The canonical L-moment functions are a tool for establishing strong laws of large numbers for L-additive Markov chains. In particular, the following results hold for a given L-additive Markov chain {Sn } constructed as in (4.48): I.  If {rn }n∈N is a sequence of positive  numbers such that limn rn = ∞ and ∞ 1 2 < ∞, then (X )] − E[ϕ (X )] E[ϕ 2 n 1 n n=1 rn  1  lim √ ϕ1 (Sn ) − E[ϕ1 (Sn )] = 0 n rn

π-a.s.

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4 Generalized Convolutions for Sturm-Liouville Operators

II. If {Sn } is an L-random walk such that E[ϕ2 (X1 )θ/2 ] < ∞ for some 1 ≤ θ < 2, then E[ϕ1 (X1 )] < ∞ and lim

1 

n n1/θ

 ϕ1 (Sn ) − nE[ϕ1 (X1 )] = 0

π-a.s.

III. Suppose that ϕ1 ≡ 0. If 1{rn }n∈N is a sequence of positive numbers such that limn rn = ∞ and ∞ n=1 rn E[ϕ2 (Xn )] < ∞, then lim n

1 ϕ2 (Sn ) = 0 rn

π-a.s.

IV. Suppose that ϕ1 ≡ 0. If {Sn } is an L-random walk such that E[ϕ2 (X1 )θ ] < ∞ for some 0 < θ < 1, then lim n

1 ϕ2 (Sn ) = 0 n1/θ

π-a.s.

The above statements can be proved exactly as in the hypergroup framework, see [19, Section 7.3]. In addition, one can show that the modified moments E[ ϕk (X)] can be computed via the L-transform of measures and that a martingale property holds for L-moment functions applied to L-Lévy processes (these results are proved as in Propositions 3.16 and 3.17). Under additional assumptions on the coefficients of , one can also establish a Lévy-type characterization similar to that of Theorem 3.4 for the diffusion process associated with the Sturm-Liouville operator. In particular, an adaptation of the x 1 y proof of Theorem 3.4 yields the following result: Set η1 (x) := a p(y) r(ξ )dξ dy a x 1 y and η2 (x) := a p(y) a η1 (ξ )r(ξ )dξ dy. Suppose that a > −∞, Assumption MP∞ holds and one of the following conditions is satisfied: • η1 (x) = c1 (x −a)+c2 (x −a)2 +o((x −a)2) and η2 (x) = c3 (x −a)2 +o((x −a)2 ) as x ↓ a, with c1 , c3 > 0; • η1 (x) = c1 (x−a)2 +c2 (x−a)4 +o((x−a)4) and η2 (x) = c3 (x−a)4 +o((x−a)4) as x ↓ a, with c1 , c3 > 0. Let X = {Xt }t ≥0 be an [a, b)-valued Markov process with a.s. continuous paths. Then the following assertions are equivalent: (i) X is the diffusion process generated by the Neumann realization (L(2) , D(L(2) )) of ; 2 (ii) {η1 (Xt ) − t}t ≥0 and {η2 (Xt ) − t η1 (Xt ) + t2 }t ≥0 are martingales (or local martingales); t s)

2 (iii) {η1 (Xt ) − t}t ≥0 is a local martingale with [η1 (X)]t = 2 0 p(X r(Xs ) (η1 (Xs )) ds.

4.6 Sturm-Liouville Hypergroups

165

4.6 Sturm-Liouville Hypergroups Our purpose here is to discuss whether the convolution algebra structure constructed in the previous section satisfies the hypergroup axioms H1–H8 introduced in Definition 2.4. We will determine a sufficient condition that leads to an existence theorem for Sturm-Liouville hypergroups which is more general than that of Zeuner (stated above in Theorem 4.1). In addition to this, we will introduce a notion of degenerate hypergroup which includes the Whittaker convolution and many other Sturm-Liouville convolutions whose associated hyperbolic Cauchy problems are also parabolically degenerate.

4.6.1 The Nondegenerate Case We saw in Proposition 4.9 that the L-convolution satisfies the hypergroup axioms H1–H5 (with K = [a, b) and e = a as the identity element; H5 holds for the identity involution xˇ = x). In order to verify axioms H6–H8, one needs to determine the support of νx,y = δx ∗ δy . A detailed study of supp(νx,y ) was carried out by Zeuner in [210]. The next proposition shows that the results of Zeuner can be applied to the L-convolution, provided that the differential operator (4.1) has coefficients p = r = A defined on R+ , and there exists η ∈ C1 (R+ 0 ) satisfying the conditions given in Assumption MP. Proposition 4.17 Let

=−

1 d  d  A , A dx dx

x ∈ R+ ,

where A(x) > 0 for all x ≥ 0. Suppose that there exists η ∈ C1 (R+ 0 ) such that η ≥ 0, the functions φη , ψ η are both decreasing on R+ and limx→∞ φη (x) = 0. Let x0 = sup{x ≥ 0 | ψ η (x) = ψ η (0)} and x1 = inf{x > 0 | φη (x) = 0}. Then: (a) If x0 = ∞, x1 = 0 and η(0) = 0 then supp(δx ∗ δy ) = {|x − y|, x + y} for all x, y ≥ 0. (b) If 0 < x0 < ∞, x1 = 0 and η(0) = 0 then

supp(δx ∗ δy ) =

⎧ ⎪ ⎪ ⎨{|x − y|, x + y},

x + y ≤ x0 ,

{|x − y|} ∪ [2x0 − x − y, x + y], x, y < x0 < x + y, ⎪ ⎪ ⎩[|x − y|, x + y], max{x, y} ≥ x0 .

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4 Generalized Convolutions for Sturm-Liouville Operators

(c) If x0 = ∞, 0 < x1 < ∞ and η(0) = 0 then supp(δx ∗ δy ) =

⎧ ⎨[|x − y|, x + y],

min{x, y} ≤ 2x1 ,

⎩[|x − y|, 2x + |x − y|] ∪ [x + y − 2x , x + y], 1 1

min{x, y} > 2x1 .

(d) If 0 < 3x1 < x0 < ∞ and η(0) = 0 then supp(δx ∗ δy ) = ⎧ ⎪ ⎨[|x − y|, x + y], = [|x − y|, 2x + |x − y|] ⎪ 1 ⎩ ∪ [x + y − 2x1 , x + y],

min{x, y} ≤ 2x1 or max{x, y} ≥ x0 − x1 , min{x, y} > 2x1 and max{x, y} < x0 − x1 .

(e) If x0 ≤ 3x1 or η(0) > 0 then supp(δx ∗ δy ) = [|x − y|, x + y] for all x, y ≥ 0. The proof depends on the following lemma which ensures that the existence and uniqueness theorems for the associated hyperbolic Cauchy problem (Theorems 4.3– 4.4) are also valid for initial conditions f ∈ D(2,0).   Lemma 4.9 If f ∈ D(2,0), then there exists a unique solution h ∈ C2 (a, b)2 of the Cauchy problem (4.22) satisfying conditions (α)–(β) of Sect. 4.3.1, and this unique solution is given by (4.23). Proof The fact that there exists at most one solution of (4.22) satisfying the given requirements is proved in the same way. Let f ∈ D(2,0) and consider the function h(x, y) defined by (4.23). The limit limy↓a h(x, y) = f (x) follows from Lemma 2.4(b) and dominated convergence. Similarly, we have lim(∂y[1] h)(x, y) = lim y↓a



y↓a R+ 0

(Ff )(λ) wλ (x) wλ[1] (y) ρL (dλ) = 0

(the absolute and uniform convergence of the differentiated integral justifies the differentiation under the integral sign). Now fix y ∈ (a, b). By (4.43), we have h(·, y) = Ty f . Using (2.24) and Lemma 4.10(ii), we obtain F( x (Ty f ))(λ) = λ F(Ty f )(λ) = λ wλ (y) (Ff )(λ) = wλ (y) F( (f ))(λ) = F(Ty (f ))(λ), hence x (Ty f )(x) = (Ty (f ))(x) for almost every x. Since (by the weak continuity of (x, y) → νx,y , see Proposition 4.9) (x, y) → (Ty (f ))(x) is continuous, it follows that

x h(x, y) = (Ty (f ))(x),

for all x, y ∈ (a, b).

4.6 Sturm-Liouville Hypergroups

167

Exactly the same reasoning shows that y h(x, y) = (Ty (f ))(x), hence h ∈  2 C (a, b)2 is a solution of x u = y u. It remains to check that the function (4.23) satisfies conditions (α)–(β). As seen above we have F(h(·, y))(λ) = wλ (y) (Ff )(λ) and F[ y h(·, y)](λ) = F[Ty (f )](λ) = λ wλ (y) (Ff )(λ), hence (4.26)–(4.27) hold. Moreover, it is immediate from (2.23) that h(·, y) ∈ D(L(2) ), and therefore (α) holds. Proof of Proposition 4.17 Fix z ≥ 0, and let {fε } ⊂ D(2,0) be a family of functions such that fε (ξ ) > 0

for z − ε < ξ < z + ε,

fε (ξ ) = 0

for ξ ≤ z − ε and ξ ≥ z + ε.

(4.51)

 Observe that z ∈ supp(δx ∗ δy ) if and only if R+ fε d(δx ∗ δy ) > 0 for all ε > 0. 0 Now, we know from Lemma 4.9 that the function   hfε (x, y) := fε d(δx ∗ δy ) = wλ (x) wλ (y) (Ffε )(λ) ρL (dλ) R+ 0

[σ 2 ,∞)

(the second equality is due to (4.43)) is a nonnegative solution of the Cauchy x problem (4.22) with f ≡ fε ; writing B(x) := exp( 12 0 η(ξ )dξ ), it follows that ufε (x, y) = B(x)B(y)hfε (x, y) is a solution of ℘ x u − ℘ y u = 0, where 2

∂ ∂ ℘ x := − ∂x 2 − φη (x) ∂x + ψ η (x). Applying Lemma 4.3 with c > 0 and ℘ 1 (v) =



℘ 2 (v) = −v − φη v + ψ η v and then letting c ↓ 0, we deduce that the following integral equation holds:

A(x)A(y) uf (x, y) = H + I0 + I1 + I2 + I3 , B(x)2 B(y)2 ε

(4.52)

 A(x+y) η(0)  x+y A(s) where H = 12 A(0) A(x−y) x−y B(s) fε (s) B(x−y) fε (x −y)+ B(x+y) fε (x +y) , I0 = 4 ds and I1 , I2 , I3 are given by (4.11)–(4.13) with c = 0 and v = ufε . Since fε and hfε are nonnegative, all the terms in the right-hand side of (4.52) are nonnegative; consequently, we have z ∈ supp(δx ∗ δy ) if and only if at least one of the terms in the right-hand side of (4.52) is strictly positive for all ε > 0. In order to ascertain whether this holds or not, one needs to perform a thorough analysis of the integrals I0 , I1 , I2 and I3 . This has been done by Zeuner in [210, Proposition 3.9]; his results lead to the conclusion stated in the proposition. Theorem 4.6 (Existence Theorem for Sturm-Liouville Hypergroups) Let be a differential expression of the form (4.1). Suppose that γ (a) > −∞ and that there 1 exists η ∈ C [γ (a), ∞) satisfying the conditions given in Assumption MP. Then [a, b), ∗ is a hypergroup.

168

4 Generalized Convolutions for Sturm-Liouville Operators

Proof We need to check that the L-convolution satisfies axioms H6–H8. Assume first that satisfies the assumptions of Proposition 4.17. Then the explicit expressions for supp(δx ∗ δy ) show that (in each of the cases (a)–(e)) supp(δx ∗ δy ) is compact, depends continuously on (x, y) and contains e = 0 if and only if x = y, so that H6–H8 hold. (Verifying the continuity is easy after noting that the topology in the space of compact subsets can be metrized by the Hausdorff metric, cf. [106, Subsection 4.1].) In the general case of an operator of the form (4.1), the hypothesis that γ (a) > r(y) is integrable near a, and thus we may assume that γ (a) = 0 −∞ means that p(y) (otherwise, replace the interior point c by the endpoint a in the definition of the d d (A dξ ) defined function γ ). By assumption the transformed operator 

= − A1 dξ via (4.2) satisfies the assumptions of Proposition 4.17; by the above, the associated convolution, which we denote by  ∗, satisfies H6–H8. From the product formulas for the solutions wλ (x) and w λ (ξ ) = wλ (γ −1 (ξ )) we deduce that

 [a,b)

wλ d(δx ∗ δy ) = wλ (x)wλ (y) =w λ (γ (x)) wλ (γ (y))    = wλ (γ −1 (z)) δγ (x) ∗ δγ (y) (dz) R+ 0

and, consequently, δx ∗ δy = γ −1 (δγ (x) ∗ δγ (y)). In particular, supp(δx ∗ δy ) = γ −1 supp(δγ (x) ∗ δγ (y)) ; since γ is a continuous bijection, we immediately conclude that the convolution ∗ also satisfies axioms H6–H8. Recalling the definition of hypergroup isomorphism given in Sect. 2.3, we see    that the hypergroups [a, b), ∗ and R+ ∗ considered above (associated with the 0 , differential operators and 

respectively) are isomorphic.

For Sturm-Liouville operators on R+ of the form (u) = −u

− AA u , the assumption of the theorem above can be re-expressed in terms of conditions SL0– SL2 introduced in Sect. 4.1: Corollary 4.5 Suppose that A satisfies SL0 and SL2. For f ∈ D(2,0), denote by vf the unique solution of x vf = y vf , vf (x, 0) = vf (0, x) = f (x), of Sect. 4.3.1 (∂y[1] vf )(x, 0) = (∂x[1] vf )(0, y) = 0 such that conditions  (α)–(β)  hold for h = vf . Define the convolution ∗ via (4.3). Then R+ , ∗ is a hypergroup. 0 Proof Just notice that, by (4.43) and Lemma 4.9, the definition of convolution given in the statement of the corollary is equivalent to Definition 4.5. The statement of Corollary 4.5 strongly resembles that of Zeuner’s existence theorem for Sturm-Liouville hypergroups (Theorem 4.1), but its assumptions do not include condition SL1. The corollary therefore shows that it is natural to modify the definition of Sturm-Liouville hypergroup (Definition 4.1) by replacing the space

4.6 Sturm-Liouville Hypergroups

169

(2,0) and replacing ∂ by ∂ [1] in the initial condition, because in this C∞ y y c,even by D way we are able to extend the class of Sturm-Liouville hypergroups to all functions A satisfying conditions SL0 and SL2. We emphasize that condition SL1 imposes a great restriction on the behaviour

of the Sturm-Liouville operator (u) = −u

− AA u near zero: in the singular case

(x) A(0) = 0, SL1 requires that AA(x) ∼ αx0 . Therefore, as shown in the next example, Corollary 4.5 leads, in particular, to a considerable extension of the class of singular operators for which an associated hypergroup exists:

Example 4.1 If A satisfies SL0 and the function AA is nonnegative and decreasing, then SL2 is satisfied with η := 0. Therefore, Corollary 4.5 ensures that there exists

a hypergroup associated with the operator (u) = −u

− AA u . Notice that this

(x) existence result holds without any restriction on the growth of AA(x) as x ↓ 0. This class of examples includes the following special cases: 2

d (a) = − dx 2 (0 < x < ∞); here A(x) ≡ 1. As noted in Example 2.1, the Sturm-Liouville solutions are wλ (x) = cos(τ x) 2 (λ∞ = τ ) and the L-transform is the cosine Fourier transform (Fh)(τ ) = 0 h(x) cos(τ x)dx. By elementary trigonometric identities, wτ (x)wτ (y) = 1 2 [wτ (|x − y|) + wτ (x + y)], hence the L-convolution is given by

δx ∗ δy =

1 (δ|x−y| + δx+y ), 2

x, y ≥ 0.

In other words, ∗ is (up to identification) the ordinary convolution of symmetric measures. d2 1−2α (0 < x < ∞, α > 0). (b) = −x 2−2α dx 2 − (1 − α)x This operator is of the form (4.1) with p(x) = x 1−α and r(x) = x α−1 , and it d2 α is transformed into the operator − dx 2 via the change of variable ξ = γ (x) = x (cf. Remark 4.1). Accordingly, it follows from (a) that the L-convolution is given by δx ∗ δy =

1 (δ α α 1/α + δ(x α +y α )1/α ), 2 |x −y |

x, y ≥ 0.

This is the so-called (α, 1)-convolution [190], which is a generalized convolution satisfying Urbanik’s axioms (cf. Definition 2.3). 2η+1 d d2 1 2η+1 . (c) = − dx 2 − x dx (0 < x < ∞, η > − 2 ); here A(x) ≡ x As noted in Sect. 2.4, the Sturm-Liouville solutions are wλ (x) = J η (τ x) (λ = τ 2 , J η the normalized Bessel functionof the first kind) and the L∞ transform is the Hankel transform (Fh)(τ ) = 0 h(x) J η (τ x) x 2η+1dx. The product formula for the Bessel function given in Theorem 2.6 shows that the

170

4 Generalized Convolutions for Sturm-Liouville Operators

L-convolution is 21−2η (η + 1) (xy)−2η × (δx ∗ δy )(dξ ) = √ π (η + 12 )

η−1/2 × (ξ 2 − (x − y)2 )((x + y)2 − ξ 2 ) 1[|x−y|,x+y](ξ ) ξ dξ. This is the  Kingman convolution (cf. Definition 2.6). The corresponding  hypergroup R+ 0 , ∗ , which is known as the Bessel-Kingman hypergroup, plays a special role in the context of Sturm-Liouville hypergroups; in particular, it appears as the limit distribution in central limit theorems on hypergroups [19, Section 7.5]. 1 d2 1−2α d (d) = −x 2−2α dx 2 − (2αη + 1)x dx (0 < x < ∞, η > − 2 , α > 0). This operator is of the form (4.1) with p(x) = x 2αη+1 and r(x) = x 2α(η+1)−1; similar to (b) above, the change of variable ξ = γ (x) = x α 2η+1 d d2 transforms into the operator − dx 2 − x dx . The L-convolution is thus given by (δx ∗ δy )(dξ ) =

η−1/2

α 21−2η (η + 1) (xy)−2αη (ξ 2α − (x α − y α )2 )((x α + y α )2 − ξ 2α ) × √ 1 π (η + 2 ) × 1[|x α −y α |1/α ,(x α +y α )1/α ] (ξ ) ξ 2α−1 dξ.

This is the so-called (α, β)-convolution [190]; as in (b), one can check that it satisfies axioms U1–U6 of Urbanik convolutions. d2 d 1 (e) = − dx 2 − [(2α + 1) coth x + (2β + 1) tanh x] dx (0 < x < ∞, α ≥ β ≥ − 2 , α = − 12 ); here A(x) = (sinh x)2α+1(cosh x)2β+1 . As solutions are wλ (x) =  1 noted in 1Example 2.5, the Sturm-Liouville  2 (η = α + β + 1, λ = τ 2 + η2 , 2 F1 2 (η − iτ ), 2 (η + iτ ); α + 1; −(sinh x) 2 F1 the hypergeometric function) and the L-transform is the (Fourier-)Jacobi transform (2.43). By a deep result of Koornwinder [61, 105], the measures of the product formula wλ (x) wλ(y) = [a,b) wλ d(δx ∗ δy ) are given by 2−2σ (α + 1)(cosh x cosh y cosh ξ )α−β−1 √ π (α + 12 )(sinh x sinh y sinh ξ )2α   × (1 − Z 2 )α−1/2 2 F1 α + β, α − β; α + 12 ; 12 (1 − Z) 1[|x−y|,x+y](ξ )A(ξ )dξ,

(δx ∗ δy )(dξ ) =

+(cosh y) +(cosh ξ ) −1 where Z := (cosh x) ; the corresponding hypergroup is the 2 cosh x cosh y cosh ξ so-called Jacobi hypergroup. For half-integer values of the parameters α, β, this hypergroup structure has various group theoretic interpretations; in particular, it is related with harmonic 2

2

2

4.6 Sturm-Liouville Hypergroups

171

analysis on rank one Riemannian symmetric spaces [105]. Moreover, a remarkable property of the Jacobi hypergroup is that it admits a positive dual convolution structure, i.e. there exists a family {θλ1 ,λ2 } offinite positive measures such ∞ that the dual product formula wλ1 (x) wλ2 (x) = 0 wλ3 (x) θλ1 ,λ2 (dλ3 ) holds, and this permits the construction of a generalized convolution which trivializes the inverse Jacobi transform [12]. α d d2 α 2μx . (f) = − dx 2 − x + 2μ dx (0 < x < ∞, α, μ > 0); here A(x) = x e As noted in Example 2.3, the solutions of the Sturm-Liouville initial value α α problem are wλ (x) = (2iτ )− 2 e−μx x − 2 M− αμ , α−1 (2iτ x) (λ = τ 2 + μ2 , 2iτ 2 Mκ,ν (·) the Whittaker function of the first kind) and the L-transform is the α ∞ α index transform (Fh)(τ ) = (2iτ )− 2 0 h(x) M− αμ , α−1 (2iτ x) x 2 eμxdx. It 2iτ 2 follows from Corollary 4.5 that there exists a family of probability measures {νx,y }x,y≥0 with support [|x − y|, x + y] such that the Whittaker function of the first kind satisfies the product formula (2iτ xy)− 2 e−μ(x+y) M− αμ , α−1 (2iτ x)M− αμ , α−1 (2iτy) = 2iτ 2 2iτ 2  α = ξ − 2 e−μξ M− αμ , α−1 (2iτ ξ ) νx,y (dξ ). α

2iτ

(4.53)

2

Unlike in cases (a)–(e) above, here the function A does not satisfy condition SL1 of Zeuner’s existence theorem for Sturm-Liouville hypergroups; the existence of the product formula (4.53) is, as far as we know, a novel result. It is natural to wonder whether one can determine the closed-form expression of each of the measures νx,y in terms of classical special functions. We leave this as an open problem. The convolutions discussed in cases (a)–(d) above are the only known examples of Sturm-Liouville convolutions which satisfy axioms U1– U6 of Urbanik.

4.6.2 The Degenerate Case: Degenerate Hypergroups of Full Support Definition 4.11 Let K be a locally compact space and ∗ a bilinear operator on MC (K). The pair (K, ∗) is said to be a degenerate hypergroup of full support if it satisfies the hypergroup axioms H1– H5, together with the following axiom: DH.

supp(δx ∗ δy ) = K for all x, y ∈ K \ {e}.

In order to determine the conditions under which the Sturm-Liouville convolution algebra ([a, b), ∗) is a degenerate hypergroup of full support, we need to know when the solution of the associated hyperbolic Cauchy problem (4.22) is strictly positive inside (a, b)2 . Our starting lemma provides an integral inequality which proves to be useful for studying the strict positivity of solution.

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4 Generalized Convolutions for Sturm-Liouville Operators

 1 x Lemma 4.10 Write R(x) := A(x) B(x) , where B(x) = exp( 2 β η(ξ )dξ ) (with β > γ (a) arbitrary). Take h ∈ D(2,0) such that h ≥ 0. Let u(x, y) := h(γ −1 (x), γ −1 (y)), where h ∈ C2 (a, b)2 is the solution (4.23) of the Cauchy problem (cf. Lemma 4.9). Then the following inequality holds: R(x)R(y)u(x, y)  y

 ≥ 12 R(s)R(x − y + s) φη (s) + φη (x − y + s) u(x − y + s, s) ds  +

1 2

 +

γ (a)

 R(s)R(x + y − s) φη (s) − φη (x + y − s) u(x + y − s, s) ds

y γ (a)

 R(ξ )R(ζ ) ψ η (ζ ) − ψ η (ξ ) u(ξ, ζ ) dξ dζ,

1 2

(4.54) ≡

where

γ (a),x,y

= {(ξ, ζ ) ∈ R2 | ζ ≥ γ (a), ξ + ζ ≤ x + y, ξ − ζ ≥ x − y}.

Proof Let {am }m∈N be a sequence b > a1 > a2 > . . . with lim am = a. For m ∈ N, set a˜ m = γ (am ) and define um (x, y) := hm (γ −1 (x), γ −1 (y)), where hm is the function defined in (4.29). The function vm (x, y) = B(x)B(y)um (x, y) is a solution of (℘ x vm )(x, y) = (℘ y vm )(x, y),

x, y > a˜ m ,

vm (x, a˜ m ) = B(x)B(a˜ m )h(γ −1 (x)),

x > a˜ m ,

(∂y vm )(x, a˜ m ) = 12 η(a˜ m )B(x)B(a˜ m )h(γ −1 (x)),

x > a˜ m ,

2

∂ ∂ ˜ m ), (∂y vm )(x, a˜ m ) ≥ 0. where ℘ x := − ∂x 2 − φ η (x) ∂x + ψ η (x). Clearly, vm (x, a By Lemma 4.3 (with ℘ 1 (v) = ℘ 2 (v) = −v

− φη v + ψ η v), the integral equation (4.8) holds with v = vm and c = am . It is clear that we have H ≥ 0, I0 ≥ 0 and I4 = 0 in the right hand side of (4.8); moreover, it follows from Proposition 4.5 and Assumption MP that the integrands of I1 , I2 and I3 are nonnegative. Consequently, for α ∈ [a˜ m , y] we have

R(x)R(y)um (x, y)  y

 R(s)R(x − y + s) φη (s) + φη (x − y + s) um (x − y + s, s) ds ≥ 12 + +



α y



α

1 2

 R(s)R(x + y − s) φη (s) − φη (x + y − s) um (x + y − s, s) ds

 R(ξ )R(ζ ) ψ η (ζ ) − ψ η (ξ ) um (ξ, ζ ) dξ dζ,

1 2 α,x,y

(4.55)

4.6 Sturm-Liouville Hypergroups

173

where α,x,y = {(ξ, ζ ) ∈ R2 | ζ ≥ α, ξ + ζ ≤ x + y, ξ − ζ ≥ x − y}. Since by Proposition 4.4 limm→∞ um (x, y) = u(x, y) pointwise for x, y ∈ (γ (a), ∞), by taking the limit we deduce that for each fixed α ∈ (γ (a), y] the inequality (4.55) holds with um replaced by u. If we then take the limit α ↓ γ (a), the desired integral inequality follows. The next lemma will help us in verifying the strict positivity of the integrands in the integral inequality (4.54). Lemma 4.11 If γ (a) = −∞, then at least one of the functions φη , ψ η defined in Assumption MP is non-constant on every neighbourhood of −∞. Proof Suppose by contradiction that γ (a) = −∞ and φη , ψ η are both constant on an interval (−∞, κ] ⊂ R. Recall from the proof of Proposition 4.3 that L is d2 unitarily equivalent to a self-adjoint realization of − dξ 2 + q, where q is given by (4.5). Clearly, q(ξ ) = q∞ := 14 φ2η (κ) + ψ η (κ) < ∞ for all ξ ∈ (−∞, κ). Using the theorem on the spectral properties of Sturm-Liouville operators stated in [201, Theorem 15.3], we deduce that the essential spectrum of any self-adjoint realization of restricted to an interval (a, c) (for a < c < b) contains [q∞ , ∞). However, we know from the proof of Proposition 4.3 that self-adjoint realizations of restricted to (a, c) have purely discrete spectrum. This contradiction proves the lemma. We are now ready to prove that in the case γ (a) = −∞ the solution of the (nontrivial) Cauchy problem (4.22) always has full support on (a, b)2, even when the initial condition is compactly supported: Theorem 4.7 (Strict Positivity of Solution for the Cauchy Problem (4.22)) Suppose that γ (a) = −∞. Take h ∈ D(2,0) . If f ≥ 0 and f (τ0 ) > 0 for some τ0 ∈ (a, b), then the function h given by (4.23) is such that h(x, y) > 0

for x, y ∈ (a, b).

Proof Let u(x, y) := h(γ −1(x), γ −1(y)) and τ˜0 = γ (τ0 ). Fix x0 ≥ y0 > −∞. Since limy→−∞ u(τ˜0 , y) = f (τ0 ) > 0, there exists κ ∈ (−∞, min{y0 , τ0 }) such that u(τ˜0 , y) > 0 for all y ≤ κ. Suppose φη is non-constant on every neighbourhood of −∞. Choosing a smaller κ if necessary, we may assume that φη (κ) > φη (ξ ) for all ξ > κ. For each x > τ˜0 and y ≤ κ we have by Lemma 4.10 R(x)R(y)u(x, y) ≥ 12

 y −∞

 R(s)R(x − y + s) φη (s) + φη (x − y + s) u(x − y + s, s) ds,

and the integrand in the right hand side is continuous and strictly positive at s = y − x + τ˜0 , so the integral is positive and therefore u(x, y) > 0 for all x ≥ τ˜0 and

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4 Generalized Convolutions for Sturm-Liouville Operators

y ≤ κ. Again by Lemma 4.10, R(x0 )R(y0 )u(x0 , y0 )  y0

 ≥ 12 R(s)R(x0 + y0 − s) φη (s) − φη (x0 + y0 − s) u(x0 + y0 − s, s) ds, −∞

with the integrand being strictly positive for s < min{κ, x0 + y0 − τ˜0 }, thus u(x0 , y0 ) > 0. Suppose now that ψ η is non-constant on every neighbourhood of −∞ and that κ is chosen such that ψ η (κ) > ψ η (ξ ) for all ξ > κ. The integral inequality of Lemma 4.10 yields  R(x0 )R(y0 )u(x0 , y0 ) ≥

1 2

 R(ξ )R(ζ ) ψ η (ζ ) − ψ η (ξ ) u(ξ, ζ ) dξ dζ,

where = {(ξ, ζ ) ∈ R2 | ξ + ζ ≤ x0 + y0 , ξ − ζ ≥ x0 − y0 }. Clearly, the integrand is continuous and > 0 on {(τ0 , ζ ) | ζ ≤ min(y0 − |x0 − τ0 |, κ)} ⊂ , and it follows at once that u(x0 , y0 ) > 0. By Lemma 4.11 it follows that u(x0 , y0 ) > 0. Since x0 ≥ y0 > −∞ are arbitrary we conclude that h(x, y) > 0 for b > x ≥ y > a and, by symmetry, for x, y ∈ (a, b). Corollary 4.6 (Existence Theorem for Degenerate Hypergroups of Full Support) Let be a Sturm-Liouville expression of the form (4.1) which satisfies  Assumption MP, and suppose that γ (a) = −∞. Then [a, b), ∗ is a degenerate hypergroup of full support.   Proof By Proposition 4.9, the pair [a, b), ∗ satisfies axioms H1–H5. As in the if proof of Proposition 4.17, z ∈ [a, b) belongs to supp(δx ∗ δy ) if and only (2,0) w (x) w (y) (Ff )(λ) ρ (dλ) > 0 for all ε > 0, where {f } ⊂ D λ ε ε L [σ 2 ,∞) λ is a family of  functions satisfying (4.51). But it follows from Theorem 4.7 that hfε (x, y) = [σ 2 ,∞) wλ (x) wλ (y) (Ffε )(λ) ρ L (dλ) > 0 for all x, y ∈ (a, b). Hence each z ∈ [a, b) belongs to all the sets supp(δx ∗ δy ), x, y ∈ (a, b); therefore,  [a, b), ∗ satisfies axiom DH. This corollary confirms that, as anticipated in Sect. 4.1, the Whittaker convolution studied in Chap. 3 is a particular case of a general family of Sturm-Liouville convolutions which do not satisfy the compactness axiom, but which also allow us to develop the basic notions and facts from probabilistic harmonic analysis. Example 4.2 Let ζ ∈ C1 (R+ ) be a nonnegative decreasing function and let κ > 0. The differential expression

= −x 2

  d d2 , − κ + x 1 + ζ(x) 2 dx dx

0 0.

4.7 Harmonic Analysis on Lp Spaces Finally, we turn to the mapping properties of the L-convolution of functions, defined in the natural way (compare with Sect. 3.7): Definition 4.12 Let f, g : [a, b) −→ C. If the integral  (f ∗ g)(x) =

b



b

(Ty f )(x) g(y) r(y)dy =

a

a

 [a,b)

f (ξ ) (δx ∗ δy )(dξ ) g(y) r(y)dy

exists for almost every x ∈ [a, b), then we call it the L-convolution of the functions f and g. As usual, the convolution is trivialized by the L-transform and commutes with both the Sturm-Liouville operator and the associated translation operator: Proposition 4.18 Let y ∈ [a, b) and λ ≥ 0. Then:   (a) If f ∈ L2 (r) and g ∈ L1 (r), then F(f ∗ g) (λ) = (Ff )(λ) (Fg)(λ); (b) If f ∈ L2 (r) and g ∈ L1 (r), then Ty (f ∗ g) = (Ty f ) ∗ g; (c) If f ∈ D(L(2) ) and g ∈ L1 (r), then f ∗ g ∈ D(L(2) ) and (f ∗ g) = ( f ) ∗ g. Proof We know from Proposition 4.10 that F(Tμ f )(λ) = $ μ(λ) (Ff )(λ), hence these properties can be proved using the same reasoning as in Proposition 3.20. The L-convolution also satisfies a Young inequality analogous to those of the previous chapters: Proposition 4.19 (Young Inequality for the L-Convolution) Let p1 , p2 ∈ [1, ∞] such that p11 + p12 ≥ 1. For f ∈ Lp1 (r) and g ∈ Lp2 (r), the L-convolution f ∗ g is well-defined and, for s ∈ [1, ∞] defined by

1 s

=

1 p1

+

1 p2

− 1, it satisfies

f ∗ gs ≤ f p1 gp2 (in particular, f ∗ g ∈ Ls (r)). Consequently, the L-convolution is a continuous bilinear operator from Lp1 (r) × Lp2 (r) into Ls (r).

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4 Generalized Convolutions for Sturm-Liouville Operators

Proof Identical to that of Proposition 3.18.

4.7.1 A Family of L1 Spaces We will study the L-convolution as an operator acting on the  family of Lebesgue spaces {L1κ }−∞ 0, a < x < b and ψ ∈ L1 (r), the integral equation 

b

h(y) +

h(ξ ) qt (x, y, ξ )r(ξ )dξ = ψ(y)

a

has a unique solution h ∈ L1 (r) which can be written in the form (4.64) for some function g ∈ L1 (r). Proof Let us justify that this result is obtained by setting f = ft,x := p(t, x, ·) in the statement of Theorem 4.9. Notice first that by Corollary 2.2 we have ft,x ∈ L10 ≡ L1 (r). Moreover, we have (Fft,x )(λ) = e−t λwλ (x) (cf. (2.33)), thus 1 + (Fft,x )(0) = 2 and |1 + (Fft,x )(λ)| ≥ 1 − e−t Re λ |wλ (x)| > 0,

λ ∈ !0 \ {0}

(this is easily seen to hold by setting λ = τ 2 + σ 2 and recalling the estimate (4.17)). Recalling that     F(Ty ft,x ) (λ) = (Fft,x )(λ) wλ (y) = e−t λwλ (x)wλ (y) = Fqt (x, y, ·) (λ), where we used Proposition 4.10(ii), we see that (Ty ft,x )(ξ ) = qt (x, y, ξ ), so that the corollary is a particular case of the theorem.

Chapter 5

Convolution-Like Structures on Multidimensional Spaces

After having constructed convolution-like operators associated with the Shiryaev process and with a family of one-dimensional diffusion processes generated by Sturm–Liouville operators, we devote this final chapter to a more general discussion of the problem formulated in the Introduction: the construction of a generalized convolution associated with a given Feller process on a generally multidimensional state space. We start this chapter by describing the desired properties of such a generalized convolution structure; these properties will be seen to lead to strong restrictions in the behaviour of the eigenfunctions of the Feller generator. In Sects. 5.2–5.3, after reviewing some basic notions and facts from spectral theory and differential operators, we show that such restrictions fail to hold for reflected Brownian motions on bounded smooth domains of Rd (d > 2) and also for certain one-dimensional diffusions, leading to negative results on the existence of associated convolution-like structures. Finally, in Sect. 5.4 we propose the notion of a family of convolutions associated with a given Feller semigroup as a natural way of overcoming the difficulties in constructing convolutions on multidimensional spaces; such families of convolutions will be shown to exist for a general class of two-dimensional manifolds endowed with cone-like metrics.

5.1 Convolutions Associated with Conservative Strong Feller Semigroups Our work in Chaps. 3 and 4 indicates that none of the standard axiomatic definitions in the literature on generalized harmonic analysis—hypergroups, hypercomplex systems, Urbanik generalized convolutions, stochastic convolutions, etc. (see Sect. 2.3)—is fully satisfactory in identifying the essential requirements for © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Sousa et al., Convolution-like Structures, Differential Operators and Diffusion Processes, Lecture Notes in Mathematics 2315, https://doi.org/10.1007/978-3-031-05296-5_5

183

184

5 Convolution-Like Structures on Multidimensional Spaces

constructing a convolution-like operator associated with a given transition semigroup. In light of the results of the previous chapters, we will instead consider the following notion of convolution-like structure: Definition 5.1 Let E be a locally compact separable metric space, and let {Tt }t ≥0 be a conservative strong Feller semigroup on E. We say that a bilinear operator  on MC (E) is a Feller-Lévy trivializable convolution (FLTC) for {Tt } if the following conditions hold: I. (MC (E), ) is a commutative Banach algebra over C (with respect to the total variation norm) with identity element δa (a ∈ E), and (μ, ν) → μ  ν is continuous in the weak topology of measures; II. P(E)  P(E) ⊂ P(E); III. There exists a family  ⊂ Bb (E)\{0} such that for μ, μ1 , μ2 ∈ P(E) we have μ = μ1  μ 2

if and only if

μ(ϑ) = μ1 (ϑ)·μ2 (ϑ) for all ϑ ∈ ,

 where μ(ϑ) := E ϑ(ξ )μ(dξ ); IV. The transition kernel {pt,x }t ≥0,x∈E of the semigroup {Tt } is of the form pt,x = μt  δx , where {μt }t ≥0 ⊂ P(E) is a family of measures such that μt +s = μt  μs for all t, s ≥ 0. Conditions I and II in the above definition can be interpreted as basic axioms that allow us to interpret (MC (E), ) as a probability-preserving convolution-like structure. Condition III requires the existence of an integral transform with bounded kernels which determines uniquely a given measure μ ∈ MC (E) (in the sense that if μ(ϑ) = ν(ϑ) for all ϑ ∈ , then μ = ν) and trivializes the convolution in the same way as the Fourier transform trivializes the ordinary convolution. As noted in [197], it is possible, in principle, to study infinite divisibility of probability measures on measure algebras not satisfying Condition III; however, it is natural to require Condition III to hold, not only because, to the best of our knowledge, all known examples of convolution-like structures are constructed from a product formula of the form ϑ(x)ϑ(y) = (δx  δy )(ϑ) (and therefore possess such a family of trivializing functions) but also because this trivialization property leads to a richer theory. Lastly, condition IV expresses the motivating goal discussed in the Introduction: the Feller semigroup {Tt } should have the convolution semigroup property with respect to the operator  or, in other words, the Feller process {Xt }t ≥0 determined by {Tt } is a Lévy process with respect to  in the sense that we have

P Xt ∈ ·|Xs = x = μt −s  δx for every 0 ≤ s ≤ t and x ∈ E. The problem of existence of an associated FLTC is meaningful for any given strong Feller semigroup on a locally compact separable metric space. Before we present some notable examples of this class of semigroups, let us recall some prerequisite notions from the theory of Dirichlet forms. Let μ be a σ -finite measure

5.1 Convolutions Associated with Conservative Strong Feller Semigroups

185

on E. We say that (E, D(E)) is a Dirichlet form on L2 (E, μ) if D(E) is a dense subspace of L2 (E, μ) and E is a nonnegative, closed, Markovian symmetric sesquilinear form defined on D(E) × D(E). The associated non-positive self-adjoint operator (G(2), D(G(2))) is defined as u ∈ D(G(2))

if and only if

∃φ ∈ L2 (E, μ) such that E(u, v) = −φ, vL2 (E,μ) for all v ∈ D(E), (5.1) and G(2) u := φ for u ∈ D(G(2) ). The semigroup determined by E is defined by (2) (2) := et G (where the latter is obtained by spectral calculus); one can show T t

(2)

[33, Theorem 1.1.3] that {Tt } is a strongly continuous, sub-Markovian contraction semigroup on L2 (E, μ). The Dirichlet form (E, D(E)) is said to be strongly local if E(u, v) = 0 whenever u ∈ D(E) has compact support and v ∈ D(E) is constant on a neighbourhood of supp(u). It is said to be regular if D(E) ∩ Cc (E) is dense both  in D(E) with respect to the norm uD(E) = E(u, u) + uL2 (E,μ) and in Cc (E) with respect to the sup norm. A well-known result [66, Theorem 7.2.1] states that if (2) (E, D(E)) is a regular Dirichlet form on L2 (E, μ) with semigroup {Tt }t ≥0 , then there exists a Hunt process with state space E whose transition semigroup {Pt }t ≥0 is such that Pt u is, for all u ∈ Cc (E), a quasi-continuous version of Tt(2) u. (A Hunt process is essentially a strong Markov process whose paths are right-continuous and quasi-left-continuous; see [66, Appendix A.2] for details.) We refer to the textbooks of Fukushima, Oshima and Takeda [66] and of Chen and Fukushima [33] for further background on Dirichlet forms and related objects. Example 5.1 (a) Let (E, g) be a complete Riemannian manifold, let m be the Riemannian volume on E and let ∇ denote the Riemannian gradient on (M, g). The sesquilinear form E(u, v) =

1 2

 ∇u, ∇vg dm,

u, v ∈ D(E)

E

with domain 1 D(E) = closure of C∞ c (E) in the Sobolev space H (E)

≡ {u ∈ L2 (E) | |∇u| ∈ L2 (E)} is a strongly local regular Dirichlet form on L2 (E) ≡ L2 (E, m). The Hunt diffusion process {Xt }t ≥0 with state space E associated with this Dirichlet form is the Brownian motion on (E, g). One can show that the strongly  continuous contraction semigroup {Tt } determined by E is such that Tt Bb (E) ⊂ Cb (E),

186

5 Convolution-Like Structures on Multidimensional Spaces

so that the Brownian motion {Xt } is a strong Feller process [182, Section 6]. Moreover, it is shown in [66, Example 5.7.2] that the Feller semigroup {Tt } is conservative provided that the Riemannian volume m is such that lim inf r→∞

1 log m(B(x0 ; r)) < ∞ for some fixed x0 ∈ E. r2

Let G(0) : D(G(0) ) ⊂ C0 (E) −→ C0 (E) be the infinitesimal generator of the (0) Brownian motion {Xt }. Then G(0) u = 12 u for u ∈ C∞ c (E) ⊂ D(G ), where is the Laplace–Beltrami operator on the Riemannian manifold (E, g). (b) Let E = Rd , m a positive function such that m, m1 ∈ Cb (Rd ) and A = (aj k ) a symmetric d × d matrix-valued function such that aj k ∈ C(Rd ) (for each j, k ∈ {1, . . . , d}) and c−1 |ξ |2 ≤

d 

aj k (x)ξj ξk ≤ c|ξ |2 ,

(x, ξ ) ∈ Rd × Rd

(5.2)

j,k=1

for some constant c > 0. The sesquilinear form E(u, v) =

d  1  ∂u ∂v aj k (x) m(x)dx, 2 ∂xj ∂xk Rd

u, v ∈ D(E)

j,k=1

with domain d D(E) = closure of C∞ c (R ) under the inner product E(·, ·) + ·, ·L2 (m)

is a strongly local regular Dirichlet form on the space L2 (m) ≡ L2 (Rd , m(x)dx) [66, Section 3.1]. The Hunt diffusion {Xt }t ≥0 associated with the Dirichlet form E is conservative [66, Example 5.7.1]. The process {Xt }, which is called the (A, m)-diffusion on Rd , is a strong Feller process, cf. [183, Example 4.C and Proposition 7.5]. If, in addition, we have ∂(majk ) ∈ L2loc (Rd ) for each j, k ∈ {1, . . . , d}, then the infinitesimal ∂xj generator G(0) of the Feller semigroup is the elliptic operator (G(0) u)(x) =  ∂ 1 d ∂u (0) 2 d j,k=1 ∂xj m(x)aj k (x) ∂xk (for u ∈ Cc (R ) ⊂ D(G )). 2m(x) ˚ ⊂ Rd and, as usual, (c) Let E be the closure of a bounded Lipschitz domain E k k let H (E) be the Sobolev space defined as H (E) := {u ∈ L2 (E, dx) | ∂ α u ∈ L2 (E, dx) for all α = (α1 , . . . , αd ) with |α| ≤ k}. Let m ∈ H 1 (E) be a positive function such that m, m1 ∈ C(E) and let A = (aj k ) be a symmetric bounded d × d matrix-valued function such that aj k ∈ H 1 (E) for j, k ∈ {1, . . . , d} and the uniform ellipticity condition (5.2) holds for

5.1 Convolutions Associated with Conservative Strong Feller Semigroups

187

(x, ξ ) ∈ E × Rd . The sesquilinear form d  1  ∂u ∂v E(u, v) = aj k (x) m(x)dx, 2 ∂x j ∂xk E

u, v ∈ D(E) = H 1 (E)

j,k=1

is a strongly local regular Dirichlet form on L2 (E, m) ≡ L2 (E, m(x)dx) whose associated Hunt diffusion process is a conservative Feller process, cf. [32, 34]. The process {Xt } is called the (A, m)-reflected diffusion on E. The infinitesimal ˚ ⊂ D(G(0) ) and generator G(0) of the Feller process {Xt } is such that C2c (E)   1 d ∂u ∂ (0) ˚ In the special (G u)(x) = 2m(x) j,k=1 ∂xj m(x)aj k (x) ∂xk for u ∈ C2c (E). case aij = δij and m = 1, the (A, m)-reflected diffusion is known as the reflected Brownian motion on E, whose infinitesimal generator G(0) u = 12 u is the so-called Neumann Laplacian on E. (d) Let E be a locally compact separable metric space with distance d and let m be a locally finite Borel measure on E with m(U ) > 0 for all nonempty open sets U ⊂ E. Suppose that the triplet (E, d, m) satisfies the measure contraction property introduced in [183, Definition 4.1]; roughly speaking, this means that there exists a family of quasi-geodesic maps connecting almost every pair of points x, y ∈ E and which satisfy a contraction property which controls the distortions of the measure m along each quasi-geodesic. It was proved in [183] that the family of Dirichlet forms defined as    u(z) − u(x) 2 m(dx)    m(dz)  , 2 E B(x;r)\{x} d(z, x)  m(B(z; r)) m(B(x; r))

1 Er (u, u) =

 

r >0

(and Er (u, v) defined via the polarization identity) converges as r ↓ 0 (in a suitable variational sense, see [183]) to a strongly local regular Dirichlet form on L2 (E, m). The associated Hunt diffusion {Xt }t ≥0 is a strong Feller process with state space E. If the growth of the volumes m(B(x; r)) satisfies the condition stated in [66, Theorem 5.7.3], then the Feller process {Xt } is conservative. This class of strong Feller processes includes, as particular cases, the diffusions of Examples (a) and (b) above, diffusions on manifolds with boundaries or corners, spaces obtained by gluing together manifolds, among others. As discussed above, our motivating problem is that of determining, for a given Feller semigroup {Tt }, necessary or sufficient conditions for the existence of an associated FLTC satisfying the requirements of Definition 5.1. The following proposition shows that the converse problem of constructing a (strong) Feller semigroup associated with a given convolution-like semigroup of measures on the space E has a straightforward solution: Proposition 5.1 Let E be a locally compact separable metric space with the HeineBorel property (i.e. where each closed bounded subset is compact), and let  be a bilinear operator on MC (E) satisfying conditions I and II of Definition 5.1. Let

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5 Convolution-Like Structures on Multidimensional Spaces

{μt } ⊂ P(E) be a family of measures such that μt +s = μt  μs for all t, s ≥ 0,

w

μt −→ μ0 as t ↓ 0,

μ0 = δ a .

 For ν ∈ M+ (E), define (Tν f )(x) := E f d(ν  δx ), and assume that Tδy f ∈ C0 (E) for all f ∈ C0 (E) and y ∈ E. Then the operators Tt : C0 (E) −→ C0 (E),

Tt f := Tμt f

constitute a conservative Feller semigroup. If, in addition, we have (μt  δx )(dy) = pt (x, y)m(dy) (t > 0, x ∈ E) for some measure m ∈ M+ (E) and some density function pt (x, y) which is locally bounded on E × E for each t > 0, then {Tt }t ≥0 is a strong Feller process. Proof It is trivial that each operator Tt is positivity-preserving and conservative. The semigroup property Tt +s = Tt Ts and the strong continuity limt ↓0 Tt f − f ∞ = 0 follow as in the proof of Proposition 3.8. Next, let ν ∈ M+ (E) and {xn } ⊂ E a sequence such that d(xn , a) → ∞ as n → ∞. By assumption Tδy f ∈ C0 (E), thus by dominated convergence we have  |(T f )(xn )| ≤

|(Tδy f )(xn )| ν(dy) −→ 0

ν

as n → ∞,

E

showing that for each ε > 0 the space {x : |(Tν f )(x)| ≥ ε} is bounded and,   ν therefore, compact (by the Heine-Borel property). This shows that T (E) ⊂ C 0  C0 (E) for all bounded measures ν; in particular, Tt (C0 (E) ⊂ C0 (E). The fact that {Tt } is strong Feller under the stated absolute continuity condition follows from Theorem 2.6. We now prove an important fact concerning the family  of trivializing functions in the definition of an FLTC, namely that each ϑ ∈  is an eigenfunction of the Cb -generator of the associated Feller semigroup. (As in (2.4), the notion of Cb (b) (b) generator of   a semigroup {Tt }t ≥0 refers(b)to the operator G with domain D(G ) = Rη Cb (E) (η > 0) and given by (G u)(x) = ηu(x) − g(x) for u = Rη g, g ∈ ∞ Cb (E); recall that Rη f = 0 e−ηt Tt f dt denotes the resolvent of the semigroup {Tt }.) Proposition 5.2 Let {Tt } be a conservative strong Feller semigroup on a locally compact separable metric space E, and let  be a bilinear operator on MC (E) satisfying conditions I, II and IV of Definition 5.1. Suppose that ϑ ∈ Bb (E), ϑ ≡ 0 is a function such that (δx  δy )(ϑ) = ϑ(x)·ϑ(y)

for all x, y ∈ E.

(5.3)

5.1 Convolutions Associated with Conservative Strong Feller Semigroups

189

Then ϑ(a) = ϑ∞ = 1. Moreover, ϑ is an eigenfunction of the Cb -generator (G(b) , D(G(b) )) associated with an eigenvalue of nonpositive real part, in the sense that we have ϑ ∈ D(G(b) ) and G(b) ϑ = −λϑ for some λ ∈ C with Re λ ≥ 0. Proof Clearly, ϑ(x) = δx (ϑ) = (δx  δa )(ϑ) = ϑ(x)ϑ(a) for all x ∈ E. Since ϑ ≡ 0, this implies that ϑ(a) = 1. Next, pick ε > 0 and choose x0 ∈ E such that |ϑ(x0 )| > ϑ∞ − ε. Then (ϑ∞ − ε)2 < |ϑ(x0 )|2 = |(δx0  δx0 )(ϑ)| ≤ ϑ∞ (by condition II, δx0  δx0 ∈ P(E), which justifies the last step). Since ε is arbitrary, ϑ2∞ ≤ ϑ∞ , hence ϑ∞ ≤ 1. Since  is bilinear   and weakly continuous, a straightforward argument yields that (μ  ν)(dξ ) = E E (δx  δy )(dξ )μ(dx)ν(dy) for μ, ν ∈ MC (E). Consequently, (5.3) implies that (μ  ν)(ϑ) = μ(ϑ)·ν(ϑ) for all μ, ν ∈ MC (E). In particular, (Tt ϑ)(x) ≡ pt,x (ϑ) = (μt  δx )(ϑ) = μt (ϑ)·ϑ(x) due to condition IV. Given that {Tt } is strong Feller, we have Tt ϑ ∈ Cb (E) and ϑ therefore ϑ = μTt t(ϑ) ∈ Cb (E). Moreover, the fact that {Tt } is a conservative Feller semigroup ensures that μt (ϑ) = (Tt ϑ)(a) is a continuous function of t which, by condition IV, satisfies the functional equation μt +s (ϑ) = μt (ϑ)μs (ϑ). Therefore μt (ϑ) = e−λt for some λ ∈ C, and the fact that |μt (ϑ)| ≤ ϑ∞ = 1 implies that ϑ Re λ ≥ 0. We thus have Tt ϑ = e−λt ϑ and Rη ϑ = λ+η for η > 0, so we conclude that ϑ ∈ D(G(b) ) and G(b) ϑ = −λϑ. Corollary 5.1 Let {Tt } be a conservative strong Feller semigroup on a locally compact separable metric space E. Let  be an FLTC for {Tt } and  the corresponding family of trivializing functions. Then     ⊂ ω ∈ D(G(b) ) 

 ω(a) = ω∞ = 1, . G(b) ω = −λω for some λ ∈ C with Re λ ≥ 0

In particular, each μ ∈ MC (E) is uniquely determined by(b)the integrals μ(ω) ≡ E ω(x)μ(dx), where ω belongs to the set of solutions of G u = −λu (Re λ ≥ 0) satisfying ω(a) = ω∞ = 1. It is worth noting that if the strong Feller semigroup  {Tt } is symmetric with respect to a finite measure m ∈ M (E) (i.e. if + E (Tt f )(x) g(x)m(dx) =  f (x) (T g)(x)m(dx) for f, g ∈ C (E)), then the space Cb (E) is contained in t c E 2 L (E, m); accordingly, the Feller semigroup {Tt }t ≥0 and the Cb -generator G(b) (2) extend, respectively, to a strongly continuous semigroup {Tt } of symmetric 2 operators on L (E, m) and to the corresponding infinitesimal generator G(2) . In this setting, the trivializing functions ϑ ∈  are eigenfunctions of the L2 -generator G(2) . Applying spectral-theoretic results for self-adjoint operators on Hilbert spaces, we can deduce further properties for the trivializing functions: Proposition 5.3 Let {Tt } be a conservative Feller semigroup on a locally compact separable metric space E. Suppose that the corresponding transition kernels

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5 Convolution-Like Structures on Multidimensional Spaces

{pt,x (·)}t >0,x∈E are of the form pt,x (dy) = pt (x, y)m(dy) for some finite measure m ∈ M+ (E) and some density function pt (x, y) which is bounded and symmetric on E × E for each t > 0. Then {Tt } is strong Feller, is symmetric with respect to m, and admits an extension {Tt(2) } which is a strongly continuous semigroup on the space L2 (E, m). There exists a sequence 0 ≤ λ1 ≤ λ2 ≤ λ3 < . . . with λj → ∞ and an orthonormal basis {ωj }j ∈N of L2 (E, m) such that Tt(2)ωj = e−λj t ωj

(t ≥ 0),

G(2) ωj = −λj ωj ,

where G(2) stands for thegenerator of the L2 -semigroup {Tt(2) }. The sequence of −λj t < ∞ for each t > 0 (so that, in particular, eigenvalues is such that ∞ j =1 e limj →∞ λj = ∞). Assume also that  is an FLTC for {Tt } and that  is the family of trivializing functions for . Write Sk = {j ∈ N | λj = λk } and mk = |Sk | (k ∈ N). Then each function ϑ ∈  is a solution of G(2)ϑ = −λj ϑ for some j ∈ N. Furthermore, there exist functions {ϑj }j ∈N ⊂  such that span({ωj }j ∈Sk ) = span({ϑj }j ∈Sk ) and, consequently, span() = L2 (E, m). Proof The strong Feller property follows from Theorem 2.6. The symmetry with respect to m is obvious, and it is straightforward to show that for f ∈ Cc (E) we have Tt f L2 (E,m) ≤ f L2 (E,m) and Tt f − f L2 (E,m) → 0 as t ↓ 0, so that the extension {Tt(2) } is a strongly continuous semigroup on L2 (E, m). Let ·, · be the inner product on L2 (E, m). By the spectral theorem for compact self-adjoint operators (cf. e.g. [185, Theorem 6.7]), the operator T1(2) can be written  (2) (2) as T1 = ∞ j =1 μj ωj , · ωj , where μ1 ≥ μ2 ≥ . . . are the eigenvalues of T1 and {ωj }j ∈N is an orthonormal basis of L2 (E, m) such that each ωj is an eigenfunction (2) (2) of T1 associated with the eigenvalue μj ; in addition, we have μ1 ≤ T1  and μj ↓ 0 as j → ∞. If we define λj = − log μj , then it follows that Tt(2) =  ∞ −λj t ω , · ω . (This can be justified as follows, cf. [72, pp. 463–464] for j j j =1 e 1/2

1/2

(2) (2) further details: we know that (T1(2) − μj )ωj = (T1/2 + μj )(T1/2 − μj )ωj = 0, 1/2

1/2

(2) (2) and all the eigenvalues of (T1/2 + μj ) are positive, hence T1/2 ωj = μj ωj ;

similarly Tt ωj = e−λj t ωj for all t = m/2k and thus, by strong continuity, for all (2) t > 0.) Consequently, G(2) ωj = limt ↓0 1t (Tt ωj − ωj ) = −λj ωj . Since m is a (2) Tt is, for each finite measure and the densities pt (·, ·) are bounded, the operator ∞ −λ t > 0, a Hilbert-Schmidt operator, and therefore we have j =1 e j t < ∞ for all t > 0. By Corollary 5.1 each ϑ ∈  is such that G(2)ϑ = −λϑ for some λ ∈ C. Given that  ⊂ L2 (E, m) and eigenfunctions associated with different eigenvalues are orthogonal, we have λ = λj because otherwise we get a contradiction with the basis property of {ωj k }. (2)

5.1 Convolutions Associated with Conservative Strong Feller Semigroups

191 (2)

For the last part, fix t > 0, k ∈ N and let k := {ϑ ∈  | Tt ϑ = e−λk t ϑ} ⊂ L2 (E, m). Given that {ωj }j ∈Sk is a basis of the eigenspace associated with λk , we have span(k ) ⊂ span({ωj }j ∈Sk ). To prove the reverse inclusion, let h ∈ span({ωj }j ∈Sk ) ∩ span(k )⊥ , write νh (dx) := h(x)m(dx) and observe that (since m is a finite measure) νh ∈ MC (E). Then the integral  νh (ϑ) =

ϑ(x)h(x)m(dx) E

is equal to zero for ϑ ∈ k because h ∈ span(k )⊥ , and is also equal to zero (2) for ϑ ∈  \ k because then h and ϑ are eigenfunctions of Tt associated with different eigenvalues. Since measures ν ∈ MC (E) are uniquely determined by the integrals {ν(ϑ)}ϑ∈ , it follows that νh = 0 and therefore h = 0; this shows that span(k ) = span({ωj }j ∈Sk ). It follows at once that there exist linearly independent functions {ϑj }j ∈N ⊂  such that span({ωj }j ∈Sk ) = span({ϑj }j ∈Sk ). The conclusions of Proposition 5.3 are valid, in particular, for the Feller semigroups associated with the Brownian motion on a compact Riemannian manifold or with an (A, m)-reflected diffusion on a bounded Lipschitz domain, cf. Examples 5.1(a) and (c) respectively. (Indeed, it follows from e.g. [183, Theorem 7.4] that in both cases we have pt,x (dy) = pt (x, y)m(dy) with pt (x, y) bounded and symmetric; recall also that compact Riemannian manifolds have finite volume, cf. e.g. [75, Theorem 3.11].) It is worth emphasizing that, by Corollary 5.1 and Proposition 5.3, the existence of an FLTC for a Feller semigroup {Tt } satisfying the assumptions above implies that the following common maximizer property holds: CM.

There exists a set {ϑj }j ∈N of eigenfunctions of G(2) which is dense in L2 (E, m) and such that ϑj (a) = ϑj ∞ = 1 for some point a ∈ E.

(The functions ϑj associated with a common eigenvalue need not be orthogonal in L2 (E, m).) This necessary condition will play a fundamental role in the proof of the inexistence results established in Sect. 5.2. The next proposition and corollary show that the existence of an FLTC is closely related with the positivity of a regularized kernel which resembles the density (4.39) of the time-shifted product formula for Sturm–Liouville convolutions. Proposition 5.4 In the conditions of the first paragraph of Proposition 5.3, assume that the metric space E is compact. Let 0 ≤ λ1 ≤ λ2 ≤ λ3 < . . . be the eigenvalues of −G(2) and let {ϕj }j ∈N ⊂ L2 (E, m) be an orthogonal set of functions such that Tt ϕj = e−λj t ϕj , (2)

ϕ1 = 1,

supϕj 2 < ∞, j

where  · 2 denotes the norm of the space L2 (E, m). Then ϕj ∈ C(E) for all  1 −λj t ϕ (x) ϕ (y) ϕ (ξ ) is absolutely convergent j ∈ N, and the series ∞ j j j 2e j =1 ϕj 2

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5 Convolution-Like Structures on Multidimensional Spaces

for all t > 0 and x, y, ξ ∈ E. Moreover, the following assertions are equivalent: (i) We have ∞ 

qt (x, y, ξ ) :=

j =1

1 e−λj t ϕj (x) ϕj (y) ϕj (ξ ) ≥ 0 ϕj 22

(5.4)

for all t > 0 and x, y, ξ ∈ E. (ii) For each x, y ∈ E there exists a measure νx,y ∈ P(E) such that the product ϕj (x) ϕj (y) admits the integral representation  ϕj (x) ϕj (y) =

ϕj (ξ ) νx,y (dξ ),

x, y ∈ E, j ∈ N.

(5.5)

E

If these equivalent conditions hold and, in addition, there exists a ∈ E such that ϕj (a) = 1 for all j ∈ N, then the bilinear operator  on MC (E) defined as   νx,y (dξ ) μ(dx) ν(dy)

(μ  ν)(dξ ) = E

(5.6)

E

is an FLTC for {Tt } with trivializing family  = {ϕj }j ∈N . Proof Denote the inner product of the space L2 (E, m) by ·, ·. For each ε > 0 we have  |ϕj (x)| = eλj ε |(Tε(2) ϕj )(x)| = eλj ε ϕj , pε (x, ·) ≤ cε m(E) eλj ε ϕj 2 < ∞ (5.7) for m-almost every x ∈ E, where cε = sup(x,y)∈E×E pε (x, y). This shows that the function ϕj belongs to the space Bb (E) (possibly after redefining ϕj on a m-null set). Since {Tt } is strong Feller (Proposition 5.3), it follows that ϕj = eλj ε Tε ϕj ∈ C(E). The assumption that supj ϕj 2 < ∞, together with the estimate (5.7),  1 −λj t ϕ (x) ϕ (y) ϕ (ξ ) is absolutely convergent. ensures that the series ∞ j j j j =1 ϕ 2 e j 2

Suppose that (5.4) holds and fix x, y ∈ E. For t > 0, let νt,x,y ∈ M+ (E) be the measure defined by νt,x,y (dξ ) = qt (x, y, ξ )m(dξ ). We have 

 ϕj (ξ ) νt,x,y (dξ ) = E

ϕj (ξ ) E

=

∞  k=1

∞  k=1

1 e−λk t ϕk (x) ϕk (y) ϕk (ξ ) m(dξ ) ϕj 22

1 e−λk t ϕk (x) ϕk (y) ϕj , ϕk  ϕj 22

= e−λj t ϕj (x) ϕj (y). (5.8)

5.1 Convolutions Associated with Conservative Strong Feller Semigroups

193

It then follows from (5.8) (with j = 1) that νt,x,y (E) = 1, so that νt,x,y ∈ P(E)

for all t > 0, x, y ∈ E.

Now, let {tn }n∈N be an arbitrary decreasing sequence with tn ↓ 0. Since any uniformly bounded sequence of finite positive measures contains a vaguely convergent subsequence, there exists a subsequence {tnk } and a measure νx,y ∈ M+ (E) such v that νtnk ,x,y −→ νx,y as k → ∞. Let us show that all such subsequences {νtnk ,x,y } have the same vague limit. Suppose that tk1 , tk2 are two different sequences with v

tks ↓ 0 and that νtks ,x,y −→ νsx,y as k → ∞ (s = 1, 2). Recalling that E is compact, it follows that for all h ∈ C(E) and ε > 0 we have   (Tε h)(ξ ) νsx,y (dξ ) = lim (Tε h)(ξ ) νtks ,x,y (dξ ) k→∞ E

E

= lim

∞ 

k→∞

=

∞  j =1

j =1

s 1 e−λj (tk +ε) ϕj (x) ϕj (y) h, ϕj  2 ϕj 2

1 e−λj ε ϕj (x) ϕj (y) h, ϕj , ϕj 22

where the second equality follows from the identities qt (x, y, ·), ϕj  = e−λj t ϕj (x) ϕj (y) and Tε h, ϕj  = h, Tε ϕj  = e−λj ε h, ϕj . Consequently, we have   (Tε h)(ξ ) ν1x,y (dξ ) = (Tε h)(ξ ) ν2x,y (dξ ) for all ε > 0. (5.9) E

E

Since h ∈ C(E), by strong continuity of the Feller semigroup {Tt } we have limε↓0 Tε h − h∞ = 0, so by taking the limit ε ↓ 0 in both sides of (5.9) we deduce that ν1x,y (h) = ν2x,y (h), where h ∈ C(E) is arbitrary; therefore, ν1x,y = ν2x,y . Thus all subsequences have the same vague limit, and from this we conclude that v νt,x,y −→ νx,y as t ↓ 0. The product formula (5.5) is then obtained by taking the limit t ↓ 0 in the leftmost and rightmost sides of (5.8). Conversely, suppose that (5.5) holds for some measure νx,y ∈ M+ (E). Noting that for h ∈ C(E) we have (

)

h, pt (x, ·) = (Tt h)(x) =

∞  j =1

=

∞  j =1

1 Tt h, ϕj  ϕj (x) ϕj 22 1 e−λj t h, ϕj  ϕj (x) ϕj 22

∞ *  = h, j =1

+ 1 −λj t e ϕ (x)ϕ (·) , j j ϕj 22

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5 Convolution-Like Structures on Multidimensional Spaces

we see that pt (x, y) = x, y ∈ E we have qt (x, y, ξ ) =

∞  j =1

∞

1 −λj t ϕ (x) ϕ (y). j j j =1 ϕj 2 e 2

1 e−λj t ϕj (x) ϕj 22

Consequently, for t > 0 and



 ϕj (z) νy,ξ (dz) =

E

pt (x, z) νy,ξ (dz) ≥ 0, E

because both the density pt (x, ·) and the measures νy,ξ are nonnegative. Finally, assume that ϕj (a) = 1 for all j and that (ii) holds. Let  be the operator defined by (5.6). To prove that  = {ϕj }j ∈N satisfies condition III in Definition 5.1, it only remains to show that each μ ∈ MC (E) is uniquely characterized by {μ(ϕj )}j ∈N . Indeed, if we take μ ∈ MC (E) such that μ(ϕj ) = 0 for all j , then for h ∈ C(E) and t > 0 we have  (Tt h)(x) μ(dx) = E

  ∞ E j =1

=

∞ 

1 e−λj t h, ϕj  ϕj (x) μ(dx) ϕj 22

e−λj t h, ϕj  μ(ϕj )

j =1

= 0, and this implies that μ(h) = 0 for all h ∈ C(E), so that μ ≡ 0. Using the fact that  satisfies condition III, we can easily check that  is commutative, associative, bilinear and has identity element δa . It is also straightforward that μν ≤ μ·ν and that P(E)  P(E) ⊂ P(E). If xn → x and yn → y, then (δxn  δyn )(ϕj ) = ϕj (xn )ϕj (yn ) −→ ϕj (x)ϕj (y) = (δx  δy )(ϕj )

(j ∈ N)

and therefore (by compactness of E and a vague convergence argument similar to w that of Remark 4.3.II) δxn δyn −→ δx δy ; arguing as in the proof of Proposition 4.9 it follows that (μ, ν) → μ  ν is continuous in the weak topology. Noting that pt,x (ϕj ) = e−λj t ϕj (x) = pt,a (ϕj )δx (ϕj ), we conclude that  is an FLTC for {Tt }. Corollary 5.2 In the conditions of Proposition 5.4, assume that the operator (2) T1 has simple spectrum (i.e. all the eigenvalues e−λj have multiplicity 1). Let {(λj , ωj )}j ∈N be the eigenvalue-eigenfunction pairs defined in Proposition 5.3. Then the following are equivalent: (i) There exists an FLTC for {Tt }t ≥0; (ii) There exists a ∈ E such that |ωj (a)| = ωj ∞ for all j ∈ N, and the positivity ω (x) condition (5.4) holds for the nonnormalized eigenfunctions ϕj (x) := ωjj (a) . Proof The implication (ii) ⇒ (i) follows from the final statement in Proposition 5.4.

5.1 Convolutions Associated with Conservative Strong Feller Semigroups

195

Conversely, if (i) holds then the common maximizer property discussed above ω (x) implies that  = {ϕj }j ∈N where ϕj (x) := ωjj (a) ; from this it follows (by condition III of Definition 5.1) that the product formula (5.5) holds with νx,y = δx  δy and therefore (by Proposition 5.4) the ϕj satisfy the positivity condition (5.4). (2)

We note here that the assumption that Tt (or, equivalently, the generator G(2) ) has no eigenvalues with multiplicity greater than 1 is known to hold for many strong Feller semigroups of interest. In fact, it is proved in [81, Example 6.4] that the property that all the eigenvalues of the Neumann Laplacian are simple is a generic property in the set of all bounded connected C2 domains E ⊂ Rd . (The meaning of this is the following: given a bounded connected C2 domain E, consider the collection of domains M3 (E) = {h(E) | h : E −→ Rd is a C3 -diffeomorphism}, which is a separable Banach space, see [81] for details concerning the appropriate  ∈ M3 (E) such that all the topology. Let Msimp ⊂ M3 (E) be the subspace of all E  eigenvalues of the Neumann Laplacian on E are simple. Then Msimp can be written as a countable intersection of open dense subsets of M3 (E).) Similar results hold for the Laplace–Beltrami operator on a compact Riemannian manifold: it was proved in [189] that, given a compact manifold M, the set of Riemannian metrics g for which all the eigenvalues of the Laplace–Beltrami operator on (M, g) are simple is a generic subset of the space of Riemannian metrics on M. However, one should not expect the property of simplicity of spectrum to hold for Euclidean domains or Riemannian manifolds with symmetries. For instance, if a bounded domain E ⊂ R2 is invariant under the natural action of the dihedral group D n , then one can show (see [79]) that the Dirichlet or Neumann Laplacian on E has infinitely many eigenvalues with multiplicity ≥ 2. Remark 5.1 (Connection with Ultrahyperbolic Equations) Assume that there exists a dense orthogonal set of eigenfunctions {ϕj }j ∈N such that ϕ1 = 1 and ϕj (a) = ϕj ∞ = 1 for all j ∈ N. By the above, in order to prove the existence of an FLTC for {Tt } we need to ensure that Qt,h (x, y) :=

∞  j =1

1 e−t λj ϕj (x) ϕj (y) h, ϕj  ≥ 0 ϕj 22

for all h ∈ C(E), t > 0.

(0) (0) The function Qt,h (x, y) is a solution of G(0) is the x u = Gy u (where Gx (0) Feller generator G acting on the variable x) satisfying the boundary condition Qt,h (x, a) = (Tt h)(x). Since the point a is a maximizer of all the eigenfunctions ϕj , the function Qt,h (x, y) also satisfies (at least formally) the boundary condition (∇y Qt,h )(x, a) = 0. (This could be justified e.g. by proving that the series can be differentiated term by term. In the next section we will see that this argument can be applied to the Neumann Laplacian on suitable bounded domains of Rd .) This

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5 Convolution-Like Structures on Multidimensional Spaces

indicates that, as in Sect. 4.3.1, the (positivity) properties of the boundary value problem (0) G(0) x u = Gy u,

u(x, a) = u0 (x),

(∇y u)(x, a) = 0

(5.10)

are related with the problem of constructing a convolution associated with the given strong Feller semigroup. Consider Examples 5.1(b)–(c) or, more generally, any example of a strong Feller semigroup generated by a uniformly elliptic differential operator on E ⊂ Rd (0) (d > 2). In this context, the principal part of the differential operator G(0) x − Gy ∂2 ∂2 has d terms 2 with positive coefficient and d terms 2 with negative coefficient. ∂xj

∂yj

Such partial differential operators are often said to be of ultrahyperbolic type (cf. e.g. [150, §I.5] and [161, Definition 2.6]). According to the results of [41] and [40, §VI.17], the solution for the boundary value problem (5.10) is, in general, not unique. The existing theory on well-posedness of ultrahyperbolic boundary value problems is, in many other respects, rather incomplete; in particular, as far as we know, no maximum principles have been determined for such problems. Adapting the integral equation technique which was used in Chap. 4 to problems determined by Feller semigroups on multidimensional spaces is, therefore, a highly nontrivial problem.

5.2 Nonexistence of Convolutions: Diffusion Processes on Bounded Domains The results of the previous section show that the existence of an FLTC for a given Feller process depends on two conditions—the common maximizer property and the positivity of an ultrahyperbolic boundary value problem—for which there are no reasons to hope that they can be established other than in special cases. In particular, the common maximizer property becomes less natural when we move from onedimensional diffusions to multidimensional diffusions: while in the first case it is natural that the properties of a differential operator enforce one of the endpoints of the interval to be a common maximizer, this is no longer the case on a bounded domain of Rd with differentiable boundary because one no longer expects that one of the points of the boundary will play a special role. In fact, as we will see in Sect. 5.2.3, under certain conditions one can prove that (reflected) Brownian motions on bounded domains of Rd or on compact Riemannian manifolds (cf. Examples 5.1(a) and (c) respectively) do not satisfy the common maximizer property and, therefore, it is not possible to construct an associated FLTC. In Sect. 5.2.1 we start by presenting some examples which allow us to understand the geometrical properties of the eigenfunctions that are usually encountered in the multidimensional case. Section 5.2.2 contains some useful auxiliary results.

5.2 Nonexistence of Convolutions: Diffusion Processes on Bounded Domains

197

5.2.1 Special Cases and Numerical Examples The first example illustrates the fact that the construction of the convolution becomes trivial if the generator of the multidimensional diffusion decomposes trivially (via separation of variables) into one-dimensional Sturm–Liouville operators for which an associated FLTC exists. Example 5.2 (Neumann Eigenfunctions of a d-Dimensional Rectangle) Consider the d-dimensional rectangle E = [0, β1 ] ×. . .×[0, βd ] ⊂ Rd . The (nonnormalized) eigenfunctions of the Neumann Laplacian on E and the associated eigenvalues are given by ϕj1 ,...,jd (x1 , . . . , xd ) =

d ,

=1

 x  cos πj

, β

λj1 ,...,jd = π 2

d  j 2

=1

β 2

(j1 , . . . , jd ∈ N0 ). These eigenfunctions constitute an orthogonal basis of L2 (E, dx). The point (0, . . . , 0) is, obviously, a maximizer of all the functions ϕj1 ,...,jd , thus the common maximizer property holds. Moreover, we can trivially construct an FLTC for the (reflected Brownian) semigroup generated by the Neumann Laplacian: indeed, it is easy to check that the product of the hypergroups ([0, β1 ],  ), . . . , ([0, βd ],  ) satisfies all the requirements of Definition 5.1. (The β1

βd

product of the hypergroups is taken as in Sect. 2.3; recall also that ([0, β ], ) is the β

Sturm–Liouville hypergroup of compact type associated with the operator [0, β ], cf. Remark 4.2.)

d2 dx 2

on

Example 5.3 (Neumann Eigenfunctions of Disks and Balls) Let E ⊂ R2 be the closed disk of radius R. It is well-known that the eigenfunctions of the Neumann Laplacian on E are given, in polar coordinates, by

r ϕ0,k (r, θ ) = J0 (j0,k R ),

r ϕm,k,1 (r, θ ) = Jm (jm,k R ) cos(mθ ),

r ϕm,k,2 (r, θ ) = Jm (jm,k R ) sin(mθ ),

where m, k ∈ N and jm,k stands for the k-th (simple) zero of the derivative of the Bessel function of the first kind Jm (·) (see [104, Section 7.2] and [80, Proposition

/R)2 (with multiplicity 1) and 2.3]). The corresponding eigenvalues are λ0,k = (j0,k

2 λm,k = (jm,k /R) (with multiplicity 2). A classical result on the Bessel function

)| > |J (x)| for all [200, pp. 485, 488] ensures that for m ≥ 1 we have |Jm (jm,k m

, hence the eigenfunctions ϕ x > jm,k m,k,1 and ϕm,k,2 (m ≥ 1) attain their global   j

maximum on the circle r = jm,1

R . This shows, in particular, that no orthogonal m,k

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5 Convolution-Like Structures on Multidimensional Spaces

basis of L2 (E, dx) composed of Neumann eigenfunctions can satisfy the common maximizer property. More generally, if E ⊂ Rd is a closed d-ball with radius R, then the eigenfunctions of the Neumann Laplacian on E are d

ϕm,k (r, θ ) = r 1− 2 Jm−1+ d (cm,k Rr ) Hm (θ ), 2

where (r, θ ) are hyperspherical coordinates, m ∈ N0 , k ∈ N, Hm is a spherical harmonic of order m (see [104]) and cm,k is the k-th zero of the function ξ → 2 (1 − d2 )Jm−1+ d (ξ ) + ξ J

d (ξ ). The corresponding eigenvalues are λm,k = cm,k , 2

m−1+ 2

whose multiplicity is equal to the dimension of the space of spherical harmonics of order m. By similar arguments we conclude that the common maximizer property does not hold. Example 5.4 (Neumann Eigenfunctions of a Circular Sector) Let E ⊂ R2 be the sector of angle πq , E = {(r cos θ, r sin θ ) | 0 ≤ r ≤ 1, 0 ≤ θ ≤ πq }, where q ∈ N. The eigenfunctions of the Neumann Laplacian on E and the associated eigenvalues (which have multiplicity 1, cf. [20]) are given by

r ϕm,k (r, θ ) = cos(qmθ )Jqm (jqm,k R ),

λm,k = (jqm,k /R)2 .

As in the previous example it follows that the global maximizer of ϕm,k lies in the   j

arc r = jqm,1 R , so that the common maximizer property does not hold.

qm,k

Example 5.5 (Neumann Eigenfunctions of a Circular Annulus) If E ⊂ R2 is the annulus {(r, θ ) | r0 ≤ r ≤ R, 0 ≤ θ < 2π}, where 0 < r0 < R < ∞, then the Neumann eigenfunctions on E are ϕ0,k (r, θ ) = J0 (c0,k Rr )Y0 (c0,k ) − J0 (c0,k )Y0 (c0,k Rr ),   ϕm,k,1 (r, θ ) = Jm (cm,k Rr )Ym (cm,k ) − Jm (cm,k )Ym (cm,k Rr ) cos(mθ ),   ϕm,k,2 (r, θ ) = Jm (cm,k Rr )Ym (cm,k ) − Jm (cm,k )Ym (cm,k Rr ) sin(mθ ),

(5.11)

where m, k = 1, 2, . . ., Ym (·) is the Bessel function of the second kind [145, §10.2] and cm,k is the k-th zero of the function ξ → Jm ( rR0 ξ )Ym (ξ ) − Jm (ξ )Ym ( rR0 ξ ). The 2 /R 2 ). Figure 5.1 presents the contour plots associated eigenvalues are λm,k = (cm,k of some of the Neumann eigenfunctions, obtained in two different ways: in panel (a) using the explicit representations (5.11), where the constants cm,k are computed numerically with the help of the NSolve function of Wolfram Mathematica; and in panel (b) using a numerical approximation of the eigenvalues and eigenfunctions which was computed via the NDEigensystem routine of Wolfram Mathematica. Since the eigenvalues λm,k with m ≥ 1 have multiplicity 2, the plots obtained by these two approaches differ by a rotation. The results indicate that some of the eigenfunctions (those associated with the first zero cm,1 ) attain their maximum at

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199

the outer circle {r = R}, while other eigenfunctions (those associated with the higher zeros cm,k , k ≥ 2) attain their maximum either at the inner circle {r = r0 } or at the interior of the annulus. It is therefore clear that the Neumann eigenfunctions do not satisfy the common maximizer property. There are few other examples of domains of Rd for which the Neumann eigenfunctions can be computed in closed form. However, in the general case of an arbitrary domain E ⊂ R2 it is still possible to assess whether the common maximizer property holds by analysing the contour plots of the eigenfunctions; these can be computed, for a given bounded domain of R2 , by the same procedure which was used to produce the plots in panel (b) of Fig. 5.1. This is illustrated in Figs. 5.2 and 5.3, which present the contour plots of the first eigenfunctions of two non-symmetric bounded regions of R2 with smooth boundary. As we can see, the eigenfunctions attain their maximum values at different points which lie either on the boundary or at the interior of the domain. Note also that the

(a)

(b) Fig. 5.1 Contour plots of the Neumann eigenfunctions of a circular annulus with inner radius r0 = 0.3 and outer radius R = 1. (a) Closed form expressions. (b) Numerical approximation. In panel (b), the notation ωk refers to the orthogonal eigenfunction associated with the k-th largest eigenvalue λk . In both panels the eigenfunctions were normalized so that their L2 norm equals 1. Similar results were obtained for other values of rR0

Fig. 5.2 Contour plots of some eigenfunctions of a region obtained by a non-symmetric deformation of an ellipse. (As above, we denote by ωk the Neumann eigenfunction associated with the k-th largest eigenvalue λk , and the plots were produced using the NDEigensystem function of Wolfram Mathematica)

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5 Convolution-Like Structures on Multidimensional Spaces

Fig. 5.3 Contour plots of the Neumann eigenfunctions of a region obtained by a non-symmetric deformation of a pentagon with smoothed corners

associated eigenvalues are simple, which is unsurprising since the domain has no symmetries (cf. comment after Corollary 5.2). Remark 5.2 (Connection with the hot spots conjecture) All the examples presented above have the property that if ϕ2 is an eigenfunction associated with the smallest nonzero Neumann eigenvalue λ2 , then the maximum and minimum of ϕ2 are attained at the boundary ∂E. This is the so-called hot spots conjecture of J. Rauch, which asserts that this property should hold on any bounded domain of Rd . The physical intuition behind this conjecture is that, for large times, the hottest point on an insulated body with a given initial distribution should converge towards the boundary of the body. The hot spots conjecture has been extensively studied in the last two decades: it has been shown that the conjecture holds on convex planar domains with a line of 2 symmetry [4, 148], on convex domains E ⊂ R2 with diam(E) Area(E) < 1.378 [142] and on any Euclidean triangle [95] (for further positive results see [95] and references therein). On the other hand, some counterexamples have also been found, namely certain domains with holes [28]. The common maximizer property can be interpreted as an extended hot spots conjecture: instead of requiring that the maximum of (the absolute value of) the second Neumann eigenfunction is attained at the boundary, one requires that the maximum of all the eigenfunctions is attained at a common point of the boundary. The negative result of Corollary 5.3 below shows that the location of the hottest point in the limiting distribution (as time goes to infinity) of the temperature of an insulated body depends on the initial temperature distribution. The common maximizer property and the hot spots conjecture are subtopics of the more general problem of understanding the topological and geometrical structure of Laplacian eigenfunctions, which is the subject of a huge amount of literature. We refer to [74, 90] for a survey of known facts, applications and related references.

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201

5.2.2 Some Auxiliary Results Before moving on to a discussion of the common maximizer property on more general spaces, we collect some results from functional analysis and PDE theory which will be needed in the ensuing analysis. We begin by stating a version of the Sobolev embedding theorem and of the trace (and inverse trace) theorem. Theorem 5.1 (a) [203, Corollary 6.1] Let E ⊂ Rd be the closure of a bounded convex domain, and let j, k ∈ N0 with k − j > d2 . Then any function in H k (E) is j times continuously differentiable, and the embedding H k (E) $→ Cj (E) is continuous. (b) [75, Theorem 7.1] Let (E, g) be a compact Riemannian manifold of dimension d, and let j, k ∈ N with k > j2 + d4 . Then any function f ∈ H 2k (E) := {u | u, u, . . . , k u ∈ L2 (E, m)} belongs to Cj (E), and the embedding H 2k (E) $→ Cj (E) is continuous. Theorem 5.2 ([203, Theorems 8.7 and 8.8]) Let k ∈ N and let E ⊂ Rd be the closure of a boundedCk+1 domain. There exists a linear continuous trace operator k−j −1/2 (∂E) such that Tr∂E : H k (E) −→ k−1 j =0 H  ∂f  ∂ k−1 f  Tr∂E f = f |∂E , , . . . , k−1  ∂n ∂E ∂n ∂E

for f ∈ C2k−1 (E),

where n denotes the unit outer normal vector orthogonal to ∂E, and H k−j −1/2(∂E) is the fractional Sobolev space defined as the set of functions ϕ ∈ H k−j −1 (∂E) such that for each α = (α1 , . . . , αd ) with |α| = k − j − 1 we have  Rd ×Rd

|∂ α ϕ(x) − ∂ α ϕ(y)|2 dx dy < ∞. |x − y|d+1

In  addition, there exists a linear continuous k−j −1/2 (∂E) −→ H k (E) such that Z : k−1 j =0 H Tr∂E (Zϕ) = ϕ

for all ϕ ∈

k−1 ,

extension

operator

H k−j −1/2(∂E).

j =0

We will also make use of the following upper bounds on the heat kernel (i.e. the transition density of the semigroup {Tt(2) } generated by the Neumann Laplacian or the Laplace–Beltrami operator) and on its gradient: Proposition 5.5 Assume that E ⊂ Rd is the closure of a bounded Lipschitz domain, and let pt (x, y) be the heat kernel determined by the Neumann Laplacian on E.

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(a1)

5 Convolution-Like Structures on Multidimensional Spaces

[8, Theorem 3.1] There exist constants c1 , c2 > 0 such that  |x − y|2 pt (x, y) ≤ c1 t −d/2 exp − c2 t

(a2)

for all t > 0.

[199, Lemma 3.1] There exist constants c3 , c4 > 0 such that  |x − y|2 |∇y pt (x, y)| ≤ c3 t −(d+1)/2 exp − c4 t

for all t > 0.

Assume now that (E, g) is a compact Riemannian manifold of dimension d, let pt (x, y) be the heat kernel determined by the Laplace–Beltrami operator on E, and let d be the Riemannian distance function. (b1)

[75, Corollary 15.17] There exist constants c5 , c6 > 0 such that pt (x, y) ≤ c5 t

(b2)

−d/2

 d(x, y)2 exp − c6 t

for all t > 0.

[87] There exist constants c7 , c8 > 0 such that |∇y pt (x, y)| ≤ c7 t

−(d+1)/2

 d(x, y)2 exp − c8 t

for all t > 0.

Lastly, we present a version of Mercer’s theorem, which provides an uniformly convergent eigenfunction expansion representation for positive-definite integral kernels: Theorem 5.3 ([170, Theorem 3.11.9]) Let E be a compact metric space endowed with a positive Borel measure μ. Assume L2 (E, μ) is infinite-dimensional. Let K ∈ C(E × E), let AK : L2 (E, μ) −→ L2 (E, μ) be the operator defined as  (AK f )(x) =

K(x, y)f (y) μ(dy), E

and suppose that AK is a positive operator. Then the eigenvalues {λj }j ∈N of AK are associated with continuous eigenfunctions {ϕj }j ∈N ⊂ C(E) ∩ L2 (E, μ), and we have K(x, y) =

∞ 

λj ϕj (x)ϕj (y),

j =1

where the series converges absolutely and uniformly on E × E.

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203

5.2.3 Eigenfunction Expansions, Critical Points and Nonexistence Theorems We are now ready to discuss the (failure of the) common maximizer property for reflected Brownian motions on general bounded domains of Rd (d ≥ 2). We begin with two important observations, the first of which is quite obvious: if a is a common maximizer for the eigenfunctions {ϕj }j ∈N , then it is a common critical point, i.e. we have (∇ϕj )(a) = 0 for all j . The second observation is that the usual eigenfunction expansion f =

∞ 

1 f, ϕj  ϕj ϕj 22 j =1

(5.12)

suggests that the point a will also be a critical point of any function f which is sufficiently regular so that the expansion (5.12) is convergent in the pointwise sense and can be differentiated term by term. Thus if we prove that such pointwise convergence and differentiation is admissible for a class of functions whose derivatives are not restricted to vanish at any given point, then the common maximizer property cannot hold. The next proposition and corollary make this rigorous. Proposition 5.6 Let E ⊂ Rd be the closure of a bounded convex domain. Let {Xt } be the reflected Brownian motion on E, let {ωj }j ∈N be an orthonormal basis of L2 (E) ≡ L2 (E, dx) consisting of eigenfunctions of the Neumann Laplacian −G(2) ≡ − N : D( N ) −→ L2 (E) and let 0 ≤ λ1 ≤ λ2 ≤ λ3 ≤ . . . be the associated eigenvalues. Let m ∈ N, m > d2 + 1 and let h ∈ H m (E) be a function such that k h ∈ D( N ) for k = 0, 1, . . . , m − 1. Then h(x) =

∞  j =0

h, ωj  ωj (x)

and

(∇h)(x) =

∞ 

h, ωj (∇ωj )(x)

for all x ∈ E,

j =0

(5.13) where both series converge absolutely and uniformly on E. Proof Let {Tt(2) }t ≥0 and {R(2) η }η>0 be, respectively, the strongly continuous semi2 group and resolvent on L (E) generated by the Neumann Laplacian and let pt (x, y) (2) be the Neumann heat kernel, i.e. the transition density of the semigroup {Tt }. Using the Sobolev embedding theorem (Theorem 5.1(a)), one can prove (cf. [43, proof of Theorem 5.2.1]) that the heat kernel is C∞ jointly in the variables (t, x, y) ∈ (0, ∞) × E × E. Denote by ∂v) ,x the directional derivative with respect to the variable x ∈ Rd in a given direction v) ∈ Rd \ {0}. By Proposition 5.5, there

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5 Convolution-Like Structures on Multidimensional Spaces

exist constants c1 , c2 , c3 , c4 > 0 such that the following estimates hold: pt (x, y) ≤ c1 t |∂v) ,y pt (x, y)| ≤ c3 t

−d/2

 |x − y|2 exp − , c2 t

−(d+1)/2

 |x − y|2 exp − . c4 t

(5.14)

(5.15)

Using the basic semigroup identity for the heat kernel, we obtain  |∂v) ,x ∂v) ,y pt (x, y)| ≤

E

≤ c5 t

|∂v) ,x pt /2 (x, ξ ) ∂v) ,y pt /2(ξ, y)|dξ −(d+1)

 |x − y|2 . exp − 2c4 t

(5.16)

Next we recall that [42, Problem 2.9] k (R(2) α ) h ≡ (α −

=

−k N) h

1 (k − 1)!



∞ 0

(h ∈ L2 (E), α > 0, k = 1, 2, . . .),

e−αt t k−1 Tt h dt (2)

k and therefore the k-th power (R(2) α ) of the resolvent is an integral operator with kernel  ∞ 1 Gα,k (x, y) = e−αt t k−1 pt (x, y) dt. (5.17) (k − 1)! 0

If k = 2m > d +1, then using the estimate (5.14) we see that Gα,2m (x, x) < ∞ and, furthermore, Gα,2m is a continuous function of (x, y) ∈ E × E. Since E is compact 2m : L2 (E) −→ L2 (E) is nonnegative and has a continuous kernel, an and (R(2) α ) application of Mercer’s theorem (Theorem 5.3) yields that the kernel Gα,2m can be represented by the absolutely and uniformly convergent spectral expansion Gα,2m (x, y) =

∞  ωj (x)ωj (y) j =1

(α + λj )2m

,

(5.18)

series converges absolutely and uniformly in (x, y) ∈ E × E. (Note that  where the −2m , ωj are the eigenvalue-eigenfunction pairs for R(2) (α + λj ) α .) In addition, it follows from (5.17) and the estimates (5.15)–(5.16) that 1 ∂v) ,x ∂v) ,y Gα,2m (x, y) = (2m − 1)!



∞ 0

e−αt t 2m−1 ∂v) ,x ∂v) ,y pt (x, y) dt,

5.2 Nonexistence of Convolutions: Diffusion Processes on Bounded Domains

205

where the integral converges absolutely and uniformly and defines a continuous function of (x, y) ∈ E × E. (The function ∂v) ,y Gα,2m is also continuous on E × E.) Using standard arguments (cf. [144, §21.2, proof of Corollary 3]), one can then deduce from (5.18) that (x, y) = ∂v) ,x ∂v) ,y G(2m) α

∞  (∂v) ωj )(x)(∂v) ωj )(y)

(α + λj )2m

j =1

,

(5.19)

again with absolute and uniform convergence in (x, y) ∈ E × E. Let h ∈ H m (E) be such that k h ∈ D( N ) for k = 0, 1, . . . , m − 1, and write d m 2 1 h = Rm α g where g := (α − N ) h ∈ L (E). Since m > 2 + 1, we have h ∈ C (E) by the Sobolev embedding theorem. We thus have ∞ ∞     |g, ωj | h, ωj  ωj (x) = |ωj (x)| (α + λj )m j =0

j =0

≤

 ∞ j =0

1  1 ∞ 2 |ωj (x)|2 2 |g, ωj | · (α + λj )2m 2

j =0



= g· G(2m) (x, x) α

1/2

< ∞,

and similarly ∞      h, ωj (∂v) ωj )(x) ≤ g· ∂v) ,x ∂v) ,y G(2m) (x, x)1/2 < ∞. α j =0

This shows that the series in the right-hand sides of (5.13) converge absolutely and uniformly in x ∈ E, and the result immediately follows. Corollary 5.3 (Nonexistence of Common Critical Points) Let m ∈ N, m > d2 +1 and let E ⊂ Rd (d ≥ 2) be the closure of a bounded convex domain with C2m+2 boundary ∂E. Let {ωj }j ∈N be an orthonormal basis of L2 (E) consisting of eigenfunctions of − N . Then for each x0 ∈ E there exists j ∈ N such that (∇ωj )(x0 ) = 0. ˚ it is clearly possible to choose h ∈ C∞ (E) ⊂ {u ∈ H m (E) | Proof If x0 ∈ E, c m−1 u, u, . . . , u ∈ D( N )} such that (∇h)(x0 ) = 0. If x0 belongs to ∂E, let v) ∈ T∂E (x0 ) \ {0} and choose ϕ ∈ C∞ (∂E) such that dϕx0 ()v ) = 0. Combining the trace theorem (Theorem 5.2) with the Sobolev embedding theorem, we find that that there exists h ∈ H 2m (E) ⊂ C1 (E) such that h|∂E = ϕ

and

Tr∂E

 ∂j h  ∂nj

= 0,

j = 1, 2, . . . , 2m − 1.

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5 Convolution-Like Structures on Multidimensional Spaces

m−1 h ∈ D( Consequently, N) =  h is such  ∂h  that (∇h)(x0 ) = 0 and h, h, . . . ,  2 u ∈ H (E)  Tr∂E ∂n = 0 . (This characterization of D( N ) is well-known, see [167, Section 10.6.2].) Therefore, given any x0 ∈ E we can apply Proposition 5.6 to the function h defined above to conclude that ∞  h, ωj (∇ωj )(x0 ) = (∇h)(x0 ) = 0, j =0

which implies that (∇ωj )(x0 ) = 0 for at least one j . The conclusions of Proposition 5.6 and Corollary 5.3 are also valid for the eigenfunctions of the Laplace–Beltrami operator on a compact Riemannian manifold (Example 5.1(a)): Proposition 5.7 (Nonexistence of Common Critical Points on Compact Riemannian Manifolds) Let (E, g) be a compact Riemannian manifold (without boundary) of dimension d and {ωj }j ∈N an orthonormal basis of L2 (E, m) consisting of eigenfunctions of the Laplace–Beltrami operator on (E, g). Then (5.13) holds for all functions h ∈ H 2m (E) (m ∈ N, m > d4 + 12 ), with the series converging absolutely and uniformly. Furthermore, for each x0 ∈ E there exists j ∈ N such that (∇ωj )(x0 ) = 0. Proof It is well-known that the heat kernel pt (x, y) for the Laplace–Beltrami operator is C∞ jointly in the variables (t, x, y) ∈ (0, ∞) × E × E [43, Theorem 5.2.1]. By Proposition 5.5, the heat kernel pt (x, y) and its gradient satisfy, for 0 < t ≤ 1 and x, y ∈ E, the upper bounds  d(x, y)2 pt (x, y) ≤ c1 t −d/2 exp − , c2 t

 d(x, y)2 |∇y pt (x, y)| ≤ c3 t −(d+1)/2 exp − . c4 t

Let U ⊂ E be a coordinate neighbourhood. Arguing as in the proof of Proposition 5.6, we find that for x, y ∈ U the kernel of the 2m-th power of the resolvent admits the spectral representation (5.18) and can be differentiated term by term as in (5.19). (The directional derivatives are defined in local coordinates.) By the Sobolev embedding theorem (Theorem 5.1(b)), H 2m (E) ⊂ C1 (E); therefore, the estimation carried out above yields that the expansions (5.13) hold. We have ∂E = ∅, thus for each x0 ∈ E we can choose h ∈ C∞ (E) such that (∇h)(x0 ) = 0. As in the proof of Corollary 5.3 it follows that (∇ωj )(x0 ) = 0 for at least one j . As noted above, the existence of a common critical point is a necessary condition for the common maximizer property to hold; in turn, this is (under the assumption that the spectrum is simple, cf. Corollary 5.2) a necessary condition for the existence of an FLTC. Therefore, the following nonexistence theorem is a direct consequence of the preceding results. Theorem 5.4 Let {Tt }t ≥0 be either the Feller semigroup on a bounded domain E ⊂ Rd with C2m+2 boundary (m > d2 +1) associated with the reflected Brownian motion

5.3 Nonexistence of Convolutions: One-Dimensional Diffusions

207

on E or the Feller semigroup associated with the Brownian motion on a compact Riemannian manifold. Assume that the operator T1(2) has simple spectrum. Then there exists no FLTC for the semigroup {Tt }. This theorem is not applicable to regular polygons and other domains which are invariant under reflection or rotation (i.e. under the natural action of a dihedral group), as this invariance enforces the presence of eigenvalues with multiplicity greater than 1. On the other hand, we know that the eigenspaces on such symmetric domains can be associated to the different symmetry subspaces of the irreducible representations of the dihedral group [79]. In most cases, the multiplicity of all the eigenspaces corresponding to the one-dimensional irreducible representations is equal to 1 [134]; therefore, an adaptation of the proofs presented above should allow us to establish the nonexistence of common critical points among the eigenfunctions associated to the one-dimensional eigenspaces. The nonexistence theorem established above strongly depends on the discreteness of the spectrum of the generator of the Feller process. Extending Theorem 5.4 to Brownian motions on unbounded domains on Rd or on noncompact Riemannian manifolds is a challenging problem, as these diffusions generally have a nonempty continuous spectrum. We leave this topic for future research.

5.3 Nonexistence of Convolutions: One-Dimensional Diffusions As we saw in the previous sections, the construction of an FLTC is a difficult problem for which there is little hope of finding a solution unless the generator can be decomposed into a product of one-dimensional operators. Motivated by this, we now return to the one-dimensional setting in order to demonstrate that the necessary conditions for the existence of an FLTC determined in Sect. 5.1 also give rise to nonexistence theorems for a class of one-dimensional diffusion processes. The following result shows that a necessary and sufficient condition for existence of an FLTC similar to that of Corollary 5.2 holds for Sturm–Liouville operators whose spectrum is not necessarily discrete: Proposition 5.8 Consider a Sturm–Liouville operator of the form (4.1) whose coefficients are such that p(x), r(x) > 0 for all x ∈ (a, b), p, p , r, r ∈ c  c dx ACloc (a, b) and a y p(x) r(y)dy < ∞. Let wλ (·) (λ ∈ C) be the unique solution of (2.12), ρL the spectral measure of Theorem 2.5,  = supp(ρL ), and {Tt }t ≥0 the Feller semigroup generated by the realization of defined in (2.34). Assume that the endpoint b is not exit and that e−t · ∈ L2 (; ρL ) for all t > 0. Set I = [a, b) if b is natural and I = [a, b] if b is regular or entrance. Then the following are equivalent: (i) There exists an FLTC for {Tt }t ≥0 with trivializing family  = {wλ }λ∈.

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5 Convolution-Like Structures on Multidimensional Spaces

(ii) We have wλ ∈ Cb (I ) for all λ ∈ , and the function  qt (x, y, ξ ) :=



e−t λ wλ (x) wλ (y) wλ (ξ ) ρL (dλ)

  t > 0, x, y, ξ ∈ (a, b)

(5.20)

is well-defined as an absolutely convergent integral; moreover, the measures defined as νt,x,y (dξ ) = qt (x, y, ξ ) r(ξ )dξ are such that {νt,x,y }0 0, x, y ∈ (a, b)

and it follows that for g ∈ Cc (I )   g(ξ ) (μt  δx  δy )(dξ ) = lim (Ts g)(ξ ) (μt  δx  δy )(dξ ) s↓0

I

= lim s↓0

 =



I



(Fg)(λ) e−(t +s)λ wλ (x) wλ (y) ρ(dλ)

g(ξ ) qt (x, y, ξ ) r(ξ )dξ, I

where we used Fubini’s theorem, Proposition 2.14 and the isometric property of F. Since g is arbitrary, this shows that the measures (μt  δx  δy )(dξ ) and νt,x,y (dξ ) := qt (x, y, ξ ) r(ξ )dξ coincide. Consequently, νt,x,y ∈ P(I ) for all t > 0 and x, y ∈ (a, b). Since {Tt }t ≥0 is a Feller process, the mapping (t, x) → pt,x = μt  δx is continuous on R+ 0 × I with respect to the weak topology of measures, and therefore the family {νt,x,y }0 0, x, y ∈ (a, b), λ ∈  ,

I

(5.21) where the integral converges absolutely. Since {νt,x,y }0 0 for all x ∈ (a, b),p, p , r, r ∈  c  c dx  b r(y) ACloc (a, b) and a y p(x) r(y)dy < ∞. Suppose that γ (b) = c p(y) dy = ∞ and the function A defined in (4.2) is such that for some c˜ > γ (a) and limξ →∞ wλ (·) of (2.12) is such that

A (ξ ) 2A(ξ )

is of bounded variation on [c, ˜ ∞)

= σ ∈ (−∞, 0). Then the unique solution

sup |wλ (x)| = ∞ x∈[a,b)

A

A

for all λ > σ 2 .

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5 Convolution-Like Structures on Multidimensional Spaces

Consequently, there exists no FLTC for the Feller semigroup {Tt } generated by the realization of defined in (2.34). Proof The Sturm–Liouville equation − A1 (Au ) = λu is of the form (pu ) − qu = 0, where the coefficients p = A and q = −λA are such that √ (pq)

= κi λ(1 + φ) pq

4σ i with κ := − √ , λ

φ :=

A

− 1, 2σ A



so that φ(ξ ) = o(1) as ξ → ∞ and φ = ( 2σAA ) is integrable near +∞ (this follows

from the assumption that AA is of bounded variation, cf. [62, Proposition 3.30]). After applying a result on the asymptotic behaviour of solutions of second-order differential equations stated in [52, Theorem 2.6.1], we conclude that for λ = σ 2 the equation − A1 (Au ) = λu has two linearly independent solutions u+ and u− such that  

 ξ √ 1 1 A (z) A (z) iλ 2 u± (ξ ) ∼ [−λA(ξ )2 ]− 4 ± 4 1−λ/σ exp ± √ − 1 dz . λ − σ 2 c˜ 2σ A(z) 2σ A(z)  ξ A (z)   The function A(ξ )−1/2 = A(c)−1/2 exp − 12  c A(z) dz is clearly unbounded as ξ → ∞; therefore, u+ and u− are both unbounded for λ > σ 2 . Since wλ (x) is a real-valued linear combination of u− (γ (x)) and u+ (γ (x)), it follows that supx∈[a,b) |wλ (x)| = ∞ for λ > σ 2 . Combining the above with Proposition 5.8, we find that {wλ }λ>σ 2 cannot belong to any trivializing family for an FLTC, hence by Corollary 5.1 the trivializing family must be contained in {wλ }λ∈[0,σ 2] ∩ Cb [a, b). On the other hand, it follows from Theorem 2.5 that there exist nonzero measures μ ∈ MC [a, b) such that μ(wλ ) = 0 whenever λ ∈ [0, σ 2 ] and wλ is bounded. (Indeed, we have (σ 2 , ∞) ⊂  by the same argument in the proof of Proposition 4.3; if we let ϕ ∈ L2 (; ρL ) \ {0} with supp(ϕ) ⊂ (σ 2 , ∞), then μ(dx) = (F−1 ϕ)(x)r(x)dx defines a measure μ ∈ MC [a, b), and we have μ(wλ ) = 0 for λ ∈ / (σ 2 , ∞).) Consequently, no family  ⊂ {wλ }λ∈[0,σ 2 ] ∩Cb [a, b) can satisfy condition III of Definition 5.1. This contradiction shows that there exists no FLTC for {Tt }. Example 5.6 Proposition 5.9 shows, in particular, that the following operators do not admit an associated (positivity-preserving) Sturm–Liouville convolution structure: α d d2 (a) = − dx 2 − x + 2μ dx , with α > 0 and μ < 0. This is the generator of a mean-reverting Bessel process with negative drift (Example 2.3). d (b) = − dxd 2 − [(2α + 1) coth x + (2β + 1) tanh x] dx , with α > −1 and α + β + 1 < 0. This is the Jacobi operator, which is the generator of the hypergeometric diffusion (Example 2.5). d2 d 1 (c) = −x 2 dx 2 − (c + 2(1 − α)x) dx , with c > 0 and α > 2 .

5.3 Nonexistence of Convolutions: One-Dimensional Diffusions

211

This is the Whittaker operator, which is the generator of a nonstandardized Shiryaev process (Example 2.4 and Remark 3.11). Sturm–Liouville operators with two natural endpoints were excluded from the discussion in Chap. 4 because, given the absence of a natural candidate for the identity element, the construction of generalized convolutions would require a different approach. The ordinary convolution is a Sturm–Liouville convolution d2 for the operator dx 2 on R; as far as we know, this is the only known example of a convolution associated with a Sturm–Liouville operator with two natural boundaries. Let us note some examples in which the nonexistence of an associated FLTC follows from the results above: Example 5.7 (Sturm–Liouville Operators with Two Natural Endpoints) (a) Let θ > 0 and c ∈ R. The differential operator

=−

d2 d − (c − θ x) , dx 2 dx

−∞ < x < ∞

is the generator of the Ornstein–Uhlenbeck process [3, 122]. Both endpoints x = −∞ and x = +∞ are natural. It is well-known that the self-adjoint realization of has a purely discrete spectrum, with eigenvalues λn = nθ and orthogonal eigenfunctions ϕn (x) = Hn(θ,c) (x) (n ∈ N0 ), where Hn(θ,c) are the Hermite polynomials defined as θ

Hn(θ,c) (x) := e−cx+ 2 x

2

d n  −n cx− θ x 2  2 θ e . dx n

Since each Hn(θ,c) is a polynomial of degree n, it is clear that the L2 -extension of the Feller semigroup generated by (and therefore the Feller semigroup itself) has no bounded eigenfunctions other than ϕ0 ≡ 1. Consequently, by Corollary 5.1, one cannot construct a Sturm–Liouville convolution for the transition semigroup of the Ornstein–Uhlenbeck process. (b) Let κ > 0 and α ∈ R. The differential operator

= −(1 + x 2 )

d2 d − κ(α − x) , 2 dx dx

−∞ < x < ∞

is the generator of the Student diffusion process [3, 122]. Both endpoints x = ±∞ are natural, and the self-adjoint  realization of  has a purely absolutely continuous spectrum on the interval (κ + 1)2 /2, ∞ , together with a finite set of eigenvalues below (κ + 1)2 /2. The operator can be transformed, via the A (ξ ) d d2 change of variables ξ = arcsinh(x), into the standard form − dξ 2 − A(ξ ) dξ , where

A (ξ ) A(ξ )

=

κα cosh(ξ )

− (κ + 1) tanh(x). Since limξ →∞

A (ξ ) A(ξ )

= −(κ + 1) < 0

212

5 Convolution-Like Structures on Multidimensional Spaces

(ξ ) and limξ →−∞ AA(ξ ) = κ + 1 > 0, it follows from the proof of Proposition 5.9 that if λ > (κ + 1)2 /2 then the equation (u) = λu has no nonzero bounded solutions. Arguing in the same way we conclude that the transition semigroup of the Student diffusion does not admit an associated Sturm–Liouville convolution.

The next examples, related with the Laguerre and Jacobi polynomials, show that Sturm–Liouville operators which do not admit an associated FLTC may, nevertheless, admit a convolution structure satisfying a weaker set of axioms: Example 5.8 Let α, κ > 0. The Laguerre differential operator

= −x

d2 d − κ(α − x) , 2 dx dx

0