Semi-Dirichlet Forms and Markov Processes 9783110302066, 9783110302004

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De Gruyter Studies in Mathematics 48 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, Missouri, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Yoichi Oshima

Semi-Dirichlet Forms and Markov Processes

De Gruyter

Mathematics Subject Classification 2010: Primary: 60J45; Secondary: 31C25, 60J60, 60J75, 35J20, 35K05

ISBN 978-3-11-030200-4 e-ISBN 978-3-11-030206-6 Set-ISBN 978-3-11-030207-3 ISSN 0179-0986 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: P T P-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com

Dedicated to the late Professor Heinz Bauer, whose kind advice and warm encouragements were the origin of this work.

Preface

The organization of this book follows the lines in the author’s unpublished lecture notes at University of Erlangen-Nürnberg [123] partly combined with other lecture notes [127]. But the basic setting has been changed from non-symmetric Dirichlet forms to lower bounded semi-Dirichlet forms. In this book, we intend to extend those results for symmetric Dirichlet forms given in [55] to lower bounded semi-Dirichlet forms and to time dependent semi-Dirichlet forms as well. In Chapter 1 and Chapter 2, we present basic analytic properties related to the lower bounded semi-Dirichlet forms. In particular, the Markov property and the related potential theory similar to the symmetric Dirichlet forms are formulated. Although the dual semigroups are only positivity preserving and not Markovian in general, many of the results can be obtained with minor modifications of the arguments for the non-symmetric Dirichlet forms presented in [123] and [104]. Furthermore, by changing the basic measure using a coexcessive function, we can obtain a dual Markov resolvent relative to the measure. For this dual pair of the Markov resolvents, we introduce an auxiliary bilinear form. Although it does not satisfy the sector condition in general, it works efficiently in carrying out the corresponding stochastic calculus. From Chapter 3 to Chapter 5, properties of the Markov processes associated with regular lower bounded semi-Dirichlet forms are studied. In particular, stochastic calculus related to the associated Hunt process M is investigated. Usually the stochastic calculus for Markov processes is developed in connection with their dual Markov processes. For a regular semi-Dirichlet form, by changing the basic measure using a suitable family of coexcessive functions, we can obtain a family of dual Hunt processes of M relative to the changed measures. Furthermore, we can construct a pseudo Hunt process which is in duality with M relative to the original basic measure. Since the pseudo Hunt process can be treated as if it is an ordinary Hunt process, the stochastic calculus related to the semi-Dirichlet forms can be performed by modifying the calculus for symmetric Dirichlet forms. But, due to the lack of the excessiveness of the basic measure, some results need to be changed from the symmetric or non-symmetric Dirichlet forms. The essential difference is to use the weak sense energy instead of the energy. The symmetric or non-symmetric cases correspond to the case that the constant function 1 can be taken as a coexcessive function. The contents of Chapter 6 are essentially taken from Chapter 5 of the author’s note [127]. For a given time dependent family of semi-Dirichlet forms possessing a common domain and a common basic measure, an associated space-time Markov pro-

viii

Preface

cess is constructed and a related parabolic potential theory is developed via stochastic calculus. Contrary to the time independent case, the space-time Markov process involves nonexceptional semipolar sets so that a partially different approach to the related stochastic calculus is required. Although only certain basic parts of it will be presented in this section, it is possible to modify most of the arguments of Chapter 5 to get the parallel results in the time dependent cases. We intend this to be a self-contained textbook. Most results are stated accompanied by their proof. References are given for those statements without proof. The author would like to express his hearty thanks to Professor N. Jacob for his constant and warm encouragement in writing this book. He is grateful to Professor M. Fukushima for the kind advice to change the frameworks from the non-symmetric Dirichlet forms to the semi-Dirichlet forms and the valuable comments on the organization of this book. He also thanks Professor R. Schilling for his kind suggestion to publish this book and Professor K. Kuwae, Professor T. Uemura and Mr. R. Kinoshita for their kind suggestions improving the proofs. He is also grateful to Professor Z. M. Ma and W. Sun, who kindly gave the author many valuable comments on some essential parts. Thanks are also due to Dr. Y. Tawara for making the first version of the TeX file of the original lecture note. Thanks are due to the editorial staff of De Gruyter for their pleasant cooperation.

Contents

Preface 1

vii

Dirichlet forms

1

1.1 Semi-Dirichlet forms and resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Closability and regular Dirichlet forms . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3 Transience and recurrence of Dirichlet forms . . . . . . . . . . . . . . . . . . . . 12 1.4 An auxiliary bilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5.1 Diffusion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5.2 Jump type case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2

Some analytic properties of Dirichlet forms

43

2.1 Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2 Quasi-Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.3 Potential of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.4 An orthogonal decomposition of the Dirichlet forms . . . . . . . . . . . . . . 60 3

Markov processes

72

3.1 Hunt processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2 Excessive functions and negligible sets . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.3 Hunt processes associated with a regular Dirichlet form . . . . . . . . . . . 80 3.4 Negligible sets for Hunt processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.5 Decompositions of Dirichlet forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4

Additive functionals and smooth measures

112

4.1 Positive continuous additive functionals . . . . . . . . . . . . . . . . . . . . . . . . 112 4.2 Dual PCAFs and duality relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.3 Time changes and killings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5

Martingale AFs and AFs of zero energy

149

5.1 Fukushima’s decomposition of AFs . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.1.1 AFs generated by functions of F . . . . . . . . . . . . . . . . . . . . . . . 151

x

Contents

5.1.2 5.1.3

Martingale additive functionals of finite energy . . . . . . . . . . . . 152 CAFs of zero energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.2 Beurling–Deny type decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.3 CAFs of locally zero energy in the weak sense . . . . . . . . . . . . . . . . . . . 179 5.4 Martingale AFs of strongly local Dirichlet forms . . . . . . . . . . . . . . . . . 190 5.5 Transformations by multiplicative functionals . . . . . . . . . . . . . . . . . . . 203 5.6 Conservativeness and recurrence of Dirichlet forms . . . . . . . . . . . . . . . 208 6

Time dependent Dirichlet forms

215

6.1 Time dependent Dirichlet forms and associated resolvents . . . . . . . . . . 215 6.2 A parabolic potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 6.3 Associated space-time processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 6.4 Additive functionals and associated measures . . . . . . . . . . . . . . . . . . . 248 6.5 Some stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Notes

267

Bibliography

272

Index

283

Chapter 1

Dirichlet forms

In this chapter, basic settings throughout this book are presented. The semi-Dirichlet forms .E, F / that this book concerns itself with are bilinear forms satisfying .E.1/  .E.4/. For a semi-Dirichlet form, associated semigroups and resolvents are constructed and their Markov property is established in Section 1.1. Since we mainly consider the semi-Dirichlet forms, we call them Dirichlet forms for short. Closability to generate regular Dirichlet forms is explained in Section 1.2. Irreducibility, transience and their related properties are studied in Section 1.3. As a preparation for the stochastic calculus developed after Chapter 3, we consider in Section 1.4 a dual Markov resolvent relative to a basic measure being changed by a suitable coexcessive function b hı to.ı/ gether with an associated auxiliary bilinear form A . We introduce condition .E.5/ on the original Dirichlet form E so that A.ı/ is well controlled by E. Furthermore, an additional condition .E.6/ and its consequence are stated also in Section 1.4. Condition .E.5/ will be assumed for Dirichlet form E in most parts after Chapter 3, while .E.6/ will only be used in Theorem 5.1.4 and its consequences. In the final Section 1.5, we give typical examples of diffusion type as well as jump type Dirichlet forms satisfying .E.5/ and .E.6/.

1.1

Semi-Dirichlet forms and resolvents

Let X be a locally compact separable metric space and B.X / the Borel  -algebra on X . We shall fix an everywhere dense positive Radon measure m on X and denote by ., / and k  k the inner product and the norm in L2 .X ; m/, respectively. Let C .X / be the space of all continuous functions on X . For a family of functions D, let us denote by D0 , D C and Db the sub-family of compact support, non-negative and bounded functions of D, respectively. For a dense linear subspace F of L2 .X ; m/, if a bilinear form E defined on F  F satisfies the following conditions .E.1/, .E.2/ and .E.3/, then we call .E, F / a closed form on L2 .X ; m/. .E.1/ E is lower bounded: There exists a non-negative constant ˛0 such that E˛0 .u, u/  0 for all u 2 F ,

(1.1.1)

where E˛ .u, v/ D E.u, v/ C ˛.u, v/. .E.2/ E satisfies the sector condition: There exists a constant K  1 such that jE.u, v/j  KE˛0 .u, u/1=2 E˛0 .v, v/1=2 for all u, v 2 F .

(1.1.2)

2

Chapter 1 Dirichlet forms

.E.3/ F is a Hilbert space relative to the inner product 1 E˛.s/ .u, v/ D .E˛ .u, v/ C E˛ .v, u// for all ˛ > ˛0 . 2 For ˛ > ˛0 , put K˛ D K C ˛=.˛  ˛0 /. Since .u, u/ 

1 ˛˛0 E˛ .u, u/,

(1.1.2) implies

jE˛ .u, v/j  K˛ E˛ .u, u/1=2 E˛ .v, v/1=2 for all u, v 2 F .

(1.1.3)

If equation (1.1.1) holds for ˛0 D 0, then .E, F / is called non-negative. .E, F / is called symmetric if E.u, v/ D E.v, u/ for all u, v 2 F . If .E, F / is non-negative and symmetric, then equation (1.1.2) holds for ˛0 D 0 and K D 1 by the Schwarz inequality. For later use, we shall first show the following Stampacchia’s theorem. Theorem 1.1.1. Let  be a non-empty closed convex subset of F . If J is a continuous linear functional on F with respect to E˛ for ˛ > ˛0 , then there exists a unique function v 2  satisfying E˛ .v, w  v/  J.w  v/ for all w 2 .

(1.1.4)

Proof. Let us fix ˛ > ˛0 . Uniqueness: Assume that v1 and v2 satisfy equation (1.1.4). Then E˛ .v1 , v2  v1 /  J.v2  v1 / and E˛ .v2 , v1  v2 /  J.v1  v2 /. Hence E˛ .v2  v1 , v2  v1 / D E˛ .v2 , v2  v1 /  E˛ .v1 , v2  v1 /  J.v2  v1 /  J.v2  v1 / D 0, which implies that v2 D v1 m-a.e. Existence: We shall first prove equation (1.1.4) under the assumption that E is symmetric. Let I.v/ D E˛ .v, v/  2J.v/ and let d D infv2 I.v/. Since I.v/  E˛ .v, v/  2kJ kE˛ .v, v/1=2 D .E˛ .v, v/1=2  kJ k/2  kJ k2  kJ k2 , it follows that d > 1. Let vn 2  be a sequence satisfying d  I.vn / < d C n1 . Then E˛ .vn  vm , vn  vm / D 2E˛ .vn , vn / C 2E˛ .vm , vm / vn C vm v n C vm  4E˛ . , / 2 2 vn C vm / D 2I.vn / C 2I.vm /  4I. 2 1 1 < 2.d C / C 2.d C /  4d n m ! 0, n, m ! 1.

Section 1.1 Semi-Dirichlet forms and resolvents

3

Hence ¹vn º converges to some v 2  relative to E˛ and limn!1 I.vn / D I.v/ D d . For any w 2  and 0 <  < 1, since .1  /v C w D v C .w  v/ 2 , 0  I.v C .w  v//  I.v/ D 2E˛ .v, w  v/  2J.w  v/ C  2 E˛ .w  v, w  v/. This implies equation (1.1.4). In the general case, put 1 E˛.s/ .u, v/ D .E˛ .u, v/ C E˛ .v, u// 2 1 .a/ E˛ .u, v/ D .E˛ .u, v/  E˛ .v, u// . 2 .ˇ /

.s/

.a/

Further put B˛ .u, v/ D E˛ .u, v/CˇE˛ .u, v/. Then the theorem holds for ˇ D 0. .ˇ / Hence, assuming that there exists a function v 2  such that B˛ 0 .v, w  v/  J.w  v/ for all w 2 J , it is enough to show the same inequality replaced ˇ0 by ˇ satisfying ˇ0  ˇ < ˇ0 C 1=K˛ . By the assumption, for any fixed u 2 F and ˇ0  ˇ < .a/ ˇ0 C 1=K˛ , since J.v/  .ˇ  ˇ0 /E˛ .u, v/ is a continuous linear functional relative .ˇ / .ˇ / to B˛ 0 , there exists unique function T .u/ 2  such that B˛ 0 .T .u/, w  T .u//  .a/ J .w  T .u//  .ˇ  ˇ0 /E˛ .u, w  T .u// for all w 2 . In particular, by putting v1 D T .u1 /, v2 D T .u2 / for u1 , u2 2 F , it holds that B˛.ˇ0 / .v1 , v1  v2 /  J .v1  v2 /  .ˇ  ˇ0 /E˛.a/ .u1 , v1  v2 / B˛.ˇ0 / .v2 , v1  v2 /  J .v1  v2 /  .ˇ  ˇ0 /E˛.a/ .u2 , v1  v2 / . Therefore, noting that E .a/ .v1  v2 , v1  v2 / D 0 and using equation (1.1.3) we obtain that E˛ .v1  v2 , v1  v2 / D B˛.ˇ0 / .v1  v2 , v1  v2 /  .ˇ  ˇ0 /E˛.a/ .u2  u1 , v1  v2 /  .ˇ  ˇ0 /K˛ E˛ .u2  u1 , u2  u1 /1=2  E˛ .v1  v2 , v1  v2 /1=2 . This implies that E˛ .v1  v2 , v1  v2 /  .K˛ .ˇ  ˇ0 //2 E˛ .u1  u2 , u1  u2 /, that is, T is a contraction operator relative to E˛ if K˛ .ˇ ˇ0 / < 1. Hence there exists a fixed point v 2  such that T .v/ D v, that is B˛.ˇ0 / .v, w  v/  J.w  v/  .ˇ  ˇ0 /E˛.a/ .v, w  v/. This yields the desired relation B˛.ˇ / .v, w  v/  J.w  v/ for all w 2 .

4

Chapter 1 Dirichlet forms

Theorem 1.1.2. Suppose that .E, F / is a closed form on L2 .X ; m/. Then there exist b t º t>0 on L2 .X ; m/ such that kT t k  strongly continuous semigroups ¹T t º t>0 and ¹T ˛ t ˛ t 0 0 b b b ˛ given by whose resolvents G˛ and G e , kT tRk  e , .T t f , g/ D .f , T t g/ R 1and 1 ˛t ˛t b b G˛ f D 0 e T t f dt and G ˛ f D 0 e T t f dt satisfy b ˛ f /, E˛ .G˛ f , u/ D .f , u/ D E˛ .u, G

(1.1.5)

for all f 2 L2 .X ; m/, u 2 F and ˛ > ˛0 . Proof. For any ˇ > 0 and f 2 L2 .X ; m/, applying Theorem 1.1.1 to E˛0 Cˇ , J.v/ D .f , v/ and  D F , we obtain a unique function G˛0 Cˇ f 2 F satisfying E˛0 Cˇ G˛0 Cˇ f , w  G˛0 Cˇ f  .f , w  G˛0 Cˇ f / for all w 2 F . By putting w D G˛0 Cˇ f ˙ u for any u 2 F , it holds that E˛0 Cˇ .G˛0 Cˇ f , u/ D .f , u/. b ˛ Cˇ g 2 F satisfying Similarly, for any g 2 L2 .X ; m/ and ˇ > 0, there exists G 0 b ˛ Cˇ g/ D .u, g/ for all u 2 F . Obviously E˛0 Cˇ .u, G 0 b ˛ Cˇ g/ D .f , G b ˛ Cˇ g/, .G˛0 Cˇ f , g/ D E˛0 Cˇ .G˛0 Cˇ f , G 0 0 for all f , g 2

L2 .X ; m/.

(1.1.6)

Furthermore, for any ˇ,  > 0, since

E˛0 Cˇ .G˛0 C f  .ˇ   /G˛0 Cˇ G˛0 C f , v/ D E˛0 C .G˛0 C f , v/ C .ˇ   /.G˛0 C f , v/  .ˇ   /.G˛0 C f , v/ D .f , v/, for any v 2 F , it follows that G˛0 C f  .ˇ   /G˛0 Cˇ G˛0 C f D G˛0 Cˇ f , that is ¹G˛0 C˛ º satisfies the resolvent equation G˛0 Cˇ f  G˛0 C f C .ˇ   /G˛0 Cˇ G˛0 C f D 0.

(1.1.7)

Since ˇkG˛0 Cˇ f k2  E˛0 Cˇ .G˛0 Cˇ f , G˛0 Cˇ f / D .f , G˛0 Cˇ f /  kf k  kG˛0 Cˇ f k, it follows that kG˛0 Cˇ f k 

1 kf ˇ

k.

b ˛ Cˇ ºˇ >0 such that ˇkG b ˛ Cˇ f k  kf k and Similarly, there exists a resolvent ¹G 0 0 b ˛ Cˇ f / E˛0 Cˇ .G˛0 Cˇ f , u/ D .f , u/ D E˛0 Cˇ .u, G 0

for all f 2 L2 .X ; m/ and u 2 F . Define the generator L of ¹G˛0 Cˇ º by D.L/ D ¹G˛0 Cˇ f : f 2 L2 .X ; m/º and Lu D ˇu  f for u D G˛0 Cˇ f 2 D.L/ and ˇ > 0. Since G˛0 C f D G˛0 Cˇ f C .ˇ   /G˛0 C f by equation (1.1.7), D.L/ is independent of the choice of ˇ. If a function g 2 L2 .X ; m/ satisfies .G˛0 Cˇ f , g/ D 0 for all f 2 L2 .X ; m/, b ˛ Cˇ g D 0. This implies that .u, g/ D E˛ Cˇ .u, G b ˛ Cˇ g/ D 0 for all then G 0 0 0 u 2 F . Hence g D 0 by the denseness of F in L2 .X ; m/. Therefore, ¹G˛0 Cˇ f : f 2 L2 .X ; m/º is dense in L2 .X ; m/. In particular, since k G Gˇ f  Gˇ f k D 1 kf  ˇGˇ f k and the right-hand side converges to kG f  ˇG Gˇ f k  ˛ 0

5

Section 1.1 Semi-Dirichlet forms and resolvents

zero as  increases to infinity, it follows that lim!1  G u D u in L2 .X ; m/ for all u D Gˇ f 2 D.L.˛0 / / and hence for all u 2 L2 .X ; m/. Furthermore, since the domain D.L.˛0 / / of the generator L.˛0 / is dense in L2 .X ; m/, the Hille– Yoshida theorem implies that there exists a strongly continuous contraction semi.˛ / b ˛ Cˇ º is also a regroup T t 0 on L2 .X ; m/ with resolvent ¹G˛0 Cˇ º. Similarly, ¹G 0 .˛ / 0 b t . Put T t f D e ˛0 t T t.˛0 / f solvent of a strongly continuous contraction semigroup T 0/ b.˛ b t f D e ˛0 t T f . Then they are strongly continuous semigroups such that and T t ˛ t 0 b kT t k  e and kT t k  e ˛0 t . Furthermore, their resolvents are respectively given b˛ f D G b ˛ C.˛˛ / f for ˛ > ˛0 . Equation (1.1.5) is by G˛ f D G˛0 C.˛˛0 / f and G 0 0 clear from this. Define the approximating form E ˛ of E by E ˛ .u, v/ D ˛.u  ˛G˛ u, v/ for u, v 2 and put Eˇ˛ .u, v/ D E ˛ .u, v/ C ˇ.u, v/.

L2 .X ; m/

Lemma 1.1.3. For any ˛ > ˛0 and ˇ  ˛0 .˛=.˛  ˛0 //2 , the following inequalities hold: E.˛G˛ u, ˛G˛ u/  E ˛ .u, u/; ˛

jE .u, v/j  jE ˛ .u, u/j 

KEˇ˛ .u, u/1=2 E˛0 .v, v/1=2 ; p K 2 E˛0 .u, u/ C K ˇkukE˛0 .u, u/1=2 .

(1.1.8) (1.1.9) (1.1.10)

In particular, if ˛0 D 0 then, for any ˛ > 0 and u 2 F , E.˛G˛ u, ˛G˛ u/  E ˛ .u, u/  K 2 E.u, u/.

(1.1.11)

Proof. Let ˛ > ˛0 . Then equation (1.1.8) follows from E ˛ .u, ˛G˛ u/ D ˛.u  ˛G˛ u, ˛G˛ u/ D E.˛G˛ u, ˛G˛ u/ D E ˛ .u, u/  ˛.u  ˛G˛ u, u  ˛G˛ u/  E ˛ .u, u/. To show equation (1.1.9), by using .E.2/ we have jE ˛ .u, v/j D jE.˛G˛ u, v/j  KE˛0 .˛G˛ u, ˛G˛ u/1=2 E˛0 .v, v/1=2 . In the right-hand side, using equation (1.1.8), it holds that  2 ˛ ˛ kuk2  Eˇ˛ .u, u/, 0  E˛0 .˛G˛ u, ˛G˛ u/  E .u, u/ C ˛0 ˛  ˛0 which shows equation (1.1.9). By putting v D u in equation (1.1.9) it holds that   E ˛ .u, u/2  K 2 jE ˛ .u, u/j C ˇ.u, u/ E˛0 .u, u/. Hence

 1=2 K2 K4 2 2 2 jE .u, u/j  . E˛0 .u, u/ C K ˇkuk E˛0 .u, u/ C E˛0 .u, u/ 2 4 ˛

6

Chapter 1 Dirichlet forms

p p p By noting the inequality a C b  a C b holding for a, b  0, equation (1.1.10) follows easily from this. Equation (1.1.11) follows from equation (1.1.10) by putting ˛0 D 0 and ˇ D 0. b ˛ v/, by a similar argument to the proof of Lemma Since E ˛ .u, v/ D ˛.u, v  ˛ G 1.1.3, we have b ˛ u/  E ˛ .u, u/, b ˛ u, ˛ G .1.1.8/0 E.˛ G jE ˛ .u, v/j  KE˛0 .u, u/1=2 Eˇ˛ .v, v/1=2 .

.1.1.9/0

Theorem 1.1.4. (i) A function u 2 L2 .X ; m/ belongs to F if and only if ˛ lim˛!1 E .u, u/ < 1. (ii)

If u, v 2 F , then lim˛!1 E ˛ .u, v/ D E.u, v/.

(iii) If u 2 F , then for any ˇ > ˛0 , lim˛!1 Eˇ .˛G˛ u  u, ˛G˛ u  u/ D 0. Proof. Since .u  ˛G˛C˛0 u, u/  0, E ˛ .u, u/ D ˛.u  ˛G˛C˛0 u, u/  ˛0 ˛ 2 .G˛ G˛C˛0 u, u/  

˛0 ˛ .u, u/. ˛  ˛0

Hence lim˛!1 E ˛ .u, u/  ˛0 .u, u/. To show the assertion in (i), assume that u 2 F . Then lim˛!1 E ˛ .u, u/ < 1 by equation (1.1.10). Suppose conversely that lim˛!1 E ˛ .u, u/ < 1. Then lim˛!1 Eˇ .˛G˛ u, ˛G˛ u/ < 1 for any ˇ > ˛0 . Hence there exists a subsequence ¹˛n G˛n uº converging weakly to some v 2 F rela.s/ tive to the inner product Eˇ . By the continuity of the resolvent, limn!1 ˛n G˛n u D u in L2 .X ; m/ and consequently u D v 2 F which gives the if part of (i). Since   lim E ˛ .u, v/ D lim Eˇ .˛G˛ u, v/  ˇ.˛G˛ u, v/ D E.u, v/, ˛!1

˛!1

(ii) follows from (iii). To prove (iii), as in the proof of Theorem 1.1.2, let us introduce the generator L of ¹Gˇ º with domain D.L/ D ¹Gˇ f : f 2 L2 .X ; m/º which is independent of ˇ > ˛0 . If it is shown that D.L/ is dense in F , then any function u 2 F can be approximated by a sequence of functions of D.L/ relative to Eˇ . Then, by virtue of equations (1.1.8) and (1.1.10), ˛G˛ u is also approximated by functions of the form ˛G˛ un with un 2 D.L/ uniformly for ˛ relative to Eˇ . If u D Gˇ f 2 D.L/ for f 2 L2 .X ; m/, then (iii) is obvious from lim˛!1 Eˇ .˛G˛ u  u, ˛G˛ u  u/ D lim˛!1 .˛G˛ f  f , ˛G˛ u  u/ D 0. To show the denseness of D.L/ in F , take any function u 2 F . By equation (1.1.8) and (i), Eˇ .nGn u, nGn u/ is bounded relative to n  1 for ˇ > ˛0 . Hence we can apply the Banach–Saks theorem to obtain a sequence of CesJaro means of ¹nGn uº which converges to u relative to Eˇ . More precisely, choose a subsequence uk D nk Gnk u .s/ which converges weakly to u as k ! 1 relative to Eˇ . By choosing a subsequence, .s/

we may assume that max¹jEˇ .u  u` , u  uk /j : 1  `  k  1º < 1=k for any k.

7

Section 1.1 Semi-Dirichlet forms and resolvents

Then

  1 1 Eˇ u  .u1 C    C uk /, u  .u1 C    C uk / k k k k X 1 1 X .s/ Eˇ .u  ui , u  ui / C 2 Eˇ .u  ui , u  uj / D 2 k k iD1



i¤j

k 4 2 Xi 1 sup Eˇ .un , un / C 2 . k n k i

Pk

iD2

This shows that ¹ k1 iD1 ui º is an Eˇ -Cauchy sequence converging to u in L2 .X ; m/. P Hence it converges strongly in .E, F / to u. Since .1=k/ kiD1 ui belongs to D.L/, this shows the denseness of D.L/ in F . Theorem 1.1.5. Suppose that .E, F / is a closed form and let ¹T t º t>0 be the semigroup corresponding to .E, F / by Theorem 1.1.2. Then the following conditions are mutually equivalent. .E.4/ For all u 2 F and a  0, u ^ a 2 F and E.u ^ a, u  u ^ a/  0. .E.4a/ For all u 2 F , uC ^ 1 2 F and E.uC ^ 1, u  uC ^ 1/  0. .E.4b/ For all u 2 F , uC ^1 2 F and E.uCuC ^1, uuC ^1/  ˛0 kuuC ^1k2 . .E.4c/ ¹T t º is sub-Markov: If f 2 L2 .X ; m/ satisfies 0  f  1 m-a.e., then 0  T t f  1 m-a.e. b t º is positivity preserving and contractive in L1 .X ; m/: If f 2 L1 .X ; m/ .E.4d / ¹T b t f kL1  kf kL1 . b t f  0 m-a.e. and kT satisfies f  0 m-a.e., then T Proof. .E.4/ ) .E.4a/: For any u 2 F , uC D .u/ ^ 0 2 F . Hence by noting that uC ^ 1 D .u ^ 1/C and u D .u ^ 1/ , we obtain .E.4a/ by E.uC ^ 1, u  uC ^ 1/ D E..u ^ 1/C , u ^ 1  .u ^ 1/C / C E.uC ^ 1, u  u ^ 1/ D E..u ^ 1/C , u ^ 1  .u ^ 1/C / C E.uC ^ 1, uC  uC ^ 1/  0. .E.4a/ ) .E.4b/ follows from E.u C uC ^ 1, u  uC ^ 1/ D E.u  uC ^ 1, u  uC ^ 1/ C 2E.uC ^ 1, u  uC ^ 1/  ˛0 ku  uC ^ 1k2 .

8

Chapter 1 Dirichlet forms

.E.4b/ ) .E.4c/: Put u D ˛G˛ f for f 2 L2 .X ; m/ such that 0  f  1 m-a.e. and ˛ > ˛0 . Then 1 ˛0 ku  uC ^ 1k2   ¹E.u C uC ^1, u  uC ^1/ C E.u  uC ^1, u  uC ^1/º 2 D E.u, u  uC ^ 1/ D ˛.u  f , u  uC ^ 1/ D ˛ku  uC ^ 1k2 C ˛.uC ^ 1  f , u  uC ^ 1/. Since 0  f  1,

.uC ^ 1  f , u  uC ^ 1/  0,

which implies that .˛  ˛0 /ku  uC ^ 1k  0. Hence u  uC ^ 1 D 0, which implies that 0  ˛G˛ f  1. The sub-Markov property of the associated semigroup is clear from this. .E.4c/ ) .E.4/: Suppose that .E.4c/ holds. For any u 2 F and a  0, since ˛G˛ .u ^ a/  a, .u  a/C  .˛G˛  I /.u ^ a/  0. Hence   E ˛ .u ^ a, u  u ^ a/ D ˛ .˛G˛  I /.u ^ a/, .u  a/C  0. Hence, using .1.1.9/0 we get that   E ˛ .u  a/C , .u  a/C D E ˛ .u, .u  a/C /  E ˛ .u ^ a, .u  a/C /  E ˛ .u, .u  a/C /  KE˛0 .u, u/1=2 Eˇ˛ ..u  a/C , .u  a/C /1=2 for ˇ D ˛0 .˛=.˛  ˛0 //2 . This yields that lim˛!1 E ˛ ..u  a/C , .u  a/C / < 1 and hence .u  a/C 2 F . Since u ^ a D u  .u  a/C , u ^ a 2 F and E.u ^ a, u  u ^ a/ D lim E ˛ .u ^ a, u  u ^ a/  0. ˛!1

.E.4c/ , .E.4d /: If .E.4c/ holds. Then for any f 2 L2 .X ; m/ such that 0  f  1 and g 2 L1C .X ; m/\L2 .X ; m/, since 0  .T t f , g/  kgkL1 .X;m/ , it follows that 0  b t g/  kgkL1 .X;m/ . This implies .E.4d /. Similarly .E.4d / implies .E.4c/. .f , T By virtue of .E.4c/, since T t f1  T t f2 for any f1 , f2 2 L2 .X ; m/ such that f1  2 f2 , for any f 2 L1 C .X ; m/, by taking a sequence fn 2 L .X ; m/ such that fn " f , T t f is well defined by T t f D limn!1 T t fn . In fact, if ¹fn1 º, ¹fn2 º  L2 .X ; m/ are two increasing sequences converging to f a.e., then the limits T t1 f D limn!1 T t fn1 and T t2 f D limn!1 T t fn2 exist. For any g 2 L2C .X ; m/, b t g, fni / D .T b t g, f / .g, T ti f / D lim .g, T t fni / D lim .T n!1

n!1

D . For any f 2 D f C f  , C  T t f is well defined by T t f D T t f  T t f . By the sub-Markov property, the extended resolvent on L1 .X ; m/ satisfies kT t f k1  kf k1 . Also, its resolvent ¹G˛ º can be extended to a sub-Markov resolvent on L1 .X ; m/ satisfying k˛G˛ f k1  kf k1 .

which implies T t1 f

T t2 f

L1 .X ; m/, by considering f

9

Section 1.2 Closability and regular Dirichlet forms

b t g/  0 for any non-negative funcBy the duality relation, since .T t f , g/ D .f , T 2 b b t and G b ˛ are tions f , g 2 LC .X ; m/, T t g  0 for any g 2 L2C .X ; m/. Hence T 1 extended to a semigroup and a resolvent on L .X ; m/. By the duality relation, the exb ˛ f kL1 .m/  kf kL1 .m/ b t f kL1 .m/  kf kL1 .m/ and k˛ G tended operators satisfy kT for any f 2 L1 .X ; m/. b ˛ be those in Theorem 1.1.2. Then, similarly to Theorem 1.1.5, they b t and G Let T are sub-Markov if and only if O .E.4/

E.u  u ^ a, u ^ a/  0

for all u 2 F and a  0. Definition 1. A bilinear form .E, F / is called a lower-bounded semi-Dirichlet form if it satisfies .E.1/, .E.2/, .E.3/, and .E.4/. In this monograph, we call the lower bounded semi-Dirichlet form simply as Dirichlet form. In particular, if ˛0 D 0, then .E, F / is called non-negative Dirichlet form. Furthermore, if a non-negative DirichO let form .E, F / satisfies the dual sub-Markov property .E.4/, then .E, F / is called a non-symmetric Dirichlet form. A non-symmetric Dirichlet form is called a symmetric Dirichlet form if E is symmetric. For a Dirichlet form .E, F /, since ˛ .juj  ˛G˛ juj, juj/  ˛ .u  ˛G˛ u, u/ , Theorem 1.1.4 implies that juj 2 F and E.juj, juj/  E.u, u/

(1.1.12)

for any u 2 F . In particular, if u, v 2 F , then u ^ v D 12 .u C v  ju  vj/ 2 F .

1.2

Closability and regular Dirichlet forms

Let E be a bilinear form defined on D.E/D.E/ satisfying .E.1/ and .E.2/ for a dense linear subspace D.E/ of L2 .X ; m/. We say that .E, D.E// is closable on L2 .X ; m/ if the following condition holds: If un 2 D.E/ satisfies

lim E.un  um , un  um / D 0

m,n!1

and lim .un , un / D 0, then lim E.un , un / D 0. (1.2.1) n!1

n!1

Suppose that E is closable on L2 .X ; m/. Denote by F the family of functions u 2 L2 .X ; m/ for which there exists an E-Cauchy sequence ¹un º such that limn!1 un D u in L2 .X ; m/. In this case, we call the sequence ¹un º an approximating sequence of u and .E, F / the smallest closed extension of .E, D.E//.

10

Chapter 1 Dirichlet forms

Theorem 1.2.1. Suppose that .E, D.E// is closable on L2 .X ; m/. For u, v 2 F , and its approximating sequences ¹un º and ¹vn º of u and v, respectively, E.u, v/ D lim E.un , vn / n!1

(1.2.2)

exists independently of the choice of the approximating sequences. Furthermore, the smallest closed extension .E, F / of .E, D.E// is a closed form. Proof. Let ¹un º be an approximating sequence of u 2 F . Since E˛0 .un , un /  0 for any un 2 D.E/, it satisfies the triangle inequality: jE˛0 .un , un /1=2  E˛0 .um , um /1=2 j  E˛0 .un  um , un  um /1=2 .

(1.2.3)

Hence limn!1 E˛0 .un , un / and hence limn!1 E.un , un / exists. Since jE.un , vn /  E.um , vm /j  KE˛0 .un , un /1=2 E˛0 .vn  vm , vn  vm /1=2 CKE˛0 .vm , vm /1=2 E˛0 .un  um , un  um /1=2 , the uniform boundedness of ¹E˛0 .un , un /º and ¹E˛0 .vm , vm /º yields the existence of equation (1.2.2). .E.1/ and .E.2/ of .E, F / follows easily from the corresponding properties of .E, D.E//. If ¹wn º  F is an E˛ -Cauchy sequence, then for the approximating sequences ¹wn,k º  D.E/ of wn , the diagonal sequence ¹wn,n º converges to w 2 F . This implies .E.3/ for .E, F /. Theorem 1.2.2. Let .E, D.E// be a closable form satisfying .E.4/ for D.E/ instead of F . Then the smallest closed extension .E, F / also satisfies .E.4/, that is, .E, F / is a Dirichlet form on L2 .X ; m/. Proof. Let ¹G˛ º be the resolvent associated with .E, F /. Then it is enough to show that the resolvent ¹G˛ º is sub-Markov. Take a function f 2 L2 .X ; m/ such that 0  f  1 m-a.e. and put u D ˛G˛ f for ˛ > ˛0 . Since u 2 F , there exists an approximating sequence ¹un º  D.E/ such that limn!1 un D u relative to E˛ . By virtue of Theorem 1.1.5, if .E, D.E// satisfies .E.4/, then it satisfies .E.4a/ and .E.4b/. In particular uC n ^ 1 2 D.E/ and 2 C C C ˛0 kuC n ^ 1k  E.un ^ 1, un ^ 1/  E.un ^ 1, un / C 1=2  KE˛0 .uC E˛0 .un , un /1=2 . n ^ 1, un ^ 1/ C This implies that E.uC n ^1, un ^1/ is bounded relative to n  1. Since equation (1.1.2) holds for un , vn 2 D.E/, it also holds for u, v 2 F by the definition of .E, F /. Hence C the E˛0 -boundedness of ¹un uC n ^1º implies that limn!1 E.un u, un un ^1/ D 0.

11

Section 1.2 Closability and regular Dirichlet forms

Therefore, by using .E.4a/ and equation (1.1.1), 2 ˛0 ku  uC ^ 1k2 D ˛0 lim kun  uC n ^ 1k n!1

C   lim E.un  uC n ^ 1, un  un ^ 1/ n!1 ® ¯ C C C   lim E.un  uC n ^ 1, un  un ^ 1/ C E.un ^ 1, un  un ^ 1/ n!1

C D  lim E.un , un  uC n ^ 1/ D  lim E.u, un  un ^ 1/ n!1

D lim ˛.u  f , un  n!1

n!1

uC n

^ 1/ D ˛.u  f , u  uC ^ 1/

D ˛ku  uC ^ 1k2 C .uC ^ 1  f , u  uC ^ 1/. Since .uC ^ 1  f , u  uC ^ 1/  0, this shows that u D uC ^ 1. Definition 2. A Dirichlet form .E, F / on L2 .X ; m/ is called a regular Dirichlet form with core C1 if C1 is a sub-family of C0 .X / such that F \ C1 is E˛ -dense in F and uniformly dense in C0 .X /. R For a regular Dirichlet form .E, F / and its associated resolvent ¹G˛ º, since X G˛ g.x/f .x/m.dx/ is a non-negative bilinear form relative to f , g 2 C0 .X /, it is represented as Z Z Z G˛ g.x/f .x/m.dx/ D f .x/g.y/G˛ .dxdy/ (1.2.4) X

X

X

for a positive Radon measure G˛ .dxdy/ on X  X . Note that G˛ .dxdy/ does not charge any set of zero m ˝ m-measure. As a particular case, suppose that .E, F / is a symmetric regular Dirichlet form. Then the approximating form E ˛ can be written as Z Z ˛2 E ˛ .u, u/ D .u.x/  u.y//2 G˛ .dxdy/ 2 X X Z u2 .x/ .1  ˛G˛ 1.x// m.dx/. (1.2.5) C˛ X

By virtue of Theorem 1.1.4, u 2 L2 .X ; m/ belongs to F if and only if E ˛ .u, u/ remains bounded as ˛ increases to infinity. In particular, for any u, v 2 Fb , it follows from equation (1.2.5) and the inequalities juv.x/j  ju.x/jjv.x/j and juv.x/  uv.y/j  kvk1 ju.x/  u.y/j C kuk1 jv.x/  v.y/j m-a.e. x, y, that uv 2 F and E.uv, uv/1=2  kvk1 E.u, u/1=2 C kuk1 E.v, v/1=2 .

(1.2.6)

12

Chapter 1 Dirichlet forms

1.3 Transience and recurrence of Dirichlet forms b t º t>0 be the semiFor a given Dirichlet form .E, F / on L2 .X ; m/, let ¹T t º t>0 and ¹T groups on L2 .X ; m/ associated with .E, F / by Theorem 1.1.2. A Borel measurable subset B of X is called an invariant set relative to ¹T t º if, for all t > 0 and f 2 L2 .X ; m/, (1.3.1) 1B T t f D 1B T t .1B f / m-a.e. b t º is defined similarly. If B is an invariant set relative An invariant set relative to ¹T 2 to ¹T t º, then for any g 2 L .X ; m/, b t .1B g//. b t .1B g// D .1B T t f , g/ D .1B T t .1B f /, g/ D .1B f , T .f , T b t .1B g/ D 1B T b t .1B g/ and hence Hence T   bt g  T b t .1XnB g/ D T b t .1B g/ D 1B T b t .1XnB g/ . bt g  T T This implies b t g D 1XnB T b t .1XnB g/ for all t > 0 and g 2 L2 .X ; m/, 1XnB T

(1.3.2)

b t º. Since the converse assertion also that is X n B is an invariant set relative to ¹T holds, B is an invariant set relative to ¹T t º if and only if X n B is an invariant set b t º. relative to ¹T Lemma 1.3.1. For any non-negative Borel function f and ˛ > 0, ¹x : G˛ f .x/ > 0º is an invariant set relative to ¹T t º. Proof. Put B D ¹x : G˛ f .x/ D 0º. For any non-negative function g 2 L1 .X ; m/ such that g D 0 m-a.e. on X n B,   b t .1B g/ 0 D .G˛ f , g/  e ˛t .T t G˛ f , g/ D e ˛t G˛ f , T   b t .1B g/  e ˛t 1XnB  G˛ f , T b t .1B g/ D 1B T b t .1B g/ D 0 m-a.e. and hence T b t .1B g/. This implies Hence 1XnB T b that B is an invariant set of ¹T t º and hence X n B is an invariant set of ¹T t º. b t º if and only if Theorem 1.3.2. B is an invariant set relative to both ¹T t º and ¹T IB u, IB v 2 F for all u, v 2 F and satisfies E.u, v/ D E.1B u, 1B v/ C E.1XnB u, 1XnB v/.

(1.3.3)

Section 1.3 Transience and recurrence of Dirichlet forms

13

b t º. Then, as we Proof. Suppose that B is an invariant set relative to both ¹T t º and ¹T b t º. noted before Lemma 1.3.1, X n B is also an invariant set relative to ¹T t º and ¹T Hence, for all u 2 F , 1B G˛ u D 1B G˛ .1B u/ and 1XnB G˛ u D 1XnB G˛ .1XnB u/. Therefore,   .u, u  ˛G˛ u/ D .1B u, 1B u  ˛G˛ .1B u// C 1XnB u, 1XnB  ˛G˛ .1XnB u/ . Hence, by Theorem 1.1.4 (i) and (ii), 1B u, 1XnB u 2 F and equation (1.3.3) holds. Conversely, suppose that 1B u, 1B v 2 F and equation (1.3.3) hold for all u, v 2 F . By putting 1B u and 1XnB v instead of u, v, respectively, it holds that E.1B u, 1XnB v/ D 0 u, v 2 F .

(1.3.4)

For any ˛ > ˛0 , since     E˛ G˛ f , 1XnB v D .f , 1XnB v/ D E˛ G˛ .1XnB f /, 1XnB v ,   it follows that E˛ G˛ f  G˛ .1XnB f  /, 1XnB v D 0. In particular, by putting v D G˛ f  G˛ .1XnB f /, we get that E˛ v, 1XnB v D 0. Hence, by equation (1.3.4) we get that   E˛ 1XnB v, 1XnB v D 0, which shows that 1XnB v D 0, that is 1XnB G˛ f D 1XnB G˛ .1XnB f /. Similarly, by taking B in place of X n B, we have 1B G˛ f D 1B G˛ .1B f /. A Dirichlet form .E, F / (or semigroup ¹T t º) is called irreducible if any invariant set B relative to ¹T t º satisfies m.B/ D 0 or m.X n B/ D 0. As noted before Lemma b t º. 1.3.1, if B is a ¹T t º-invariant set, then X n B is an invariant set relative to ¹T b t º satisfies Hence .E, F / is irreducible if and only if any invariant set B relative to ¹T m.B/ D 0 or m.X n B/ D 0. As we have seen after Theorem 1.1.5, ¹G˛ º can be considered as a sub-Markov L1 .X ; m/-resolvent. For any f 2 L1 C .X ; m/, since G˛ f is increasing as ˛ decreases, the potential operator Gf D lim G1=n f n!1

(1.3.5)

b ˛ º can be considered as a contractive L1 .X ; m/-resolis well defined. Similarly, ¹G 1 b vent. If f 2 LC .X ; m/, G ˛ f is also increasing as ˛ decreases. Hence we can define the copotential operator b 1=n f . b D lim G (1.3.6) Gf n!1

A Dirichlet form .E, F / is called transient if there exists a strictly positive function f 2 L1 .X ; m/ such that Gf < 1 m-a.e. (1.3.7) An irreducible Dirichlet form is called recurrent if it is non-transient.

14

Chapter 1 Dirichlet forms

If .E, F / is transient, then there exists a strictly positive function f 2 L1 .X ; m/ satisfying equation (1.3.7). Let ¹Bn º be an increasing sequence of compact sets such that [BnPD X and Gf .x/  n for a.e. x 2 Bn . Then the function g defined by n g.x/ D 1 nD1 .1=nm.Bn //2 1Bn .x/ is a strictly positive function satisfying g 2 1 1 L .X ; m/ \ L .X ; m/ and 1 X 1 b D .f , Gg/ .Gf , 1Bn / < 1. n nm.B n /2 nD1 In particular, b 0º. Then be the set defined by E˛ D ¹x 2 X : supn G Z b ˛ f .x/m.dx/  0. (1.3.9) G E˛

b ˛=k f .x/  0º. If x 2 E˛n , then Proof. Let E˛n D ¹x 2 X : max1kn G b ˛=k f  G b ˛ f /C .x/ b ˛ f .x/ C max .G G 1kn

b ˛=k f /C .x/. b ˛=k f .x/ D max .G  max G 1kn

Noting that

1kn

 k1 b˛ G b ˛=k f /C .x/ ˛.G k 1kn b ˛ max .G b ˛=k f /C .x/,  ˛G

b ˛=k f  G b ˛ f /C .x/ D max max .G

1kn



1kn

weZhave

b ˛ f .x/m.dx/ G  Z  C C b b b max .G ˛=k f / .x/  ˛ G ˛ . max .G ˛=k f / /.x/ m.dx/  n 1kn 1kn ZE˛ Z b ˛=k f /C .x/m.dx/  b ˛ . max .G b ˛=k f /C /.x/m.dx/ D max .G ˛G 1kn X 1kn E˛n Z Z b ˛=k f /C .x/m.dx/  b ˛ . max .G b ˛=k f /C /.x/m.dx/  max .G ˛G

E˛n

X 1kn

 0.

X

1kn

Section 1.3 Transience and recurrence of Dirichlet forms

15

Theorem 1.3.4. .E, F / is transient if and only if there exists a strictly positive funcb < 1 m-a.e. tion g 2 L1 .X ; m/ such that Gg b < 1 m-a.e. For Proof. Suppose that g 2 L1 .X ; m/ satisfies g > 0 m-a.e. and Gg 1 any f 2 LC .X ; m/, define E˛ as in Lemma 1.3.3 by taking f  cg in place of f . R b D 1º is b ˛ .f  cg/.x/m.dx/  0. Since the set B D ¹Gf Then it holds that E˛ G contained in E˛ up to a negligible set, Z Z Z 1 b˛ f d m  c b ˛ gd m. fdm  G G ˛ X E˛ E˛ R R b Hence, for any compact set K and integer N , .1=c/ X f d m  ˛ B\K R G ˛ .g ^ 2 .X ; m/, letting ˛ tend to infinity, we have .1=c/ N /d m. Since g ^ N 2 L X fdm  R R B\K .g ^ N /d m. Then let K " X , N " 1 and c " 1 to get B gd m D 0 which implies m.B/ D 0. For a non-negative measurable function g 2 L1 .X ; m/ \ L1 .X ; m/, define a bilinear form E g by (1.3.10) E g .u, v/ D E.u, v/ C .u, v/gm , R where .u, v/gm D X u.x/v.x/g.x/d m.x/. Since E.u, u/  E g .u, u/  E˛ .u, u/ for ˛  kgk1 , .E g , F / satisfies .E.1/, .E.2/ and .E.3/. Hence there corresponds a semigroup ¹T tg º and its resolvent ¹G˛g º on L2 .X ; m/ satisfying kT tg f k  e ˛0 t kf k and (1.3.11) E˛g .G˛g f , u/ D .f , u/ for all f 2 L2 .X ; m/, u 2 F and ˛ > ˛0 . Furthermore, if E satisfies .E.4/, then so does E g . In particular, ¹G˛g º can be extended to ˛ > 0 as a sub-Markov resolvent on L1 .X ; m/. Denote by g  G˛ the operator defined by .g  G˛ / f .x/ D g.x/G˛ f .x/. Lemma 1.3.5. Let g be a strictly positive function of L1 .X ; m/ \ L1 .X ; m/. Then, for all f 2 L2 .X ; m/ and ˛ > ˛0 C kgk1 , G˛g f D

1 X

.1/n G˛ .g  G˛ /n f .

(1.3.12)

nD0

If f 2 L1 .X ; m/ \ L2 .X ; m/, then for any ˛ > 0, G˛ f D G˛g f C G˛g .g  G˛ /f D G˛g f C G˛ .g  G˛g /f .

(1.3.13)

Moreover, G g g  1 m-a.e and .E g , F / is transient. If .E, F / is irreducible, then .E g , F / is also irreducible. In particular, if .E, F / is recurrent, then G gRg D 1 m-a.e. for any non-negative function g 2 L1 .X ; m/ \ L1 .X ; m/ such that X g.x/m.dx/ > 0.

16

Chapter 1 Dirichlet forms

Proof. Since k.g  G˛ /f k  kgk1 kf k=.˛  ˛0 /, the right-hand side of equation (1.3.12) is well defined and belongs to F because it can be written as G˛ h for h D P 1 n n 2 nD0 .1/ .g  G˛ / f 2 L .X ; m/. Let us denote the right-hand side of .1.3.12/ g N by G˛ f . Then, for any u 2 F ,  X 1 E˛g .GN ˛g f , u/ D E˛ .1/n G˛ .g  G˛ /n f , u nD0

X 1

C

 n

n

.1/ G˛ .g  G˛ / f , u gm

nD0

D

1 X

  .1/n .g  G˛ /n f , u 

nD0

1 X

  .1/n .g  G˛ /n f , u

nD1

D .f , u/. This implies GN ˛g f D G˛g f for ˛ > ˛0 C kgk1 . By the sub-Markov property, G˛g g g g and GN ˛ can be extended to the operators on L1 .X ; m/ and GN ˛ f D G˛ f for any 1 f 2 L .X ; m/ and ˛ > 0. Equation (1.3.13) is clear from equation (1.3.12). To show the inequality G g g  1, for ˇ > ˛0 , note that .E, F / can be considered as a Dirichlet form on L2 .X ; .ˇ C g/  m/ for which .E.1/, .E.2/ and .E.3/ hold by ˇ taking ˛0 =ˇ < 1 instead of ˛0 . Let us denote by ¹K˛ º˛>0 the resolvent of .E, F / on L2 .X ; .ˇ C g/  m/ given by E.K˛ˇ f , u/ C ˛.K˛ˇ f , u/.ˇ Cg/m D .f , u/.ˇ Cg/m for all u 2 F . Since

    g g E Gˇ .ˇf C gf /, u C Gˇ .ˇf C gf /, u .ˇ Cg/m   g g D Eˇ Gˇ .ˇf C gf /, u D .ˇf C gf , u/ D .f , u/.ˇ Cg/m ˇ

ˇ

for all u 2 F , Gˇg .ˇf C gf / D K1 f . Since ¹K˛ º is a sub-Markov resolvent by ˇ

Theorem 1.1.5, 0  K1 f  1 for all ˇ > ˛0 and f satisfying 0  f  1. Furtherˇ more, since ¹Gˇg º can be extended to ˇ > 0 as a resolvent on L1 .X ; m/, K1 can be extended to ˇ > 0 as a bounded linear operator on L1 .X ; m/. If  > ˇ > ˛0 and f 2 L1 .X ; m/ \ L2 .X ; m/, since     ˇ  ˇ  E K1 f  K1 f , u C K1 f  K1 f , u .ˇ Cg/m    D .  ˇ/ K1 f  f , u m   1  D .  ˇ/ , .K1 f  f /, u ˇCg .ˇ Cg/m

Section 1.3 Transience and recurrence of Dirichlet forms

it follows that ˇ



ˇ

K1 f  K1 f D .  ˇ/K1



17

 1  .K1 f  f / . ˇCg

By putting .ˇ C g/f =. C g/ instead of f , it follows that       ˇ  ˇCg ˇ  ˇCg ˇ K1 f D K1 f C K1 K1 f  Cg ˇCg  Cg     k1 X  ˇ  n ˇCg  D K1 K1 f ˇCg  Cg nD0      ˇ  k ˇCg ˇ CK1 K1 f . ˇCg  Cg This relation also holds for 0 < ˇ  ˛0 <  . Hence, it holds that  X   1   ˇ  n ˇCg ˇ  K1 f D K1 K1 f .  Cg  Cg nD0 

ˇ

Since K1 f  1 for any  > ˛0 and f 2 L2 .X ; m/ with f  1, for 0 < ˇ  ˛0 , K1 ˇ can be considered as an operator on L1 .X ; m/ satisfying K1 1  1. Therefore, ˇ

G g g D lim Gˇg g  lim K1 1  1. ˇ !0

ˇ !0

This implies the transience of .E g , F /. g If ¹G˛ º is irreducible, then equation (1.3.12) implies the irreducibility R of ¹G˛ º. If .E, F / is recurrent, then for any non-negative function g such that X gd m > 0, G g g  G˛ g for any ˛  kgk1 . Put B D ¹x : G˛ f .x/ > 0º for a non-negative function f . For a.e. x 2 X n B, since 0 D G˛ f .x/  .ˇ  ˛/Gˇ .1B G˛ f /.x/ for ˇ > ˛, it follows that Gˇ .1B G˛ f / D 0 and hence Gˇ 1B D 0 a.e. on X n B. Hence B is an invariant set of ¹T t º and hence B D X a.e., that is G g g > 0 by irreducibility. Furthermore, it holds that G g g  1 for non-negative function g. Then by letting ˛ tend to 0 in equation (1.3.13), we obtain that   G g.1  G g g/ D G g g  1. Hence, the recurrence of .E, F / yields that G g g D 1 a.e. By using Lemma 1.3.5, we have the following maximum principle. Corollary 1.3.6. Suppose that .E, F / is transient and let f 2 L1 .X ; m/\L1 .X ; m/ be a non-negative function such that Gf .x/ D lim˛!0 G˛ f .x/ < 1 a.e. For any Borel set B, if f .x/ D 0 for a.e. x 2 X n B, then kGf k1 D ess.sup ¹Gf .x/ : x 2 Bº .

18

Chapter 1 Dirichlet forms

In particular, there exists a strictly positive m-integrable function g such that kGgk1 < 1. Proof. Put g D ˇ1B and ˛ D 0 in equation (1.3.13). Then, by the last result of Lemma 1.3.5, Gf .x/ D G ˇIB f .x/ C G ˇIB .ˇIB  Gf / .x/  G ˇIB f .x/ C ess. sup Gf .y/. y2B

Furthermore, since G ˇ 1B .ˇ1B /  1, it follows that lim G ˇIB f D lim G ˇIB .IB f / D 0,

ˇ !1

ˇ !1

which yields the first assertion. To show the latter assertion, take a strictly positive bounded m-integrable function h such that Gh < 1 a.e. Put Bn D ¹x : Gh.x/  nº n n and gn .x/ D 2n .1=n/1 PB1n h.x/. Then Ggn  2 .1=n/G.1Bn h/  2 . Hence it is enough to put g.x/ D nD1 gn .x/. Theorem 1.3.7. .E, F / is recurrent R if and only if Gf D 1 m-a.e. for all f 2 1 1 LC .X ; m/ \ L .X ; m/ such that X f d m > 0. Proof. Suppose that .E, F / is recurrent and let B D ¹Gf D 1º. Then X n B is an invariant set of ¹T t º. In fact, for all t > 0 and g 2 L1C .X ; m/ such that g D 0 outside of ¹Gf  nº, b t g/, 1 > .Gf , g/  .T t Gf , g/  .T t .IB Gf /, g/ D .IB Gf , T b t g/ D 0, that b t g D 0 m-a.e. on B. Hence .T t .IB u/, g/ D .IB u, T which implies that T is IXnB T t .IB u/ D 0. This implies 1XnB T t u D 1XnB T t .1XnB u/, that is X n B is an invariant set of ¹T t º. According to irreducibility, this implies that m.B/ D 0 or m.X n B/ D 0. Suppose that m.B/ D 0. By virtue of Lemma 1.3.1, since ¹x : Gf .x/ > 0º is a non-trivial invariant set of ¹T t º, Gf > 0 m-a.e. and hence G˛ f > 0 m-a.e. for any ˛ > 0. Then, by the resolvent equation, ˛GG˛ f D Gf  G˛ f  Gf < 1 m-a.e. which contradicts the hypothesis of recurrence. If ˛G˛ 1 D 1 m-a.e. for all ˛ > 0, then .E, F / is called conservative. Corollary 1.3.8. If .E, F / is recurrent, then it is conservative. Proof. Letting ˇ tend to 0 in Gˇ .1  .˛  ˇ/G˛ 1/ D G˛ 1  ˛1 , we obtain that G.1  ˛G˛ 1/  ˛1 m-a.e. Since 1  ˛G˛ 1  0, Theorem 1.3.7 implies that ˛G˛ 1 D 1. Let us define the extended Dirichlet form .E, Fe / of .E, F / as follows: Fe is the family of functions u for which there exists an E-Cauchy sequence ¹un º  F such that limn!1 un D u a.e. and E.u, u/ D limn!1 E.un , un / exists.

Section 1.3 Transience and recurrence of Dirichlet forms

19

The sequence ¹un º is called an approximating sequence of u. Generally, E.u, u/ depends on the choice of the approximating sequence. But, if ˛0 D 0, then, for any function u 2 Fe , E.u, u/ D limn!1 E.un , un / is well defined independently of the choice of the approximating sequence ¹un º. In fact, the symmetric part E .s/ becomes non-negative and hence the triangle inequality holds. Hence, by ˇ ˇ ˇ 1=2 1=2 ˇ , u /  E.u , u / E.u ˇ  E.un  um , un  um /1=2 , ˇ n n m m E.u, u/ exists. Furthermore, by the sector condition, for any u, v 2 Fe and their approximating sequences ¹un º, ¹vn º respectively, E.u, v/ D limn,m!1 E.un , vm / is well defined. Theorem 1.3.9. Suppose that .E, F / is transient and ˛0 D 0. If .jf j, Gjf j/ < 1, then Gf 2 Fe and satisfies Z f .x/u.x/m.dx/ (1.3.14) E.Gf , u/ D X

for all u 2 Fe . In particular, .E .s/ , Fe / is a Hilbert space and there exists a strictly positive bounded integrable function g and a constant Kg depending on g such that Z juj.x/g.x/m.dx/  Kg E.u, u/1=2 (1.3.15) X

for all u 2 Fe . Proof. Let .EN .s/ , FN / be the Hilbert space determined as the abstract completion of .E .s/ , F /, that is uN 2 FN is an equivalence class of E-Cauchy sequences ¹un º N u, N u, N v/ N D and E. N u/ N D limn!1 E.un , un /. By the sector condition .E.2/, E. limn,m!1 E.un , vn / is well defined for uN D ¹un º and vN D ¹vn º. As in the last part of the proof of Theorem 1.1.4, for any EN .s/ -bounded sequence ¹un º  F , there exists an element uN 2 FN such that a subsequence of CesJaro means of ¹un º converges to uN N relative to E. Assume that g 2 L1 .X ; m/ \ L1 .X ; m/ is a strictly positive function satisfying kGgk1 < 1. Since E.G˛ g, G˛ g/ D .g  ˛G˛ g, G˛ g/  .g, Gg/, there exists a sequence ¹un º  F constituting a CesJaro sum of G˛n g for ˛n # 0 which is E-Cauchy. Put uN 2 FN the equivalence class containing ¹un º. For any v 2 Fb , N u, N v/. On the other hand, since ¹un º is a convex combilimn!1 E.un , v/ D E. nation of G˛n g, limn!1 E.un , v/ D limn!1 E.G˛n g, v/ D .g, v/ and hence N u, E. N v/ D .g, v/. In particular, Z jvj.x/g.x/m.dx/  Kg E.v, v/1=2 (1.3.16) X

N u, N u/ N 1=2 . Since any function of F can be for any v 2 Fb and a constant Kg D K0 E. approximated by the functions of Fb , equation (1.3.16) holds for any v 2 F . If ¹vn º

20

Chapter 1 Dirichlet forms

is an E-Cauchy sequence corresponding to vN 2 FN , then equation (1.3.16) implies that it converges to a function v in L1 .X ; g  m/. By the definition of Fe , this implies that v 2 Fe . Hence we can identify any element vN 2 FN with a function of Fe . In particular, .E, Fe / is a Hilbert space and Gg 2 Fe . Assume that a non-negative function f satisfies .f , Gf / < 1. Then, for the function g given above, Gfn 2 Fe and satisfies E.Gfn  Gfm , Gfn  Gfm / D .fn  fm , Gfn  Gfm / for fn D f ^ .ng/. Hence limn,m!1 E.Gfn  Gfm , Gfn  Gfm / D 0 which yields that Gf 2 Fe and that the relation (1.3.14) holds. For a general function f , it is enough to consider f C and f  separately.

1.4 An auxiliary bilinear form As we have seen after Theorem 1.1.5, for any ˛ > 0, G˛ can be extended to a subMarkov resolvent on L1 .X ; m/. Hence, G˛ f can be further extended to all nonnegative measurable functions f by G˛ f D limk!1 G˛ .f ^ kg/ by using a strictly b ˛ f is well defined for any positive function g 2 L1 .X ; m/ \ L1 .X ; m/. Similarly G u) is called f 2 L1 .X ; m/. Under this extension, a non-negative function u (resp. b ˛-excessive (resp. ˛-coexcessive) if ˇG˛Cˇ u  u

b ˛Cˇ b .resp. ˇ G u b u/

m-a.e.

(1.4.1)

for all ˇ. The 0-excessive function and 0-coexcessive function are called excessive function and coexcessive function, respectively. Theorem 1.4.1. The following conditions are equivalent to each other for u 2 F (resp. b u 2 F ) and ˛ > ˛0 . (i)

u is ˛-excessive (resp. b u is ˛-coexcessive).

(ii)

E˛ .u, v/  0 (resp. b u  0 and E˛ .v,b u/  0) for all v 2 F C .

(iii) E˛ .u, v/  0 (resp. b u  0 and E˛ .v,b u/  0) for all v 2 F C \ C0 .X /. Proof. Since limˇ !1 ˇGˇ u D u in L2 .X ; m/, (i) ) (ii) follows from E˛ .u, v/ D lim ¹ˇ.u  ˇGˇ u, v/ C ˛.u, v/º ˇ !1

D lim ¹ˇ.u  ˇG˛Cˇ u, v/ C ˛.u  ˇGˇ G˛Cˇ u, v/º ˇ !1

D lim ¹ˇ.u  ˇG˛Cˇ u, v/ C ˛.u  ˇGˇ u, v/º ˇ !1

 0. The equivalence of (ii) and (iii) is obvious.

21

Section 1.4 An auxiliary bilinear form

b ˛Cˇ v  0 for v  0, (ii) ) (i) Since G .u  ˇG˛Cˇ u, v/ D E˛ .G˛ u  ˇG˛ G˛Cˇ u, v/ b ˛Cˇ v/ D E˛ .G˛Cˇ u, v/ D E˛ .u, G  0. Hence u  ˇG˛Cˇ u m-a.e. To show the non-negativity of u, note that E˛ .u, uC  u/  0. Hence by using .E.4/, E˛ .uC  u, uC  u/ D E˛ .uC , uC  u/  E˛ .u, uC  u/  E˛ .uC , uC  u/ D E˛ ..u/ ^ 0, .u/  .u/ ^ 0/  0. Thus u D uC  0. The assertion concerning the coexcessive function except the non-negativity statement follows similarly. Lemma 1.4.2. Suppose that both of u1 2 F and u2 2 L2 .X ; m/ are ˛-excessive or ˛-coexcessive functions. Then u1 ^ u2 2 F and  2 2˛ E˛ .u1 , u1 /. (1.4.2) E˛ .u1 ^ u2 , u1 ^ u2 /  K C ˛  ˛0 Proof. Since equation (1.4.1) holds for u1 and u2 , clearly ˇG˛Cˇ .u1 ^ u2 /  u1 ^ u2 for all ˇ > 0. Hence, by using equation (1.1.9) and noting that ˇGˇ G˛Cˇ .u1 ^ u2 /  G˛C.ˇ ˛/ .u1 ^ u2 / 

1 .u1 ^ u2 /, ˇ˛

we get for  D ˛0 .ˇ=.ˇ  ˛0 //2 that E ˇ .u1 ^ u2 , u1 ^ u2 /     D ˇ u1 ^ u2  ˇG˛Cˇ .u1 ^ u2 /, u1 ^ u2  ˇ 2 ˛ Gˇ G˛Cˇ .u1 ^ u2 /, u1 ^ u2      ˇ u1 ^ u2  ˇG˛Cˇ .u1 ^ u2 /, u1  ˇ 2 ˛ Gˇ G˛Cˇ .u1 ^ u2 /, u1 ^ u2   D E ˇ .u1 ^ u2 , u1 / C ˇ 2 ˛ Gˇ G˛Cˇ .u1 ^ u2 /, u1  u1 ^ u2 ˛ˇ .u1 ^ u2 , u1  u1 ^ u2 / .  KEˇ .u1 ^ u2 , u1 ^ u2 /1=2 E˛0 .u1 , u1 /1=2 C ˇ˛

22

Chapter 1 Dirichlet forms

Since .u, u/  .1=.˛  ˛0 // E˛ .u, u/ for ˛ > ˛0 and limˇ !1  D ˛0 , it follows that limˇ !1 E ˇ .u1 ^ u2 , u1 ^ u2 / < 1 and E˛ .u1 ^ u2 , u1 ^ u2 / D lim E˛ˇ .u1 ^ u2 , u1 ^ u2 / ˇ !1

 KE˛0 .u1 ^ u2 , u1 ^ u2 /1=2 E˛0 .u1 , u1 /1=2 C 2˛.u1 ^ u2 , u1 /   2˛  KC E˛ .u1 ^ u2 , u1 ^ u2 /1=2 E˛ .u1 , u1 /1=2 . ˛  ˛0 Equation (1.4.2) follows easily from this. In the remainder of this section, we fix ı > 0 and a strictly positive ı-coexcessive function b hı . If .E, F / is transient, then we may consider that ı D 0. For any ı > 0, hı D there exists a strictly positive ı-coexcessive function b hı , in fact it is enough to put b 1 b ı g for a strictly positive function g 2 L .X ; m/\L1 .X ; m/. Then b hı 2 L1 .X ; m/. G hı is bounded from below In the later chapters, it is convenient to choose b hı such that b by a positive constant on each compact set. The existence of such a function will be given by equation (2.4.22). As a particular case, if a positive constant is ı-coexcessive, then we can take b hı D 1. A sufficient condition for this is .b E.4/ as we have seen after Theorem 1.1.5. More weakly, 1 is ı-coexcessive if the following condition is satisfied. .b E ı .4/: For all u 2 F and non-negative constant a, Eı .u  u ^ a, u ^ a/  0.

(1.4.3)

But we do not assume this condition in general. If we use this condition, we shall mention it explicitly. .ı/ b b .ı/ b Let G˛ D G˛Cı and G ˛ be the hı -transformation of the resolvent G ˛Cı , that is

b .ı/ G ˛ f .x/ D

1 b hı .x/

b ˛Cı .b G hı f /.x/.

(1.4.4)

b b .ı/ b .ı/ D G b be the In particular, put G 0 . Fixing such a ı-coexcessive function hı , let m b measure defined by m b.dx/ D hı .x/m.dx/. .ı/ b .ı/ Lemma 1.4.3. The resolvents ¹G˛ º˛>0 and ¹G ˛ º˛>0 are sub-Markov resolvents 2 b/. Furthermore, they are in dual relative to m b , that is for all f , g 2 on L .X ; m b / and ˛ > 0, L2 .X ; m b .ı/ .G˛.ı/ f , g/b D .f , G . (1.4.5) ˛ g/b m m

b ˛ º are strongly continuous on L2 .X ; m b /. If b hı 2 L1 .X ; m/, then ¹G˛ º and ¹G .ı/

.ı/

23

Section 1.4 An auxiliary bilinear form .ı/

Proof. The sub-Markov property of ¹G˛ º and hence of ¹G˛ º have already been b ˛ º, shown in Theorem 1.1.5. Since b hı is a ı-coexcessive function relative to ¹G .ı/ .ı/ b b b b b ˛ G ˛ 1 D .1=hı .x//˛ G ˛Cı hı .x/  1 which means that ¹G ˛ º is also sub-Markov. b ˛Cı º relaEquation (1.4.5) is a consequence of the duality relation of ¹G˛Cı º and ¹G tive to m, in fact         .ı/ b ˛Cı .gb b .ı/ D G˛Cı f , gb hı D f , G hı / D f , G . G˛ f , g b ˛ g b m m b /, since For any f 2 L2 .X ; m Z k˛G˛.ı/ f k2 2  ˛G˛Cı f 2 .x/b hı .x/m.dx/ L .X;b m/ X Z Z b ˛Cıb hı .x/m.dx/  f 2 .x/˛ G f 2 .x/b m.dx/, D X

X

.ı/

¹G˛ º is a contraction operator on L2 .X ; m b /. .ı/ b /, let f be a non-negative To prove the strong continuity of ¹G˛ º on L2 .X ; m 2 b /. For any strictly positive bounded .m C m b /-integrable function function of L .X ; m .ı/ .ı/ g, since ˛G˛ .f ^ ng/ is uniformly bounded relative to ˛ and limk!1 ˛k G˛k .f ^ ng/ D f ^ ng in L2 .X ; m/, for any increasing sequence ˛k " 1, we can choose .ı/ a subsequence ˛k0 " 1 such that limk!1 ˛k0 G˛0 .f ^ ng/ D f ^ ng a.e. and k boundedly. Hence, by the Lebesgue theorem, Z  2 .ı/ ˛k0 G˛0 .f ^ ng/  .f ^ ng/ d m lim bD0 k!1 X

k

R for all n. Therefore, for any  > 0, by taking n such that X .f  .f ^ ng//2 d m b < , we have Z  Z  2 2 .ı/ 0 .ı/ ˛k G˛0 f  f d m ˛k0 G˛0 .f  f ^ ng/ d m b3 b k k X X Z  Z 2 .ı/ C3 ˛k0 G˛0 .f ^ ng/  f ^ ng d m b C 3 .f  f ^ ng/2 d m b k X X Z  2 .ı/ ˛k0 G˛0 .f ^ ng/  f ^ ng d m b.  6 C 3 X

Hence

k

Z  2 .ı/ lim ˛k0 G˛0 f  f d m b D 0.

k!1 X

k

.ı/

Since the sequence is arbitrary, this yields the strong continuity of ¹G˛ º. Using b .ı/ b .ı/ the sub-Markov property of ¹G ˛ º, the proof of the strong continuity of ¹G ˛ º is similar.

24

Chapter 1 Dirichlet forms

In the remainder of this section,we assume that b hı is bounded from below by a posb/ itive constant on each compact set. Let G .ı/ be the family of functions u 2 L2 .X ; m such that A.ı/ .u, u/ exists, where   A.ı/ .u, u/ D lim˛!1 ˛ u  ˛G˛.ı/ u, u b . m

(1.4.6)

. Then, A.ı/ .u, u/ D A.u, u/ C Also put A.u, u/ D lim˛!1 ˛.u  ˛G˛ u, u/b m . By virtue of Theorem 1.1.4, if u 2 F and ub hı 2 F , then u 2 G .ı/ . For ı.u, u/b m .ı/ u, v 2 G , put   A.ı/ .u, v/ D lim ˛ u  ˛G˛.ı/ u, v b m ˛!1

if the limit exists. Let AN .ı/,˛ be an approximating form of the symmetric part AN .ı/ of A.ı/ given by     ˛  AN .ı/,˛ .u, v/ D C v  ˛G˛.ı/ v, u u  ˛G˛.ı/ u, v m m b b 2 Z Z ˛2 D .u.x/  u.y//.v.x/  v.y//hı .x/G˛.ı/ .dxdy/ 2 X X Z ˛ uv.x/.1  ˛G˛.ı/ 1/.x/hı .x/m.dx/ C 2 X Z ˛ b .ı/ ˛ 1/.x/hı .x/m.dx/, C uv.x/.1  ˛ G (1.4.7) 2 X .ı/

where G˛ .dxdy/ D G˛Cı .dxdy/. This clearly implies that A.ı/ .u, u/  0. Furthermore, for any u, v 2 G .ı/ , since   AN .ı/,˛ .u C v, u C v/  2 A.ı/,˛ .u, u/ C A.ı/,˛ .v, v/ , it holds that u C v 2 G .ı/ and jAN .ı/ .u, v/j  A.ı/ .u, u/1=2 A.ı/ .v, v/1=2 .

(1.4.8)

In particular the following triangle inequality holds: A.ı/ .u C v, u C v/1=2  A.ı/ .u, u/1=2 C A.ı/ .v, v/1=2 . Furthermore, for u, v 2 G .ı/ , if limn,m!1 A.ı/ .un , vm / exists independently of the approximating sequences ¹un º and ¹vm º of u and v respectively, then we put .ı/ b .ı/ A.ı/ .u, v/ D limn,m!1 A.ı/ .un , vm /. The sub-Markov resolvents ¹G˛ º and ¹G ˛ º .ı/ .ı/ 2 b / in the following sense. are associated with .A , G / on L .X ; m

25

Section 1.4 An auxiliary bilinear form

.ı/ b .ı/ Lemma 1.4.4. If ˛ C ı > ˛0 and f 2 L2 .X ; m b /, then G˛ f and G ˛ f belong to G .ı/ and satisfy .ı/ .ı/ b .ı/ A.ı/ ˛ .G˛ f , v/ D A˛ .v, G ˛ f / D .f , v/b m

(1.4.9)

.ı/

for all v 2 G .ı/ , where A˛ .u, v/ D A.ı/ .u, v/ C ˛.u, v/b . m .ı/

b /, G˛ f 2 L2 .X ; m b /. By virtue of the resolvent equaProof. For any f 2 L2 .X ; m 2 b/, tion, for any v 2 L .X ; m .ı/

.ı/

.ı/

ˇ.G˛.ı/ f  ˇGˇ G˛.ı/ f , v/b D ˇ.Gˇ f  ˛Gˇ G˛.ı/ f , v/b . m m .ı/

b/ by Lemma 1.4.3, the right-hand Since ¹Gˇ º is strongly continuous on L2 .X ; m .ı/

side converges to .f  ˛G˛ f , v/b as ˇ tends to infinity. This implies that m .ı/

.ı/

.ı/

A˛ .G˛ f , v/ D .f , v/b . Furthermore, since this holds for v D G˛ f , it follows m .ı/ b .ı/ that G˛ f 2 G .ı/ . The assertion concerning G ˛ f follows similarly.

As we have seen in Lemma 1.4.4, G˛Cı f 2 G .ı/ for any ˛ C ı > ˛0 and f 2 To show the existence of other functions belonging to G .ı/ , we introduce the following condition .E.5/:

L2 .X ; m/.

.E.5/: If some Borel versions of u, v 2 F and w 2 L2 .X ; m/ satisfy jw.x/w.y/j  ju.x/  u.y/j C jv.x/  v.y/j and jw.x/j  ju.x/j C jv.x/j, then w 2 F and jE.w, w/j  K0 .E˛0 .u, u/ C E˛0 .v, v// for some constant K0 depending on kuk1 and kvk1 . Lemma 1.4.5. Under the assumption .E.5/, the following results hold. hı 2 F is bounded on the support of (i) If u, v 2 Fb , then uv 2 F . In particular, if b u 2 Fb , then u 2 G .ı/ . (ii) If u, v 2 Fb and v is bounded from below by a positive constant on the support of u, then uv 2 F . Proof. The first assertion of (i) is clear from .E.5/. If b hı 2 F is bounded by a constant b b hı ^ K 2 Fb . K on the support of u, then uhı D u  .hı ^ K/ 2 F , because u, b (ii) If u 2 Fb and v 2 F is bounded from below by a positive constant k on suppŒu, then for any x, y 2 suppŒu, w D u=v satisfies jw.x/  w.y/j 

kuk1 1 jv.x/  v.y/j ju.x/  u.y/j C k k2

and jw.x/j  k1 ju.x/jC.kuk1 =k 2 /jv.x/j. These inequalities also hold for x, y which do not belong to suppŒu. Since u=k, .kuk1 =k 2 /v 2 F , .E.5/ yields that w 2 F .

26

Chapter 1 Dirichlet forms

Assume that, for any compact set F , there exists a function b hF 2 F such that b hF on F and b hF is bounded from below by a positive constant on F . If .E.5/ hı D b hı is bounded from below by a positive holds then, for any u 2 F \ C0 .X /, since b b hı 2 G .ı/ and constant on the support of u, u=hı 2 F by Lemma 1.4.5. Hence u=b A.u=b hı , v=b hı / D E.u=b hı , v/ for any u, v 2 F \ C0 .X /. An important class of Dirichlet form .E, F / satisfying .E.5/ is a Dirichlet form possessing the reference form .Q, F /. A symmetric Dirichlet form .Q, F / on L2 .X ; m/ is called a reference form of .E, F / if, for each fixed ˛ > ˛0 , there exist positive constants 1 , 2 such that 1 Q1 .u, u/  E˛ .u, u/  2 Q1 .u, u/,

for all u 2 F .

(1.4.10)

In fact, for the symmetric Dirichlet form .Q, F /, .E.5/ is satisfied as we noted at the end of Section 1.2. Theorem 1.4.6. Assume .E.5/. Suppose that ¹un º is a sequence of uniformly bounded functions of F supported by a Borel set B on which b hı coincides with a function of Fb . If ¹un º is an E˛ -Cauchy sequence for ˛ > ˛0 , then there exists a function u 2 F \ G .ı/ such that limn!1 A.ı/ .un  u, un  u/ D 0. Proof. By the assumption, ¹un º is a Cauchy sequence in .E, F /. Hence there exists a function u 2 F such that limn!1 un D u relative to E˛ . Clearly ¹un º and u are uniformly bounded and supported by B  ¹x : b hı .x/  C º for some C < 1. By the assumption, we may assume that there exists a bounded function w b 2 F such that b hı D w b on B. In particular u 2 L2 .X ; m b / because Z Z 2 u dm b  kb w k1 u2 d m < 1. X

X

hı D ub w 2 F by .E.5/. Therefore A.ı/ .u, u/ D Since u, w b 2 Fb , it follows that ub E.u, ub hı / exists and hence u 2 G .ı/ . By .E.5/, since   jE.uv, uv/j  K0 kvk21 E˛0 .u, u/ C kuk21 E˛0 .v, v/ , for all u, v 2 Fb , ˇ  ˇˇ ˇ b hı .un  u/ ˇ D jE .b b.un  u//j w .un  u/, w ˇE hı .un  u/, b   2  K0 kb w k1 E˛0 .un  u, un  u/ C kun  uk21 E˛0 .b w, w b/ is uniformly bounded relative to n. Hence hı .un  u//j lim jA.un  u, un  u/j D lim jE.un  u, b n!1  1=2 hı .un  u/, b hı .un  u/ D 0.  lim E˛0 .un  u.un  u/1=2 E˛0 b

n!1

n!1

Section 1.4 An auxiliary bilinear form

27

In the last part of Section 1.3, we defined the extended Dirichlet form .E, Fe / of .E, F /. As we mentioned there, the existence of E.u, u/ for u 2 Fe is not sure unless h0 , ˛0 D 0. If ˛0 > 0 and .E, F / is transient, then by using a 0-coexcessive function b .0/ .0/ .0/ define an extended form .A , Ge / similarly to .E, Fe /, that is, u 2 Ge if there exists an A.0/ -Cauchy sequence ¹un º  G .0/ such that limn!1 un D u a.e. Since A.0/ is non-negative, for any A.0/ -Cauchy sequence ¹un º, limn!1 A.0/ .un , un / D A.0/ .u, u/ exists. Theorem 1.4.7. Suppose that .E, F / is a transient Dirichlet form satisfying .E.5/. .0/ For any measurable function f such that .jf j, Gjf j/ < 1, Gf belongs to Ge . If h0 2 F \ f 2 L1 .X ; m/ additionally and v is a measurable function such that vb L1 .X ; m/, then . (1.4.11) A.0/ .Gf , v/ D .f , v/b m b/ \ Proof. We may assume that f  0. Assume furthermore that f 2 L1 .X ; m P 1 1 1 k kC1 1 b/ and Gf 2 L .X ; m b /. Since kD0 ˛ G˛ f D Gf 2 L .X ; m b /, L .X ; m k G k f D 0 in L1 .X ; m ˛ b /. If we consider the partial sum G f D lim 0,n k!1 ˛ Pn Pn k G kC1 f , then G f D G f 2 F \ G .0/ for f D k Gk f 2 ˛ ˛ 0,n ˛ n n ˛ ˛ kD0 kD0 b / with ˛ > ˛0 . Hence L2 .X ; m A.0/ .G0,n f , v/ D E.G˛ fn , vb h0 / D .fn  ˛G˛ fn , v/b m   nC1 nC1 D f ˛ G˛ f , v b m

(1.4.12)

for any v such that vb h0 2 F \L1 .X ; m/. Hence limn!1 A.0/ .G˛,n f , v/ D .f , v/b . m .0/

For the proof of Gf 2 Ge , put v D G0,n f  G0,m f in equation (1.4.12). Then we obtain for n  m that A.0/ .G0,n f  G0,m f , G0,n f  G0,m f /   D ˛ mC1 G˛mC1 f  ˛ nC1 G˛nC1 f , G0,n f  G0,m f b m   mC1 b .0/  f , ˛ mC1 .G / .G f  G f / . 0,n 0,m ˛ m b

mC1 .G f  G b .0/ m, f i < 1 and Since k˛ mC1 .G 0,n 0,m f /k1  kGf k1 < 1, hb ˛ / mC1 .G f G b .0/ limm!1 limn!1 ˛ mC1 .G / f / D 0, we obtain by the Lebesgue 0,n 0,m ˛ theorem that the right-hand side of the above inequality converges to zero as m, n ! b -a.e. 1. Hence ¹G0,n f º  G .0/ is an A.0/ -Cauchy sequence converging to Gf m .0/ Therefore Gf 2 Ge . Equation (1.4.11) is clear from equation (1.4.12). < 1, put fn D f ^ .ng/ for For any non-negative function f such that .f , Gf /b m b / \ L1 .X ; m b / such that kGgk1 < 1. Exisa strictly positive function g 2 L1 .X ; m tence of such a function g is a consequence of the transience. Since limn!1 Gfn D

28

Chapter 1 Dirichlet forms

Gf and .fn , Gfn /b  .f , Gf /b < 1, m m lim A.0/ .Gfn  Gfk , Gfn  Gfk / D

k,n!1

.0/

Hence Gf 2 Ge

lim .fn  fk , Gfn  Gfk /b D 0. m

k,n!1

and D .f , v/b A.0/ .Gf , v/ D lim .fn , v/b m m n!1

for any f 2 L1 .X ; m b / and vb h0 2 F \ L1 .X ; m/. By the lack of the sector condition of A.ı/ , it is not clear that any function of G .ı/ can be approximated by the functions of the form G˛ f with f 2 L2 .X ; m b/. To make such an approximation possible for a wide class of functions of G .ı/ , we introduce the following condition .E.6/ which is weaker than the sector condition of A.ı/ . hı 2 F , there exists a .E.6/ For some fixed ˛ > ˛0 and ı-coexcessive function b constant K1 depending on K in equation (1.1.2) and a constant K2 depending hı , b hı / such that on E˛ .b   1=2 jA.ı/ .u, v/j  K1 K2 kvk1 C A.ı/ ˛0 .v, v/   (1.4.13)  E˛ .u, u/1=2 C A.ı/ .u, u/1=2 for all u, v 2 Fb \ G .ı/ . Since A.ı/ .u, u/  0, its symmetric part A.ı,s/ satisfies the triangle inequality. In particular, limn!1 A.ı,s/ .un  u, v/ D 0 if limn!1 A.ı/ .un  u, un  u/ D 0. .ı/ Furthermore, under the condition .E.6/, limn!1 A.ı/ .un u, v/ D 0 for all v 2 Gb if limn!1 un D u relative to E˛ CA.ı/ . Hence it also holds that limn!1 A.ı/ .v, un  u/ D 0. Theorem 1.4.8. Suppose that .A.ı/ , G .ı/ / satisfies .E.5/ and .E.6/ for some ˛ > hı 2 F . If a sequence ¹un º of uniformly bounded ˛0 and ı-coexcessive function b .ı/ functions of F \ G is Cauchy relative to A.ı/ C E˛ , then there exists u 2 F \ G .ı/ such that limn!1 un D u relative to A.ı/ C E˛ . b /º is A.ı/ -dense in Fb \ G .ı/ . In particular, ¹G˛ f : f 2 L2 .X ; m C m .ı/

Proof. For u, v 2 G .ı/ , let A.ı/,ˇ .u, v/ D ˇ.u  ˇGˇ u, v/b be an approximating m .ı/ .ı/,ˇ .ı/,ˇ N .u, u/ coincides with A .u, u/ given by equation form of A .u, v/. Then A (1.4.7). Similar to equation (1.1.8), we can see that b .ı/ u, ˇ G b .ı/ u/ D ˇ.ˇ G b .ı/ u, u  ˇ G b .ı/ u/ A.ı/ .ˇ G ˇ ˇ ˇ ˇ m b b .ı/ u/ D A.ı/,ˇ .u, u/  ˇ.u, u  ˇ G ˇ b m

(1.4.14)

Section 1.4 An auxiliary bilinear form

29

b .ı/ uk1  kuk1 < 1 and hence, by for all u 2 G .ı/ . If u 2 Fb \ G .ı/ , then kˇ G ˇ .E.6/ and equation (1.4.14), b u/ A.ı/,ˇ .u, u/ D A.ı/ .u, ˇ G ˇ   b .ı/ u, ˇ G b .ı/ u/1=2  K1 K2 kuk1 C A.ı/ .ˇ G ˇ ˇ    E˛ .u, u/1=2 C A.ı/ .u, u/1=2     K1 K2 kuk1 C A.ı/,ˇ .u, u/1=2 E˛ .u, u/1=2 C A.ı/ .u, u/1=2 . .ı/

From this we obtain similarly to equation (1.1.10) that   A.ı/,ˇ .u, u/  K3 E˛ .u, u/ C A.ı/ .u, u/   CK4 E˛ .u, u/1=2 C A.ı/ .u, u/1=2

(1.4.15)

for some constants K3 and K4 depending on K1 , K2 and kuk1 . Therefore, the corresponding inequalities to equation (1.1.8) and equation (1.1.10) hold for A.ı/ . Suppose that ¹un º  F \ G .ı/ is an E˛ C A.ı/ -Cauchy sequence of uniformly bounded functions. Then there exists a function u 2 F such that limn!1 un D u b /. By equation (1.4.15), relative to E˛ and in L2 .X ; m C m lim A.ı/,ˇ .un  u, un  u/    lim lim K3 A.ı/ .un  um , un  um / C K4 A.ı/ .un  um , un  um /1=2

n!1

n!1 m!1

D 0, uniformly relative to ˇ. In particular, for any  > 0, by taking large n such that A.ı/,ˇ .un  u, un  u/ <  for any ˇ, it holds that limˇ !1 jA.ı/,ˇ .u, u/1=2  p A.ı/,ˇ .un , un /j <  which yields that u 2 F \ G .ı/ and limn!1 un D u relative to A.ı/ . .ı/ .ı/ If u 2 F \ G .ı/ , then similarly to equation (1.4.14), A.ı/ .ˇGˇ u, ˇGˇ u/  A.ı/,ˇ .u, u/. Hence, ¹ˇG .ı/ uº is uniformly E˛ C A.ı/ -bounded relative to ˇ and b /. Hence, as in the proof of Theorem 1.1.4 (iii), Cesàro converges to u in L2 .X ; m C m .ı/ means of ¹ˇGˇ uº contain a subsequence converging to u relative to A.ı/ . Since each .ı/

function ˇGˇ u can be written as G˛ un with un 2 L2 .X ; m C m b/, the last assertion holds.

30

Chapter 1 Dirichlet forms

1.5 Examples 1.5.1 Diffusion case   Let D be a domain of the d -dimensional Euclidean space Rd and aij .x/ i,j D1,:::,d be a symmetric family of locally bounded measurable functions on D satisfying d X

aij .x/i j  

i,j D1

d X

i2

(1.5.1)

iD1

for all .1 ,    , d / 2 Rd and a positive constant . Let m be a measure on D such that m.dx/ D m.x/dx for a measurable function m.x/ satisfying 0 < inf¹m.x/ : x 2 F º  sup¹m.x/ : x 2 F º < 1 for any compact set F  D. For a non-negative locally bounded measurable function c, define the symmetric form Qc .u, v/ for u, v 2 C01 .D/ by c

Q .u, v/ D

d Z X i,j D1

@u @v aij .x/ dx C @xi @xj D

Z c.x/u.x/v.x/m.dx/.

(1.5.2)

D

Lemma 1.5.1. The symmetric form .Qc , C01 .D// is closable on L2 .D; m/. Proof. Suppose that ¹un º  C01 .D/ is a Q1c -Cauchy sequence converging to zero in L2 .D; m/. Then limn!1 un D 0 in L2 .D; c  m/. Since ¹@un =@xi º is a Cauchy sequence in L2 .D; dx/, it converges to some 'i in L2 .D; dx/. To show the assertion of the lemma, it is enough to show that 'i D 0 for all i . This follows from Z Z @un 'i .x/v.x/dx D lim v.x/dx n!1 D @xi D Z @v D  lim un .x/ dx n!1 D @xi Z 1 @v D  lim un .x/ m.dx/ D 0 n!1 D m.x/ @xi for all v 2 C01 .D/. Clearly .Qc , C01 .D// satisfies the Markov property .E.4/. Hence its smallest closed extension .Qc , F / defines a Dirichlet form on L2 .D; m/. Any function u 2 F has a distribution sense derivative @u=@xi 2 L2 .D; dx/, 1  i  d .

31

Section 1.5 Examples

Let E be the bilinear form defined by E.u, v/ D Qc .u, v/ C

d Z X iD1 D

bi .x/

@u v.x/dx @xi

(1.5.3)

for all u, v 2 C01 .D/. Put A D .aij / and assume that one of the following conditions holds for b D .b1 , b2 ,    , bd /. (i) .b, A1 b/.x/  k1 m.x/ for some positive constant k1 . Pd (ii) d  3, r  b D iD1 @bi =@xi  k2 m.x/ for some positive constant k2 and kbkd < 1, where k  kp is the Lp .D; dx/-norm.

Theorem 1.5.2. Assume that b satisfies one of the conditions (i) or (ii) stated above. Then there exists a constant ˛0  0 such that .E.1/ and .E.2/ hold for C01 .D/. If (i) holds, then for any ˛ > ˛0 , there also exist positive constants 0 < 1  2 < 1 such that (1.5.4) 1 Q1c .u, u/  E˛ .u, u/  2 Q1c .u, u/ for all u 2 C01 .D/. If (ii) holds, then the symmetric part E .s/ .u, v/ is a symmetric Dirichlet form.

Proof. Suppose that condition (i) is satisfied. For any u 2 C01 .D/,

  D 

d Z X

@u u.x/dx @xi iD1 D 1=2 Z Qc .u, u/  Qc .u, u/1=2 .b, A1 b/u2 dx D   Z 1 Qc .u, u/  2 Qc .u, u/ C .b, A1 b/u2 dx  D Z 2 .1  2/ Qc .u, u/  .b, A1 b/u2 dx  D ı0 Qc .u, u/  ı1 kuk2

E.u, u/ D Qc .u, u/ C

bi .x/

for ı0 D 1  2 and ı1 D 2k1 = with any fixed 0 <  < 1=2. Hence, if we let ˛0 D ı0 C ı1 and 1 D ı0 , E˛ .u, u/  ı0 Q1c .u, u/ C .˛  ı0  ı1 /.u, u/  1 Q1c .u, u/ for ˛  ˛0 .

32

Chapter 1 Dirichlet forms

p p p p Since p C q  2 p C q and Qk1 .u, u/  .k1 _ 1/Q1 .u, u/, for any u, v 2 C01 .D/ and ˛  ˛0 , 1=2 Z jE.u, v/j  Qc .u, u/1=2 Qc .v, v/1=2 C Qc .u, u/1=2 .b, A1 b/v 2 dx D    p p c 1=2 1=2 c 1=2 Q .v, v/ C k1 .v, v/1=2  Q .u, u/ C k1 .u, u/  2Qkc 1 .u, u/1=2 Qkc 1 .v, v/1=2 2  .k1 _ 1/E˛ .u, u/1=2 E˛ .v, v/1=2 . 1 Therefore, .E.1/ and .E.2/ hold for ˛0 and K D 2.k1 _ 1/=1 . By putting v D u in the above inequality, it holds for ˛ > ˛0 that E˛ .u, u/  2Qkc 1 C˛=2 .u, u/  2 ..k1 C ˛=2/ _ 1/ Q1c .u, u/  2 Q1c .u, u/ for 2 D 2 ..k1 C ˛=2/ _ 1/. Next assume that (ii) is satisfied. For any u 2 C01 .D/, since Z 1 k2 c E.u, u/ D Q .u, u/  r  bu2 dx  Qc .u, u/  kuk2 , 2 D 2 .E.1/ holds for ˛0 D k2 =2. On the other hand, by the Hölder inequality, for a positive number p satisfying 1=p C 1=d D 1=2, kb  .ru/vk  kruk  kbvk  kruk  kbkd  kvkp . Since kvkp  k3 krvk for a constant k3 depending on d by the Sobolev inequality, we arrive to the estimate jE.u, v/j  Qc .u, u/1=2 Qc .v, v/1=2 C k3 kruk  krvk  kbkd  .1 C k3 kbkd =/ Qc .u, u/1=2 Qc .v, v/1=2  KE˛0 .u, u/1=2 E˛0 .v, v/1=2 for K D .1 C k3 kbkd =/. This yields .E.2/. By equation (1.5.4), F is E˛0 -closure of C01 and hence .E.3/ holds for .E, F / by Theorem 1.2.1. Furthermore, .E.1/, .E.2/ and equation (1.5.4) also hold for u, v 2 F . Theorem 1.5.3. Assume that either (i) or (ii) stated in front of Theorem 1.5.2 is satisfied. Then .E, F / is a Dirichlet form on L2 .D; m/. It also has the property .E.5/ under condition (i). Furthermore, if c  r  b is bounded from below, there exists a constant   0 such that .E, F / satisfies the dual Markov property given by

33

Section 1.5 Examples

equation (1.4.3), that is, E .u  u ^ a, u ^ a/  0 for any u 2 F and a non-negative constant a. Proof. To show that .E, F / is a Dirichlet form, it remains only to prove the subMarkov property .E.4/. For any non-negative constant a, by putting Da D ¹x 2 D : u.x/ < aº, we have d Z X @u c bi .x/ u.x/dx E.u ^ a, u ^ a/ D Q .u ^ a, u ^ a/ C @x i iD1 Da  E.u ^ a, u/. This implies the sub-Markov property .E.4/ of .E, F /. Similarly Z b.x/  r.u  u ^ a/.u ^ a/dx Z c.x/.u  u ^ a/.u ^ a/dx C ZD .c.x/  r  b.x//.u  a/dx. Da

E.u  u ^ a, u ^ a/ D

D

D\¹uaº

Since c  r  b is bounded from below, the right-hand side is greater than ı u ^ a/.u ^ a/dx for ı D kc  r  bk1 which gives equation (1.4.3).

R

D .u 

Corollary 1.5.4. If (i) or (ii) stated in front of Theorem 1.5.2 holds, then the resolvent ¹G˛ º satisfies 0  ˛G˛ f  1 for any f 2 L2 .D; m/ such that 0  f  1. Furthermore, if c  r  b is bounded from below, then there exists a constant   0 b ˛C f  1 for any f 2 L2 .D; m/ b ˛ º satisfies 0  ˛ G such that the coresolvent ¹G such that 0  f  1. Fixing a strictly positive function 2 F , define a symmetric form E ./ by d Z X @u @v E ./ .u, v/ D aij .x/ .x/dx (1.5.5) @x i @xj D i,j D1

C01 .D/.

for u, v 2 Then .E ./ , C01 .D; m// is closable on L2 .D;  ./ an E -Cauchy sequence ¹un º  C01 .D/ converging to 0 in L2 .D n i exists in L2 .D;  dx/. Then limn!1 @u @xi Z

Z

m/. In fact, if :  m/, then

@un

i .x/v.x/ .x/dx D lim v.x/ .x/dx n!1 D D dxi  Z Z @v @ un .x/ .x/dx C un .x/v.x/ dx D  lim n!1 @xi @xi D Z D @ D  lim un .x/v.x/ dx. n!1 D @xi

34

Chapter 1 Dirichlet forms

In particular, if we take v as a function given by v.x/ D  . .x// for any smooth function  .t / supported by Œ, M  for some 0 <  < M < 1, the absolute value of the last term is dominated by ˇ ˇ Z ˇ @ ˇˇ ˇ lim un .x/v.x/ dx n!1 ¹.x/M º ˇ @xi ˇ  Z 1=2 Z   @ 2 1 u2n .x/ .x/dx /1=2  dx  k k1 lim n!1 .x/ D ¹.x/M º @xi 1=2 Z  Z 2 1=2 k k1 @  u2n .x/ .x/dx dx D 0. lim n!1  D D @xi This implies that i .x/ D 0 on ¹x :  < .x/ < M º and hence for a.e. m. We shall denote the symmetric Dirichlet form determined by the closure of .E ./ , C01 .D// on L2 .D;  m/ by .E ./ , F ./ /. hı 2 F be a strictly positive m-integrable ı-coexcessive function For ı > ˛0 , let b which is bounded from below by a positive constant on each compact set. As we shall exists and there exists a positive see by equation (2.4.22), such a function b hı always R hı / D X w.x/db ı for all w 2 C0 .X / \ F . Radon measure b ı such that Eı .w, b b In the above result, by putting D hı , we can define a symmetric Dirichlet form hı / , F .b hı / / on L2 .X ; m .E .b b / for m b Db hı  m. For any u, v 2 C01 .D/, since vb hı and 2 b b r.v hı / belongs to L .D; dx/, v hı 2 F and the bilinear form A.u, v/ is well defined by A.u, v/ D E.u, vb hı / and satisfies d Z X @u @b hı b A.u, v/ D E .hı / .u, v/ C aij .x/ v.x/dx @x @x i j D

C Since Eı .w, b hı / D .ı/

A

i,j D1

d Z X

R X

bi .x/

iD1 D

@u v.x/b hı .x/dx C @xi

Z

cuv.x/b hı .x/dx.

D

wdb ı and A.ı/ D Aı , we have

Z 1 1 2 b hı dx .u, u/ D E .u, u/ C Eı .u , hı / C .c C ı/u2b 2 2 D Z Z 1 1 b u2 db ı C .c C ı/u2 d m b. D E .hı / .u, u/ C 2 D 2 D .b hı /

(1.5.6)

hı / \L2 .D; b For any bounded function u 2 F .b ı C.cCı/b m/, there exists a uniformly 1 hı / \ L2 .D; b bounded sequence ¹u º  C .D/ such that lim u D u in F .b  C n

0

n!1

n

ı

.c C ı/  m b /. Hence the above inequality implies that ¹un º also converges to u relative .ı/ to A . Then, under the condition .E.6/, Theorem 1.4.8 implies that u belongs to

35

Section 1.5 Examples

G .ı/ . On the other hand, since C0 .D/ is dense in F , we have b ı C .c C ı/  m b/. G .ı/ \ Fb D Fb \ F .hı / \ L2 .D; b In particular, C0 .D/  F \ G .ı/ . Theorem 1.5.5. If b satisfies condition (i) in front of Theorem 1.5.2, then .E, F / satisfies .E.6/. Proof. It is enough to show equation (1.4.13) for any u, v 2 C01 .D/. Since .b h / Eı=2ı .u, u/  A.ı/ .u, u/ by equation (1.5.6), using equations (1.5.4) and (1.5.6), we have ˇ d Z ˇ ˇ X ˇ @u @b hı .ı/ .b hı / 1=2 .b hı / 1=2 ˇ .u, u/ E .v, v/ C ˇ aij .x/ v.x/dx ˇˇ jA .u, v/j  E @xi @xj i,j D1 D ˇ ˇ ˇZ  ˇ Z  ˇ ˇ ˇ ˇ 1=2 1=2 b b ˇ ˇ ˇ A b, A ru .x/v.x/hı .x/dx ˇ C ˇ .c C ı/uv hı d mˇˇ Cˇ D







D

.b hı /

.u, u/ E .v, v/ C kvk1 Q .u, u/ Qc .b hı , b hı /1=2 Z 1=2 Z 1=2 .b, A1 b/ 2b v hı d m C .ru, Aru/b hı dx C ı.u, v/b m m D D b b E .hı / .u, u/1=2 E .hı / .v, v/1=2 C kvk1 Qc .u, u/1=2 Qc .b hı , b hı /1=2  Z 1=2 p b 1=2 1=2 C k1 v2d m b E .hı / .u, u/1=2 C ı.u, u/ .v, v/ b m b m D   b 1=2 E .hı / .u, u/1=2 C Qc .b hı , b hı / C ı.u, u/ b m   p p b 1=2 hı , b hı /1=2 C . k1 C ı/.v, v/  E .hı / .v, v/1=2 C kvk1 Qc .b b m   1 .b h / 2Eı=2ı .u, u/1=2 C E˛ .u, u/1=2 1   .b hı / 1=2 1=2 b b  E .v, v/ C kvk1 Q.hı , hı /    K1 A.ı/ .u, u/1=2 C E˛ .u, u/1=2 A.ı/ .v, v/1=2 C K2 kvk1 ,

E



.b hı /

1=2

1=2

c

1=2

p p where D . ı C k1 /2 , K1 D .2 C .1=1 //  ..2 =ı/ _ 1/1=2 and K2 D hı , b hı /1=2 ..2 =ı/ _ 1/1=2 . Q1c .b

36

Chapter 1 Dirichlet forms

1.5.2 Jump type case Let X be a separable metric space and m a positive Radon measure on X . Consider a non-negative measurable function j.x, y/ on X  X n d for d D ¹.x, x/ : x 2 X º. Following [61], we assume that j.x, y/ satisfies Z 2 .1 ^ d 2 .x, y//j .s/ .x, y/m.dy/, (1.5.7) Ms 2 Lloc for Ms .x/ D y¤x

where j .s/ .x, y/ D 12 .j.x, y/ C j.y, x// and d.x, y/ is a metric on X . We also put j .a/ .x, y/ D 12 .j.x, y/  j.y, x// and assume that j .a/ satisfies for some  2 .0, 1 and a constant C Z jj .a/ .x, y/jm.dy/ < 1, (1.5.8) c1 D sup x2X d.x,y/1 Z c2 D sup jj .a/ .x, y/j m.dy/ < 1 (1.5.9) x2X

d.x,y/ 0º and let x be the x-section of . Then, for m-a.e. x, w.x, y/ D lim .unk .x/  unk .y// D lim unk .x/ k!1

k!1

for m-a.e. y 2 x . Therefore w.x, y/ D 0 a.e. relative to j.x, y/m.dx/m.dy/. This implies that limk!1 Q.unk , unk / D 0. Since ¹un º is Q-Cauchy, we have limn!1 Q.un , un / D 0 which implies the closedness of .Q, F / on L2 .X ; m/. 1

A weaker condition yielding the following results is given in [143].

37

Section 1.5 Examples lip

Clearly the family C0 .X / of all Lipschitz continuous functions with compact suplip port is a subspace of F . We denote by F0 the Q1 -closure of C0 .X /. Then .Q, F0 / is a symmetric regular Dirichlet form on L2 .X ; m/ in view of Theorem 1.2.2. Further, lip lip the bilinear form E on C0 .X /  C0 .X / is well defined by “ E.u, v/ D lim

n!1

¹d.x,y/>1=nº

.u.x/  u.y//v.x/j.x, y/m.dx/m.dy/

(1.5.11)

and the following results hold.2 For any u, v 2 C lip .X /, 1 E.u, v/ D Q.u, v/C 2

“ .u.x/u.y//v.y/j .a/ .x, y/m.dx/m.dy/ (1.5.12) XXnd lip

and, for some constant C independent of u, v 2 C0 .X /, “

ˇ ˇ p ˇ ˇ ˇ.u.x/  u.y//v.y/j .a/ .x, y/ˇ m.dx/m.dy/  C kvk Q.u, u/.

XXnd lip

Accordingly we have for any u, v 2 C0 .X / p 1 1 3˛ Q˛ .u, u/ C Q.u, u/ C kuk2  C kuk Q.u, u/ 4 4 4 1  Q˛ .u, u/ 4

E˛ .u, u/ 

for ˛  4C 2 , and moreover p 1 jQ.u, v/j C C kvk Q.u, u/ 2 p 1 p Q.v, v/ C 2C kvk Q.u, u/  2p p 2p Q˛ .v, v/ E˛ .u, u/.  2

jE.u, v/j 

Therefore E extends to a bilinear form on F0  F0 so that .E, F0 / is a closed form on L2 .X ; m/.

2

See [61].

38

Chapter 1 Dirichlet forms lip

To show its Markov property, put U.u/ D uC ^ 1. Then, for any u 2 C0 .X /, E.U.u/, u  U.u// “ D lim n!1

¹d.x,y/>1=nº

.U.u/.x/  U.u/.y//

 .u.x/  U.u/.x// j.x, y/m.dx/m.dy/ “ .1  U.u/.y// .u.x/  1/ j.x, y/m.dx/m.dy/ D lim n!1

¹d.x,y/>1=nº\¹u.x/1º

“

 lim

n!1

¹d.x,y/>1=nº\¹u.x/1=n

40

Chapter 1 Dirichlet forms

it holds that

b A.u, v/ C A.v, u/ D E .hı / .u, v/ C E.uv, b hı /.

hı / D Note that the second term of the right-hand side can be written as Eı .uv, b R .ı/ D b b uvdb  for a measure b  such that h D U b  . In particular, since A ı ı ı ı ı X satisfies A C ı. , /b m Z 1 .b 1 hı / .ı/ u2 .x/b ı .dx/, (1.5.16) A .u, u/ D Eı .u, u/ C 2 2 X .b h / it holds that Eı ı .u, u/  2A.ı/ .u, u/. We also have the expression that

A.u, v/  A.v, u/ D “ lim n!1

  u.x/v.y/ b hı .x/j.x, y/  b hı .y/j.y, x/ m.dx/m.dy/.

d.x,y/>1=n

Therefore, 2A.u, v/ D .A.u, v/ C A.v, u// C .A.u, v/  A.v, u// “ b D E .hı / .u, v/ C lim .u.x/  u.y// n!1

d.x,y/>1=n

   v.x/ b hı .x/  b hı .y/ j.x, y/m.dx/m.dy/  “ .a/ b C2 .u.x/  u.y// v.x/hı .y/j .x, y/m.dx/m.dy/ XXnd

b E .hı / .u, v/ C I C 2 II. In the right-hand side, hı , b hı /1=2 jIj  kvk1 Q.u, u/1=2 Q.b  4kvk1 E˛ .u, u/1=2 Q.b hı , b hı /1=2 . For any n  1, by putting ƒn D ¹.x, y/ : 1=n < d.x, y/  1º, we have ˇ ˇZ Z ˇ ˇ .a/ b ˇ .u.x/  u.y// v.x/hı .y/j .x, y/jm.dx/m.dy/ˇˇ ˇ ƒn

1=2

Z Z 

ƒn

.u.x/ 

hı .y/jj .a/ .x, y/j2 m.dx/m.dy/ u.y//2 b

Z Z

1=2 b hı .y/jj .a/ .x, y/j m.dx/m.dy/ kvk1 ƒn Z Z p b  C E .hı / .u, u/1=2 kvk1 hm, b hı i1=2 sup jj .a/ .x, y/j m.dx/. 

y

d.x,y/ ı, D 2Q.b hı , b hı /1=2 C2K3 and K1 D 2. Since C0 .X / is dense in Fb \G .ı/ , we obtain .E.6/.

As a typical example which satisfies the condition of this subsection, there is the stable-like process on Rd with the generator Z   w.x/ u.x C h/  u.x/  ru.x/  h1B1 .0/ .h/ dh, (1.5.17) Lu.x/ D jhjd C˛.x/ Rd for u 2 C02 .Rd /, where B1 .0/ is the ball of center 0 and radius 1 and w.x/ D d /. ˛.x/Cd / sin  ˛.x/ is a function chosen so that Le ihx D 2˛.x/1  2 1 . 1C˛.x/ 2 2 2 ˛.x/ ihx jhj e . For any n  1, since “ w.x/ ru.x/  h1B1 .0/ .h/ d C˛.x/ v.x/dhdx D 0, jhj jhj>1=n

42

Chapter 1 Dirichlet forms

it holds that “ 

jhj>1=n

  u.x C h/  u.x/  ru.x/  h1B1 .0/ .h/ v.x/

“

D

jhj>1=n

.u.x C h/  u.x// v.x/

“ D

jxyj>1=n

.u.y/  u.x// v.x/

w.x/ dhdx jhjd C˛.x/

w.x/ dhdx jhjd C˛.x/

w.x/ dxdy. jx  yjd C˛.x/

By letting n ! 1, the associated Dirichlet form is given by “ E.u, v/ D lim .u.x/  u.y// v.x/j.x, y/dxdy n!1

(1.5.18)

¹jxyj¤1=nº

for u, v 2 C01 .Rd / and j.x, y/ D w.x/=jx  yjd C˛.x/ . N M and ı satisfying 0 < ˛  ˛.x/  ˛N < 2, ˛N < 1C˛=2, If there are constants ˛, ˛, j˛.x/  ˛.y/  M jx  yjı and .1=2/.2˛N  ˛/ < ı < 1, then by using the estimates of w given by c1 ˛.x/.2  ˛.x//  w.x/  c2 .2  ˛.x// and jw.x/  w.y/j  c3 j˛.x/  ˛.y/j, we can obtain that   j .s/ .x, y/  M jx  yjd ˛N _ jx  yjd ˛ N .log jx  yj1 /1 . jj .a/ .x, y/j  M 0 jx  yjd ˛Cı

From this, equations (1.5.7), (1.5.8) and (1.5.9) hold for  satisfying .d C 2˛N  2ı  ˛/=.d C ˛N  ı/ <  < d=.d C ˛N  ı/.

Chapter 2

Some analytic properties of Dirichlet forms

In this chapter, we study the analytic potential theory related to Dirichlet forms. The capacity of any open set is defined by using the equilibrium and co-equilibrium potentials of the set and then extending to any Borel sets in Section 2.1. Some quasinotions related to this capacity are presented in Section 2.2. In particular, existence of a quasi-continuous version of a function belonging to the domain of the Dirichlet form is shown. The notion of potentials of measures is introduced in Section 2.3. In Section 2.4, we give a decomposition of the domain of the Dirichlet form into the orthogonal families of functions. Since our Dirichlet form is not symmetric, we need to consider the orthogonality relative to the Dirichlet form and its dual form separately. Some related notions concerning the auxiliary bilinear forms introduced in Section 1.4 are also studied in this section.

2.1

Capacity

In this chapter, we fix a regular Dirichlet form .E, F / on L2 .X ; m/. Definition 3. Let  be a non-empty closed convex subset of F . For a given function u 2 F and ˛ > ˛0 , a function v D ˛ .u/ 2  (resp. vO D O ˛ .u/ 2 ) is called an ˛-projection (resp. ˛-coprojection) of u on  if it satisfies O u  v/ O  0/ E˛ .u  v, w  v/  0 .resp. E˛ .w  v,

for all w 2 .

(2.1.1)

Intuitively, equation (2.1.1) means that the angle between u  v and w  v is obtuse. The existence and the uniqueness of the ˛-projection (resp. ˛-coprojection) is a consequence of Theorem 1.1.1 applied to J.w/ D E˛ .u, w/ (resp. J.w/ D E˛ .w, u/). For any Borel set A of X , let LA D ¹u 2 F : u  1 m-a.e. on Aº.

(2.1.2)

If LA ¤ ;, then it is a closed convex subset of F . Hence there exist a unique ˛˛ ˛ .0/ 2 LA and an ˛-coprojection b eA Db

˛LA .0/ 2 LA of 0 on projection eA˛ D L A LA satisfying E˛ .eA˛ , eA˛ /  E˛ .eA˛ , u/, ˛ ˛ ˛ E˛ .b eA ,b eA /  E˛ .u,b eA /

for all u 2 LA .

(2.1.3)

44

Chapter 2 Some analytic properties of Dirichlet forms

˛ We call eA˛ (resp.b eA ) the ˛-equilibrium potential (resp. ˛-coequilibrium potential ) of A. By .E.2/, for any u 2 LA ,

E˛ .eA˛ , eA˛ /  E˛ .eA˛ , u/  K˛ E˛ .eA˛ , eA˛ /1=2 E˛ .u, u/1=2 . ˛ . Hence, for all u 2 LA , A similar inequality also holds for b eA

E˛ .eA˛ , eA˛ /  K˛2 E˛ .u, u/,

˛ ˛ E˛ .b eA ,b eA /  K˛2 E˛ .u, u/.

(2.1.4)

˛ eA , Lemma 2.1.1. (i) If LA ¤ ;, then 0  eA˛  1, eA˛ D 1 a.e. on A and 0  b ˛ 1 b eA a.e. on A.

(ii)

If u 2 F and u D 1 m-a.e. on A, then ˛ ˛ E˛ .eA˛ , u/ D E˛ .eA˛ , eA˛ /, E˛ .u,b eA / D E˛ .eA˛ ,b eA /.

(2.1.5)

˛ ˛ eA ,b eA / D 0, then m.A/ D 0. (iii) If E˛ .eA˛ , eA˛ / D 0 or E˛ .b

(iv) If A  B, then

˛ eA˛  eB ,

˛ ˛ b eA b eB

m  a.e.

(2.1.6)

Proof. Since .eA˛ /C ^ 1 is ˛-excessive and .eA˛ /C ^ 1 2 LA , by using Theorem 1.4.1 and .E.4a/ we have E˛ ..eA˛ /C ^ 1  eA˛ , .eA˛ /C ^ 1  eA˛ / D E˛ ..eA˛ /C ^ 1, .eA˛ /C ^ 1  eA˛ / C E˛ .eA˛ , eA˛  .eA˛ /C ^ 1/  E˛ .eA˛ , eA˛  .eA˛ /C ^ 1/  0. Hence .eA˛ /C ^ 1 D eA˛ and hence eA˛ D 1 m-a.e. on A. By equation (2.1.3), for any function v 2 F such that v  0 a.e. on A, E˛ .eA˛ , v/  ˛ eA /  0. In particular, if v D 0 a.e. on A, then E˛ .eA˛ , v/ D 0 and 0 and E˛ .v,b ˛ eA / D 0. Hence, if u D 1 a.e. on A, then by putting v D eA˛  u, it holds that E˛ .v,b ˛ ˛ eA / D E˛ .eA˛ ,b eA /. E˛ .eA˛ , u/ D E˛ .eA˛ , eA˛ / and E˛ .u,b ˛ ˛ eB are ˛-coexcessive by Lemma Suppose that A  B are open sets. Since b eA and b ˛ ˛ ˛ ˛ ˛ eB is also ˛-coexcessive. Since b eA  b eA ^ b eB  0, it then holds from 1.4.2, b eA ^ b ˛ ˛ ˛ ˛ ˛ ˛ ˛ eA b eA ^b eB ,b eA ^b eB /  0. Furthermore, sinceb eA ^b eB 2 Theorem 1.4.1 (ii) that E˛ .b ˛ ˛ ˛ ˛ ˛ eA ^b eB ,b eA /  E˛ .b eA ,b eA / by equation (2.1.3). By these inequalities, we have LA , E˛ .b ˛ ˛ ˛ ˛ ˛ ˛ E˛ .b eA b eA ^b eB ,b eA b eA ^b eB / ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ D E˛ .b eA b eA ^b eB ,b eA /  E˛ .b eA b eA ^b eB ,b eA ^b eB /  0. ˛ ˛ ˛ ˛ ˛ ˛ Therefore b eA ^b eB D b eA which shows that b eA  b eB . The proof of eA˛  eB is similar.

45

Section 2.1 Capacity ˛ Since b eA 2 LA , it holds that ˛ ˛ ˛ 1=2 eA /  K˛ E˛ .eA˛ , eA˛ /1=2 E˛ .b eA ,b eA / . E˛ .eA˛ , eA˛ /  E˛ .eA˛ ,b

Hence ˛ ˛ E˛ .eA˛ , eA˛ /  K˛2 E˛ .b eA ,b eA /.

Also, since eA˛ 2 LA and eA˛ D 1 a.e on A, equations (2.1.4) and (2.1.5) yield that ˛ ˛ ˛ eA / D E˛ .b eA ,b eA /  K˛2 E˛ .eA˛ , eA˛ /. E˛ .eA˛ , eA˛ /  E˛ .eA˛ ,b

(2.1.7)

Definition 4. For any open set A of X , define the ˛-capacity Cap.˛/ .A/ of A by ´ ˛ eA / if LA ¤ ; E˛ .eA˛ ,b .˛/ Cap .A/ D 1 if LA D ;. Then equation (2.1.7) gives the following inequality: E˛ .eA˛ , eA˛ /  Cap.˛/ .A/  K˛2 E˛ .eA˛ , eA˛ /. Lemma 2.1.2. (i) (ii)

(2.1.8)

If A  B are open sets, then Cap.˛/ .A/  Cap.˛/ .B/.

If A and B are open sets, then Cap.˛/ .A [ B/  Cap.˛/ .A/ C Cap.˛/ .B/.

(iii) If ¹An º is an increasing sequence of open sets, then Cap.˛/ .[n An / D supn Cap.˛/ .An /. ˛ ˛ ˛ Proof. (i) is clear from equation (2.1.6), in fact since eA˛  eB andb eA b eB , it follows from Theorem 1.4.1 that ˛ ˛ ˛ ˛ eA /  E˛ .eA˛ ,b eB /  E˛ .eB ,b eB / D Cap.˛/ .B/. Cap.˛/ .A/ D E˛ .eA˛ ,b ˛ ˛ ˛ b eA Cb eB . To show it, note that Theorem (ii) We first show the inequality b eA[B ˛ ˛ ˛ ˛ ˛ ˛ Cb ˛/  eA Cb eB / is ˛-coexcessive andb eA[B b eA[B ^.b eA eB 1.4.1 (ii) yields thatb eA[B ^.b 0. Hence  ˛  ˛ ˛ ˛ ˛ ˛ ˛ eA[B b eA[B ^ .b eA Cb eB /,b eA[B ^ .b eA Cb eB /  0. E˛ b ˛ ˛ ˛ ^ .b eA Cb eB / 2 LA[B , Moreover, since b eA[B  ˛   ˛  ˛ ˛ ˛ ˛ eA[B ^ .b  E˛ b eA[B ,b . eA Cb eB /,b eA[B eA[B E˛ b

Therefore  ˛  ˛ ˛ ˛ ˛ ˛ ˛ ˛ eA[B ^ .b  0. eA Cb eB / b eA[B ,b eA[B ^ .b eA Cb eB / b eA[B E˛ b ˛ ˛ Cb ˛/ Db ˛ ˛ ˛ Cb ˛. ^ .b eA eB eA[B and hence b eA[B b eA eB This implies that b eA[B

46

Chapter 2 Some analytic properties of Dirichlet forms

For the proof of (ii), it is enough to assume that Cap.˛/ .A/ C Cap.˛/ .B/ < 1. ˛ ˛ Cb ˛ b ˛ D 1 on A [ B andb eA eB eA[B , combining the results shown above, Since eA[B we have ˛ ˛ ˛ eA / C E˛ .eB ,b eB / Cap.˛/ .A/ C Cap.˛/ .B/ D E˛ .eA˛ ,b ˛ ˛ ˛ D E˛ .eA[B ,b eA Cb eB / ˛ ˛  E˛ .eA[B ,b eA[B / D Cap.˛/ .A [ B/.

(iii) It remains only to show the inequality Cap.˛/ .[n An /  supn Cap.˛/ .An /. FureAn ,b eAn / is bounded and ther, we may assume that supn Cap.An / < 1. Then E˛ .b hence by choosing a subsequence if necessary, there exists a function eO 2 F such that ˛ eA D eO weakly in F and increasingly. Clearly eO 2 L[n An . Furthermore, limn!1 b n ˛ ˛ ˛ eA /  E˛ .b eA ,b eA /, for any u 2 L[n An , by letting n ! 1 in the inequality E˛ .u,b n n n ˛ O  E˛ .e, O e/. O This implies that eO D b e [n An . Therefore we have E˛ .u, e/ ˛ ˛ O e/ O  lim E˛ .b eA ,b eA / D lim Cap.˛/ .An /. Cap.˛/ .[n An / D E˛ .e, n n n!1

n!1

For any set B  X , define the ˛-capacity Cap.˛/ .B/ of B by Cap.˛/ .B/ D inf¹Cap.˛/ .A/ : A is open, A Bº.

(2.1.9)

Theorem 2.1.3. Cap.˛/ ./ has the following properties: (i)

A  B ) Cap.˛/ .A/  Cap.˛/ .B/;

(ii)

An is increasing ) Cap.˛/ .[n An / D supn Cap.˛/ .An /.

Proof. (i) follows from Lemma 2.1.2 (i). For any n  1, there exists an open set Bn An such that Cap.˛/ .An /  Cap.˛/ .Bn /  1=n. We may assume that ¹Bn º is increasing. Then, for any n  1, lim Cap.˛/ .An /  lim Cap.˛/ .Bn / D Cap.˛/ .[n Bn /  Cap.˛/ .[n An /.

n!1

n!1

This implies that Cap.˛/ .[n An / D limn!1 Cap.˛/ .An /. For any ˛  0 and a Borel set A with m.A/ < 1, define the ˛-equilibrium potential eA˛ of a Borel set A by eA˛ D min¹u : ˛-excessive, u  1 m-a.e. on Aº.

(2.1.10)

If ˛ > ˛0 , this coincides with the ˛-equilibrium potential given by equation (2.1.3). To prove the existence of the equilibrium potential for 0  ˛  ˛0 , we use the transient resolvent G˛g corresponding to g D 1A given by Lemma 1.3.5. More ˛ f D G pg .gf / and U ˛ f D U ˛ f . generally, for ˛  0 and p > 0, put Up,g ˛ g 0,g

47

Section 2.1 Capacity

˛ 1  1=p for any ˛  0, p > 0 by Lemma 1.3.5 and kU ˛ 1k 2 Then Up,g p,g L .m/  kG˛ gkL2 .m/  m.A/=.˛  ˛0 / for any p  0, ˛ > ˛0 . If ˛ > ˛0 , then for any ˇ > p and u 2 F , since

E˛Cˇ .G˛pg g, u/ D E˛pg .G˛pg g, u/ C ˇ.G˛pg g, u/  .pgG˛pg g, u/   D .g, u/ C .ˇ  pg/G˛pg g, u , it holds that

  G˛pg g D G˛Cˇ g C .ˇ  pg/G˛pg g ,

that is



 ˛ 1 D G˛Cˇ g. I  G˛Cˇ .ˇ  pg/ Up,g

From this we obtain that ˛ 1D Up,g

1 X

 n G˛Cˇ .ˇ  pg/G˛Cˇ g.

nD0

The right-hand side converges for all ˛ > 0, and hence   ˛ ˛ Up,g 1 D G˛Cˇ g C G˛Cˇ .ˇ  pg/Upg 1 ˛ ˛ 1 C Ug˛Cˇ .1  pUp,g 1/. D ˇG˛Cˇ Up,g

(2.1.11)

˛ 1  G pg .pg/  1 by Lemma 1.3.5, this implies that U ˛ 1  Since pUp,g p,g ˛ ˛ 1 is ˛-excessive. Similarly, for any p < q, U ˛ 1 D ˇG˛Cˇ Up,g 1, that is Up,g p,g ˛ 1 C .q  p/U ˛ U ˛ 1 because both sides satisfy E pg .u, v/ D .g, v/ for all Uq,g ˛ p,g q,g v 2 F and ˛ > ˛0 . Hence qp ˛ ˛ ˛ ˛ ˛ ˛ 1 D Uq,g 1 C .q  p/Up,g Uq,g 1  Uq,g 1C (2.1.12) Up,g 1, Up,g q ˛ 1 is increasing relative to p. In particular, eN ˛ which implies that pUp,g A ˛ limp!1 pUp,g 1 exists as an increasing limit and becomes an ˛-excessive function. Furthermore, if .E, F / is transient, then eNA˛ is defined for ˛  0.

Lemma 2.1.4. For any ˛ > 0, there exists a unique function eA˛ satisfying equation (2.1.10). If .E, F / is transient, then the assertion also holds for ˛ D 0. Proof. We need to show the result only in the case of 0  ˛  ˛0 . Put K D ¹h : ˛-excessive h  1 m-a.e. on Aº. Noting that G˛Cˇ is a bounded linear operator on L1 .X ; m/, by letting p ! 1 in equation (2.1.11), we obtain that ˛ 1/. 0 D Ug˛Cˇ .1  lim pUp,g p!1

By multiplying ˇf and integrating, we obtain that   ˛ ˛ b ˛Cˇ f , g.1  lim pUp,g 0 D lim ˇ G 1/ D .fg, 1  lim pUp,g 1/. ˇ !1

p!1

p!1

48

Chapter 2 Some analytic properties of Dirichlet forms

˛ 1 exists. Since f is As noted before the statement of the lemma, eNA˛ D limp!1 pUp,g ˛ arbitrary and g D 1A , it follows that eNA D 1 m-a.e. on A. Consequently, eNA˛ 2 LA . It holds that ˛ G˛Cˇ u, v/ D E˛pg .G˛pg .gG˛Cˇ u/, v/ D .gG˛Cˇ u, v/. E˛pg .Up,g pg

pg

Similarly, since E˛ .G˛ G˛Cˇ u, v/ D .G˛Cˇ u, v/ and E˛pg .G˛Cˇ u, v/ D .u, v/  ˇ.G˛Cˇ u, v/ C .G˛Cˇ u, v/pgm , we can see that   ˛ E˛pg G˛Cˇ u  pUp,g G˛Cˇ u C ˇG˛pg G˛Cˇ u, v D .u, v/ pg

pg

˛ G for any v 2 F . Hence G˛Cˇ upUp,g ˛Cˇ uCˇG˛ G˛Cˇ u D G˛ u. In particular put u D h for an ˛-excessive function h such that h  1 m-a.e. on A. Then ˛ G˛Cˇ h D ˇG˛Cˇ h C ˇG˛pg .ˇG˛Cˇ h  h/ ˇpUp,g

 ˇG˛Cˇ h  h. By letting ˇ ! 1 and then p ! 1, the left-hand side converges to ˛ ˛ ˛ lim lim ˇpUp,g G˛Cˇ h D lim pUp,g h  lim pUp,g 1 D eNA˛ .

p!1 ˇ !1

p!1

p!1

Hence h  eNA˛ m-a.e. which implies that eNA˛ is the minimal ˛-excessive function dominating 1A a.e. For any non-negative measurable function f , ˛ > 0 and 0 < p < q, by replacing 1 with f in equation (2.1.12), we obtain that ˛ G˛qg f , G˛pg f D G˛qg f C .q  p/Up,g pg

XnA

In particular, G˛ f is decreasing as p increases. Put G˛ ˛ > ˛0 and f 2 L2C .X ; m/, then

(2.1.13) pg

f D limp!1 G˛ f . If

E˛ .G˛pg f , G˛pg f /  E˛pg .G˛pg f , G˛pg f / D .f , G˛pg f /  .f , G˛ f /. XnA

By letting p tend to infinity, we obtain that G˛

f 2 F and satisfies

E˛ .G˛XnA f , G˛XnA f /  .f , G˛XnA f /.

(2.1.14)

Furthermore, by letting p tend to infinity in ˇ

ˇ

˛ 1 D pUp,g 1 C p.ˇ  ˛/G˛pg Up,g 1, pUp,g

we have from Theorem 2.1.4 that ˇ

ˇ

eA˛ D eA C .ˇ  ˛/G˛XnA eA .

(2.1.15)

49

Section 2.1 Capacity

Lemma 2.1.5. For any ˛ > 0 and a decreasing sequence ¹An º of open sets, eA˛n # 0 m-a.e. if and only if Cap.ˇ / .An / # 0 for any ˇ > ˛0 . In particular, if .E, F / is transient, then the assertion also holds for ˛ D 0. Proof. We may assume that Cap.ˇ / .A1 / < 1. Suppose that Cap.ˇ / .An / # 0. Then ˇ ˇ ˇ equation (2.1.8) implies that Eˇ .eAn , eAn / # 0. In particular, limn!1 eAn D 0 XnA

ˇ

ˇ

m-a.e. and decreasingly. Since G˛ n eAn  G˛ eAn , equation (2.1.15) implies that limn!1 eA˛n D 0 m-a.e. Conversely, assume that limn!1 eA˛n D 0. Then ˇ

limn!1 eAn  limn!1 eA˛n D 0 m-a.e. Since Cap.ˇ / .A1 / < 1 and hence ˇ

ˇ

ˇ

eAn  eA1 2 L2 .X ; m/, it holds that limn!1 eAn D 0 in L2 .X ; m/. Hence ˇ

ˇ

limn!1 Cap.ˇ / .An /  Kˇ2 limn!1 Eˇ .eAn , eAn / D 0 by closability. In the transient case, by the similar argument, we can show that the result also holds for ˛ D 0. b g˛ be the dual To show the corresponding results for the coexcessive functions, let G g resolvent of G˛ satisfying       g b g˛ h D E˛g G˛g f , G b g˛ h (2.1.16) G˛ f , h D f , G for ˛ > ˛0 and f , h 2 L2 .X ; m/. If f and h are non-negative, then       b g˛ h D E˛ G˛ f , G b g˛ h  E˛g G˛ f , G b g˛ h f ,G b ˛ h/, D .G˛ f , h/ D .f , G b g˛ h  G b ˛ h. Hence ¹G b g˛ º can be extended to an L1 .X ; m/which implies that G resolvent for any ˛ > 0 and for any ˛  0 if .E, F / is transient. ˛ f DG b pg b˛ b bp,g Put U ˛ .gf / and U g f D G ˛ .gf /. Then they satisfy bˇ f  G b pg b b pg b ˇ b pg G ˛ f C .ˇ  ˛/G ˇ G ˛ f  p U g G ˛ f D 0.

(2.1.17)

This holds for all ˛, ˇ > 0 if f 2 L1 .X ; m/. In particular, if .E, F / is transient, then b ˛XnA f D limp!1 G b XnA º b pg it also holds for ˛, ˇ  0. Put G ˛ f for g D 1A . Since ¹G ˛ b ˛XnA f  G b ˛ f for any f 2 L1 .X ; m/, we can is an L1 .X ; m/-resolvent and 0  G C b XnA f for any non-negative measurable function f . For any p > 0 and also define G ˛ > ˛0 , by b pg b pg b pg b pg b pg E˛ .G ˛ f , G ˛ f / C .G ˛ f , G ˛ f /pgm D .f , G ˛ f / 

1 kf k2 ˛  ˛0

b ˛XnA f , G b ˛XnA f /gm D 0 and hence G b ˛XnA f D 0 a.e. on A. If f 2 we get that .G b ˛XnA f D 0 for all L1 .X ; m/, then by the resolvent equation, it also holds that G ˛  ˛0 .

50

Chapter 2 Some analytic properties of Dirichlet forms

ˇ Suppose that b h is an integrable ˇ-coexcessive function. For ˇ  ˛, define b hA by ˇ ˇ b b bp,g h. hA D lim p U p!1

(2.1.18)

ˇ Lemma 2.1.6. If b h is an integrable ˛-coexcessive function then, for any ˇ  ˛, b hA is an integrable ˇ-coexcessive function satisfying ˇ ˛ ˛ b b XnAb hA hA  .ˇ  ˛/G . hA D b ˇ

(2.1.19)

Proof. Suppose that b h is an integrable ˛-coexcessive function. Then, for any ˇ  ˛, b b b h b h, b h is ˇ-coexcessive. Then, for any p > 0 and  > 0, since  G Cˇ h   G C˛b since ˇ bp,g b Cˇ b b pgb b pg G b Cˇ b b Cˇ b pU h/ D G hG h C G h .G ˇ ˇ

b pgb b pgb b Cˇ b hG hCG h  G ˇ ˇ 1 b Cˇ b DG h b h,  ˇ bp,g b Cˇ b h/  b h. By letting  ! 1, it follows that we have p U . G ˇ b bp,g h b h. pU

(2.1.20)

Hence, similarly to equation (2.1.11), ˇ b ˇ b ˇ b b ˇ C U b ˇg C .b bp,g bp,g bp,g h D G hCU h/ h  pU U

(2.1.21)

ˇ b b ˇ C U bp,g  G h. ˇ ˇ b bp,g h exists Therefore, by a similar manner to equation (2.1.12), b hA D limp!1 p U as an increasing limit and becomes an integrable ˇ-coexcessive function. Equation (2.1.19) follows from ˇ b ˛ b ˛ b bp,g b pg U bp,g bp,g h  pU h C p.ˇ  ˛/G h D 0. pU ˇ

˛ Theorem 2.1.7. For any integrable ˛-coexcessive function b h with ˛ > 0, b hA is the ˛ ˛ b b b eA for ˛ > ˛0 , minimal ˛-coexcessive function such that hA  h a.e. on A. If h D b ˛ ˛ ˛ b eA . In particular, b eA is the minimal ˛-coexcessive function dominating 1 then hA D b a.e. on A and satisfies ˇ ˛ ˛ b XnAb eA  .ˇ  ˛/G eA (2.1.22) b eA D b ˇ

for any ˇ > ˛ > ˛0 .

51

Section 2.2 Quasi-Continuity

˛ b bp,g h  limp!1 b h=p D 0 by equation (2.1.20), by letting Proof. Since limp!1 U ˛ / D 0. As in the proof of b ˛Cˇ p ! 1 in equation (2.1.21), it holds that U .b h b hA g ˛ hb hA /gm D 0 for any bounded integrable Lemma 2.1.4, this implies that .ˇG˛Cˇ f , b ˛ h a.e. on A. function f and hence b hA D b h0  b h on A a.e. Since Let b h0 be an integrable ˛-coexcessive function such that b ˛ ˛ bp,gb bp,gb h  pU h0  b h0 . Letting p ! 1, it follows equation (2.1.20) holds for b h0 , p U ˛ ˛ that b hA b h0 which yields the minimality of b hA . ˛ ˛ ˛ ˛ is an ˛-coexcessive b b b eA . Conversely, since b hA If h D b eA for ˛ > ˛0 , then hA  h D b ˛ ˛ ˛ ˛ b˛ ˛ b eA , E˛ .b eA b hA , hA /  0. On the other hand, since b hA Db h D function and b hA ˛ ˛ ˛ ˛ ˛ ˛ ˛ b b eA  hA ,b eA / D 0 by equation (2.1.5). Therefore E˛ .b eA  hA ,b eA  b eA on A, E˛ .b ˛ /  0 and hence b ˛ D b ˛ . For any ˛-coexcessive function b b eA hA h 2 F such that hA ˛ ˛ ˛ b h  1 a.e. on A, E˛ .b h ^b eA b eA ,b eA /  0 by equation (2.1.3). Furthermore, since ˛ ˛ ˛ b ˛ ˛ ˛ b b b eA  h ^ b eA , h ^b eA /  0. Hence E˛ .b h ^b eA b eA ,h ^ h ^b eA is ˛-coexcessive, E˛ .b ˛ ˛ ˛ ˛ b eA /  0 and hence b eA  h. This yields the minimality of b eA among the ˛b eA  b coexcessive functions which dominates 1 a.e. on A. Equation (2.1.22) is a consequence of equation (2.1.19).

2.2

Quasi-Continuity

In the preceding section, we defined the notion of the capacity Cap.˛/ for ˛ > ˛0 of subsets of X . Using this capacity, if a statement holds for all x 2 A n N for some set N with Cap.˛/ .N / D 0, we shall say that the statement holds quasi-everywhere (q.e.) on A. For a function u defined q.e. on X , we say that u is quasi-continuous (q.c. in abbreviation) if, for any  > 0, there exists an open set A  X such that Cap.˛/ .A/ <  and the restriction ujXnA of u on X n A is continuous. An increasing sequence ¹Fn º of closed sets is called a nest if it satisfies limn!1 Cap.˛/ .X nFn / D 0. If in addition, the nest ¹Fn º satisfies suppŒIFn  m D Fn , then it is called an m-regular nest. For a given nest ¹Fn º, let C .¹Fn º/ D ¹u : ujFn is continuous for each nº.

(2.2.1)

Lemma 2.2.1. (i)

Let S be a countable family of quasi-continuous functions on X . Then there exists a regular nest ¹Fn º on X such that S  C .¹Fn º/.

(ii)

Let ¹Fn º be a regular nest and u 2 C .¹Fn º/. If u  0 m-a.e., then u.x/  0 on [1 nD1 Fn . .k/

Proof. (i) Let S D ¹uk º. For any k, there exists a nest ¹Fn º such that Cap.˛/ .X n T .k/ .k/ .k/ Fn / < 1=.2k n/ and uk 2 C .¹Fn º/. Let FNn D k Fn . Then Cap.˛/ .X nFNn / < n1

52

Chapter 2 Some analytic properties of Dirichlet forms

and uk 2 C .¹FNn º// for all k. Let Fn D suppŒIFNn  m, then it satisfies (i). (ii) Suppose u.x/ < 0 for some x 2 Fn , then there exists a neighborhood U.x/ of x such that u.y/ < 0 on Fn \ U.x/. Since m.Fn \ U.x// > 0, this contradicts the hypothesis. By virtue of Lemma 2.2.1, if u is quasi-continuous and u  0 m-a.e., then u  0 q.e. For a given function u, if there exists a quasi-continuous function uQ such that uQ D u m-a.e., then uQ is said to be a quasi-continuous (q.c.) modification of u. In Theorem 2.2.3, we shall show that any function u 2 F has a q.c. modification u. Q To show it, we prepare the following lemma. Lemma 2.2.2. If u 2 F \ C .X /, then  Cap

.˛/

.¹x 2 X : ju.x/j > º/ 

K˛ 

2 E˛ .u, u/.

(2.2.2)

Proof. Let u 2 F \ C .X / and A D ¹x : ju.x/j > º. Then A is an open set and 1 juj 2 LA . Hence  ˛ Cap.˛/ .A/ D E˛ .eA˛ ,b eA /

1 ˛ eA / E˛ .juj,b 

K˛ ˛ ˛ 1=2 eA ,b eA / E˛ .juj, juj/1=2 E˛ .b  K˛  Cap.˛/ .A/1=2 E˛ .u, u/1=2 .  

This yields equation (2.2.2). Theorem 2.2.3. If u 2 F , then there exists a q.c. modification uQ of u. Proof. Since .E, F / is regular, there exists a sequence ¹un º  F \ C0 .X / such that limn!1 un D u relative to E˛ . By taking a subsequence, we may assume that E˛ .unC1  un , unC1  un / < 23n . Hence, equation (2.2.2) implies that Cap.˛/ .¹x : junC1 .x/  un .x/j > 2n º/  K˛2 2n . T k º. Then Cap.˛/ .X nF /  K 2 2N C1 Put Fn D 1 N ˛ kDn ¹x : jukC1 .x/uk .x/j  2 by equation (2.2.2) and, for x 2 FN and m, n  `  N , jun .x/  um .x/j 

1 X

jukC1 .x/  uk .x/j  2`C1 .

kD`

Q D limn!1 un .x/ for ThusS¹un º converges uniformly on FN as n ! 1. Set u.x/ 1 x 2 nD1 Fn , then uQ belongs to C .¹Fn º/ and uQ D u m-a.e.

Section 2.2 Quasi-Continuity

53

Lemma 2.2.4. The inequality (2.2.2) holds for any q.c. modification uQ of u 2 F . Proof. Let u 2 F and un 2 C0 .X / \ F be an approximating sequence of u. Consider the q.c. modification u.x/ Q D limn!1 un .x/ of u defined in Theorem 2.2.3. As we have seen in the proof of Theorem 2.2.3, for any  > 0, there exists an open set A with Cap.˛/ .A/ <  such that un converges to uQ uniformly on X n A. Since ¹x 2 X : ju.x/j Q > º  ¹x 2 X : jun .x/j >   2 º [ A for large n, applying inequality (2.2.2) to un 2 C0 .X /, we have   2K˛ 2 Cap.˛/ .¹x 2 X : ju.x/j Q > º/  E˛ .un , un / C . 2   Letting n ! 1 and  ! 0, we have the result. We say that ¹un º converges to u q.e. uniformly if, for any  > 0, there exists an open set B such that Cap.˛/ .B/ <  and ¹un º converges uniformly to u on X n B. Theorem 2.2.5. If ¹un º  F is an E˛ -Cauchy sequence for ˛ > ˛0 , then there exists a subsequence ¹uQ nk º of q.c. modifications of ¹un º and a q.c. function uQ 2 F such that limk!1 uQ nk D uQ q.e. uniformly and relative to E˛ . Proof. By virtue of Lemma 2.2.4 and the proof of Theorem 2.2.3, there exist a decreasing sequence of open sets ¹Bn º and a subsequence ¹unk º such that Cap.˛/ .Bn / # 0 and un 2 F their q.c. modifications ¹uQ nk º converge relative to E˛ and uniformly to some e un D e unC1 q.e. on X n Bn , put e u De un on X n Bn . For any on each X n Bn . Since e .˛/  > 0, take an open set A1 satisfying Cap .A1 / < =2 and A1 Bn for all large n. Lemma 2.2.1 (i) yields that there exists an open set G2 with Cap.˛/ .G2 / < =2 such that any function of ¹uQ nl º is continuous on X n G2 . Let G D G1 [ G2 . Then u on X n G. This implies the asserCap.˛/ .G/ <  and ¹uQ nl º converges uniformly to e tion of the theorem. By virtue of Theorem 2.2.3, for any f 2 L2 .X ; m/ and ˇ > ˛0 , there exists a q.c. modification Rˇ fPof Gˇ f 2 F . If f 2 L2 .X ; m/ \ L1 .X ; m/, then for any n nC1 f and the right-hand side converges in ˛ > 0, since G˛ f D 1 nD0 .ˇ  ˛/ .Gˇ / 1 L .X ; m/, R˛ f defined by the right-hand side with Rˇ instead of Gˇ is a q.c. modification of G˛ f . Theorem 2.2.6. For any ˛ > 0 and f 2 L1 .X ; m/ (resp. f 2 L1 .X ; m/), G˛ f b ˛ f ) has a q.c. modification. In particular, if .E, F / is transient, then for any (resp. G b kL1 < 1), function f 2 L1 .X ; m/ \ L1 .X ; m/ such that kGf k1 < 1 (resp. kGf b Gf (resp. Gf ) has a q.c. modification. Proof. We may assume that f is non-negative. For any ˇ > ˛0 and f 2 L1 .X ; m/\ L2 .X ; m/, Gˇ f has a q.c. modification Rˇ f satisfying kˇRˇ f k1  kf k1 . For any

54

Chapter 2 Some analytic properties of Dirichlet forms

f 2 L1 .X ; m/, by approximating f by functions ¹fn º  L1 .X ; m/ \ L2 .X ; m/ in L1 .X ; m/, Rˇ fn converges q.e. uniformly to a q.c. modification Rˇ f of Gˇ f . Hence Gˇ f has a q.c. modification for any f 2 L1 .X ; m/. For any 0 < ˛  ˛0 , by taking ˇ > ˛0 , the resolvent equation gives us the relation G˛ f D

1 X

.ˇ  ˛/n GˇnC1 f

(2.2.3)

nD0

P n nC1 f converges for all f 2 L1 .X ; m/. Hence the function R˛ f D 1 nD0 .ˇ  ˛/ Rˇ q.e. uniformly and gives a q.c. modification of G˛ f . If .E, F / is transient and Gf is bounded q.e., then it satisfies Gf D G˛ f C ˛G˛ Gf . for any ˛ > 0. Since the right-hand side has a q.c. modification, so does Gf . If f 2 L1 .X ; m/ and 0 < ˛  ˛0 , then by taking a strictly positive function b ˛ f ^.nR bˇ g/ is a ˇ-excessive function belonging g 2 L2 .X ; m/ and ˇ > ˛0 , un G b b ˇ f . Hence un has a to F by Lemma 1.4.2, where Rˇ g is a q.c. modification of G q.c. modification e un . In particular, there exists a nest ¹Fk º such that all functions of u D limn!1 e un is a ¹e un º are continuous and increasing on Fk for all n. Hence e b ˛ f and lower semi-continuous on each Fk . Put Bk D ¹x 2 Fk : modification of G ˛  .1=k/e u.x/ C e u.x/ > kº [ .X n Fk /. Then Bk is an open set and satisfies eB k ˛ .ˇ / eXnF by Theorem 2.1.4. This implies that limk!1 Cap .Bk / D 0 for ˇ > ˛0 by k Lemma 2.1.5. In particular, e u < 1 q.e. Furthermore, since e u.x/ coincides with the bˇ g.x/º which increases to un .x/ < nR q.c. function e un .x/ on the quasi-open set ¹x : e X q.e., e u is quasi-continuous. Remark 1. By using an inverse Laplace transformation, it is possible to show that, for any t > 0 and f 2 L2 .X ; m/, T t f belongs to the domain of the generator and hence to F .1 Hence T t f has a q.c. modification. The result of Theorem 2.2.6 follows easily from this. For an open set A, the ˛-capacity Cap.˛/ .A/ of A is defined by Cap.˛/ .A/ D ˛ ˛ ˛ eA / D E˛ .b eA ,b eA / in the definition after equation (2.1.7). We shall show that E˛ .eA˛ ,b this relation also holds for any Borel set B. Theorem 2.2.7. For any Borel set B  X such that LB ¤ ; and ˛ > ˛0 , ˛ ˛ ˛ ˛ ,b eB / D E˛ .b eB ,b eB /. Cap.˛/ .B/ D E˛ .eB

(2.2.4)

Proof. For any  > 0, there exists an open set A such that B  A and Cap.˛/ .A/ < ˛ ˛ eA b eB and eA˛ D 1 a.e. on A, equations (2.1.3) and (2.1.5) imply Cap.˛/ .B/. Since b 1

See e.g. [81, 104, 111].

55

Section 2.3 Potential of measures

that ˛ ˛ ˛ Cap.˛/ .B/ > E˛ .b eA ,b eA /   D E˛ .eA˛ ,b eA / ˛ ˛ ˛  E˛ .eA˛ ,b eB /   D E˛ .b eB ,b eB /  . ˛ ,b ˛ /. Hence Cap.˛/ .B/  E˛ .b eB eB ˛ is quasi-continuous already. To show the converse inequality, we assume that b eB ˛ is Then, for any  > 0, there exists an open set G such that Cap.˛/ .G / <  and b eB ˛ eB .x/ > 1  º [ G . Then A is an continuous on X n G . Put A D ¹x 2 X n G : b ˛ ˛ eB Cb e G  1   on A . Hence noting that eA˛ D 1 open set satisfying A B and b m-a.e. on B and G , it follows by equation (2.1.5) that ˛ Cap.˛/ .B/  Cap.˛/ .A /  E˛ .eA˛ ,b eA /  1 ˛ eB Cb e ˛G / E˛ .eA˛ ,b  1  1  ˛ ˛ ˛ ˛ ,b eB / C E˛ .eG ,b e / E˛ .eB D G   1  1  ˛ ˛ E˛ .eB ,b eB / C  .  1 This gives the desired inequality.

2.3

Potential of measures

In this section, we also assume that .E, F / is a regular Dirichlet form on L2 .X ; m/. Since E˛0 .u, u/  0, we have .u, u/  1=.˛  ˛0 /E˛ .u, u/ for ˛ > ˛0 . Hence Eˇ .u, u/ D E˛ .u, u/ C .ˇ  ˛/.u, u/  Therefore

ˇ  ˛0 E˛ .u, u/. ˛  ˛0

˛  ˛0 Eˇ .u, u/  E˛ .u, u/  Eˇ .u, u/ ˇ  ˛0

(2.3.1)

for any ˇ > ˛ > ˛0 and u 2 F . This implies that the following definition of the measure of the finite energy integral is independent of ˛ > ˛0 . Definition 5. A positive Radon measure  on X is said to be a measure of the finite energy integral if for any ˛ > ˛0 , there exists a positive constant C such that Z (2.3.2) jv.x/j.dx/  C E˛ .v, v/1=2 , for all v 2 F \ C0 .X /. Let S0 be the family of all positive Radon measures on X with a finite energy integral.

56

Chapter 2 Some analytic properties of Dirichlet forms

R Applying Theorem 1.1.1 to J.v/ D v.x/.dx/ Q and  D F , we can see that a positive Radon measure  belongs to S0 if and only if, for any ˛ > ˛0 there exists a unique function U˛  2 F such that Z (2.3.3) E˛ .U˛ , v/ D v.x/.dx/ for all v 2 C0 .X / \ F . b ˛  2 F such that This is also equivalent to the existence of U Z b E˛ .v, U ˛ / D v.x/.dx/ for all v 2 C0 .X / \ F .

(2.3.4)

b ˛  the ˛-potential and ˛-copotential of , respectively. We shall call U˛  and U Theorem 2.3.1. The following conditions are equivalent to each other for u 2 F and ˛ > ˛0 : (i)

u is an ˛-potential (resp. u  0 and ˛-copotential);

(ii)

u is ˛-excessive (resp. ˛-coexcessive).

Proof. If u is an ˛-potential, then E˛ .u, v/  0 for any v 2 F such that v  0 by equation (2.3.3). Hence u is ˛-excessive. Conversely, for an ˛-excessive function u 2 F , put `.v/ D E˛ .u, v/ for v 2 F \C0 .X /. For any compact set F take a non-negative function vF 2 F \C0 .X / such that vF  1 on F . If the support of v is contained in F , then jvj  kvk1 vF . Since u is ˛-excessive, this yields that j`.v/j  `.jvj/  kvk1 `.vF /. Furthermore, for any w 2 C0 .X / such that w  0, we can choose a function v 2 F \ C0 .X / such that kwvk1 < . Put v  D v./_.v^/, then v  2 F \C0 .X /, suppŒv    suppŒw and v  ! w uniformly as  ! 0. Hence `.w/ lim!0 `.v  / defines a positive linear functional on C0 .X /. Then there exists a positive Radon measure  such that R `.w/ D X w.x/.dx/ for all w 2 C0 .X /. This implies that u is an ˛-potential. The proof of the dual assertion is similar. Corollary 2.3.2. If both u and v are ˛-potentials or non-negative ˛-copotentials, then so is u^v. If u is an ˛-potential, then so is u^a for any non-negative constant a. Lemma 2.3.3. For  2 S0 and ˛ > ˛0 , let gn D n.U˛   nG˛Cn U˛ /.

(2.3.5)

Then gn  m converges vaguely to  and G˛ gn converges E˛ -weakly to U˛ . Proof. By virtue of Theorem 2.3.1, gn  0 m-a.e. Since .gn , v/ D E˛ .G˛ gn , v/ D n.U˛   nG˛Cn U˛ , v/ D E˛ .nGnC˛ U˛ , v/

57

Section 2.3 Potential of measures

and limn!1 E˛ .nGRnC˛ U˛ , v/ D E˛ .U˛ , v/ by Theorem 1.1.4 (iii), it follows that limn!1 .gn , v/ D X v.x/.dx/ for v 2 F \ C0 .X /. Lemma 2.3.4. Any measure of S0 charges no set of zero capacity. Proof. Suppose that  2 S0 . By virtue of Lemma 2.3.3, for any open set A and a function f 2 C0 .X / satisfying 0  f  1 with suppŒf   A, we have Z Z Z ˛ f .x/.dx/ D lim gn .x/f .x/m.dx/  lim inf gn .x/b eA .x/m.dx/ n!1 X

X



˛ eA / lim inf E˛ .G˛ gn ,b n!1

n!1

A

˛ D E˛ .U˛ ,b eA /

˛ ˛ 1=2  K˛ E˛ .U˛ , U˛ /1=2 E˛ .b eA ,b eA / .

Letting f " IA , we have .A/  K˛ E˛ .U˛ , U˛ /1=2 Cap.A/1=2

(2.3.6)

This shows the lemma. Theorem 2.3.5. Suppose that ˛ > ˛0 ,  2 S0 and uQ 2 F is quasi-continuous. Then uQ is -integrable and Z b Q D E˛ .u, Q U ˛ / D u.x/.dx/. Q (2.3.7) E˛ .U˛ , u/ b ˛  are quasi-continuous In particular, for any ,  2 S0 , assuming that U˛  and U already, it holds that Z Z b U˛ .x/.dx/. (2.3.8) U ˛ .x/.dx/ D X

X

Proof. For uQ 2 F , there exists a sequence ¹un º  F \ C0 .X / which converges to uQ relative to E˛ . Then a subsequence ¹unk º converges to uQ q.e. by Theorem 2.2.5. Using Fatou’s lemma, we have Z Z ju.x/ Q  un .x/j.dx/  lim inf junk .x/  un .x/j.dx/ k!1

 K˛ E˛ .U˛ , U˛ /1=2 lim inf E˛ .unk  un , unk  un /1=2 k!1

 K˛ E˛ .U˛ , U˛ /

1=2

E˛ .u  un , u  un /1=2 .

Thus un converges to uQ in L1 .X ; /. Hence, by taking un in place of v in equation (2.3.3) and letting n tend to infinity, we obtain equation (2.3.7). Equation (2.3.8) follows from equation (2.3.7).

58

Chapter 2 Some analytic properties of Dirichlet forms

For any Borel set A  X such that LA ¤ ;, the ˛-equilibrium potential eA˛ and the ˛ are ˛-excessive and ˛-coexcessive, respectively. Hence ˛-coequilibrium potential b eA ˛

A 2 S0 satisfying there are corresponding positive Radon measures A˛ 2 S0 and b eA˛ D U˛ A˛ ,

˛ ˛ b˛b b eA DU

A .

(2.3.9)

˛ The measures A˛ and b

A are respectively called the ˛-equilibrium measure and the ˛-coequilibrium measure of A. Suppose that u 2 F \ C0 .X / is a non-negative function such that suppŒu  N where AN is the closure of A. Since e ˛ 2 LA , equation (2.1.5) implies that X n A, A R ˛ N Similarly, b

A is sup0 D E˛ .eA˛ , u/ D u.x/ A˛ .dx/. Hence A˛ is supported by A. N ported by A. RIf u 2 F satisfies u D 1 m-a.e. on A, then uQ D 1 q.e. on A. Hence ˛ ˛ ˛ N eA / D X u.x/b Q A .dx/ D b

A .A/. In particular, if A is open, since eQA˛ .x/ D 1 E˛ .u,b q.e. on A, Z ˛ ˛ ˛ N eA /D eQA˛ .x/b

A .dx/ D b

A .A/. (2.3.10) Cap.˛/ .A/ D E˛ .eA˛ ,b X

Similarly, for any open set A and u 2 F such that u D 1 a.e. on A, ˛ .˛/ N D E˛ .e ˛ , u/  E˛ .e ˛ ,b .A/.

A˛ .A/ A A eA / D Cap

(2.3.11)

Theorem 2.3.6. If A is a compact set, then for any ˛ > ˛0 , ˛ ˛ ˛ Cap.˛/ .A/ D E˛ .b eA ,b eA / D b

A .A/.

(2.3.12)

Proof. For any compact set A, let ¹An º be a decreasing sequence of relatively compact .˛/ .A / # Cap.˛/ .A/. For N open sets such that ANnC1  An , \1 n nD1 An D A and Cap ˛ ˛ ˛ b˛b each n, the ˛-coequilibrium potential b eAn can be written as b eA D U

A . Since n n ˛ ˛ .˛/ N N Cap .An / D b

An .An /, ¹b

An .An /º is uniformly bounded. Hence, by choosing a ˛ º converges vaguely to a measure b

0 on A. subsequence, we may assume that ¹b

A n ˛ converges Furthermore, after choosing a further subsequence, we can assume thatb eA n to some b e 0 2 LA weakly in .E˛ , F /. Hence, for all u 2 F \ C0 .X /, Z ˛ ˛ e 0 / D lim E˛ .u,b eA / D lim u.x/b

A .dx/ E˛ .u,b n n n!1 n!1 X Z D u.x/b

0 .dx/. X

˛ b˛b

0 . Hence, it suffices to show that b e0 D b eA . Clearly b e0  1 This implies that b e0 D U on A. Suppose that u 2 C0 .X / \ F satisfies u  1 on A. For any  > 0, there exists n such that u  1   on Ak for all k  n. Hence

1 ˛ ˛ ˛ eA /  E˛ .b eA ,b eA /. E˛ .u,b k k k 1

59

Section 2.3 Potential of measures

By letting k ! 1, and then  ! 0, we have E˛ .u,b e 0 /  E˛ .b e 0 ,b e 0 /. This shows that ˛ eA . b e0 D b Theorem 2.3.7. The following conditions are equivalent for any compact set A  X and ˛ > ˛0 . (i)

Cap.˛/ .A/ D 0.

(ii)

.A/ D 0 for all  2 S0 .

(iii) .A/ D 0 for all  2 S00 D ¹ 2 S0 : kU˛ k1 < 1º. Proof. (i) ) (ii) has been shown by Lemma 2.3.4. Conversely, suppose that Cap.˛/ .A/ > 0 for some compact set A. By virtue of Theorem 2.3.6, it then holds ˛ .A/ > 0. This gives the equivalence of (i) and (ii). that b

A (ii) ) (iii) is clear. To show the converse assertion, we first note that, for a compact e ˛  is q.e. bounded by a constant k on F , then U˛ .1F  set F , if a q.c. modification U /  k m-a.e. on X . In fact, by putting u D U˛ .1F  /, it holds that Z Z E˛ .u, u ^ k/ D .u.x/ ^ k/ 1F .x/.dx/ D u.x/1F .x/.dx/ D E˛ .u, u/ . X

X

From this we get that E˛ .u ^ k  u, u ^ k  u/ D E˛ .u ^ k, u ^ k  u/ which is non-positive because u^k is ˛-excessive. Hence we obtain that u D u^k  k m-a.e. Suppose that .A/ > 0 for some  2 S0 . Then by taking a compact set F on which U˛  is bounded and .A \ F / > 0, we can see that U˛ .1A\F  / is bounded. Hence 1A\F   2 S00 and 1A\F  .A/ > 0. For ˛  ˛0 we extended G˛ to a resolvent on L1 .X ; m/. By the resolvent equation, G˛ f D Gˇ f C .ˇ  ˛/Gˇ G˛ f for any f 2 L1 .X ; m/ and ˇ > ˛0 . Furthermore, for any g 2 L1 .X ; m/, lim .G˛ f   G G˛ f , g/ D lim .G f  ˛G G˛ f , g/

!1

!1

D .f  ˛G˛ f , g/.

(2.3.13)

This can be extended to any potential of measures. For any  2 S00 , since Uˇ  2 b ˇ  2 L1 .X ; m/ \ F , for any ˛ > 0, we can define a bounded L1 .X ; m/ \ F and U b ˛  respectively by ˛-excessive function U˛  and integrable ˛-coexcessive function U U˛  D Uˇ  C .ˇ  ˛/G˛ Uˇ  b˛  D U b ˇ  C .ˇ  ˛/G b˛ U b ˇ . U

(2.3.14)

60

Chapter 2 Some analytic properties of Dirichlet forms

Then, similarly to equation (2.3.13), we get lim .U˛    G U˛ , g/ D h, gi  ˛.U˛ , g/

!1

b˛   G b U b ˛ / b ˛ / D h, f i  ˛.f , U lim .f , U

(2.3.15)

!1

for any q.c. functions f 2 L1 .X ; m/ \ F and g 2 L1 .X ; m/ \ F . If .E, F / is transient, then these results also hold for ˛ D 0.

2.4 An orthogonal decomposition of the Dirichlet forms In the preceding section, for ˛ > ˛0 , we defined the ˛-equilibrium potential eA˛ of a Borel set A which is given as the minimal ˛-excessive function among the ˛-excessive functions which dominate 1 on A a.e. Generalizing this, for any ˛-excessive function u 2 F and a Borel set A  X , consider the family Lu,A D ¹v 2 F : vQ  uQ q.e. on Aº.

(2.4.1)

Then Lu,A is a convex closed subset of .E˛ , F /. Theorem 1.1.1 then shows that there ˛ 2L exists a unique function uA u,A such that ˛ ˛ , v  uA /0 E˛ .uA

(2.4.2)

˛ is called the ˛-reduced function of u on A. for all v 2 Lu,A . The function uA Similarly, for a given ˛-coexcessive function uO 2 F , we can defined the ˛-co˛ 2 Lb satisfying reduced function uOA u,A ˛ ˛ , uOA /  0 E˛ .v  uOA

(2.4.3)

˛ O and uOA are quasifor all v 2 Lu,A O . For convenience of notation, we assume that u , equations (1.1.3) and (2.4.3) imply continuous already. For any v 2 Lu,A O ˛ ˛ uA ,b uA /  K˛2 E˛ .v, v/. E˛ .b

(2.4.4)

Lemma 2.4.1. For any ˛ > ˛0 , let u 2 F and uO 2 F be a q.c. ˛-excessive and a q.c. ˛ and the ˛-coexcessive function, respectively. Assume that the ˛-reduced function uA ˛ ˛ ˛ D u ˛-coreduced function b uA are quasi-continuous already. Then uA  u q.e., uA q.e. on A, and ˛ ˛ , uA /  K˛2 E˛ .v, v/ (2.4.5) E˛ .uA ˛ ˛ uA also holds, that is b uA b u q.e., for any v 2 Lu,A . The corresponding result for b ˛ u q.e. on A and b uA D b ˛ ˛ , uOA /  K˛2 E˛ .v, v/. (2.4.6) E˛ .uOA

. for all v 2 Lb u,A

Section 2.4 An orthogonal decomposition of the Dirichlet forms

61

˛ Proof. For any w  0, w 2 F , by putting v D uA C w in equation (2.4.2), it ˛ ˛ holds that E˛ .uA , w/  0 which means that uA is an ˛-excessive function. Then, by ˛ is an ˛-excessive function belonging to F . In particular, virtue of Lemma 1.4.2, u^uA ˛ ˛ ˛ ˛ ^u 2 Lu,A in equation E˛ .uA ^u, uA uA ^u/  0. Furthermore, by putting v D uA ˛ ˛ ˛ ˛ ˛ ˛ ˛ , uA u^uA / (2.4.2) we have E˛ .u^uA , uA ^uuA /  0. Therefore E˛ .uA u^uA ˛ ˛ ˛ ˛ D uA m-a.e. Thus uA  u and uA D u q.e. on A. Equation (2.4.5) 0 and hence u ^ uA follows from equations (2.4.2) and (1.1.3), in fact, ˛ ˛ ˛ ˛ ˛ 1=2 E˛ .uA , uA /  E˛ .uA , v/  K˛ E˛ .uA , uA / E˛ .v, v/1=2 . ˛ which gives equation (2.4.5). The assertion concerning b uA follows similarly. ˛ ˛ If w 2 Lu,A satisfies w D u q.e. on A, then by putting v D uA ˙ .w  uA / in equation (2.4.2), we have ˛ ˛ , w  uA / D 0. (2.4.7) E˛ .uA

For general function u 2 F , we can also define the corresponding notion to the ˛ . For a Borel set A  X , put ˛-reduced function uA F XnA D ¹u 2 F : uQ D 0, q.e. on Aº and

HA˛ D ¹v 2 F : E˛ .v, w/ D 0, for all w 2 F XnA º b ˛ D ¹v 2 F : E˛ .w, v/ D 0, for all w 2 F XnA º. H A

(2.4.8)

(2.4.9)

b ˛ ) is called ˛-harmonic (resp. ˛-coharmonic) on u2H Any function u 2 HA˛ (resp. b A X n A. Theorem 2.4.2. For any u 2 F , ˛ > ˛0 and Borel set A, there exist unique functions ˛ ˛ b ˛ ) such 2 HA˛ (resp. b

˛F XnA .u/ 2 F XnA and uOA 2H

F˛ XnA .u/ 2 F XnA and uA A that ˛ ˛ Db

˛F XnA .u/ C uOA . (2.4.10) u D F˛ XnA .u/ C uA Proof. Let ˛ > ˛0 and u 2 F . Since F XnA is a closed convex subset of .E˛ , F /, applying Theorem 1.1.1 to J.v/ D E˛ .u, v/, we can show that there exists a unique projection F˛ XnA .u/ 2 F XnA satisfying E˛ .u  F˛ XnA .u/, w  F˛ XnA .u//  0,

for all w 2 F XnA .

By replacing w with F˛ XnA .u/ ˙ w it is easy to see that F˛ XnA .u/ is the unique function of F XnA satisfying E˛ .u  F˛ XnA .u/, w/ D 0, ˛ D u  ˛ ˛ 2 H ˛. Put uA .u/. Then uA A F XnA

for all w 2 F XnA .

(2.4.11)

62

Chapter 2 Some analytic properties of Dirichlet forms

To show the uniqueness of the decomposition, suppose that u D u1 C v1 D u2 C v2 for u1 , u2 2 F XnA and v1 , v2 2 HA˛ . Then E˛ .v1  v2 , w/ D 0 for all w 2 F XnA . Since v1  v2 D u2  u1 2 F XnA , by putting w D v1  v2 we obtain that v1 D v2 . The proof of the dual decomposition is similar. ˛ If u 2 F is ˛-excessive, then equation (2.4.11) implies that the function uA in Theorem 2.4.2 satisfies equation (2.4.7). Hence it is equal to the ˛-reduced function of u on A. Let u D U˛  for  2 S0 , then there exists a measure A 2 S0 such that ˛ D U˛ A . We call A the ˛-sweeping out of  on A. Similarly, for uO D UO ˛ , O uA ˛ O define the ˛-cosweeping out of O on A by the measure O A satisfying uOA D U˛ O A . In the rest of this section, for any  2 S0 , we assume that its potentials U˛  b ˛  are quasi-continuous already. Denote by U˛XnA  and U b ˛XnA  the quasiand U b ˛ /, respectively.

˛F XnA .U continuous modifications of F˛ XnA .U˛ / and b We say that a set B is quasi-open if, for any  > 0, there exists an open set O such that O B and Cap.˛/ .O n B/ < . The complement of any quasi-open set is called quasi-closed.

Lemma 2.4.3. For any ˛ > ˛0 , ,  2 S0 and Borel set A, the following properties hold. (i) (ii)

2

˛ b ˛  D .U b ˛ /˛ q.e. If .X n A/ D 0, then U˛  D .U˛ /A q.e. and U A b ˛ /˛ D U b ˛ 1 , then 1 .X n A/ N D 0. If .U˛ /˛ D U˛ 1 or .U A

A

(iii) If .A/ D .A/ D 0, then b ˛XnA i D h, U˛XnA i D E˛ .U˛XnA , U b ˛XnA /. h, U

b

b ˛ D .U b ˛ /˛ Proof. (i) Suppose that .X n A/ D 0. Then, for any 2 S0 , since U A q.e. on A and  does not charge any set of zero capacity, by using Lemma 2.4.1 we obtain that Z Z Z b ˛ .x/.dx/ D .U b ˛ /˛ .x/.dx/ U˛ .x/ .dx/ D U A X X X     b˛ b ˛ /˛ D E˛ .U˛ /˛ , U D E˛ U˛ , .U A A Z ˛ .U˛ /A .x/ .dx/. D

b

b

X

˛ and we are assuming that the potentials of measures are quasiSince U˛   .U˛ /A ˛ q.e. The dual assertion follows similarly. continuous, this implies that U˛  D .U˛ /A ˛ (ii) If U˛ 1 D .U˛ /A , then for any function ' 2 C0 .X / \ F such that suppŒ'  N X n A, Z   ˛ '.x/1 .dx/ D E˛ .U˛ 1 , '/ D E˛ .U˛ /A , ' D 0. X

This gives the assertion of (ii).

63

Section 2.4 An orthogonal decomposition of the Dirichlet forms

(iii) follows from

    b ˛XnA  D E˛ U˛XnA , U b ˛XnA  b ˛XnA i D E˛ U˛ , U h, U   b ˛  D h, U˛XnA i. D E˛ U˛XnA , U

Using Lemma 2.4.3 as another condition which is equivalent to the conditions of Theorem 2.3.7, we have the following result. Lemma 2.4.4. For any compact set F , Cap.˛/ .F / D 0 if and only if .F \.X nA// D XnA 0 for any  2 S0 and quasi-closed set A such that kU˛ k1 < 1. Proof. By virtue of Theorem 2.3.7, the only if part is clear. To show the if part, assume that Cap.˛/ .F / > 0. Then there exists a measure  2 S0 such that .F / > 0. By considering the restriction of  to a set of finite -measure if necessary, we may assume that  is of finite measure. For a q.c. version U˛  of the ˛-potential of , let An D ¹x : U˛   nº. Since limn!1 .An / D 0, there exists n such that .F \ .X n An // > 0. As in the proof of Theorem 2.3.7, U˛ .1F \.XnAn /  / is bounded by the essential supremum of itself on F \ .X n An /, that is kU˛XnAn .1XnAn  /k1 ® ¯  ess.sup U˛ .1F \.XnAn /  /.x/ : x 2 F \ .X n An /  n. XnA

Hence, A D An and  D 1F \.XnA/   satisfy .F / > 0 and kU˛ This proves the result.

k1 < 1.

If ˛ > ˛0 and u 2 F , for any Borel set B, we defined the decomposition u D ˛ ˛ C F˛ XnB .u/ into a function uB 2 HB˛ and F˛ XnB .u/ 2 F XnB . For simplicity uB of notation, we assume that any function of F is quasi-continuous already. ˛ ˛ Lemma 2.4.5. If u1 , u2 2 F satisfies u1 D u2 q.e. on B, then .u1 /B D .u2 /B q.e. for ˛ > ˛0 .

Proof. Suppose u1 D u2 q.e. on B. By equation (2.4.11),   ˛ ˛ b XnB E˛ .u1 /B  .u2 /B , U ˛ .f  m/ D 0 ˛ for any f 2 L2 .X ; m/ such that f D 0 a.e. on B. On the other hand, since .u1 /B  ˛ ˛ ˛ XnB , it holds that .u2 /B D u1  u2 C F XnB .u2 /  F XnB .u1 / 2 F ˛ ˛ b XnB ˛ ˛ E˛ ..u1 /B  .u2 /B , U ˛ .f  m// D ..u1 /B  .u2 /B , f /. ˛ ˛ D .u2 /B m-a.e. on X nB. Since they coincide m-a.e on B, it follows Therefore, .u1 /B ˛ ˛ that .u1 /B D .u2 /B m-a.e. and hence q.e.

64

Chapter 2 Some analytic properties of Dirichlet forms

˛ To extend the definition of uB to 0  ˛  ˛0 and u which does not necessarily belong to F , let us consider the resolvent G˛ f for 0 < ˛  ˛0 and f 2 L2 .X ; m/ \ L1 .X ; m/. If .E, F / is transient, then we may assume that ˛ D 0. As we have seen in equation (2.3.13), for all ˛ > 0, and f 2 L1 .X ; m/, since the limit of the righthand side of the following equation exists for v 2 L1 .X ; m/ \ F , we can define E.G˛ f , v/ by

E.G˛ f , v/ D lim ˇ.G˛ f  ˇGˇ G˛ f , v/ D .f  ˛G˛ f , v/. ˇ !1

(2.4.12)

b ˛ g/ similarly for any ˛ > 0, g 2 L1 .X ; m/ and u 2 F \ L1 .X ; m/. If Define E.u, G b similarly for any f 2 L1 .X ; m/ .E, F / is transient, define E.Gf , v/ and E.u, Gg/ 1 1 b 2 L1 .X ; m/. and g 2 L .X ; m/ satisfying Gf 2 L .X ; m/ and Gg 2 For any Borel set B and f 2 L1 C .X ; m/ \ L .X ; m/, before equation (2.1.14) we XnB

defined G˛

f by G˛XnB f D lim G˛pg f p!1

for g D 1B . Since

pg G˛ f

2 F satisfies

E˛ .G˛pg f , G˛pg f /  E˛pg .G˛pg f , G˛pg f / D .f , G˛pg f /  pg

pg

1 kf k2 , ˛  ˛0

XnB

¹G˛ f º is E˛ -bounded. Hence G˛ f converges to G˛ f 2 F in .E, F /. If w 2 pg F XnB and ˛ > ˛0 , then by noting that .G˛ f , w/pgm D 0, we have E˛ .G˛pg f , w/ D E˛pg .G˛pg f , w/ D .f , w/. XnB

Hence, by letting p ! 1, G˛

f satisfies

E˛ .G˛XnB f , w/ D .f , w/

(2.4.13)

for any w 2 F XnB . By virtue of Lemma 1.3.5, since G pg .pg/  1, by letting p ! 1 it follows that G XnB 1B D 0. Similarly, for ˛ > ˛0 and f 2 L2 .X ; m/, the dual resolvent pg pg b pg b pg b XnB f D G ˛ f of G˛ f is defined as a solution of E˛ .u, G ˛ f / D .u, f /. Put G ˛ pg b ˛ f . By letting p tend to infinity in limp!1 G 

     b pg b pg G˛pg f , h D E pg G˛pg f , G ˛ h D f , G˛ h

holding for all f , h 2 L2 .X ; m/, we obtain 

     b ˛XnB h D f , G b ˛XnB h . G˛XnB f , h D E˛ G˛XnB f , G

65

Section 2.4 An orthogonal decomposition of the Dirichlet forms

b pg Furthermore, since G ˛ g decreases as p increases and pg b pg lim p.u, G ˛ g/gm D lim .G˛ .pgu/, g/  kuk1 hm, gi,

p!1

p!1

b˛ g D for any bounded strictly positive function u 2 F , it follows that limp!1 G XnB XnB b G ˛ 1B D 0 m-a.e. Dually, this implies that G˛ f D 0 a.e. on B. Therefore, XnB for any f 2 L2 .X ; m/ and ˛ > ˛0 , since G˛ f 2 F XnB and satisfies equation XnB (2.4.13), G˛ f D F˛ XnB .G˛ f /, that is, pg

˛ D G˛ f  G˛XnB f . .G˛ f /B

If ˛ > ˛0 and f 2 L1 .X ; m/, by choosing a sequence of functions ¹fn º  \ L2 .X ; m/ such that limn!1 fn D f a.e. and boundedly, we can XnB show that limn!1 E˛ .G˛ fn , w/ D limn!1 .fn , w/ D .f , w/ for any w 2 L1 .X ; m/ \ F XnB . Hence, for such functions of f and w, we can consider that XnB G˛ f 2 L1 .X ; m/ for f 2 L1 .X ; m/, ˛ > ˛0 and satisfies

L1 .X ; m/

E˛ .G˛XnB f , w/ D lim E˛ .G˛XnB fn , w/ D .f , w/.

(2.4.14)

n!1

XnB

For 0 < ˛  ˛0 < ˇ and f 2 L1 .X ; m/, define G˛ G˛XnB f D

1 X

f by

XnB nC1

.ˇ  ˛/n .Gˇ

/

f.

nD0 XnB

Since G˛

XnB

f D Gˇ

g with g D

P1

nD0 .ˇ

XnB n / f

 ˛/n .Gˇ

2 L1 .X ; m/, equa-

tion (2.4.14) implies, for any w 2 L1 .X ; m/ \ F XnB , that E.G˛XnB f , w/ D .g, w/  ˇ.G˛XnB g, w/ D .f , w/  ˛.G˛XnB f , w/.

(2.4.15)

b ˛XnB g and .G b ˛ g/˛ similarly b ˛ g, G Define the corresponding dual notions such as G B 1 for any g 2 L .X ; m/. ˇ

˛ by For u 2 F \ L1 .X ; m/ and ˛  ˛0 , by using uB with ˇ > ˛0 , define uB ˇ

ˇ

˛ uB D uB C .ˇ  ˛/G˛XnB uB .

(2.4.16)

˛ ˛ does not belong to F in general, equation (2.4.15) implies that E.uB , w/ Although uB 1 XnB is well defined and satisfies for any w 2 L .X ; m/ \ F , ˇ

ˇ

˛ E.uB , w/ D E.uB , w/ C .ˇ  ˛/E.G˛XnB uB , w/ ˇ

ˇ

ˇ

D ˇ.uB , w/ C .ˇ  ˛/.uB , w/  ˛.ˇ  ˛/.G˛XnB uB , w/ ˛ D ˛.uB , w/.

66

Chapter 2 Some analytic properties of Dirichlet forms

˛ ˛ Therefore the function uB defined by equation (2.4.16) for ˛  ˛0 satisfies uB 2 1 L .X ; m/ and ˛ , w/ D 0 E˛ .uB

for all w 2 L1 .X ; m/ \ F XnB .

(2.4.17)

ˇ

By virtue of Lemma 2.4.5, for ˇ > ˛0 , uB depends only on the value of u on B and ˛ for ˛  ˛0 . Hence, for any function u such that u D u1 a.e. on hence so does uB ˛ independently of the B for some function u1 2 L1 .X ; m/ \ F , we can define uB ˛ ˛ v satisfies b v Db v 1 a.e. on B for some choice of u1 by uB D .u1 /B . Furthermore, if b ˛ ,b v / by b v 1 2 L1 .X ; m/ \ F , then we can define E.uB ˛ ˛ ,b v / D E˛ .uB ,b v1/ E˛ .uB

v 1 2 F \ L1 .X ; m/, Similarly, for any functions b v such that b v Db v 1 on B for some b ˛ ˛ D .vO 1 /B is well defined. Furthermore, for any u such that u D u1 on B for b vB ˛ vB / by u1 2 F \ L1 .X ; m/, define E.u,b

b

b

˛ ˛ vB / D E˛ .u1 , .vO 1 /B /. E˛ .u,b

Then we have the following theorem. Theorem 2.4.6. For any Borel set B and any functions u,b v for which there exist functions u1 2 F \ L1 .X ; m/ and b v 1 2 F \ L1 .X ; m/ such that u D u1 ,b v Db v1 ˛ ˛ a.e. on B, E.uB ,b vB / is well defined independently of the choice of u1 ,b v 1 and satisfies.

b

˛ ˛ ˛ ˛ ˛ ˛ ,b v / D E˛ .u,b vB / D E˛ .uB ,b vB / D E˛ ..u1 /B , .vO 1 /B /. E˛ .uB

(2.4.18)

This result also holds for ˛ D 0 if .E, F / is transient. If  2 S00 , then U˛  2 F \ L1 .X ; m/ for any ˛ > ˛0 . For any 0 < ˛  ˛0 , we defined U˛  by equation (2.3.14). Then it satisfies E.U˛ , v/ D E.Uˇ , v/ C .ˇ  ˛/.Uˇ   ˛G˛ Uˇ , v/ D h, vi Q  ˛.U˛ , v/ XnB

for all v 2 F \ L1 .X ; m/. Furthermore, put U˛ tions (2.4.17) and (2.4.19) imply

(2.4.19) ˛  D U˛   .U˛ /B . Then equa-

˛ , v/ D h, vi Q E˛ .U˛XnB , v/ D E˛ .U˛   .U˛ /B

(2.4.20)

for any q.c. modification vQ of v 2 F XnB \ L1 .X n B; m/. In particular, if .E, F / is transient, then equation (2.4.20) holds for all  satisfying kU XnA k1 < 1. For  2 S0 , let

E .u, v/ D E.u, v/ C .u, v/

for u, v 2 F \ C0 .X /, where ., / is the inner product in L2 .X ; /.

(2.4.21)

Section 2.4 An orthogonal decomposition of the Dirichlet forms

67

Lemma 2.4.7. .E , F \ C0 .X // is closable on L2 .X ; m/.

Proof. Let ¹un º  F \ C0 .X / be an E˛ -Cauchy sequence converging to zero in L2 .X ; m/. Then, by the closability of .E, F \ C0 .X // in L2 .X ; m/, it holds that limn!1 E.un , un / D 0. Since ¹un º is a Cauchy sequence in L2 .X ; /, limn!1 un D u in L2 .X ; / for some function u. By equation (2.3.2), it then follows that u D 0 -a.e. Hence we have limn!1 E .un , un / D 0. Denote by .E , F / the Dirichlet form determined by the smallest closed extension of .E , F \ C0 .X // on L2 .X ; m/. Obviously F  F \ L2 .X ; /. We will see in Theorem 4.3.1 that F D F \ L2 .X ; /. In Section 1.4, we introduced a bilinear form A by using a strictly positive ıhı coexcessive function b hı . In the following, we shall fix a ı-coexcessive function b such that b hı is integrable and bounded from below by a positive constant on each set Bn for some nest ¹Bn º. The existence of such a function is clear if 1 is ı-coexcessive. In the general case, if ı  ˛0 , for a strictly positive bounded integrable function f , let b ı f such that R bı f is continuous and positive on each bı f be a q.c. modification of G R bı f . Then b hı D R hı is bounded from below on every comset Fn of a nest ¹Fn º. Put b pact subset of Fn . Furthermore, in the next theorem, we shall show that, for ı > ˛0 , hı is bounded from below by a positive constant on there exists b hı 2 F such that b every compact set. Theorem 2.4.8. For any ı > 0, there exists a ı-coexcessive function b hı 2 L1 .X ; m/ b and a nest ¹Fn º such that hı is bounded from below by a positive constant on every hı can be taken as a q.e. strictly positive integrable Fn . In particular, if ı > ˛0 , then b b function of F such that hı is bounded from below by a positive constant on each compact set. Proof. For any ı > 0, the existence of b hı has already been shown above. Assume that ı > ˛0 and take an increasing sequence of relatively compact open sets ¹Bn º such that ı eB be a q.c. modification of the ı-coequilibrium [n1 Bn D X . For each n  1, let b n potential of Bn . Then it belongs to F and exceeds 1 on Bn q.e. on Bn . By virtue of equations (2.3.9) and (2.3.12), there exists the ı-coequilibrium measure b

ın such that ı bı b b eB DU

ın and b

ın .X / D b

ın .BN n / D Cap.ı/ .Bn / < 1. In particular, by using a n q.c. modification Rı f of Gı f , we have Z Z 1 ı b eB .x/m.dx/ D Rı 1.x/b

ın .dx/  Cap.ı/ .Bn /. n ı X X ı Hence b eB is integrable. Define the function b hı by n

b hı .x/ D

1 X nD1

 1 ı 2n Cap.ı/ .Bn / _ 1 b eBn .x/.

(2.4.22)

68

Chapter 2 Some analytic properties of Dirichlet forms

Then 1 X

Eı .b hı , b hı /  Kı

 1  1 2nm Cap.ı/ .Bn / _ 1  Cap.ı/ .Bm / _ 1

n,mD1 ı ı 1=2 ı ı  Eı .b eB ,b eB / Eı .b eB ,b eB /1=2 n n m m 1 X

 Kı

 1=2  1=2 2nm .Cap.ı/ .Bn / _ 1  Cap.ı/ .Bm / _ 1

n,mD1 .ı/

 Cap

.Bn /1=2 Cap.ı/ .Bm /1=2

 Kı . Therefore b hı becomes the limit of an increasing sequence of uniformly Eı -bounded hı can be considered a quasifunctions of F and hence b hı 2 F . In particular, b continuous function. Clearly, b hı is bounded from below on Bn by 2n .Cap.ı/ .Bn // _ 1/1 . Furthermore Z X

1 X

 1 1 2n Cap.ı/ .Bn / _ 1 Cap.ı/ .Bn / ı nD1 1  < 1. ı

b hı .x/m.dx/ 

From now on, if not stated otherwise, we assume that b hı is the ı-coexcessive function given by Theorem 2.4.8. In equation (1.4.6), we defined A.u, v/ by A.u, v/ D lim˛!1 ˛.u  ˛G˛ u, v/b m and G .ı/ D ¹u 2 L2 .X ; m b/ : A.u, u/ existsº. Furthermore, we define A.u, v/ simb / and v 2 L1 .X ; m b / if the limit exists. If u is ˛-excessive ilarly for u 2 L1 .X ; m ˛ of u on B is defined by equation and B is a Borel set, then the ˛-reduced function uB (2.4.2), which is the smallest ˛-excessive function dominating u on B. Furthermore, ˛ D U˛ B for some B 2 S0 . it can be written as uB Theorem 2.4.9. Suppose that u is a bounded ˛-excessive function for ˛ > ˛0 and B ˛ 2 G .ı/ and satisfies is a Borel set with compact closure. Then uB ˛ A˛ .uB , v/ D hB , b hı vi

(2.4.23)

for all v 2 F such that vb hı 2 L1 .X ; B /. ˛ ˛ Proof. By virtue of Lemma 1.4.2, since u ^ uB 2 F and uB depend only on the value of u on B by Theorem 2.4.6, we may assume that u 2 F . Furthermore, since R B ˛ bB b b b N , hı / < 1. hı coincides with a function hı 2 F q.e. on B, X hı dB D E˛ .uB hı 2 L1 .X ; m/ \ Furthermore, for any q.c. function v 2 Fb , since b hı jvj  kvk1b ˛ , for any  > 0, there exists a relatively compact L1 .X ; B / and ˇRˇ C˛ B  uB

69

Section 2.4 An orthogonal decomposition of the Dirichlet forms

open set A such that ˛ bˇ C˛ .1XnAb hı v/i  .uB hı v/  kuk1 kvk1 , 1XnAb hB , ˇ R

Z

b hı d m < .

XnA

hı v 2 L2 .X ; m/ and b hı v is quasi-continuous on A, On the other hand, since 1Ab R b b b hı v/dB limˇ !1 ˇ Rˇ C˛ .1A hı v/ D 1A hı v q.e. and hence limˇ !1 X ˇRˇ C˛ .1Ab R b b b D A hı vdB by Lebesgue theorem. Therefore, limˇ !1 hB , ˇ Rˇ C˛ .hı v/i D hı vi. This implies that hB , b .ı/

˛ ˛  ˇR˛ıCˇ uB , v/b lim ˇ.uB m   D lim ˇ U˛ B  ˇRˇ C˛ U˛ B , b hı v

ˇ !1

ˇ !1

D lim

ˇ !1



 bˇ C˛ .b ˇ.Uˇ C˛ B , v b D lim hB , ˇ R hı v/i m ˇ !1

hı vi. D hB , b ˛ ˛ , it follows that uB 2 G .ı/ . Furthermore, since vb hı 2 L1 .X ; B /, By putting v D uB ˛ A˛ .uB , v/ D hB , b hı vi.

(2.4.24)

This clearly implies equation (2.4.23). ˛ 2 F \G .ı/ and satisfies Corollary 2.4.10. If b hı 2 F and B is a compact set, then eB ˛ ˛ ˛ b A˛ .eB , eB / D E˛ .eB , hı / < 1.

(2.4.25)

˛ D U ˛ for the ˛-equilibrium measure ˛ on B, noting that Proof. Since eB ˛ B B ˛ b bˇ C˛b bˇ C˛ .e ˛ b h h /  ˇ R  h and e Q D 1 on B q.e. we have ˇR ı ı ı B B ˛ ˛ ˛ ˛ ˛b A˛ .eB , eB / D lim ˇ.eB  ˇRˇ C˛ eB , eB hı / ˇ !1

˛b hı / D lim ˇ.Uˇ C˛ B˛ , eB ˇ !1

bˇ C˛ .e ˛ b D lim ˇh B˛ , R B hı /i ˇ !1 Z Z ˛b b hı d B˛ D hı d B˛ D eQB X

X

˛ b D E˛ .eB , hı /. B For any Borel set A  B, define LA by B LA D ¹u 2 F B : u  1 m-a.e. on Aº.

(2.4.26)

70

Chapter 2 Some analytic properties of Dirichlet forms

Similarly to equation (2.1.3), by using .E, F B / instead of .E, F /, we can define the ˛˛,B of the set A relative eA equilibrium potential eA˛,B and the ˛-coequilibrium potentialb B to .E, F /. For any subset A of B of the form A D O \B for an open set O, define the ˛,B eA / ˛-capacity Cap.˛/,B .A/ of A relative to .E, F B / by Cap.˛/,B .A/ D E˛ .eA˛,B ,b B B .˛/,B .A/ D 1 if LA D ;. Furthermore, for any subset C of B if LA ¤ ; and Cap put Cap.˛/,B .C / D inf¹Cap.˛/,B .A/ : A D B \ O C , O is openº. The following lemma will be shown in Theorem 3.5.6 (iii). Lemma 2.4.11. If B is open, then a subset A of B is of zero capacity relative to .E, F / if and only if it is relative to .E, F B /. By this lemma, it holds that F A D ¹u 2 F B : e u D 0 q.e. on B n Aº.

(2.4.27)

For any Borel set B with compact closure and ˛0 < ı < ˛, since b hı is an ˛˛ of b hı on B belongs to F . coexcessive function of F , the ˛-coreduced function .b hı /B ı by Define b dB 1 ˛ ˛ b dB .b hı /B .x/ D .x/. (2.4.28) b hı .x/ ˛ b ˛ º which is equal to 1 is an ˛-coexcessive function relative to ¹G Clearly, b dB a.e. on B. .ı/

Lemma 2.4.12. Suppose that B is a Borel set with compact closure. Then, for any ˛ and b ˛ belong to G .ı/ and satisfy dB ˛0 < ı < ˛, eB ˛ b˛ ˛ A˛ .eB , d B /  A˛ .u, b dB /

(2.4.29)

for any u 2 F \ G .ı/ such that u  1B m-a.e. ˛ ˛ is quasi-continuous. By virtue of equation (2.3.9), eB Proof. We may assume that eB ˛ ˛ ˛ N Since is expressed as eB D U˛ B by means of the ˛-equilibrium measure B on B. 2 b b / such hı is an ˛-coexcessive function, for any ˇ > 0 and function u 2 L .X ; m that 0  u  1,     ˛ ˛  ˇRˇ C˛ eB ,u b D ˇ U˛ B˛  ˇRˇ C˛ U˛ B˛ , b hı u ˇ eB m   bˇ C˛b hı i hı u  h B˛ , ˇ R D ˇ Uˇ C˛ B˛ , b ˛ b  h , hı i. B

˛ , it follows that e ˛ 2 G .ı/ . By putting u D eB B

71

Section 2.4 An orthogonal decomposition of the Dirichlet forms

˛ ˛ To show the corresponding assertion of b dB , let b B be the measure on BN satisfying ˛ ˛ ˛ ˛ b˛b B . Since b B is supported by BN and .b hı /B  b hı  c q.e. on the compact .b hı /B D U N set B for some positive constant c, using equation (2.4.6) we have   Z 2 ˛ b ˛ ˛ ˛ b b b hı /B db B K˛ E˛ .hı,n , hı,n /  E˛ ..hı /B , .hı /B D .b X

˛ N  cb B .B/.

(2.4.30)

b / such that 0  u  1, Hence, for any ˇ > 0 and u 2 L2 .X ; m     ˛ ˛ dB D ˇ u  ˇRˇ C˛ u, .b hı /B ˇ u  ˇRˇ C˛ u, b b m   ˛ ˛ b˛b D ˇhb B D ˇ u  ˇRˇ C˛ u, U B , R˛ u  ˇR˛ Rˇ C˛ ui ˛ ˛ D hb B , ˇRˇ C˛ ui  b B .X /.

(2.4.31)

˛ ˛ , it follows that b dB 2 G .ı/ . For any bounded quasiIn particular, by taking u D b dB continuous function u 2 F , by letting ˇ tend to infinity in equation (2.4.31) we obtain ˛ ˛ dB / D hb B , ui which implies equation (2.4.29). that A˛ .u, b

Assume that .E, F / is transient and ˛0 D 0. Then the extended Dirichlet form .E, Fe / is complete. Hence the results of this chapter can be extended to any function u 2 Fe . In particular, the 0-order capacity Cap.0/ and the 0-equilibrium potential eA0 of an open set A is defined and satisfies equation (2.1.15) for any ˛, ˇ  0. Furthermore, for any Borel set A, Cap.˛/ .A/ D 0 if and only if Cap.0/ .A/ D 0. Similarly to Theorem u. 2.2.3, any function u 2 Fe has a q.c. modification e XnA be the family defined by For a Borel set A, let Fe u D 0 q.e. on Aº. FeXnA D ¹u 2 Fe : e

(2.4.32)

XnA

is a closed convex set, there exists a projection F XnA .u/ of u 2 Fe on Since Fe XnA .0/ Fe . As in equation (2.4.10), for uA .u/ D u  F XnA .u/ .0/

u D uA C F XnA .u/

(2.4.33) .0/

is an E-orthogonal decomposition of u 2 Fe into the elements of uA 2 HA0 D ¹u 2 XnA

Fe : E.u, v/ D 0 for all v 2 F XnA º and F XnA .u/ 2 Fe

.

Chapter 3

Markov processes

In this chapter, we construct a Hunt process corresponding to a regular Dirichlet form. In Section 3.1, we shall state some general properties of Hunt processes. In particular, some basic properties of excessive functions are shown in Section 3.2. For a regular Dirichlet form, by choosing a suitable modification ¹R˛ º of the resolvent ¹G˛ º and using the Ray process corresponding to it, we construct in Section 3.3 a Hunt process associated with the Dirichlet form. Although the dual Hunt process does not exist in general, by changing the basic measure by a coexcessive function we can obtain a dual Hunt process relative to the changed measure. By interpolating this Hunt process, we introduce a dual pseudo Hunt process relative to the original basic measure which behaves like an ordinary Hunt process. In Section 3.4, the negligible sets are characterized in terms of the hitting probability of the Hunt process. The subprocesses of the Hunt process on the Borel sets and their correspondences to the Dirichlet forms are investigated in Section 3.5. Furthermore, analytic characterizations of the killing measures and the jumping measures are also presented in this section.

3.1 Hunt processes Let X be a locally compact separable metric space and B be its Borel  -field. For a probability measure  on X , the -completion of B is denoted by B . Put B  D \B , where the intersection is taken for all probability measures  on X . An element of B  is called a universally measurable set. The family of all bounded (resp. nonnegative) B-measurable functions is denoted by Bb (resp. B C ). The quadruplet ., M, ¹X t º t0 , P / is called a stochastic process with state space .X , B/ if X t is a measurable map from a probability space ., M, P / to .X , B/ for each t  0. Let X D X [ ¹º be the one point compactification of X if X is not compact and add  as an isolated point if X is compact. Let B be the  -field generated by B and ¹º. A family of stochastic process M D ., M, ¹X t º t0 , ¹Px ºx2X / is said to be a Markov process on .X , B/ (with admissible family ¹M t º) if it satisfies the following conditions: (M.1) (i) X1 .!/ D , for all ! 2 ; (ii) X t .!/ D  if t  .!/ D inf¹t  0 : X t .!/ D º; (iii) for each t  0 there exists a map  t :  !  such that Xs ı  t D XsCt ; (iv) for each ! 2 , X t .!/ is right continuous on Œ0, 1/ and has left limit on .0, 1/ relative to t .

73

Section 3.1 Hunt processes

(M.2) P .X t 2 E/ is B-measurable for all t  0 and E 2 B. (M.3) There exists a family ¹M t º of increasing sub- -fields of M such that ¹! : X t .!/ 2 Eº 2 M t for any E 2 B and Px .X tCs 2 E j M t / D PX t .Xs 2 E/

Px -a.s.

(3.1.1)

for all x 2 E and s, t  0. (M.4) P .X t D / D 1 for all t  0. (M.5) Px .X0 D x/ D 1 for all x 2 X . Given a Markov process M, let us introduce the  -fields 0 D  ¹Xs : 0  s < 1º, F1

F t0 D  ¹Xs : 0  s  t º. (3.1.2) R For a given probability measure  on X let P .ƒ/ D X Px .ƒ/.dx/. Denote by 0 relative to P . Also, let F be the completion of F 0 in F , F the completion of F1 t T t that is the  -field generated by F t0 and P -null sets of F . Put F1 D F1 and T T 0 0 F t D F t . If M is a Markov process with admissible family F tC s>t F t , then F t and F t are right continuous. A map  :  ! Œ0, 1 is called a stopping time (with respect to ¹M t º) if ¹  t º 2 M t for all t  0. For a stopping time  , define the  -field M by M D ¹ƒ 2 M : ƒ \ ¹  t º 2 M t , for all t  0º. Let M be a Markov process on .X , B/. If it satisfies M t D M tC and P .XCs 2 E j M / D PX .Xs 2 E/ P -a.s.

(3.1.3)

for all probability measure  on X , E 2 B , s  0 and ¹M t º-stopping time  , then M is called a strong Markov process A Markov process M is said to be quasi-left continuous on .0, 1/ if, for any increasing sequence of stopping times n such that limn!1 n D  , P . lim Xn D X ,  < 1/ D P . < 1/ n!1

(3.1.4)

for all . If a strong Markov process M is quasi-left continuous on .0, 1/, then M is called a Hunt process with admissible family ¹M t º. In the definition of a Hunt process, if equation (3.1.4) is replaced by P . lim Xn D X ,  < / D P . < /, n!1

then M is called a standard process. A Markov process M is a Hunt process with admissible family ¹M t º if and only if it is a Hunt process with admissible family ¹F t º. Hence, in the following, we take ¹F t º as the admissible family of Hunt processes. For any Borel set B  X , we define

74

Chapter 3 Markov processes

the hitting time B and the first exit time B of B by B D inf¹t > 0 : X t 2 Bº, B D inf¹t > 0 : X t 2 X n Bº,

(3.1.5)

where B and B are considered as infinity if the set in the braces of the definition is empty. Obviously B D XnB . Also, put DB D inf¹t  0 : X t 2 Bº and I.B/ D E .e ˛DB /. Then I.B/ is increasing and strongly subadditive. Also it satisfies [  An is increasing ) I An D lim I.An / n

Fn is decreasing, Fn is closed ) I

\

n!1

 Fn D lim I.Fn /. n!1

n

(3.1.6)

(3.1.7) (3.1.8)

Equation (3.1.8) is a consequence of quasi-left continuity of the process. T Hence, for any closed set F , by taking decreasing sequence of open sets such that n GN n D F , we see that I.F / D limn!1 I.Gn /. Thus I./ defines a Choquet capacity. In particular any Borel set is capacitable, that is, for any Borel set A, inf¹I.G/ : A  G, openº D sup¹I.Fn / : A F , closedº. Since DG  DA  DF for F  A  G, this gives the following theorem. Theorem 3.1.1. For any probability measure  on X and a Borel set A, there exists an increasing sequence of compact sets ¹Kn º and decreasing open sets Gn such that Kn  A  G and limn!1 DKn D DA D limn!1 DGn P -a.s. Corollary 3.1.2. For any probability measure  on X and a Borel set A, there exists an increasing sequence ¹Kn º of compact subsets of A such that A D limn!1 Kn a.s. P . Proof. Let sk # 0. For each k, take an increasing sequence of compact sets ¹Knk ºn1 such that Knk  A and limn!1 DKnk D DA , P k -a.s., where k .B/ D P .Xsk 2 B/. Then limn!1 DKnk ı sk D DA ı sk P -a.s. Set Kn D Kn0 [ Kn1 [    [ Knn , then Kn  A and limn!1 DKn ı sk D DA ı sk P -a.s. Hence lim Kn  lim lim .sk C DKn ı sk / D lim .sk C DA ı sk / D A .

n!1

k!1 n!1

k!1

The converse inequality is obvious. A set B  X is called nearly Borel measurable if for any probability measure  on X there exist B1 , B2 2 B such that B1  B  B2 and P .X t 2 B2 n B1 , for some t  0/ D 0.

75

Section 3.1 Hunt processes

The family of all nearly Borel measurable subsets of X is denoted by B n . Obviously B  B n  B  and Theorem 3.1.1 as well as Corollary 3.1.2 hold for A 2 B n . In particular, the hitting time of a nearly Borel set B is an F t -stopping time. Hence by the Blumenthal 0-1 law, Px .ƒ/ D 0 or 1 for x 2 X and ƒ 2 F0 ,

(3.1.9)

n . which yields that Px .B D 0/ D 0 or 1 for any x 2 X , B 2 B

n , then a point x is said to be regular for B if Px .B D 0/ D 1, Definition 6. If B 2 B

and irregular for B if Px .B D 0/ D 0. The set of all regular (resp. irregular) points for B is denoted by B r (resp. B irr )

If x 2 X n B, then Px .DB D B / D 1. This also holds if x 2 B r , in fact 0  DB  B D 0 Px -a.s. In particular, if .B n B r / D 0, then P .DB D B / D 1. Hence, by Theorem 3.1.1, if .B n B r / D 0, then P . lim Gn D B / D 1

(3.1.10)

n!1

for any closed set B and a decreasing sequence of open sets Gn such that

T

N D B.

n Gn

n , then Lemma 3.1.3. Let B 2 B

XB 2 B [ B r a.s. on ¹B < 1º.

(3.1.11)

Proof. Since X t 2 X n B for 0 < t < B , if XB 2 X n B then B D B C B ı B . Hence, for any , P .XB 2 X n B, B < 1/ D P .XB 2 X n B, B ı B D 0, B < 1/   D E PXB .B D 0/; XB 2 X n B, B < 1 . This implies that PXB .B D 0/ D 1 for P -a.s. on ¹XB 2 X n B, B < 1º, that is XB 2 B r . A non-negative function L t .!/ is called a continuous multiplicative functional if L t .!/ is F t -measurable for all t  0 and there exists a set ƒ 2 F with Px .ƒ/ D 1 for all x 2 X such that, for all ! 2 ƒ, L t .!/ is continuous relative to t  0 and satisfies LsCt .!/ D L t .!/Ls . t !/ for all s, t  0. For a given Hunt process M D ., F , F t , X t , Px , / and a continuous multiplicative functional L t satisfying L0 .!/ D 1 and 0  L t .!/  1, pL t f .x/ D Ex .L t f .X t //

for all t > 0, x 2 X

(3.1.12)

L defines function  L La transition  on .X , B/. Let us defineL a Hunt process M D L L L L  , F , F t , X t , Px ,  with transition function p t as follows: Let ˇ./ on

76

Chapter 3 Markov processes

.Œ0, 1/, B.Œ0, 1//, .// be a random variable such that .¹ : ˇ./ > t º/ D L t .!/. Put L D Œ0, 1/, F L D F _B.Œ0, 1//, F tL D  .F , B.Œ0, 1///, PxL D Px  ,  L D  ^ ˇ and X tL .!, / D X t .!/ for t <  L , X tL .!, / D  for t   L . That is, the following result holds.1 Theorem 3.1.4. For a given Hunt process M D .X t , , Px / and a continuous multiplicative functional L t satisfying L0 D 1, there exists a Hunt process ML D .X tL ,  L , PxL / with transition function p L t .

3.2 Excessive functions and negligible sets Let ., M, P / be a probability space and ¹M t º an increasing sequence of right continuous complete sub- -fields of M. A stochastic process ¹Y t º t0 on ., M, P / is called ¹M t º-supermartingale (resp. martingale, submartingale) if (i)

Y t is ¹M t º-adapted, that is M t -measurable,

(ii)

Y t is P -integrable,

(iii)

Ys  E.Y t j Ms / (resp. Ys D E.Y t j Ms /, Ys  E.Y t j Ms /) a.s. for all 0  s < t < 1.

If ¹Y t º is a supermartingale, then there exists a modification limr#t,r2Q Yr D YNt such that YNt is right continuous, has left limit, YNt  Y t a.s. and P .Y t D YNt / D 1 for all t  0 if and only if E.Y t / is right continuous, where Q is the family of the rational numbers. We shall assume that ¹Y t º is right continuous already. If  and  are bounded stopping times such that    , then Y  E.Y j M /.

(3.2.1)

This property is called Doob’s optimal sampling theorem. The corresponding results for martingales and submartingales also hold. If Y is a submartingale, then for any  > 0 and T ,     (3.2.2) P sup Y t >   E Y t : sup Y t >  . t2Œ0,T

t2Œ0,T

From this, it follows for any submartingale ¹Y t º and p > 1 that   1 P sup jY t j >   p E.jYT jp /  t2Œ0,T   p p  p E sup jY t j  E.jYT jp /. p1 t2Œ0,T

(3.2.3) (3.2.4)

If ¹Y t º is a supermartingale such that sup E.Y t / < 1, t

then lim t!1 Y t exists a.s. 1

See [14], Section III.3 and [55], Appendix A.2.

(3.2.5)

77

Section 3.2 Excessive functions and negligible sets

We shall assume that we are given a Hunt process M D ., F , F t , X t , Px /. Let p t be the transition function of M. A B  -measurable function u is called an ˛-supermedian function if u  0 and e ˛t p t u  u for all t  0. If u is ˛-supermedian and satisfies lim t!0 e ˛t p t u D u, then u is called an ˛-excessive function of M. The 0-supermedian and 0-excessive functions are simply called supermedian and excessive functions, respectively. Let R˛ be the resolvent of p t : Z 1 R˛ .x, E/ D e ˛t p t .x, E/dt (3.2.6) 0

R

and set R˛ f .x/ D R˛ .x, dy/f .y/. From the definition, it is easy to see that, for any increasing sequence of ˛-excessive functions ¹un º of M, limn!1 un is ˛-excessive.  , since Also, for any f 2 bBC Z 1 e ˛s ps f .x/ds, (3.2.7) e ˛t p t R˛ f .x/ D t

R˛ f is ˛-excessive. Lemma 3.2.1. If u is an ˛-excessive function of M, then there exists a sequence of bounded non-negative B  -measurable functions ¹fn º such that lim˛!1 R˛ fn D u increasingly. Proof. Noting that e ˛t p t .u ^ n/ is increasing relative to t # 0 and n " 1, vn D lim t!0 e ˛t p t .u ^ n/ is a bounded ˛-excessive function of M for each n and limn!1 vn D u. Hence, it is enough to show that vn is approximated by functions of the form R˛ f . This follows from vn D limk!1 kRkC˛ vn D limk!1 R˛ fk for fk D k.vn  kRkC˛ vn /. Lemma 3.2.2. Let u be an ˛-excessive function of M and B a nearly Borel set. Then HB˛ u is ˛-excessive, where HB˛ u.x/ D Ex .e ˛B u.XB //. Proof. By virtue of Lemma 3.2.1, it is enough to show the result for u D R˛ f with a non-negative bounded B  -measurable unction f . In this case, HB˛ u.x/ D  R 1 ˛s Ex B e f .Xs /ds . Hence e

˛t

Z p t HB˛ u.x/

D Ex

1

e tCB ı t

˛s

 f .Xs /ds .

Since t C B ı  t  B and t C B ı  t # B as t # 0, the result follows. Lemma 3.2.3. Suppose that u is ˛-excessive, B 2 B n and x 2 B r . Then inf¹u.y/ : y 2 Bº  u.x/  sup¹u.y/ : y 2 Bº.

78

Chapter 3 Markov processes

Proof. Let K be a compact subset of B. Then, using Lemma 3.2.2, we have u.x/  HK˛ u.x/  inf¹u.y/ : y 2 BºEx .e ˛K /. By virtue of Corollary 3.1.2, we can choose a compact subset K of B such that Ex .e ˛K / is arbitrarily close to Ex .e ˛B / D 1. Hence we have the first inequality. For the second inequality, we shall assume that u D R˛ f for a non-negative bounded B  -measurable function f . If K is a compact subset of B, then  Z K   e ˛t f .X t /dt C Ex e ˛K u.XK / R˛ f .x/  Ex  Z0 K e ˛t f .X t /dt C sup¹u.y/ : y 2 Bº.  Ex 0

Since x 2 there also exists an increasing sequence of compact sets Kn  B such that Kn # B D 0 a.s. Px . The latter inequality follows by putting K D Kn and letting n tend to infinity in the above inequality. For a general ˛-excessive function u, approximating u by an increasing sequence of ˛-excessive function R˛ fn given by Lemma 3.2.1, the result holds. Br ,

Theorem 3.2.4. Let u be an ˛-excessive function. Then (i)

u is nearly Borel measurable,

(ii)

t 7! u.X t / is right continuous on Œ0, 1/ and has left limit on .0, 1/ a.s.,

(iii) if A D ¹u < 1º, then u.X t / < 1 for all t > A a.s. Proof. For the proof of (i), by Lemma 3.2.1, it is enough to assume that u D R˛ f for a non-negative bounded B  -measurable function f . Given a measure  on X , let R .B/ D R˛ .s, B/.dx/. Then there exist bounded B-measurable functions f1 and f2 such that f1  f  f2 and .f2  f1 / D 0. Hence R˛ f1  R˛ f  R˛ f2 and Z E .R˛ .f2  f1 /.X t // D .dy/p t R˛ .f2  f1 /.y/ Z ˛t  e .dy/R˛ .f2  f1 /.y/ D 0. Thus, it follows that R˛ f1 .X t /R˛ f2 .X t / D 0 for all rational t  0 a.s. P . Hence (i) follows once (ii) is shown for any bounded Borel measurable ˛-excessive function u. For the proof of (ii), first assume that u is bounded. Let 0 D 0, nC1 D n C  ı n for  D inf¹t : ju.X t /  u.X0 /j > º. Then Yn D u.Xn / is a bounded supermartingale and hence limn!1 u.Xn / exists. On the other hand, since  equals the hitting time of B D ¹y : ju.y/  u.x/j > º Px -a.s., X 2 B [ B r from Lemma 3.1.3. Therefore, Lemma 3.2.3 implies that ju.X0 /  u.X /j   a.s. on

79

Section 3.2 Excessive functions and negligible sets

¹ < 1º. Hence the existence of limn!1 u.Xn / implies that Px .limn!1 n D 1/ D 1 for all  > 0 and x. This shows (ii) for any bounded ˛-excessive function u. For a general excessive function u, by noting that 1  e u.x/ is a bounded ˛-excessive function by Jensen’s inequality, (ii) holds. (iii) Let A D ¹u < 1º and B D ¹u D 1º. By virtue of Lemma 3.2.3, B r  B. Also the inequality HB˛ u.x/  u.x/ implies that Px .B < 1/ D 0 for x 2 A. Let Kn be an increasing sequence of compact subsets of A such that Px .limn!1 Kn D A / D 1. Then Px .B ı A < 1/ D lim Px .B ı Kn < 1/ n!1   D lim Ex PXK .B < 1/ D 0. n!1

n

This gives the result. Corollary 3.2.5. If B 2 B n then B r 2 B n . Proof. The function .x/ D Ex .e ˛B / is ˛-excessive and hence nearly Borel measurable. Hence B r D ¹x : .x/ D 1º 2 B n . A set A  X is called a polar set if there exists B 2 B n such that A  B and Px .B < 1/ D 0 for all x. A set A  X is called thin if there exists B 2 B n such that A  B and Px .B D 0/ D 0 for all x. If A is contained in a countable union of thin sets, then A is called a semipolar set. Theorem 3.2.6. If B 2 B n then B n B r is semipolar.

S Proof. Let Bn D B \¹x : HB˛ 1.x/  11=nº. Then B nB r D n Bn . If HB˛ 1.x/ < 1, then x … B r and hence x … Bnr . On the other hand, if HB˛ 1.x/ D 1, then x 2 B r and hence x is not a regular point of Bn by Lemma 3.2.3. Therefore Bnr D ; that is, Bn is a thin set. Theorem 3.2.7. If B is semipolar, then ¹t : X t 2 Bº is at most a countable set a.s. Px for all x.

Proof. We may suppose that B is a thin set of B n . Define the set Bn as in the proof of Theorem 3.2.6. Then we can reduce the proof to the case B D Bn for some n. Let 1 D B and kC1 D k C 1 ı k . Since B r D ;, XB 2 B D Bn and hence Xk 2 B on ¹k < 1º. This shows that EXk .e B /  1  1=n and hence       Ex e ˛kC1 D Ex Ek e ˛B e ˛k   1 kC1 1 Ex .e ˛k /  1  .  1 n n Therefore limk!1 Ex .e ˛kC1 / D 0 and hence Px .limk!1 k D 1/ D 1. Since X t 2 X n B for k < t < kC1 , the assertion of the theorem holds.

80

Chapter 3 Markov processes

Theorem 3.2.8. Suppose that u is an ˛-excessive function. Then B D ¹u D 1º is polar if and only if R˛ .x, B/ D 0 for all x. Proof. Suppose that R˛ .x, B/ D 0, then there exists tn # 0 such that Px .X tn 2 B/ D 0. Hence Px .X t 2 B for some t > tn / D 0 by Theorem 3.2.4. Letting n ! 1, it follows that Px .B < 1/ D 0. The converse assertion is clear. A set A  X is called finely open if, for all x 2 A, there exists B 2 B n such that X n A  B and Px .B > 0/ D 1. The topology determined by finely open sets is called fine topology. Theorem 3.2.9. If u is ˛-excessive, then it is finely continuous. Proof. Let I be an open interval in Œ0, 1 and B D u1 .I /. Then by Lemma 3.2.3, Px .XnB D 0/ D 0 for all x 2 B. Hence B is finely open.

3.3 Hunt processes associated with a regular Dirichlet form In this section, we assume that we are given a regular Dirichlet form .E, F / on L2 .X ; m/ and see briefly the correspondence between .E, F / and its associated Hunt process. As we have seen in Chapter 1, there exits a sub-Markov resolvent G˛ f satisfying .E, F / by E.u, v/ D lim E ˛ .u, v/ D lim ˛.u  ˛G˛ u, v/ ˛!1

˛!1

(3.3.1)

and F D ¹u 2 L2 .X ; m/ : E.u, u/ existsº. Furthermore, in Chapter 2, for any f 2 L1 .X ; m/ and ˛ > 0, there corresponds a q.c. modification R˛ f of G˛ f . The purpose of this section is, by a suitable choice of the modification R˛ f of G˛ f , to show the existence of a Hunt process possessing ¹R˛ º as its resolvent. Since F \ C0 .X / is uniformly dense in C0 .X /, there exists a countable sub-family B0 of F \ C0 .X / such that (i)

B0 is a vector space over Q,

(ii)

B0 is uniformly dense in C0 .X /,

(iii)

B0 is inf-stable and, for any u 2 B0 , u ^ 1 2 B0 .

For the set Q of rational numbers and its non-negative sub-family QC , since ¹G˛ f : f 2 B0 , ˛ 2 QC º is a countable sub-family of functions possessing a q.c. modification, Lemma 2.2.1 and Theorem 2.2.5 yield that there exists a regular nest ¹Fk0 º and a family (3.3.2) H0 D ¹R˛.1/ f : f 2 B0 , ˛ 2 QC º, .1/

satisfying the following conditions, where R˛ f is a q.c. modification of G˛ f :

81

Section 3.3 Hunt processes associated with a regular Dirichlet form

(i)

H0  C1 .¹Fk0 º/;

(ii)

R˛ .f C g/.x/ D R˛ f .x/ C R˛ g.x/, ˛ 2 QC , f , g 2 B0 , x 2 Y0 D [Fk0 ;

(iii)

R˛ .qf /.x/ D qR˛ f .x/, ˛ 2 QC , q 2 Q, f 2 B0 , x 2 Y0 ;

(iv)

0  f  1 ) 0  ˛R˛ f .x/  1, ˛ 2 QC , x 2 Y0 ;

(v)

there exists a sequence ˛n 2 QC such that ˛n " 1 and limn!1 ˛n R˛n f .x/ D f .x/ for f 2 B0 , x 2 Y0 ;

(vi)

R˛ f .x/  Rˇ f .x/ C .˛  ˇ/R˛ Rˇ f .x/ D 0, ˛, ˇ 2 QC , x 2 Y0 , f 2 B0 .

.1/

.1/

.1/

.1/

.1/

.1/

.1/

.1/

.1/

.1/

.1/

.1/

According to the properties (ii), (iii) and (iv), for any x 2 Y0 and ˛ 2 QC , R˛ f .x/ can be extended to a positive continuous linear functional on C0 .X /. Hence there exists .1/ .1/ a kernel R˛ .x, dy/ on Y0  X such that ˛R˛ .x, X /  1 and Z R˛.1/ f .x/ D f .y/R˛.1/ .x, dy/ (3.3.3) X

.1/

for any x 2 Y0 and f 2 B0 . We shall extend it to all x 2 X by putting R˛ .x, / D 0 .1/ for x 2 X n Y0 . By using this kernel, define R˛ f .x/ for all ˛ 2 QC , x 2 X and measurable functions f by equation (3.3.3). .1/

Theorem 3.3.1. For all f 2 L2 .X ; m/\L1 .X ; m/ and ˛ 2 QC , R˛ f is a q.c. version of G˛ f . Proof. Assume that ˛ > ˛0 . If f 2 C0C .X /, then there exists ¹fn º  B0 such that limn!1 fn D f uniformly and in L2 .X ; m/. Then the family of q.c. versions .1/ .1/ ¹R˛ fn º of G˛ fn converges to R˛ f uniformly on X n N for some set N of zero .1/ capacity. Hence R˛ f is quasi-continuous. On the other hand, since ¹G˛ fn ºn1 converges to G˛ f in L2 .X ; m/ and hence m-a.e. by choosing a subsequence if necessary, .1/ we get that R˛ f is a modification of G˛ f . .1/ Put L D ¹f 2 L2C .X ; m/ : R˛ f is a q.c. version of G˛ f for all ˛ 2 Qº. We have shown that C0C .X /  L. Furthermore, it satisfies (i)

f1 , f2 2 L, c1 f1 C c2 f2  0 ) c1 f1 C c2 f2 2 L.

(ii)

fn 2 L, fn " f for f 2 L2 .X ; m/ ) f 2 L. .1/

In fact, (i) is clear. If fn 2 L increases to f 2 L2 .X ; m/, then R˛ fn increases to .1/ R˛ f . Since ¹G˛ fn º converges to G˛ f relative to E˛ , using Theorem 2.2.5, we ob.1/ tain that their q.c. modifications ¹R˛ fn º converges to GQ ˛ f quasi-everywhere. Hence .1/ R˛ f D GQ ˛ f q.e. This implies (ii). By the monotone class lemma, we then have L D L2 .X ; m/.

82

Chapter 3 Markov processes

If 0 < ˛  ˛0 , ˛ 2 Q and f 2 L2 .X ; m/\L1 .X ; m/, then by taking ˛0 < ˇ 2 Q, we can write 1 X .1/ .ˇ  ˛/n .Rˇ /nC1 f . (3.3.4) R˛.1/ f D nD1 .1/

Here, each term .ˇ  ˛/n .Rˇ /nC1 f of the right-hand side of equation (3.3.4) is quasi-continuous. Furthermore, since the convergence of the right-hand side is uniform quasi-everywhere, the assertion of the theorem for ˛  ˛0 holds. .1/

For simplicity of notation, we assume that ˛0 < 1 and put eA D eA . Take a countable base ¹An º of relatively compact open sets of X and let U be the collection of eA be a q.c. version the sets which can be written as finite union of ¹An º. For A 2 U, lete eA  1 on X . Let H1 be the smallest Q-cone satisfying of eA such that 0  e eA : A 2 Uº [ H0 , (H.1) H1 ¹e .1/

(H.2) ˛R˛ .H1 /  H1 , for all ˛ 2 QC , (H.3) H1 is inf-stable and u ^ 1 2 H1 for all u 2 H1 . Then H1 is countable. Hence we have the following lemma. Lemma 3.3.2. There exists a regular nest ¹Fk1 º satisfying (i)

H1  C1 .¹Fk1 º/, Fk1  Fk0 , for all k,

(ii)

e eA .x/ D 1, x 2 A \ Y1 , A 2 U, where Y1 D [Fk1 ,

(iii) A, B 2 U, A  B ) e eA .x/  e eB .x/, for all x 2 Y1 , (iv) A 2 U, ˛ 2 QC ) ˛R˛C1e eA .x/  e eA .x/, for all x 2 Y1 , (v)

.1/

.1/

.1/

.1/

f 2 H1 , ˛, ˇ 2 QC ) R˛ f .x/  Rˇ f .x/ C .˛  ˇ/R˛ Rˇ f .x/ D 0, for all x 2 Y1 , .1/

(vi) ˛n 2 QC , f 2 H1 ) ˛n R˛n f .x/ ! f .x/, for all x 2 Y1 . .1/

For any ˛ 2 QC , since G˛ IXnY1 D 0 m-a.e. and R˛ IXnY1 is a q.c. version of .1/ .1/ .1/ G˛ IXnY1 , there exists a F set Y1  Y1 such that Cap.X nY1 / D 0 and R˛ .x, X n .1/ Y1 / D 0 for all x 2 Y1 and ˛ 2 QC , where F is a set which can be written as countable union of closed sets. Similarly, there exists a decreasing sequence of F -sets .1/ .2/ .kC1/ .k/ / D 0 and R˛ .x, X n Y1 / D 0 Y1 Y1 Y1    such that Cap.X n Y1 .kC1/ .k/ for all x 2 Y1 and ˛ 2 QC . Put Y2 D \Y1 . Then Cap.X n Y2 / D 0 and .1/ R˛ .x, X n Y2 / D 0 for all x 2 Y2 and ˛ 2 QC . Define a kernel R˛ .x, dy/ by R˛ .x, A/ D

  ´ .1/ .1/ R˛ .x, A \ Y2 / C ˛1  R˛ .x, Y2 / 1A ./ 1 ˛ 1A ./

x 2 Y2 xD

(3.3.5)

Section 3.3 Hunt processes associated with a regular Dirichlet form

83

Then ¹R˛ º˛2QC is a sub-Markov resolvent on Y2 [ ¹º. Any function f on Y2 is considered a function on Y2 [¹º by putting f ./ D 0. Let J be the Q-cone generated by H1 and 1. Then it is a countable sub-family of C .¹Ek º/ which separates the points of Y2 [ ¹º. Suppose that J D ¹un ºn1 . Let YN be the completion of Y2 [ ¹º relative to the metric 1 X jun .x/  un .y/j 1  . (3.3.6) d.x, y/ D n 2 1 C jun .x/  un .y/j nD1 Then, any function of J has a continuous extension to YN . We shall denote by JN the family of functions constituting the continuous extensions of functions of J . Then JN is inf-stable, 1 2 JN and separates points of YN . Hence JN  JN D ¹u1  u2 : u1 , u2 2 JN º is dense in C .YN /. If uN 2 JN then u D uj N Y2 [¹ º , R˛ u 2 J and its continuous extension R˛ u belongs to JN for any u 2 H1 and ˛ 2 QC . Let RN ˛ be the operator from JN to N Y2 [¹ º . Then it is easy to see that RN ˛ is linear, JN defined by RN ˛ uN D R˛ u for u D uj N increasing and satisfies ˛ R˛ 1YN D 1YN . Thus RN ˛ can be extended to a positive linear operator of C .YN / ! C .YN /. Hence there exists a kernel RN ˛ .x, dy/ on YN such that Z RN ˛ u.x/ N D N RN ˛ .x, dy/u.y/. YN

For any uN 2 JN , since RN ˛ u.x/ N is uniformly continuous relative to ˛ 2 QC , RN ˛ is extended to a positive linear operator on C .YN / for any ˛ > 0. Hence RN ˛ can be considered as a kernel for any ˛ > 0 satisfying the resolvent equation. Thus we have a Ray resolvent on YN : (i) (ii)

RN ˛ .C .YN //  C .YN / for all ˛ > 0. JN separates points of YN .

N is right Hence, there exists a Markov transition function pN t on YN such that pN t u.x/ continuous with respect to t 2 .0, 1/ for all x 2 YN , f 2 C .YN / and satisfying RN ˛ D

Z

1

e ˛t pN t dt .

(3.3.7)

0

Let .x, / D lim˛!1 ˛ RN ˛ .x, / D lim t!0 pN t .x, / in the vague sense and BN D ¹x 2 YN : .x, / ¤ x º be the set of branching points. There is a corresponding Ray N D ., N FN , FNt , XN t , PN x /: process M N D ¹! : .0, 1/ ! YN , right continuous with left limitº, XN t .!/ D !.t /,  T T F 0 D  ¹XN x : 0  s  1º, F t0 D  ¹XN s : s  t º, FN D .F 0 / , FNt D .F t0 / and Z Z PN x .XN t1 2 E1 , : : : , XN tn 2En / D  pN t1 .x, dy1 /    pN tn tn1 .yn1 , dyn /. E1

En

84

Chapter 3 Markov processes

N is a strong Markov process satisfying Then FNt is right continuous and M N .M.1/ PN x .XN 0 2 E/ D .x, E/, for all x 2 YN . N is quasi-left continuous on YN n B, N that is, if ¹Tn º is an increasing sequence N .M.2/ M of stopping times increasing to T , then limn!1 XN Tn D XN T , PN x -a.s. on ¹T < N 1, limn!1 XN Tn 2 YN n Bº. N for all t  0/ D 1 for all x 2 Y2 . N .M.3/ PN x .XN t 2 YN n B, N .M.4/ If x 2 X [ ¹º, then pN t .x, / D p t .x, / and RN ˛ .x, / D R˛ .x, /. In particular, pN t .x, / and RN ˛ .x, / is supported by X [ ¹º if x 2 X [ ¹º. Lemma 3.3.3. Let DN k D inf¹t  0 : XN t 2 YN n Fk1 º and ek D eXnFk1 . Then the   N x e DN k to Y2 is a q.c. version of ek . restriction of E Proof. Let An be an increasing sequence of elements of U with union X n Fk1 . By virtue of the proof of Lemma 2.1.2 (iii), ¹eAn º converges to some ek 2 F weakly in .F , E1 / and limn!1 E1 .eAn , eAn / D E1 .ek , ek /. Hence eAn ! ek strongly in .F , E1 / N x .e DN k /. Since ˛G˛C1 .x/ D and, from Theorem 2.2.5,e eAn " e e k q.e. Let .x/ D E ˛ RN ˛C1 .x/  .x/ for a.e. x 2 X , jX is 1-excessive relative to ¹G˛ f º. Using Lemma 1.4.2, we obtain that jX ^ ek is a 1-excessive function belonging to F . By Corollary 2.3.2, we then see that jX ^ ek satisfies E1 . jX ^ ek , ek  jX ^ ek /  0. Furthermore, since jX ^ ek D 1 m-a.e. on X n Fk1 , E1 .ek , jX ^ ek  ek /  0 by equation (2.1.3). Therefore, E1 . jX ^ ek  ek , jX ^ ek  ek /  0 and hence e k .x/  ˛ RN ˛C1 .x/  .x/, ek  jX a.e. By operating ˛ RN ˛C1 , we have ˛ RN ˛C1e ek ! e e k q.e. for a suitable sequence ˛n " 1, Theorem 1.1.4 (iii) q.e. Since ˛n RN ˛n C1e and Theorem 2.2.5 implies that e e k .x/  .x/ q.e. eAn to YN and set dNk D Conversely, let dNAn be a continuous extension of e N N ek limn!1 dAn . We take a modification such that dk D 1 on X n Fk1 . Then dNk D e t N N N N q.e. on X . Set Z t .!/ N D e dk .X t .!//. Then .Z t , F t , Px / is a supermartingale for x 2 Y2 . Let T be a finite subset of the interval .a, b/ and put k .T / D min¹t 2 T : XN t 2 YN n Fk1 º with the convention that k .T / D b if ¹ º D ;. The optional sampling theorem shows that     N x .Za /  e N x Z .T /  E N x e k .T / : k .T / < b  E e k .x/, x 2 Y2 . E k Since YN n Fk1 is open in YN , by increasing T to the countable dense subset of R1C , we obtain that .x/  e e k .x/, x 2 Y2 . Since Cap.X n Fk1 / ! 0, e e k decreases to zero q.e. by Theorem 2.2.5, then Lemma 3.3.2 implies that, for q.e. x 2 Y2 , PN x .XN t 2 YN n Y2 or XN t 2 YN n Y2 , for some t 2 RC /  lim PN x .DN k < / D 0 k!1

85

Section 3.3 Hunt processes associated with a regular Dirichlet form

Modifying the set Y2 by a similar manner as the definition of Y2 , we can choose a regular nest ¹Fk º such that Fk  Fk1 and, for Y D [k Fk , PN x .XN t 2 Y , XN t 2 Y , for all t  0/ D 1,

for all x 2 Y .

(3.3.8)

N N : limk!1 D N Let  D ¹! 2  Y nFk D 1º, F D ¹ƒ \  : ƒ 2 F º, F t D ¹ƒ \  : ƒ 2 FNt º, X t .!/ D XN t .!/, ! 2 , Px D PN x j for x 2 Y . Noting that the induced topology on Y by YN coincides with initial topology, we see that ., F , F t , X t , Px / is a Hunt process on .Y , B.Y // and whose resolvent R˛ f is a q.c. modification of G˛ f for f 2 L2 .X ; m/ and ˛ > ˛0 . By letting each point of X n Y trap, we have the following theorem. Theorem 3.3.4. For a given regular Dirichlet form .E, F / on L2 .X ; m/, there exists a Hunt process M whose resolvent R˛ f is a q.c. modification of G˛ f for any f 2 L1 .X ; m/ and ˛ > 0. b ˛ f 2 F , there exists a q.c. modification For any ˛ > ˛0 and f 2 L2 .X ; m/, since G b ˛ f can be extended to b R˛ f . As we have seen after the proof of Theorem 1.1.5, G 1 b b˛ , an operator on L .X ; m/ with kG ˛ f kL1  .1=˛/kf kL1 . For the coresolvent G b b there also exists a resolvent kernel R˛ .x, dy/ such that R˛ f is a q.c. modification of b ˛ f for any f 2 L2 .X ; m/ and ˛ > ˛0 . Furthermore, there exists a, not necessarily G R1 b˛ .x, A/. p t .x, A/dt D R Markov, transition kernel b p t .x, A/ such that 0 e ˛t b As in Section 2.4, for ı > 0, we fix a strictly positive quasi-continuous ıhı is bounded coexcessive function b hı 2 L1 .X ; m/ such that, for some nest ¹Bn º, b from below by a positive constant on any compact subset of each Bn . In particular, hı is bounded from below by a positive constant on if ı > ˛0 , we may assume that b every compact set. As we have seen in Theorem 2.4.8, such a function exists. By using such a function as b hı , we defined a bilinear form .A, G .ı/ / by equation (1.4.6). b b b b.ı/ Then the sub-Markov resolvent defined by R ˛ f .x/ D .1=hı .x//R ˛Cı .hı f /.x/ 1 b /. Applying the preceding arguis also quasi-continuous for any f 2 L .X ; m .ı/ b b .1/ º ment to ¹R˛ º˛>0 instead of ¹R˛ º, we can see that there exists a nest ¹F k .ı/ bN , F bN , X b b bN , b b N N .ı/ D ., N F N P /, on  D ¹ ! O : .0, 1/ ! and a Ray process M t t x b YN , right continuous with left limitº satisfying the conditions of Lemma 3.3.2 and bN , D bN D inf¹t  0 : X bN 2 Y bN n FON 1 º and so on to repb2 , Y (M.1)(M.4). We denote by Y t k k b N .ı/ . To show the result corresponding to Theorem 3.3.4 resent the notions related to M b bN < N .ı/ .D b.ı/ º P , we need only to prove that lim for the dual resolvent ¹R ˛

˛>0

k!1

x

k

1/ D 0 q.e. For the proof, we note that a non-negative function b u is ˛-coexcessive b˛Cˇ b bˇ /, that is, it satisfies ˇ R u b u, if and only if b u=b hı is relative to the resolvent .R .ı/ b /. For any ˛-coexcessive function b .˛  ı)-coexcessive relative to .R u 2 F relative ˇ ˛ bˇ / and Borel set B, let b to .R uB be the ˛-coreduced function of b u on B defined in ˛ Section 2.4. We may assume that b u and b uB are quasi-continuous already.

86

Chapter 3 Markov processes

bˇ / and an open Lemma 3.3.5. For any ˛-coexcessive function b u 2 F relative to .R N .ı/ bN // q.e. ˛ N x .e .˛ı/DB .b .x/ D b hı .x/b E u=b hı /.X set B, b uB b NB D .ı/ NB bN //. Since b D N x .e .˛ı/b Proof. Put OB .x/ D b E .b u=b hı /.X u=b hı is .˛  ı/-coexcesbN DB

b.ı/ ºˇ >0 coincidb.ı/ , OB is also .˛  ı/-coexcessive relative to ¹R sive relative to R ˇ ˇ b b.ı/ º O ing with b u=hı on B. Furthermore, if is an .˛  ı/-coexcessive function of ¹R ˇ

.ı/ bN / is a b P x -supermartingale. Hence, coinciding with b u=b hı on B, then e .˛ı/t O .X t similarly to the proof of Lemma 3.3.3, we can obtain the inequality OB .x/  O .x/ b.ı/ / which q.e. This implies that b hı OB is a minimal .˛  ı/-coexcessive function of .R ˇ b b O coincides with b u=hı on B. This yields that hı B is a minimal ˛-coexcessive function bˇ / coinciding with b u on B, that is b hı OB D b u˛ . of .R B

For any strictly positive function g 2 L1 .X ; m/ \ L1 .X ; m/,  2 S00 and Bk D bN .1/ , it holds by equation (2.4.7) that X nF k ˛ ˛ ˛ b b˛b b˛b hı g/B hı g/B h, .R i D E˛ .U˛ , .R / D ..U˛ /B , hı g/. k k k

bN .1/ º is a nest, Since ¹F k lim

k!1

    ˛ b ˛ b .U˛ /B e , h g  kU k , h g D 0. ˛ 1 ı ı B k k

In the left-hand side, by virtue of Lemma 3.3.5, since  ˛  .˛ı/D  Bk b˛b b˛ .b bDB / hı g R .x/ D b hı .x/b E.ı/ .1=b hı /R hı g/.X x e k Bk  .˛ı/D  Bk b.ı/ bDB / E.ı/ R˛ı g.X Db hı .x/b x e k  Z 1 .ı/ .˛ı/t b b b e g.X t /dt , D hı .x/E DBk

.ı/ b N B < / D 0 q.e. Therefore, as in Theorem 3.3.4, there P x .D we obtain that limk!1 b k .ı/ b b.ı/ b .ı/ exists a Hunt process M whose resolvent R ˛ f is a q.c. modification of G ˛ f for 1 b /. any ˛ > 0 and f 2 L .X ; m Let M.ı/ be the subprocess of M relative to the multiplicative functional L t D e ıt defined by Theorem 3.1.4. Then we have the following result.

b.ı/ b .ı/ D .X bt, P Theorem 3.3.6. There exists a Hunt process M x / with resolvent .ı/ .ı/ .ı/ b b ¹R˛ º˛>0 . In particular, M is a dual Hunt process of M relative to the mea-

87

Section 3.3 Hunt processes associated with a regular Dirichlet form

sure m b Db hı  m, that is it satisfies Z Z   .ı/ b b t / g.y/b Ey.ı/ f .X Ex .g.X t // f .x/b m.dx/ D m.dy/ X

(3.3.9)

X

for all f , g 2 BC .X /. R hı d < 1. Then, for any ˛ > ı > Lemma 3.3.7. Suppose that  2 S00 satisfies X b .ı/ ˛0 , U˛  2 G and satisfies Z vQb hı d (3.3.10) A˛ .U˛ , v/ D X

for all v 2

G .ı/

\

L1 .X ; m b/.

Proof. Note that Uˇ Cı  D U˛  C .˛  ˇ  ı/Rˇ Cı U˛ , because each side satisfy Eˇ Cı .u, v/ D h, vi for any q.c. function v 2 F . We may assume that U˛  is quasi-continuous. Since limˇ !1 ˇRˇ Cı U˛  D U˛  in F , for any sequence ˇk " 1, limk!1 ˇk Rˇk Cı U˛  D U˛  q.e. and boundedly. Since R R R 1 b b˛b hı d  ˛ı b D XR X U˛ d m X hı d < 1, we have by Lebesgue theorem, D .U˛ , U˛ /b . Therefore, limk!1 ˇk .Rˇk Cı U˛ , U˛ /b m m lim ˇk .U˛   ˇk Rˇk Cı U˛ , U˛ /b m

k!1

D lim ˇk .Uˇk Cı   .˛  ı/Rˇk Cı U˛ , U˛ /b m k!1

D lim ˇk .Uˇk Cı , U˛ /b  .˛  ı/.U˛ , U˛ /b . m m k!1

Here, the first term of the right-hand side can be written as   b.ı/ U˛ i lim ˇk Uˇk Cı , U˛  b D lim ˇk hb hı  , R ˇk m k!1 k!1  Z 1 .ı/ ˇk t b b ˇk e U˛ .X t /dt D lim E hı  k!1 b  Z0 1 .ı/ t b t=ˇ /dt . E e U˛ .X D lim b k hı  k!1 b 0 Since U˛  2 F is quasi-continuous, there exists a nest ¹Fn º such that U˛  is continuous on Fn for all n. If y 2 Fn , then ! Z OFn .ı/ t b t=ˇ /dt D U˛ .y/. Ey lim b e U˛ .X k k!1

0

Then, noting that lim

Ey.ı/ lim b

n!1 k!1

Z

1

OFn

! b t=ˇ /dt e t U˛ .X k

Ey.ı/ .e  OFn / D 0  kU ˛ k1 lim b n!1

88

Chapter 3 Markov processes

for q.e. y and Z

.ı/ E lim b

.1XnFnb hı /

k!1

1

0

b t=ˇ /dt e t U˛ .X k



Z  kU˛ k1

b hı d

XnFn

for all n, we obtain that   D hb hı  , U˛ i. lim ˇk Uˇk Cı , U˛  b m

k!1

Therefore D hb hı  , U˛ i  .˛  ı/.U˛ , U˛ /b , lim ˇk .U˛   ˇk Rˇk Cı U˛ , U˛ /b m m

k!1

Z

yielding that A˛ .U˛ , U˛ / D

U˛ .x/b hı .x/.dx/.

X

The relation (3.3.10) follows similarly. b .ı/ to b˛ º˛>0 is not a sub-Markov resolvent, if we use the process M Although ¹R b define a measure P x on  by ! 1 .ı/ ıt bx ., t < b P :  \ ¹b  < tº (3.3.11) / D b hı .x/b Ex e b bt/ hı .X for  2 F t , then it holds for any f 2 L1 .X ; m/ and x 2 X that b Ex

 b t /f .X b t /dt e ˛t 1X .X 0  Z 1 b t /dt Db Ex e ˛t f .X 0  Z 1 .ı/ .˛ı/t b b b b b D hı .x/Ex e 1X .X t /f .X t /=hı .X t /dt

Z b 

0

Db hı .x/E.ı/ x

Z b  e

.˛ı/t

b t /=b b t /dt 1f .X hı .X



0

b b˛ f .x/. .f =b hı /.x/ D R Db hı .x/R ˛ı .ı/

b D .X bt, b b˛ º˛>0 as its reIn this sense, the new process M P x / defined above has ¹R solvent and so it is in duality with M D .X t , Px / relative to m: for example equation (3.3.9) can be considered as Z Z   b b t / g.x/m.dx/. Ex f .X Ex .g.X t // f .x/m.dx/ D (3.3.12) X

X

Section 3.4 Negligible sets for Hunt processes

89

Furthermore, equation (3.3.11) can be extended to any stopping time b  instead of b the fixed time t . By this extension, M satisfies the strong Markov property and is quasi-left continuous on .0, 1/. In fact, if b  is a stopping time, then for any Fb  measurable function F .!/ and m-integrable function f , the strong Markov property b .ı/ implies of M   b b b Ex F .!/f .X / : b  C t <  b Ct   .ı/  Ct/ b b b b Ex e ı.b F .!/f .X /= h . X / : b  C t <  Db hı .x/b ı b b Ct Ct     .ı/ ıb  b b b b b b b b f .X t /=hı .X t / : t <  : b  0;

90

Chapter 3 Markov processes

(iii) R˛ .x, A/ D 0 m-a.e. x 2 X for any ˛ > 0; (iv) b p t .x, A/ D 0 m-a.e. x 2 X for any t > 0; b˛ .x, A/ D 0 m-a.e. x 2 X for any ˛ > 0; (v) R Proof. (iii) , (v) follows from the duality relation. (ii) , (iii) and (iv) , (v) are obvious. Clearly, (i) implies (iii). To show (i) from (iii), suppose that R˛ .x, A/ D 0 m-a.e. Then, for any f 2 L2C .X ; m/, Z Z b˛ f .x/IA .x/m.dx/ 0 D lim ˛ R˛ .x, A/f .x/m.dx/ D lim ˛ R ˛!1 ˛!1 X X Z f .x/IA .x/m.dx/. D X

This implies that m.A/ D 0. A set N  X is called exceptional if there exists a nearly Borel set NQ N such that Pm .NQ < 1/ D 0. Lemma 3.4.2. A set N is exceptional if and only if there exists a nearly Borel set NN bm .b such that NN N and P  NN < 1/ D 0. bm .b b .b Proof. Since P  NN < 1/ D 0 if and only if P  B < 1/ D 0, it is enough to b m .ı/ .ı/ prove P .B < 1/ D b P .b  B < 1/ for any nearly Borel set B. Suppose first that b m b m B is an open set and let Tn D ¹tk : 1  k  nº be a finite set such that 0 < t1 < t2 < .ı/    < tn < t and Tn " Q \ .0, t /. Put tnC1 D t and ps .x, dy/ D e ıs ps .x, dy/. .ı/ Since limn!1 P .X tn … B, X t 2 B/ D 0 by the quasi-left continuity, by the b m duality relation in equation (3.3.9),  .ı/ .ı/  X ti 2 B for some 1  i  n P .B < t / D lim P b m m n!1 b n1 X  .ı/  X ti 2 B, X tk … B for all k  i C 1 P D lim b m n!1 .ı/

iD1

D lim

n!1

Z  D lim

n!1

n1 Z X iD1 X



B

Z

.ı/

XnB

p ti C1 ti .xi , dxiC1 /

p tn1 tn .xn1 , dxn /

iD1 XnB

XnB

m b.dx/

.ı/ p ti .x, dxi /

.ı/

XnB n Z X

Z

Z

.ı/

m b.dyn /b p tn tn1 .yn , dyn1 /

.ı/ b p ti C2 ti C1 .yiC2 , dyiC1 /

Z B

.ı/

.ı/

b p ti C1 ti .yiC1 , dyi /b p ti .yi , X /

Section 3.4 Negligible sets for Hunt processes

D lim

n!1

n X iD1

91

b.ı/ .X b tn tk … B, for all k  i C 1, X b tn ti 2 B/ P b m

b.ı/ .b DP  B < t /. b m For any nearly Borel set B, since m.B n B r / D 0, we obtain the result from equation (3.1.10). b .b Since P  < t/ D b hı .x/b E ..1=b hı /.X t / : b  NN < t /, Lemma 3.4.2 implies that m NN b b m b .b  < 1/ D 0. As in Lemma 3.2.2, HB˛ denotes N is exceptional if and only if P m NN b b .ı/,˛ u.x/ D the ˛-order hitting probability of the set B relative to M. Similarly let H B .ı/ ˛  B / and H ˛b  B u.X b b ˛ u.x/ D b b //. Also let e ˛ and b E .e e Ex .e .˛ı/b B B B be the ˛b b m B ˛ D .b ˛ =b hı /B hı equilibrium potential and ˛-coequilibrium potential respectively and b dB be the function introduced by equation (2.4.27). .ı/

Lemma 3.4.3. Assume that B is an open set of finite capacity. Then HB˛ 1 is a q.c. ver˛ ˛ b ˛ 1 (resp. H b .ı/,˛ı 1) is a q.c. version of b sion of eB for ˛ > ˛0 . Similarly, H eB B B (resp. b d ˛ ). B

˛ and HB˛ 1 are Proof. The proof is similar to Lemma 3.3.3 and Lemma 3.3.5. Since eB ˛ ˛ ˛-excessive relative to ¹Gˇ º and eB 2 F , Lemma 1.4.2 yields that eB ^HB˛ 1 2 F and ˛ ˛ ^ HB˛ 1, eB  ˛-excessive relative to ¹Gˇ º. Theorem 1.4.1 then implies that , E˛ .eB ˛ ˛ ˛ ˛ eB ^HB 1/  0. On the other hand, since eB ^HB 1  1 on B, equation (2.1.5) implies ˛ , e ˛ e ˛ ^H ˛ 1/  0. Hence E .e ˛ e ˛ ^H ˛ 1, e ˛ e ˛ ^H ˛ 1/  0. Therefore E˛ .eB ˛ B B B B B B B B B ˛ ˛ ˛ from which we obtain that eB  HB˛ 1 m-a.e. eB ^ HB˛ 1 D eB ˛ is quasi-continuous already To show the converse inequality, we assume that eB ˛ ˛t ˛ and satisfies eB D 1 on B, then Y t D e ReB .X t / is a non-negative supermartingale relative to .F t , Pm / for any > 0 with d m D 1. Since DB  B , by a similar ˛ ˛ / and hence HB˛ 1  eB argument to the proof of Lemma 3.3.3, . , HB˛ 1/  . , eB m-a.e. ˛ ˛ eB m-a.e. and hence ˇRˇ C˛ HB˛ 1 D ˛Rˇ C˛ eB q.e. Since Thus we have HB˛ 1 D e ˛ ˛ ˛ limˇ !1 ˛Rˇ C˛ HB 1.x/ D HB 1.x/ and a subsequence of ¹ˇRˇ C˛ eB º converges to ˛ q.e., we get the result. eB b ˛ 1 and H b .ı/,˛ 1 follow similarly. The assertions concerning H B

B

Theorem 3.4.4. A set N is exceptional if and only if Cap.˛/ .N / D 0 for any ˛ > ˛0 . Proof. For simplicity of notation, we assume that ˛0 < 1, ˛ D 1 and put Cap D Cap.1/ . Suppose that Cap.N / D 0. Then there exists a decreasing sequence of open sets An N such thatCap.An / # 0. Since Cap.An /  E1 .eAn , eAn /  .1 

92

Chapter 3 Markov processes

˛0 /.eAn , eAn /, we T have from Lemma 3.4.3 that limn!1 HA1n 1 D limn!1 eAn D 0 m-a.e. Then B n An N satisfies HB1 1 D 0 m-a.e. Hence N is exceptional. Conversely, assume that N is a Borel exceptional set. We may assume that N is a bounded set. By virtue of Theorem 3.1.1, there exists a decreasing sequence of relatively compact open sets ¹Gn º such that Gn N and limn!1 Gn D N a.s. Pm . 1 1 D 0 m-a.e. Hence, equation (2.1.8) and Lemma 3.4.3 Then limn!1 HG1 n 1 D HN yield that Cap.N /  lim Cap.Gn /  K12 lim E1 .HG1 n 1, HG1 n 1/. n!1

n!1

Since the right-hand side is bounded, a subsequence of Cesàro means of HG1 n 1 con1 1 D 0 m-a.e. and hence verges to a function v 2 F relative to E1 . Then v D HN Cap.N / D 0. Theorem 3.4.5. Any semipolar set is exceptional. Proof. We also assume that 0  ˛0 < ˛ D 1. It is enough to show that any thin set K is exceptional. Furthermore, since K is approximated by the hitting time of compact subsets of K® by Corollary 3.1.2, we may¯ assume that K is a thin compact set. In this  D  K K . Since Ex .e K / D lim˛!1 ˛R˛C1 uK / < Ex e case, K  x : Ex .e  D  K , the proof is reduced to the proof of the exceptionality of for uK .x/ D Ex e ¹x : lim˛!1 ˛R˛C1 uK .x/ < uK .x/º. Let ¹A Tn º be a decreasing sequence of relatively compact open sets such that An

N AnC1  , nA  n D K and limn!1 Cap.An / D Cap.K/. By virtue of Lemma 3.4.3, eAn .x/ q.e. Since DAn  DK , limn!1 DAn D  Ex e DAn D HA1n 1.x/ D e DK . On the other hand, from the quasi-left continuity of X t , XD D limn!1 XDAn 2 eAn .x/ q.e. To estimate the limit of the right\An D K. Hence uK .x/ D limn!1 e hand side of this equality, use equation (2.1.8) to get the inequality E1 .eAn , eAn /  Cap.An / which is bounded relative to n. Hence a subsequence of the Cesàro means of ¹eAn º converges to a function v 2 F relative to E1 . If w 2 F \ C0 .X / satisfies w  1 on K then, for any  > 0, since w  1   on An for all large n, we obtain that .1=.1//E1 .eAn , w/  E1 .eAn , eAn /. Letting n ! 1 and then  ! 0, it follows that v satisfies E1 .v, w/  E1 .v, v/. This implies that v D eK 2 F . By virtue of Theorem 2.2.5, q.c. modifications of a suitable subsequence of the approximating sequence of Cesàro means stated above converges to a q.c. modification eQK . Since eAn is deeK q.e. Therecreasing, it then implies that eQAn converges to eQK q.e. and hence uK D e fore, by virtue of Theorem 1.1.4 and Theorem 2.2.5, a subsequence of ¹˛R˛C1 uK º converges to a q.c. version of uK . This implies the desired assertion. Lemma 3.4.6. If ¹un º  F is a decreasing sequence of q.c. ˛-excessive functions converging to zero a.e., then limn!1 un D 0 q.e.

93

Section 3.4 Negligible sets for Hunt processes

Proof. Put v.x/ D limn!1 un .x/. For any compact subset F of ¹x : v.x/ > º,     Ex e ˛F  lim Ex e ˛F un .XF /  lim un .x/ D 0, n!1

n!1

for a.e. x. This implies that F is an exceptional set. Theorem 3.4.7. Suppose that ¹un º  F converges to u in .E, F /, then there exists a unk .X t / D e u.X t / uniformly on every compact subsequence ¹unk º such that limn!1 e t -interval a.s. Px for q.e. x. Proof. We may assume that ¹un º and u are quasi-continuous already. Let ¹Fm º be a nest such that Cap.˛/ .X n Fm / < 22m and un , n  1, u are continuous on Fm . Choose a subsequence ¹unk º satisfying E˛ .unkC1  unk , unkC1  unk / < 24k . Then by putting Bk D ¹x 2 Fk : junkC1 .x/  unk .x/j  2k º we have Cap.˛/ .X n Bk /  Cap.˛/ .¹x : junkC1 .x/  unk .x/j > 2k º/ C Cap.˛/ .X n Fk /  K˛ 22k E˛ .unkC1  unk , unkC1  unk / C 22k  .K˛ C 1/22k . Hence, for any  2 S00 ,   P junkC1 .Xs /  unk .Xs /j > 2k , for some s  t D P .XnBk  t /  e ˛t E .e ˛XnBk /  ˛  b˛  , U D e ˛t E˛ eXnB k

b ˛ , U b ˛ /1=2  K˛ e ˛t Cap.˛/ .X n Bk /1=2 E˛ .U b ˛ , U b ˛ /1=2 .  K˛ .K˛ C 1/2k e ˛t E˛ .U

This yields the desired uniform convergence as in Theorem 2.2.3. b ˛ 1) is a Theorem 3.4.8. For any nearly Borel set B of finite capacity, HB˛ 1 (resp. H B ˛ ˛ eB ). q.c. modification of eB (resp.b Proof. If B is open, then the assertion has already been shown by Lemma 3.4.3. Assume that B is a Borel set with finite capacity. As in equation (3.1.10), there exists a decreasing sequence of open sets ¹Bn º suchR that Bn B and limn!1 Bn D B a.s. Pgm for a positive function g such that X gd m D 1. Then limn!1 HB˛n 1 D ˛  e ˛ D H ˛ 1, it follows that e ˛ 1  H ˛ 1 m-a.e. and hence HB˛ 1 m-a.e. Since eB Bn Bn B B ˛ ˛ e eB  HB 1 q.e. Conversely, since B is of finite capacity, there exists a sequence of open sets ¹Bn º ˛ , e ˛ /º is such that B  Bn and Cap.˛/ .B/ D limn!1 Cap.˛/ .Bn /. Then ¹E˛ .eB Bn n

94

Chapter 3 Markov processes

bounded. Hence, by the Banach Saks theorem, a subsequence of Cesàro means of ˛ º converges strongly to e ˛ 2 F . Since e ˛ 2 L , e ˛  e ˛ . On the other hand, for ¹eB B 0 0 0 B n ˛ a q.c. modification e eB and  # 0, there exists an open set N" such that Cap.˛/ .N"n / < ˛ is continuous on X n N . Let A D ¹x : eQ ˛ > 1  " º [ N . Then A is an eB "n and e " n n " n B open set containing B. By replacing Bn with Bn \ An if necessary, we assume that ˛ eB =.1  "n / C eN 2 LBn and hence An  Bn . Then e " n

1 e e ˛ Ce e ˛N"n  e0˛ . 1  "n B ˛ By letting n ! 1 and then taking q.c. regularization, we obtain that e eB e e ˛0 . There˛ ˛ ˛ ˛ ˛ ˛ ˛ e 0 . Since e0 D limn!1 eBn  HB 1, we have e eB  HB 1 q.e. fore e eB D e

3.5 Decompositions of Dirichlet forms and their probabilistic counterparts Let .E, F / be a regular Dirichlet form on L2 .X ; m/ satisfying condition .E.5/ and M be the associated Hunt process on X . By equations (2.4.8) and (2.4.9), for any Borel set A  X , we defined the spaces F XnA and HA˛ by F XnA D ¹u 2 F : uQ D 0, q.e. on Aº HA˛ D ¹u 2 F : E˛ .u, v/ D 0, for all v 2 F XnA º. Also the dual space of HA˛ is defined by b ˛ D ¹u 2 F : E˛ .v, u/ D 0, for all v 2 F XnA º. H A Then, as in equation (2.4.10), any function u 2 F can be expressed E˛ -orthogonally as ˛ ˛ u D F˛ XnA .u/ C uA Db

˛F XnA .u/ C b uA ˛ ˛ b ˛ . As in the previous for F˛ XnA .u/, b

˛F XnA .u/ 2 F XnA and uA 2 HA˛ , b uA 2 H A b .ı/ are defined section, the ˛-order hitting distributions of A relative to M and M .ı/,˛ .ı/ ˛ ˛ ˛ b  A u.X b b //, respecu.x/ D b Ex .e by HA u.x/ D Ex .e A u.XA // and H A A b tively. Furthermore, as we noted in the last part of Section 3.3, we can also consider A u.X b by H b //. b ˛ u.x/ D b Ex .e ˛b the ˛-order hitting distribution of A relative to M A A b b .ı/,˛ı u.x/ D H b ˛ .b Then b hı .x/H hı u/.x/ q.e. x. A

A

Theorem 3.5.1. Let A be nearly Borel set and ˛ > ˛0 . Then, for any quasi-continuous ˛ b ˛ u are q.c. versions of u˛ and b uA , respectively. function u 2 F , HA˛ u and H A A

95

Section 3.5 Decompositions of Dirichlet forms

b ˛ u. Clearly H b ˛ u is linear relaProof. We shall only prove the assertion concerning H A A   ˛ ˛ u1 /A C .b tive to u. Also, for any u1 , u2 2 F , since E˛ v, .b u2 /A D 0 for all v 2 F XnA ˛  .b ˛ 2 F XnA , we have by the uniqueness of the orthogonal and u1 C u2  .b u1 /A u2 /A ˛ ˛ ˛ ˛ u2 /A D .u1 C u2 /A , that is b uA is linear relative to u. decomposition that .b u1 /A C .b Hence, we may consider that u  0. b˛ f for ˛ > ı > ˛0 and f 2 L2 .X ; m/. Then u We shall first assume that u D R C ˛ bˇ º and b is ˛-coexcessive relative to the resolvent ¹R uA is the smallest ˛-coexcessive bˇ º which dominates u on A. Similarly to Theorem 3.4.8, since function relative to ¹R b ˛ u is the smallest ˛-coexcessive function relative to ¹R bˇ º such that H b ˛ D u q.e. on H A A ˛ ˛ b u Db A, it holds that H u a.e. A A b˛ .ng/ instead of u For any non-negative function u 2 F , by considering u ^ R 1 1 for a strictly positive function g 2 L .X ; m/ \ L .X ; m/, we suppose that u is bounded by a positive ˛-coexcessive function of F \ L1 .X ; m/. For any ˇ > ˛, since b ˛ u1  R b˛ u2 for u1 D .u C .˛  ˇ/R bˇ u/C and u2 D .u C .˛  ˇ/R bˇ u/ , bˇ u D R R ˛ ˛ b .ˇ R bˇ u/ D .ˇ R bˇ u/ q.e. Since it holds that H A A

2

bˇ u  ˇR

ˇ b ˇ bˇ b h, h .ˇ  ˛/R ˇ˛ ˇ˛

bˇ u D u q.e., limˇ !1 H bˇ u/ D H b ˛ u q.e. To estimate b ˛ .ˇ R and limˇ !1 ˇ R A A bˇ u/˛ , note that E˛ .u˛ , u˛ /  K˛2 E˛ .u, u/ for any u 2 F by equation (2.4.6). .ˇ R A A A bˇ u D u relative to E˛ , it then implies that limˇ !1 .ˇ R bˇ u/˛ D u˛ Since limˇ !1 ˇ R A A relative to E˛ . Hence, by choosing a subsequence if necessary, it also converges q.e. ˛ b˛u D b uA q.e. Therefore H A b˛ .ng/ and u ^ For the general u 2 F , by applying the above result to uC ^ R b R˛ .ng/ and then letting n " 1, we obtain the result. For any Borel set B  X , let XnB

pt

.x, E/ D Px .X t 2 E, t < B /

(3.5.1)

XnB

and R˛

be the corresponding resolvent given by Z Z 1 XnB ˛t XnB R˛ f .x/ D e pt f .x/dt D Ex 0

B

e

˛t

 f .X t /dt .

(3.5.2)

0

Using the strong Markov property, the following Dynkin formula holds. R˛ f .x/ D R˛XnB f .x/ C HB˛ R˛ f .x/ f 2 BC .X /. b .ı/ , put Similarly, for the dual process M b˛.ı/,XnB f .x/ D b E.ı/ R x

Z b B e 0

˛t

 b f .X t /dt .

(3.5.3)

96

Chapter 3 Markov processes

Then, for any f 2 BC .X /, b.ı/,XnB f .x/ C H b .ı/,˛ R b.ı/ b.ı/ R ˛ f .x/ D R˛ ˛ f .x/ B b D .X bt, b P x / can be treated as a Hunt process, we can define Since M b˛XnB f .x/ D b R Ex

Z b B e

˛t

b t /dt f .X



0

  ˛b  B f .X b ˛ f .x/ D b e and H E / . Furthermore, the Dynkin’s formula also holds x B  b b that is for M,

B

b˛ f .x/ D R b˛XnB f .x/ C H b˛ R b R B ˛ f .x/

(3.5.4)

As we have seen in Theorem 3.5.1, for any nearly Borel set B, a q.c. version of ˛ coincides with HB˛ R˛ f q.e. In particular .R˛ f /B

F˛ XnB .R˛ f / D R˛XnB f

m-a.e.

Hence, the Dynkin formula is nothing but the orthogonal decomposition of R˛ f corresponding to equation (2.4.10). ˛ a.e. equation (2.4.10) implies that For any q.c. function u 2 F , since HB˛ u D uB ˛ ˛ ˛ , uB / D E˛ .uB , u/ E˛ .HB˛ u, HB˛ u/ D E˛ .uB ˛ ˛ 1=2  K˛ E˛ .uB , uB / E˛ .u, u/1=2 .

Hence E˛ .HB˛ u, HB˛ u/  K˛2 E˛ .u, u/. Theorem 3.5.2. For any nearly Borel set B, ˛ > ı and f , g 2 BC .X /,  ˛    b .ı/,˛ı R b.ı/ g HB R˛ f , g b D f ,H B ˛ı b m m  ˛    ˛b b HB R˛ f , g D f , H B R˛ g .

(3.5.5)

(3.5.6) (3.5.7)

˛ and hence Proof. By virtue of Theorem 3.5.1, if B is open, then HB˛ R˛ f D .R˛ f /B 2 b /, for any f , g 2 L .X ; m C m       ˛ b˛XnB g C E˛ H ˛ R˛ f , H b˛ R b HB R˛ f , g D E˛ HB˛ R˛ f , R B B ˛g     b˛ R b˛ b b D E˛ HB˛ R˛ f , H B ˛ g D E˛ R˛ f , H B R˛ g   b b˛ R D f ,H B ˛g .

For the general nearly Borel set B, it is enough to take a decreasing sequence of open  Bn D b B sets ¹Bn º such that B  Bn , limn!1 Bn D B a.s. Pz and limn!1 b bz for a.e. z. Hence equation (3.5.7) holds. Equation (3.5.6) is clear from equaa.s. P tion (3.5.7).

Section 3.5 Decompositions of Dirichlet forms

97

We say that a function u is finely continuous q.e. if there exists a Borel exceptional set N such that X n N is finely open and u is finely continuous on X n N . Corollary 3.5.3. If a nearly Borel and finely open set B satisfies m.B/ D 0, then B is exceptional. In particular, if u is finely continuous q.e. and u  0 m-a.e., then u  0 q.e. Proof. Let F be a compact subset of B. By applying equation (3.5.7) to B D F , it holds that     ˛ b˛ R b HF R˛ 1B , g D 1B , H F ˛g D 0 for any non-negative function g 2 C0 .X /. Since Rˇ 1B  R˛ 1B for all ˇ > ˛, this implies HF˛ .ˇRˇ 1B / D 0 m-a.e. Since limˇ !1 ˇRˇ 1B D 1 q.e. R on F , we get that HF˛ 1 D 0 m-a.e. For any strictly positive function h such that X hd m D 1, take an increasing sequence of compact subsets of B such that Fn # B Phm -a.e. The existence of such a sequence is a consequence of Corollary 3.1.2. Then Ehm .e B / D limn!1 Ehm .e Fn / D 0. This yields that B is an exceptional set. Let N be an exceptional set in the definition of the q.e. fine continuity. Then B D ¹x 2 X n N : u.x/ < 0º is a finely open set with m.B/ D 0. Then the result shown above implies that B is exceptional. A nearly Borel set B is called polar if Px .B < 1/ D 0 for all x 2 X . Theorem 3.5.4. The following conditions are equivalent. (i)

A set is polar if and only if it is exceptional.

(ii)

R˛ .x, /  m for all ˛ > 0 and x 2 X .

Proof. (i) ) (ii) Suppose that m.B/ D 0. Then, from Lemma 3.4.1, R˛ .x, B/ D 0 for m-a.e. x. By the quasi-continuity of R˛ .x, B/ there exists an exceptional set N such that R˛ .x, B/ D 0 for all x 2 X n N . Then (i) implies that N is polar. Hence R˛ .x, B/ D lim Ex .R˛ .X t , B/; X t 2 X n N / D 0 t#0

for all x 2 X . (ii) ) (i) If B is a Borel exceptional set, then Px .B < 1/ D 0 m-a.e. Hence Px .B < 1/ D lim˛!1 ˛R˛ P .B < 1/.x/ D 0 for all x 2 X . The condition of Theorem 3.5.4 is satisfied if the resolvent has the strong Feller property, that is if R˛ f is continuous for any bounded measurable function f . In fact, if R˛ is strong Feller, then m.F / D 0 implies that R˛ .x, F / D 0 for a.e. x and hence all x by the continuity of R˛ ., F /.

98

Chapter 3 Markov processes

More generally, the following condition yields the existence of a modification ¹R˛ º of ¹G˛ º such that R˛ .x, / is absolutely continuous relative to m for any x 2 X n N with a fixed exceptional set N .2 There exists p0 > 2, 0  ˛0 and a positive constant C such that kukp2 0  C E0 .u, u/

(3.5.8)

for all u 2 F , where kukp is the Lp .X ; m/-norm. Under this condition, for p > .p0 =.p0  2// _ 2 and ˛ > ˛0 , the resolvent G˛ satisfies (3.5.9) kG˛ f k1  C1 kf kp C C2 kG˛ f k  C1 kf kp C C3 kf k for all f 2 Lp .X ; m/ \ L2 .X ; m/, where C1 , C2 and C3 are constants depending on p0 , p, C , ˛0 and ˛. Hence, there exists a negligible set N and a resolvent ¹R˛ º such that R˛ f is a q.c. version of G˛ f for all f 2 L2 .X ; m/ and R˛ .x, / is absolutely continuous relative to m for all x 2 X n N . Then the Hunt process M with resolvent ¹R˛ º does not hit N if it starts from any point of X n N . Therefore, the process M considering any point of N as a trap, is well defined as a Hunt process without an exceptional set. Example 3.5.5. Let D be a domain of Rd , d  3 and .E, F / be the Dirichlet form given in Section 1.5.1, that is F D H01 .D/ and E is given by equation (1.5.3). By equation (1.5.4), E˛ .u, u/ is equivalent to Q1c .u, u/. Then the Sobolev inequality implies, for p0 > 2 satisfying 12  d1 D p10 , that there exists a constant C depending on d and p0 such that kukp2 0  C kruk22 

C c C E˛ .u, u/. Q .u, u/   1 1

(3.5.10)

Therefore, for the Dirichlet form .E, F / given by equation (1.5.3), the associated resolvent has a version R˛ .x, dy/ such that R˛ .x, / is absolutely continuous relative to m for x 2 X n N with a suitable exceptional set N . Next assume that .E, F / is a jump type Dirichlet form given by equation (1.5.12) for X D Rd and m.dx/ D dx. Furthermore, if j.x, y/ satisfies j.x, y/  C jx  yjd 1

2

See [44].

for 0 < jx  yj < 1

(3.5.11)

99

Section 3.5 Decompositions of Dirichlet forms

for some positive constant C , then the associated resolvent satisfies equation (3.5.8).3 In fact, for any u 2 C01 .Rd /, by using equation (1.4.32) in [55], we have “ Q.u, u/ D .u.x/  u.y//2 j.x, y/dxdy x¤y

“ 

.u.x/  u.y//2 j.x, y/dxdy 0 0, there exists a quasi-continuous G G b bG version R ˛ f of the coresolvent associated with .E , F /. Since hı is ı-coexcessive G G .ı/,G .ı/,G ,G / similarly to equation (1.4.6) by relative to .E , F /, we can define .A .ı/,G .ı/ .ı/,G G b bG .f b using R˛ D R˛Cı instead of R˛ . Put R˛ f .x/ D .1=b hı .x//R hı /.x/. ˛Cı .ı/,G b b b Then, similarly to Theorem 3.3.6, there exists a Hunt process M D .X t , P .ı/,G / .ı/,G .ı/,G b possessing ¹R˛ º as its resolvent, which is in duality with M relative to the b measure m b D hı  m. As we have seen after Lemma 3.3.7, there exists a pseudo Hunt G bG b G D .X bt, P bG process M x / with resolvent ¹R˛ º, which is in duality with M relative to m. Thus we have the following theorem. b .ı/,G D Theorem 3.5.8. For any open set G, there exists a Hunt process M .ı/,G .ı/,G b˛ º. Furthermore, there also exists a pseudo Hunt bt, b P x / with resolvent ¹R .X b G D .X bt, b bG º. P G / with resolvent ¹R process M x

˛

As we have seen in Theorem 2.4.8, for ı > ˛0 , we may assume that b hı 2 F \ is a ı-coexcessive function which is bounded from below by a positive constant on every compact set. As in equation (2.4.1), for u 2 F and a Borel set B, let

L1 .X ; m/

Lu,B D ¹v 2 F : v  u m-a.e. on Bº.

(3.5.13)

Lemma 3.5.9. There exists an increasing sequence of closed sets ¹Fn º satisfying the following conditions. hı .x/ coincides with a bounded (i) limn!1 XnFn D  a.s. Px for m-a.e. x and b b function hı,n 2 F on Fn . hı / for all u, v 2 FbFn . (ii) FbFn  G .ı/ and satisfies A.u, v/ D E.u, vb (iii) If u is a q.c. ˛-excessive function for ˛ > ı which coincides with a function of ˛ is the unique function of Fb on a closed subset B of Fn for some n, then uB .ı/ G \ Lu,B satisfying ˛ ˛ ˛ A˛ .uB , uB /  A˛ .uB , v/

for all bounded function v 2 Lu,B .

(3.5.14)

102

Chapter 3 Markov processes

b.ı/ (iv) If b u 2 G .ı/ is a q.c. .˛  ı/-coexcessive function relative to .R ˛ / which is ˛,ı uı D b u b hı 2 F and b uB bounded on a compact subset B of Fn , then b ˛ b ˛,ı .ı/ uB 2 L b and .b uı /B =hı is the unique function of G satisfying b u=hı b ˛,ı ˛,ı ˛,ı uB ,b uB /  Aı .v,b uB / Aı .b

(3.5.15)

for all bounded function v 2 L b . u=hı b Proof. Taking a bounded strictly positive function g 2 L1 .X ; m C m b /, put b hı,n D b b b b b b˛ g/. Then hı,n 2 F and limn!1 hı,n D hı . Since hı and b hı ^ .nR hı,n are quasihı and b hı,n are continuous continuous, there exists a nest ¹Bn º of closed sets such that b b b hı D b hı,n 2 F on Bn . Hence the set Fn D ¹x 2 Bn : hı .x/ D hı,n .x/º is closed, b on Fn and [n Fn D X q.e. Hence it is enough to show that limn!1 XnFn D  a.s. b˛ g < b hı on X n Fn , Since ˛ > ı and nR b hı .x/ 1 bı b b˛ b h .x/  H  H XnFn R˛ g.x/ n ı n n XnF  Z 1 b t /dt . b˛ g.X e ˛t R Db Ex XnFn

By letting n ! 1, the desired result follows. hı D vb hı,n for b hı,n 2 Fb , .E.5/ implies that (ii) For any u, v 2 FbFn , since vb .ı/ b b v hı 2 F and Eı .u, v hı / D A .u, v/. ˛ ˛  .u ^ kukB /B  kukB . Hence (iii) Put kukB D ess.sup¹u.x/ : x 2 Bº. Then uB ˛ ˛ is bounded. Furthermore, it can be written as uB D U˛ B for some measure uB B 2 S0 on B. Then for any non-negative bounded function v 2 F , ˛ ˛ bˇ C˛ .vb  ˇRˇ C˛ uB , v/b D ˇ.Uˇ C˛ B , vb hı / D hB , ˇ R hı /i. ˇ.uB m

hı,n 2 F on B, Since b hı D b bˇ C˛ .vb bˇ C˛b hı i hı /i  kvk1 hB , ˇ R hB , ˇ R  kvk1 hB , b hı i D kvk1 hB , b hı,n i < 1 bˇ C˛ .vb hı / D vb hı by the quasi-continuity of vb hı . Hence we obtain and limˇ !1 ˇ R by Lebesgue theorem that ˛ ˛  ˇRˇ C˛ uB , v/b D hB , vb hı i, lim ˇ.uB m

ˇ !1

103

Section 3.5 Decompositions of Dirichlet forms

.ı/ ˛ that is A˛ı .uB , v/ D hB , vb hı i. Equation (3.5.14) is clear from this. If u1 , u2 2 .ı/

.ı/

F \ Lu,B satisfies equation (3.5.14), then A˛ı .u1 , u1  u2 /  0 and A˛ı .u2 , u2  .ı/

u1 /  0. Hence A˛ı .u1  u2 , u1  u2 /  0 and hence u1 D u2 .

b.ı/ /, then (iv) If b u is a q.c. .˛  ı/-coexcessive function of G .ı/ relative to .R ˇ ˛ bˇ /. Then .b b uı b ub hı is a ı-coexcessive function of F relative to .R uı /B 2 F ˛ b is well defined and there exists a measure b B on B such that .b uı /B D U ˛ b B . ˛ ˛,ı ˛ b h˛,n , b  kb ukBb uB D .b uı /B =hı is bounded. Hence, for any bounded Since .b uı /B q.c. function v,     ˛,ı b.ı/ b˛b bˇ C˛ U b˛b lim ˇ v,b uB  ˇR b u˛,ı m D lim ˇ v, U B  ˇ R B ˇ C˛ı B b

ˇ !1

ˇ !1

B , ˇRˇ C˛ vi D hb B , vi, D lim hb ˇ !1

where, in the last equality, by noting that Z ˛ ˛ ˛ b B .B/  eQB db B D E˛ .eB , .b uı /B / < 1, X

˛,ı we used the Lebesgue theorem. In particular, by taking v D b uB , we obtain that ˛,ı .ı/ and equation (3.5.15) holds. The proof of the uniqueness is similar to b uB 2 G (iii).

Corollary 3.5.10. Assume that .E.5/ holds. For any nearly Borel set B  Fn ˛ for some n with finite capacity, eB is the unique function of G .ı/ \ LB such that ˛ ˛ ˛ A˛ .eB , eB /  A˛ .eB , v/ for all v 2 LB . We say that a regular Dirichlet form .E, F / possesses the local property if E.u, v/ D 0 whenever u, v 2 F have disjoint compact supports. Lemma 3.5.11. The following are equivalent to each other: (i)

.E, F / possesses the local property;

(ii)

˛ for each relatively compact open set A, HXnA .x, / is concentrated on @A for q.e. x 2 A.

Proof. (i) ) (ii) Assume that .E, F / possesses the local property. For a relatively compact open set A, Theorem 3.5.1 yields for any non-negative function u 2 C0 .X /\ ˛ e u is a q.c. version of Fı A .u/. On the other hand, F supported by X n AN that e u  HXnA for any v 2 F such that v is supported by A, since F˛ A .v/ D v, we have ˛ e u, v/ D E˛ .u, v/ D 0, E˛ . F˛ A .u/, v/ D E˛ .u  HXnA

104

Chapter 3 Markov processes

˛ ˛ by the local property. This implies that e u  HXnA e u D 0 q.e. Hence HXnA e u D 0 q.e. ˛ on A and HXnA .x, X n @A/ D 0 q.e. x 2 A. (ii) ) (i) Suppose that u, v 2 F have disjoint compact supports. Let A be a relatively N Then E˛ .u, v/ D compact open set such that suppŒv  A and suppŒu  X n A. ˛ ı ˛ u  HXnAe u D 0 q.e. E˛ . F A .u/, v/ D 0 because F A .u/ D e

Theorem 3.5.12. The following conditions are equivalent to each other: (i)

.E, F / possesses the local property;

(ii)

the Hunt process M associated with .E, F / is a diffusion process, that is, its trajectories are a.s. continuous for q.e. starting points.

Proof. (i) ) (ii) Let O be a countable base of relatively compact open sets of X . Let S dSD ¹! 2  : X t .!/ ¤ X t .!/ for some t 2 .0, .!//º. Then d D A2O s2QC ¹! 2  : Xs 2 A, 0 < XnA .s !/ < 1, XXnA ı s ! 2 X n @Aº. From the hypothesis and Lemma 3.5.11, there exists an exceptional set N such that ı .x, X n @A/ D 0 for all x 2 X n N , ı 2 QC and A 2 O. Since, for x 2 X n N HXnA   Px ¹! 2  : Xs 2 A, 0 < XnA .s !/ < 1, XXnA ı s ! 2 X n @Aº   D Ex PXs .XnA < 1, XXnA 2 X n @A/ : Xs 2 A   ı .Xs , X n @A/ : Xs 2 A n N D 0, D lim Ex HXnA ı!0

it follows that Px .d / D 0 for q.e. x. (ii) ) (i) is clear from Lemma 3.5.11. Fixing ı > ˛0 , consider the bilinear form .A.ı/ , G .ı/ / defined by a strictly positive ı-coexcessive function b hı which is bounded from below by a positive constant on every compact set. For any function u 2 G .ı/ and ˇ > 0, we have the following expression: Z Z ˇ2 .ı/ .ı/ hı .x/Rˇ .x, dy/m.dx/ D .u.x/  u.y//2b ˇ.u  ˇRˇ u, u/b m 2 X X Z ˇ .ı/ C u2 .x/b hı .x/.1  ˇRˇ 1.x//m.dx/ 2 X Z ˇ b.ı/ 1.y//m.dy/. (3.5.16) C u2 .y/b hı .y/.1  ˇ R ˇ 2 X Lemma 3.5.13. There exists a sequence ˇn " 1 and positive Radon measures k .ı/ and b k .ı/ such that, for any u 2 C0 .X /, Z .ı/ .ı/ u.x/.1  ˇRˇn 1.x//b hı .x/m.dx/, (3.5.17) hk , ui D lim ˇn n!1 ZX b.ı/ 1.y//m.dy/. hb k .ı/ , ui D lim ˇn u.y/b hı .y/.1  R (3.5.18) ˇn n!1

X

105

Section 3.5 Decompositions of Dirichlet forms

˛ Proof. For any compact set B, since b hı 2 F , the equilibrium potential eB belongs to ı .ı/ F \ G by Corollary 2.4.10. By putting u D eB in equation (3.5.16), it holds that Z ˇ .ı/ ı .ı/ ı ı  .1  ˇRˇ 1.x//b hı .x/m.dx/ ˇ.eB  ˇRˇ eB , eB /b m 2 B Z ˇ b.ı/ 1.x//b C .1  ˇ R hı .x/m.dx/. ˇ 2 B ı ı Since the left-hand side converges to A.ı/ .eB , eB / as ˇ ! 1, the family of measures .ı/ .ı/ b b ¹ˇ.1  ˇR 1.x//hı .x/m.dx/º and ¹ˇ.1  ˇ R 1.x//b hı .x/m.dx/º are uniformly ˇ

ˇ

bounded on any compact set B. Hence, there exists a sequence ˇn " 1 and positive k .ı/ .dx/ such that Radon measures k .ı/ .dx/ and b lim ˇn .1  ˇn Rˇn 1.x//b hı .x/m.dx/ D k .ı/ .dx/

(3.5.19)

b.ı/ 1.x//b hı .x/m.dx/ D b k .ı/ .dx/ lim ˇn .1  ˇn R ˇn

(3.5.20)

.ı/

n!1 n!1

vaguely. Theorem 3.5.14. Suppose that .E, F / satisfies .E.5/ and ı > ˛0 . Then the following results hold. (i)

There exists a positive Radon measure k charging no exceptional set such that, for any u 2 C0 .X /, Z u.x/.1  ˇRˇ 1.x//m.dx/. (3.5.21) hk, ui D lim ˇ ˇ !1

(ii)

X

For any f , g  0 and t > 0,   Eum f .X / :   t D

Z

t

hk, f b p s uids.

(3.5.22)

e ıs hb k .ı/ , fps uids.

(3.5.23)

0

(iii) For any f , u  0 and t > 0, .ı/ b b  / :   t / D E .f .X ub m

Z

t

0

Proof. (i) Put k.dx/ D .1=b hı .x//k .ı/ .dx/Cım.dx/ by using the measure k .ı/ given .ı/ .ı/ in Lemma 3.5.13. Since Rˇ 1  Rˇ ı 1 D ıRˇ Cı Rˇ 1, it holds that Z lim ˇn

n!1

u.x/.1  ˇn Rˇn 1.x//b hı .x/m.dx/ Z Z Z D u.x/k .ı/ .dx/ C ı u.x/b hı .x/m.dx/ D u.x/b hı .x/k.dx/,

X

X

X

X

106

Chapter 3 Markov processes

for any u 2 C0 .X /. For any compact set B, take a decreasing sequence of relatively compact open sets Bi B and a non-negative function u 2 F \ C0 .X / such that hı .x/ : x 2 Bº > 0, noting ui D 1 on B and ui D 0 on X n Bi . Then, for cB D inf¹b ı D 1 a.e. on B , we have that eB i i Z cB .k.B/  ım.B//  u2i .x/k .ı/ .dx/ X Z .ı/ ı 2b / hı .x/.1  ˇn Rˇn 1.x//m.dx/  lim ˇn .eB i n!1

X

.ı/

ı ı ı  ˇn Rˇn eB , eB /m  2 lim ˇn .eB i i i b

D

n!1 ı ı 2A.ı/ .eB , eB / i i

ı b  2Eı .eB , hı / i

ı ı 1=2  2Kı Eı .eB , eB / Eı .b hı , b hı /1=2 i i

 2Kı Eı .b hı , b hı /1=2 Cap.ı/ .Bi /1=2 .

In particular, if B is a compact set of zero capacity, then by choosing a decreasing sequence of relatively compact open sets ¹Bi º such that Bi B and Cap.˛/ .Bi / < 1=i , it follows that k.B/  limi!1 k.Bi / D 0. This implies that k does not charge any set of zero capacity. Furthermore, for any compact set B, k.B/ < 1, k .ı/ .B/ < R .ı/ 1 and ¹ˇn B b hı .x/.1  ˇn Rˇn 1.x//m.dx/º are uniformly bounded. For any bounded q.c. function u 2 F supported by a compact set B there exists a sequence ¹ui º  F \ C0 .X / such that li mi!1 ui D u relative to E˛ . For a function v 2 F \ C0 .X / such that v D 1 on B, since ¹ui vº is uniformly E˛ bounded and converges to u in L2 .X ; m/, it also converges relative to E˛ . Hence, by considering ui v instead of ui , we may assume that ¹ui º is uniformly bounded and their supports are contained in a fixed relatively compact open set A for all n  1. By virtue of Theorem 2.2.5, for any  > 0, there exists a closed subset F of A such that Cap.˛/ .A n F / <  hı  k  k .ı/ C ıb m and and limi!1 ui D u uniformly on F . Then, by noting that b .ı/ .ı/ 1=2 b b k .A n F /  K Cap .A n F / for K D 2Kı Eı .hı , hı /1=2 , we have Z Z jui .x/  u.x/jb hı .x/k.dx/  jui .x/  u.x/jb m.dx/ X

X

 sup¹jui .x/  u.x/j : x 2 F ºk .ı/ .A/ C K.kui kL1 C kukL1 /Cap.ı/ .A n F /1=2 . Since limi!1 ui D u in L2 .X ; m b/, this implies that hk, b hı ui D limi!1 hk, b hı ui i. R .ı/ hı .x/.1  ˇn R 1.x//m.dx/  K Cap.ı/ .A n F / and Noting that ˇn AnF b limi!1 ui D u uniformly on F , we obtain that Z b hı .x/.1  ˇRˇn 1.x//m.dx/ hk, uhı i D lim lim ˇn ui b i!1 n!1 X Z ub hı .x/.1  ˇRˇn 1.x//m.dx/ D lim ˇn n!1

X

107

Section 3.5 Decompositions of Dirichlet forms

for any bounded q.c. function u with compact support. In particular, for any u 2 hı instead of u, we obtain equation (3.5.21) for ˇ replaced C0 .X / \ F , by putting u=b hı .x/m.dx/º are uniformly bounded by ˇn . Since the measures ¹ˇn .1  ˇRˇn 1.x//b on any compact set and C0 .X / \ F is dense in C0 .X /, equation (3.5.21) also holds for any u 2 C0 .X /. Once (ii) has been shown, since k is uniquely determined independently of the choice of ¹ˇn º, we get equation (3.5.21). (ii) As we have seen above, for any bounded non-negative q.c. functions u and f such b˛ u is bounded, that f  R   b˛ u b˛ ui D lim ˇn .1  ˇn Rˇ 1/, f  R hf  k, R n n!1   D lim u, ˇn R˛ .f .1  ˇn Rˇn 1// . n!1

  By noting 1  ˇn Rˇn 1.x/ D Ex e ˇn  , we have lim ˇn R˛ .f .1  ˇn Rˇn 1//.x/  Z  e ˛t f .X t /e ˇn ı t dt D lim ˇn Ex

n!1

n!1

Z

0 

D lim ˇn Ex n!1

Z

e 0 ˇn 

˛t

f .X t /e

ˇn .t/

 dt

˛..s=ˇn //

D lim Ex e f .X.s=ˇn / /e n!1  Z 1 0 e ˛ f .X /e s ds D Ex 0   D Ex e ˛ f .X / as k ! 1. Hence

s

b˛ ui D Eum .e ˛ f .X //. hk, f R

 ds

(3.5.24)

This can be extended to any non-negative q.c. functions f and u. By inverting the .ı/ Laplace transformation, we obtain (ii). By noting that p t D e ıt p t is the transition function of M.ı/ , we obtain (iii) by a similar manner. The measure k is called the killing measure of M. Similarly we call the measure b b .ı/ . k .ı/ the killing measure of M If u, v 2 F \ C0 .X / have disjoint supports, then Z Z u.y/v.x/Gˇ .dxdy/ E.u, v/ D  lim ˇ 2 ˇ !1 X X Z Z b ˇ .dydx/, D  lim ˇ 2 u.y/v.x/G ˇ !1

X

X

108

Chapter 3 Markov processes

b ˇ .dydx/ D G b ˇ .y, dx/m.dy/. This where Gˇ .dxdy/ D Gˇ .x, dy/m.dx/ and G Pn implies that, for any function Fn .x, y/ of the form Fn .x, y/ D iD1 'i .x/ i .y/ with 'i , 2 F \ C0 .X / having disjoint supports, the limit “

n X  Fn .x, y/ ˇ Gˇ .x, dy/ m.dx/ D E.'i ,



lim

ˇ !1

2

XXnd

i/

iD1

exists, where d is the diagonal set d D ¹.x, x/ : x 2 X º. In particular, for any compact subset  of X  X n d, since there exists a function of the form of Fn such that Fn is bounded from below on  by a positive constant, ¹ˇ 2 Gˇ ./º is uniformly bounded relative to ˇ. Hence, by choosing a subsequence if necessary, we can show that there exist unique positive Radon measures J.dxdy/ and JO .dxdy/ on X  X n d such b ˇ .dxdy/ D b J .dxdy/ that limˇ !1 ˇ 2 Gˇ .dxdy/ D J.dxdy/ and limˇ !1 ˇ 2 G vaguely. As we will see in the next theorem, the limits are independent of the choice of the subsequence. Hence, for any F .x, y/ 2 C0 .X  X n d/, “

“ F .x, y/ˇ Gˇ .dxdy/ D 2

lim

ˇ !1

F .x, y/J.dxdy/.

XX

(3.5.25)

XXnd

The measure J is called the jumping measure of .E, F /. In particular, if the supports of u, v 2 F \ C0 .X / are disjoint, then Z Z E.u, v/ D 

X

Z Z

X

u.y/v.x/J.dxdy/ D 

X

u.y/v.x/b J .dydx/.

(3.5.26)

X

For the strictly positive ı-coexcessive function b hı , put b J .ı/ .dxdy/ D

1 b hı .x/

b J .dxdy/b hı .y/.

Then, similarly to equation (3.5.25), for any function F .x, y/ “ lim

ˇ !1

XX

b F .x, y/ˇ 2 G .dxdy/ D ˇ ı .ı/

“

F .x, y/b J .ı/ .dxdy/,

XXnd

b .dxdy/ D G b .x, dy/b m.dx/. Furthermore, where G ˇ ˇ .ı/

.ı/

Z Z .ı/

A

.u, v/ D  D

Z Z X

X

X

X

u.y/v.x/b hı .x/J.dxdy/ u.y/v.x/b hı .y/b J .ı/ .dydx/.

(3.5.27)

109

Section 3.5 Decompositions of Dirichlet forms

C Let G be a relatively compact open set and R u 2 F \ C0 .X / be a function satisfying N suppŒu  X n G. Set Ju .dx/ D IG .x/ u.y/J.dxdy/, then Z jv.x/jJu .dx/  jE.u, jvj/j  K˛2 E˛ .u, u/1=2 E˛ .v, v/1=2 G

for all v 2 F \ C0 .G/. Hence Ju does not charge any exceptional subset of G. SimiR .ı/ J .ı/ .dxdy/u.y/ does not charge any exceptional set. larly, b J u .dx/ D X b As we defined before equation (2.4.20), let U˛G  be a q.c. modification of the potential of measure  2 S0 relative to the part of the Dirichlet form on G given by ˛ bG U˛ . Similarly, define the dual potential of measure U U˛G  D U˛   HXnG ˛ 2 G G G G G b b F similarly. Then they satisfy h, U ˛ i D E˛ .U˛ , U ˛ / D h, U˛ i. Theorem 3.5.15. Let G be a relatively compact open set. (i)

N For any ˛ > 0, t > 0, ' 2 Bb .G/, f 2 C0 .G/ and g 2 C0 .X n G/,     E'm e ˛ G f .X G  /g.X G / D ', U˛G .f Jg / , E'm .f .X G  /g.X G / : G  t / Z tZ b pG D s '.x/f .x/g.y/J.dxdy/ds, 0

(3.5.28)

(3.5.29)

X

G where p G t is the transition function corresponding to the resolvent .R˛ /.

(ii)

N For any ˛ > ı, t > 0, ' 2 Bb .G/, f 2 C0 .G/ and g 2 C0 .X n G/,    .ı/  .˛ı/b G b b b / D ', U b .ı/,G .f  b E e f .X /g.X J gb / , ˛ı b b  m hı b G G 'b m  .ı/  b b b /;b E f .X /g.X G  t b b  G G 'b m Z tZ b p .ı/,G h.x/f .x/g.y/b J .ı/ .dxdy/ds, D s 0

.ı/,G

where b pt

(3.5.30)

(3.5.31)

X

b .ı/,G . is the transition function of the Hunt process M

N (iii) For any ˛ > ı, t > 0, ' 2 BbC .G/, f 2 C0C .G/ and g 2 C0C .X n G/,     G b b b / D ', U bG b f .X /g.X E'm e ˛b ˛ .f  J g / , G G b b   b b b /;b E'm f .X /g.X G  t b b G G Z tZ b b pG D s f .x/g.y/J .dxdy/ds, 0

X

bG where b pG t is the transition function of the pseudo Hunt process M .

(3.5.32)

(3.5.33)

110

Chapter 3 Markov processes

(iv) J.dxdy/ D b J .dydx/, that is “ “ F .x, y/J.dxdy/ D XXnd

F .y, x/b J .dxdy/

(3.5.34)

XXnd

for any non-negative function F .x, y/ on X  X n d. Proof. Since each side of equation (3.5.28) is a Laplace transformation of equation (3.5.29), equation (3.5.29) follows from equation (3.5.28) by inverting the Laplace transformation. Let f , g be as in (i) and let ' 2 C0 .G/. Then  2  bG bG .', U˛G .f  Jg // D hJg , f R ˛ 'i D lim ˇ Rˇ g, f  R˛ ' ˇ !1  2 G  D lim ˇ R˛ .f  Rˇ g/, ' . ˇ !1

Since g.Xs / D 0 for s  G a.s. Px , ˇ 2 R˛G .f  Rˇ g/.x/   Z G Z 1 D ˇ 2 Ex e ˛t f .X t /dt e ˇ s g.XsCt /ds 0  Z0 G Z 1 2 ˛. G t/ ˇ s e f .X G t /dt e g.X G tCs /ds D ˇ Ex  Z0 G Z0 1 2 ˛. G t/ ˇ s D ˇ Ex e f .X G t /dt e g.X G tCs /ds t  Z0 G Z 1 2 ˛ G .ˇ ˛/t ˇ s D ˇ Ex e e f .X G t /dt e g.X G Cs /ds/ Z D Ex

0

0 ˇ G

e



! Ex e

0 ˛ G

˛ G .ˇ ˛/t=ˇ

f .X G t=ˇ /dt  f .X G  /g.X G / as ˇ ! 1

Z

e

1

e 0

s

 g.X G Cs=ˇ /ds

for x 2 G. Hence

   .', U˛G .f Jg // D ', E e ˛ G f .X G /g.X G /

which implies equation (3.5.28). .ı/ b .ı/ , b b.ı/,G be the resolvent of the part process on G of M J g .dx/ D Similarly, let R ˇ R R b b J g .dx/ D J .ı/ .dxdy/g.y/ and b J .dxdy/g.y/. Then we have X

X



     b.ı/,G .f b bG b / D b ', R J .ı/ D ', R , f R˛G ' J gb g / b ˛ .f J gb ˛ı m hı hı   bˇ .gb hı /, f b hı R˛G ' D lim ˇ 2 R ˇ !1

111

Section 3.5 Decompositions of Dirichlet forms

  bG b b D lim ˇ 2 ', R ˛ .f  R ˇ .g hı // ˇ !1

  b.ı/,G .f  R b.ı/ g/ D lim ˇ 2 ', R ˛ı ˇ ı m b ˇ !1

 .ı/  .˛ı/b G b b / . e f .X /g. X b b G G 'b m

Db E

This implies equation (3.5.30). Equation (3.5.31) follows from equation (3.5.30) by inverting the Laplace transformation. The proof of (iii) ’ is similar. D (iv) By virtue of equation (3.5.26) XX u.y/v.x/J.dxdy/ ’ b XX u.y/v.x/J .dydx/ for all u, v 2 C0 .X / \ F with disjoint supports. Since C0 .X / \ F is dense in C0 .X /, this also holds for any u, v 2 C0 .X / with disjoint supports. For any compact sets B1 and B2 such that B1 \ B2 D ;, there exists sequences ¹ui º, ¹vi º  C0 .X / such that the supports of ui and vi are disjoint and limi!1 ui D 1B1 and limi!1 vi D 1B2 decreasingly. Hence it holds that b J .B1  B2 / D J.B2  B1 /. For any measurable function ˆ.x, y/ on X ’ X n d, define the function ˆ ’by ˆ.x, y/ D ˆ.y, x/. Then this implies b J .dxdy/ D that XXnd ˆ.x, y/b XXnd ˆ.x, y/J .dxdy/ for any function ˆ.x, y/ D 1B1 .x/1B2 .x/. Therefore it also holds for the function ˆ which can be written as a countable sum of such functions and furthermore for any non-negative Borel measurable functions ˆ on X  X n d by the monotone class lemma. This yields equation (3.5.34). By equation (3.5.25), J D 0 if and only if b J D 0. Furthermore, Theorem 3.5.14 and Theorem 3.5.15 imply the following result. Theorem 3.5.16. (i) (ii)

M is a diffusion process if and only if J D 0.

If J D 0, then k D 0 if and only if M has no killing inside, that is Px .X 2 X ,  < 1/ D 0

q.e. x.

(3.5.35)

Chapter 4

Additive functionals and smooth measures

Although the underlying measure m is not necessarily excessive in the present setting of the lower bounded semi-Dirichlet form, the Revuz measure of a positive continuous additive functional of M is well defined by making all coexcessive functions participate in the defining formula. By this, we can establish a one-to-one correspondence between the positive continuous additive functionals and smooth measures given in Section 4.1. For a smooth measure , there exists a dual pair of positive continuous additive functionals with Revuz measure . The relations between the transformed resolvents by these additive functionals are given in Section 4.2. They are used to characterize the part of the Dirichlet forms on any Borel sets. Section 4.3 is concerned with the time changes and killings of Hunt processes associated with Dirichlet forms. For semi-Dirichlet forms, similar results to those for the symmetric Dirichlet forms on time change are obtained under the assumption that the forms are non-negative.

4.1 Positive continuous additive functionals and Revuz measures Let M D .X t , Px / be a Hunt process associated with a regular Dirichlet form .E, F /. A real-valued function A t .!/ is called an additive functional (AF in abbreviation) of M if it satisfies the following conditions. (A.1) A t ./ is F t -adapted. (A.2) There exist a set ƒ and an exceptional set N satisfying Px .ƒ/ D 1, for all x 2 X nN ,  t ƒ  ƒ for all t > 0 and, for ! 2 ƒ, A t .!/ is right continuous with left limit relative to t , A0 .!/ D 0, jA t .!/j < 1, for all t < .!/, A t .!/ D A .!/, for all t  .!/ and AsCt .!/ D A t .!/ C As . t !/ for all s, t  0. The set ƒ is called a defining set of A t . A t .!/ is called continuous (resp. positive) if A t .!/ is continuous relative to t (resp. non-negative) for all ! 2 ƒ. Clearly a positive additive functional is non-decreasing. For a positive continuous additive functional (PCAF in abbreviation) A t .!/, define a kernel UA˛ by Z  UA˛ f .x/ D Ex

1

e ˛t f .X t /dA t ,

0

where and in the following we put f ./ D 0.

(4.1.1)

113

Section 4.1 Positive continuous additive functionals

For later use, we shall first give a generalized resolvent equation containing two ˛,ˇ PCAFs. For two PCAFs A and B of M, we define the kernel VA,B by Z 1  ˛,ˇ ˛A t ˇB t VA,B f .x/ D Ex e f .X t /dB t . (4.1.2) 0

0,˛ . Then V t˛ D R˛ and The AF B t D t is indicated by t . We shall set VB˛ D VA,B ˛,0 ˛ V t,A f D UA f .

Lemma 4.1.1. For any ˛, ˇ,  , ı > 0 and f 2 Bb .X /, ˛,ˇ

,ı

ˇ ,˛

,ı

˛,ˇ

,ı

VA,B f  VA,B f C .˛   /VB,A VA,B f C .ˇ  ı/VA,B VA,B f D 0.

(4.1.3)

,0

˛,0 jf j is bounded for If VAB jf j is bounded for some > 0 and f 2 Bb .X /, then VA,B all ˛ > 0 and equation (4.1.3) holds for all ˛, ˇ,  , ı  0 such that ˛ C ˇ > 0 and  C ı > 0.

Proof. We shall first assume that ˛, ˇ,  , ı > 0. In this case, ˇ ,˛

,ı

˛,ˇ

,ı

.˛   /VB,A VA,B f C .ˇ  ı/VB,A VA,B f .x/ Z 1  Z 1 D .˛   /Ex e ˇB t ˛A t dA t e Au ı t ıBu ı t f .Xu ı  t /dBu ı  t 0 0 Z  Z 1 1 ˛A t ˇB t e dB t e Au ı t ıBu ı t f .Xu ı  t /dBu ı  t C .ˇ  ı/Ex 0 Z 10 e ˛A t ˇB t dA t D .˛   /Ex 0  Z 1 .AuCt A t /ı.BuCt B t /  e f .XuCt /dBuCt Z0 1 e ˛A t ˇB t dB t C .ˇ  ı/Ex 0  Z 1 .AuCt A t /ı.BuCt B t /  e f .XuCt /dBuCt 0 Z 1 e Au ıBu f .Xu /dBu D Ex 0  Z u  e .˛/A t .ˇ ı/B t ..˛   /dA t C .ˇ  ı/dB t /  Z 10    Au ıBu .˛/Au .ˇ ı/Bu dBu e f .Xu / 1  e D Ex D

0 ,ı VA,B f .x/

˛,ˇ

 VA,B f .x/.

114

Chapter 4 Additive functionals and smooth measures ,0

If VA,B jf j is bounded, then the equation ,0

,0

˛,0 VA,B jf j D VA,B jf j  .˛  /VA˛ VA,B jf j ˛,0 yields that VA,B jf j is bounded. The remaining assertion follows from equation (4.1.3) by the limiting procedure. ˛,0 The assumption of the boundedness of V t,A jf j D UA˛ jf j in the latter part of Lemma 4.1.1 holds for a wide class of functions as is shown in the next lemma, where F denotes the first exit time from F .

Lemma 4.1.2. If A is a PCAF of M such that Px .A1 > 0/ D 1 for q.e. x, then there exists a strictly positive bounded integrable function g and a nest ¹Fn º such that, UA˛ IFn  nR˛ g for each n. In particular, sup˛>0 k˛UA˛ 1Fn k1 < 1 and sup˛˛0 C k˛UA˛ 1Fn k < 1 for any  > 0. Proof. Let f be a strictly positive bounded measurable function. Instead of .˛, ˇ,  , ı/ and B t in the resolvent equation (4.1.3), take .0, ˛, ˇ, ˛/ and t respectively. Then we have ˇ ,˛ ˇ ,˛ R˛ f .x/  VA,t f .x/  ˇUA˛ VA,t f .x/ D 0 ˇ ,˛

for q.e. x. Since R˛ f is quasi-continuous and UA˛ VA,t f is ˛-excessive, this shows ˇ ,˛

that VA,t f is quasi-upper semi-continuous, that is, there exists a nest ¹Bn º such that ˇ ,˛ .1/ VA,t f is upper semi-continuous on each Bn . In particular the set defined by Fn D ˇ ,˛

¹x 2 Bn : ˇVA,t f .x/  1=nº is closed. Furthermore, the same equation gives the estimate 1 ˇ ,˛ ˇ ,˛ ˇUA˛ VA,t f .x/ D R˛ f .x/  VA,t f .x/  kf k1 ˛ from which it follows that n ˇ ,˛ UA˛ IF .1/ .x/  nˇUA˛ VA,t f .x/  nR˛ f .x/  kf k1 . n ˛ Furthermore,  Z  Ex e ˇA t ˛t f .X t /dt

.1/ Fn

   ˇ ,˛  D Ex exp  ˇA .1/  ˛F .1/ VA,t f .X .1/ / n Fn Fn    1 Ex exp  ˇA .1/  ˛F .1/  n Fn ˇn converges to zero as n ! 1 for q.e. x. Hence limn!1 F .1/ D  a.s. Px . This .1/

combined with Lemma 3.4.2 shows that ¹Fn º is a nest.

n

Section 4.1 Positive continuous additive functionals

115

Next, take a strictly positive function g 2 L2 .X ; m/\L1 .X ; m/. Then there exists .2/ .2/ .1/ .2/ a nest ¹Fn º such that Fn  ¹x : R˛ g.x/  1=nº. Put Fn D Fn \ Fn . Then Fn is finely closed and limn!1 Fn D  a.s. Px . Furthermore, UA˛ 1Fn .x/  n.R˛ f .x/ ^ R˛ g.x//. This implies that kUA˛ 1Fn k1  ˛ > ˛0 .

n ˛ kf

k1 for ˛ > 0 and kUA˛ 1Fn k 

n ˛˛0 kgk

for

4.1.3. For a non-negative measurable function g 2 L2 .X ; m/, A t .!/ D RExample t 0 g.Xs /ds is a PCAF. In this case, UA˛ f D R˛ .fg/ and the measure d D gd m satisfies Z 1 ˇ b h.x/f .x/.dx/ D lim Eb ..fA/ t / D lim ˇ.b h, UA f / t!0 t hm ˇ !1 h, where for all f 2 B C .X / and ˛-coexcessive function b Z t .f  A/ t D f .Xs /dAs .

(4.1.4)

(4.1.5)

(4.1.6)

0

The purpose of this section is to extend the correspondence (4.1.5) to general PCAFs b˛ f a and their associated measures. As in Chapter 3, for any ˛ > 0, we denote by R 2 1 b q.c. modification of G ˛ f for f 2 L .X ; m/ or f 2 L .X ; m/. In particular, if an b˛Cnb h instead ˛-coexcessive function b h belongs to F , then by considering limn!1 nR bˇ C˛b b˛Cnb h b h and limn!1 nR h Db h of b h, we may suppose that b h satisfies b h  0, ˇ R ˛t b tb h b h m-a.e. Even if b h does not belongs to F , we q.e. Furthermore, b h satisfies e T may assume that b h is quasi-continuous. In fact, for any strictly positive  -coexcessive b f / 2 F by Lemma 1.4.2, it has a b f 2 F with  > ˛0 , since b h ^ .nR function R q.c. modification b hn . Furthermore, since b hn .x/ D b hnC1 .x/ q.e. on the quasi-open set b b h Db hn on Gn , b h can be considered as Gn D ¹x : hn .x/ < nR f .x/º, by putting b quasi-continuous. Theorem 4.1.4. For any PCAF A t there exists a unique positive  -finite measure A charging no exceptional sets such that Z  1 ˇ  f .x/b h.x/A .dx/ D lim Eb ..f  A/ t / D lim ˇ b h, UA f (4.1.7) t!0 t hm ˇ !1 X for any bounded non-negative function f and m-integrable ˛-coexcessive function b h with ˛ > 0.

116

Chapter 4 Additive functionals and smooth measures

Proof. Suppose first that b h 2 F is an m-integrable ˛-coexcessive function. For any bounded non-negative function f ,    h, E ..f  A/ t / p sb e ˛.sCt/ Eb ..f  A/sCt / D e ˛.sCt/ Eb ..f  A/s / C b hm hm ˛s  e Eb ..f  A/s / C e ˛t Eb ..f  A/ t / . hm hm Hence e ˛t Eb ..f  A/ t / is subadditive and hence .1=t /e ˛t Eb ..f  A/ t / inhm hm creases as t decreases. Furthermore, the relation Z ˇEb hm

1

e

.ˇ C˛/t

 d.f  A/ t

0

˛Cˇ D ˇ

Z

1 0

se s

  1 ˛.s=ˇ / Eb .f  A/s=ˇ ds, e hm s=ˇ

ˇ C˛ implies that limˇ !1 ˇ.b h, UA f / D lim t!0 .1=t /e ˛t Eb ..f  A/ t / exists as an hm increasing limit. Hence equation (4.1.7) exists. For any bounded non-negative nearly Borel measurable function f such that f =b h is bounded, define A .f / by

  ˇ C˛ h, UA .f =b h/ . A .f / D lim ˇ b ˇ !1

(4.1.8)

Since the right-hand limit, by changing the order of the limit, it P side is anPincreasing 1 f / D  .f holds that A . 1 nD1 n nD1 A n / for any non-negative nearly Borel measurable functions ¹fn º. Hence A .f / can be represented by a measure, R that is, there exists a measure A on the nearly Borel sets such that A .f / D X f .x/A .dx/. Clearly A charges no exceptional sets and 1 A .f / D lim Eb t!0 t hm

Z

t

e

˛s

d..f =b h/  A/s

0

 D lim

t!0

  1 ..f =b h/  A/ t . Eb hm t

h.2/ 2 F \L1 .X ; m/, let For any strictly positive q.c. ˛-coexcessive functions b h.1/ , b be the measures defined by equation (4.1.7) using b h.1/ and b h.2/ , respectively. If f is a q.e. finely continuous function such that f D 0 outside of the set ¹x : 1  h.i/ .x/  2 , i D 1, 2º for some constant 0 < 1 < 2 < 1, then .1/ .2/ A , A

Z Eb h.1/ m

t

e 0

D Eb.1/ m h

˛s

.f b h.2/ =b h.1/ /.Xs /dAs

 lim

n1 X

n!1 kD0

e

˛kdn



  .b h.2/ =b h.1/ /.Xkdn / .f  A/.kC1/dn  .f  A/kdn



117

Section 4.1 Positive continuous additive functionals

D lim

k!1

D lim

k!1

n1 X

  .2/ .1/   h =b h.1/ , pkdn .b e ˛kdn b h /E. .f  A/dn

kD0 n1  X

  h.1/ , .b e ˛kdn b p kdnb h.2/ =b h.1/ /E. .f  A/dn

kD0

 .1/ .2/ .1/   lim n b h , .b h =b h /E...f  A/dn / n!1

Dt

  1 lim Eb.2/ m .f  A/dn h n!1 dn

.2/ .2/ h f i, D t hA , b

where dn D t =n. This implies that hA , f b h.2/ i  hA , f b h.2/ i and hence .1/

.1/

.2/

.2/

hA , gi  hA , gi for any non-negative function g. Similarly, the converse inequality holds. Hence the measure A defined by equation (4.1.7) is independent of the choice of b h. Let ¹Fn º be the nest given by Lemma 4.1.2. For any Borel set h  cº for some positive constant c, since B  Fn \ ¹x : b Z   1 ˇ C˛ ˇ C˛ b b hd m, A .B/ D lim ˇ h, UA .1B =h/  sup kˇUA 1Fn k1 b c ˇ >0 ˇ !1 X A is a  -finite measure. If b h is not necessarily strictly positive, then by applying h1 and  > equation (4.1.7) to b h C b h1 for a strictly positive ˛-coexcessive function b 0 instead of b h and letting  ! 0, we obtain equation (4.1.7) for a non-negative ˛h is an ˛-coexcessive function which coexcessive function b h 2 F \ L1 .X ; m/. If b does not belong to F , by using the  -coexcessive function b hn introduced before the b statement of the lemma, equation (4.1.7) holds for hn . Hence, by letting n ! 1, we obtain the result. The measure A determined by Theorem 4.1.4 is said to be a Revuz measure associated with A t . The characterization in equation (4.1.7) is analogous to Section 5.1 of [55]. If m were excessive, it can be required only for b h D 1 (cf. [36]). Remark 2. In the proof of Theorem 4.1.4, we have shown the following result: Z  1 ˇ C˛  b f .x/b h.x/A .dx/ D sup e ˛t Eb Œ.f  A/  D sup ˇ h, U f . (4.1.9) t A hm t>0 t ˇ >0 X Lemma 4.1.5. For any non-negative measurable functions f , g and ˛ > 0, b˛ g/ A . .UA˛ f , g/ D .f , R

(4.1.10)

1 b Proof. We may assume that f , g 2 L1 C .X ; m/ \ L .X ; m/. Since R˛ g is an ˛coexcessive function, by Theorem 4.1.4 and the relation

nR˛ UAnC˛ f D nRnC˛ UA˛ f D UA˛ f  UAnC˛ f

118

Chapter 4 Additive functionals and smooth measures

we have b˛ g, U nC˛ f / D lim n.g, R˛ U nC˛ f / b˛ g/ A D lim n.R .f , R A A n!1

D

n!1

lim n.g, RnC˛ UA˛ f n!1

/ D .g, UA˛ f /,

where in the last equality we used the fine continuity of UA˛ f . Lemma 4.1.6. For any PCAF A t , there exists a nest ¹Fn º such that 1Fn  A 2 S0 and UA˛ .1Fn / is a q.c. version of U˛ .1Fn  A / for any ˛ > ˛0 .

.1/

Proof. By virtue of Lemma 4.1.2 and its proof, there exists a nest ¹Fn º and a strictly .1/ positive function g 2 L1 .X ; m/ \ L1 .X ; m/ such that .Fn / < 1 and UA˛ 1F .1/  n bˇ R˛ g D nR˛ g. In view of the dual version of Theorem 1.1.4 (iii), since limˇ !1 ˇ R .2/

R˛ g relative to E˛ , by Theorem 2.2.5, there exists a nest ¹Fn º and a sequence ˇk " bˇ R˛ g D R˛ g uniformly on each Fn.2/ . Let Fn D Fn.1/ \ 1 such that limk!1 ˇk R k .2/ Fn . Since UA˛ 1Fn is an ˛-excessive function belonging to L2 .X ; m/, by making use of Theorem 1.1.4 and its proof, it is enough to show that   (4.1.11) lim ˇk .I  ˇk Rˇk /UA˛ 1Fn , UA˛ 1Fn < 1 k!1

  R and E˛ UA˛ 1Fn , u D X u.x/1Fn .x/A .dx/ for all u 2 C0 .X / \ F . By equation (4.1.3), it holds that ˇ

UA˛ 1Fn .x/  ˇRˇ UA˛ 1Fn .x/ D UA 1Fn .x/  ˛Rˇ UA˛ 1Fn .x/. Hence, equation (4.1.10) yields     ˇ ˇ.I  ˇRˇ /UA˛ 1Fn , u D ˇ.UA˛ 1Fn  ˛R˛ UA 1Fn /, u     bˇ u  ˛ ˇRˇ U ˛ 1Fn , u D 1Fn , ˇ R A

A

for all u 2 L2 .X ; m/. In particular, if u D UA˛ 1Fn , then u  nR˛ g and bˇ R˛ g D R˛ g uniformly on Fn . Hence limk!1 ˇk R k     bˇ u b lim 1Fn , ˇk R 1  n lim , ˇ R g R Fn k ˇk ˛ k A A k!1 k!1   D n 1Fn , R˛ g A < 1. Clearly

  lim ˛ ˇRˇ UA˛ 1Fn , UA˛ 1Fn D ˛kUA˛ 1Fn k2 < 1.

ˇ !1

Section 4.1 Positive continuous additive functionals

119

Therefore equation (4.1.11) holds and hence UA˛ 1Fn 2 F . Similarly, if u 2 F \ C0 .X /, then   E.UA˛ 1Fn , u/ D lim ˇk .I  ˇk Rˇk /UA˛ 1Fn , u k!1 Z   1Fn .x/u.x/A .dx/  ˛ UA˛ 1Fn , u . D X

Hence the assertion of the lemma holds. Definition 7. Let us call a positive Borel measure  on X smooth if it satisfies the following conditions: (S.1)  charges no set of zero capacity; (S.2) there exists a nest ¹Fn º satisfying .Fn / < 1 and lim .K n Fn / D 0

n!1

(4.1.12)

for all compact set K  X . We shall denote by S the set of all smooth measures. Theorem 4.1.7. For any PCAF A t of M, the Revuz measure A defined by equation (4.1.8) is smooth. Proof. By virtue of equation (4.1.8), A satisfies condition (S.1). Also, as we have seen in Lemma 4.1.6, the nest ¹Fn º given there satisfies .Fn / < 1. Furthermore, Lemma 3.4.3 impliesthat Pm .F < 1/ D 0 for since Cap.˛/ .K n [n Fn / D 0, R 1 ˛t e 1F .X t /dA t D 0 which implies that 0 D F D K n .[n Fn /. Hence Eb hm 0 A .F / D limn!1 .K n Fn /. We next consider the converse problem of construction of a PCAF associated with a given smooth measure. Before giving the result, we prepare the following two lemmas. Lemma 4.1.8. For any u 2 F ,  2 S0 , 0 < T < 1,  > 0 and ˛ > ˛0 , it holds that P



 K 2 e ˛T b ˛ , U b ˛ /1=2 . sup je u.X t /j >   ˛ E˛ .u, u/1=2 E˛ .U 0tT 

(4.1.13)

Proof. Let E D ¹x : je u.x/j > º. Then the left-hand side of equation (4.1.13) is dominated by P .E  T /  E .e ˛E /  e ˛T . By virtue of Lemma 3.4.3 and

120

Chapter 4 Additive functionals and smooth measures

equation (2.1.4), we have Z   .˛/ .˛/ b E e ˛E D eE .x/.dx/ D E˛ .eE , U ˛ / .˛/ .˛/ b ˛ , U b ˛ /1=2  K˛ E˛ .eE , eE /1=2 E˛ .U K2 b ˛ , U b ˛ /1=2 .  ˛ E˛ .u, u/1=2 E˛ .U 

Lemma 4.1.9. Let ¹un º be an E˛ -Cauchy sequence of q.c. functions of F for ˛ > ˛0 . Then there exists a subsequence ¹unk º satisfying   unk .X t / converges uniformly relative to t P D1 on each compact interval of Œ0, 1/ for any  2 S0 . Proof. Take a sequence ¹nk º satisfying E˛ .unkC1  unk , unkC1  unk /  24k for ˛ > ˛0 . Put ƒk D ¹! : sup0tT junk .X t .!//  unkC1 .X t .!//j > 2k º, then by the preceding lemma, b ˛ , U b ˛ /1=2 P .ƒk /  K˛2 e ˛T 2k E˛ .U for every  2 S0 . By the Borel–Cantelli lemma, we then have P .limk!1 ƒk / D 0. This gives the assertion of the lemma. Theorem 4.1.10. For any  2 S0 , there exists a unique PCAF A such that UA˛ 1 is a q.c. version of U˛  for all ˛ > ˛0 . Proof. Let u be a q.c. version of U˛ . Since u is an ˛-excessive function belonging to F , there exists an exceptional set N such that nRnC˛ u " u on X n N by Lemma 3.4.6. For each n, consider the function gn defined by ´ n .u.x/  nRnC˛ u.x// x 2 X n N , gn .x/ D 0 x 2 N. Then R˛ gn .x/ D nRnC˛ u.x/ increases to u.x/ for any x 2 X n N and, as we have seen in Lemma 2.3.3, limn!1 R˛ gn D u weakly in .E˛ , F / and limn!1 gn m D  vaguely. Moreover, since E˛ .R˛ gn , R˛ gn / D .gn , R˛ gn / D n .u  nRnC˛ u, nRnC˛ u/ D n.u  nRnC˛ u, u/  n .u  nRnC˛ u, u  nRnC˛ u/  n .u  nRnC˛ u, u/ and the last term converges to E˛ .u, u/, E˛ .R˛ gn , R˛ gn / is bounded relative to n. Hence a subsequence of the CesJaro means of ¹R˛ gn º converges to u on X n N

121

Section 4.1 Positive continuous additive functionals

and strongly inP.E˛ , F /, that is, we can choose nk " 1 such that ¹R˛ fk º with k gn converges to u in the stated sense. Then, for any Borel set fk D .1=nk / nnD1 ˛ ˛ and hence RXnB f converges to U XnB  relative to B, HB R˛ fk converges to uB ˛ ˛ k E˛ . By taking a subsequence, it alsoRconverges q.e. By using this function fk  0, t define a PCAF given by AQk .t , !/ D 0 e ˛s fk .Xs .!//ds. b ˛XnB k1 < 1. Then, Take a fine open set B and a measure  2 S0 satisfying kU for any k  l,   E .AQk .B /  AQl .B //2  Z B Z B ˛s ˛u e .fk  fl /.Xs /ds e .fk  fl /.Xu /du D 2E s  Z0 B 2˛s XnB D 2E e .fk  fl /.Xs /R˛ .fk  fl /.Xs /ds 0 XnB 2h, R2˛ ..fk

 fl /R˛XnB .fk  fl //i   XnB b 2˛ , .fk  fl /R˛XnB .fk  fl / D2 U

D

b 2˛ k1 .fk , R˛XnB .fk  fl //  2kU XnB b 2˛ k1 E˛ .R˛XnB fk , R˛XnB .fk  fl // D 2kU XnB b 2˛ k1 .fk , R˛XnB fk /1=2 E˛ .R˛XnB .fk  fl /, R˛XnB .fk  fl //1=2  2kU XnB b 2˛  2kU k1 .fk , R˛ fk /1=2 E˛ .R˛ .fk  fl /, R˛ .fk  fl //1=2 . XnB

Since R˛ fk D

1 nk

Pnk

nD1 R˛ gn

D

.fk , R˛ fk /  .fk , u/ D

1 nk

Pnk

nD1 nRnC˛ u

 u,

nk 1 X .n.u  nRnC˛ u/, u/  E˛ .u, u/ nk nD1

is bounded relative to nk by Lemma 1.1.3. Hence we have   E .AQk .B /  AQl .B //2  K˛, E˛ .R˛ .fk  fl /, R˛ .fk  fl //1=2

(4.1.14)

XnB b 2˛ for K˛, D 2K˛ kU k1 E˛ .u, u/1=2 . Noting that PXB .B D 0/ D 1 a.s., let us define a .F t^B , P /-martingale Mn .t / by     Mk .t / D E AQk .B / j F t^B D AQk .t ^ B / C e ˛.t^B / EX t ^B AQk .B /

D AQk .t ^ B / C e ˛.t^B / R˛XnB fk .X t^B /.   Take a sequence ¹ki º such that E˛ R˛ .fki C1  fki /, R˛ .fki C1  fki / < 26i , then by the martingale maximal inequality in equation (3.2.3) and equation (4.1.14), we have     P max jMk .t /  Ml .t /j > 2i  22i E .AQk .B /  AQl .B //2  K 2i 0t1

122

Chapter 4 Additive functionals and smooth measures

for some constant K depending on . Hence,   P lim Mki .t / exists uniformly on Œ0, 1/ D 1. i!1

(4.1.15)

Since AQk .t ^B / D Mk .t /e ˛.t^B / R˛ fk .X t^B /, equation (4.1.15) combined with Lemma 4.1.8 implies that P . n ƒB / D 0 for ² ³ AQki .t ^ B , !/ converges uniformly in t ƒB D ! : . on each finite interval of Œ0, 1/ XnB

.1/ b ˛  > nº, where U b ˛  is assumed quasiFor general  2 S0 , let Bn D ¹x : U .2/ .2/ continuous already. Then there exists an open set Bn such that Cap.˛/ .Bn / < 1=n b ˛  is continuous on X n Bn.2/ . Put Bn D Bn.1/ [ Bn.2/ . Then, by virtue of and U Lemma 2.2.4, b ˛ , U b ˛ / C 1 ! 0, n ! 1. Cap.˛/ .Bn /  .K˛ =n/2 E˛ .U n

Hence, by Lemma 3.4.3, limn!1 Ex .e ˛Bn / D 0 for m-a.e. and hence q.e. x by Lemma 3.4.6. Therefore limn!1 Bn D 1 a.s. Px for q.e. x. This implies P . n ƒ/ D 0 for all  2 S0 and hence q.e. x, where ² ³ AQki .t , !/ converges uniformly in t ƒD !: . (4.1.16) on each finite interval of Œ0, 1/ Q , !/ D 0 for ! 2  n ƒ. Q , !/ D limi!1 AQk .t , !/ for ! 2 ƒ and A.t Put A.t Ri t ˛s Q !/. Then A is a PCAF with defining set ƒ. Define A.t , !/ by A.t , !/ D 0 e d A.s, Q It remains only to prove the relation E .A.1// D h, ui for  2 S00 . Since Q Q Bn //. On D limn!1 E .A. limn!1 Bn D 1 a.s. P , it holds that E .A.1// 2 the other hand, for any bounded non-negative function f 2 L .X ; m/, since the ˛˛ ˛ ˛ D HB˛n u of u D R˛ f satisfies uB  ˛1 kf k1 eB , by equareduced function uB n n n tion (2.4.4) we have ˛ ˛ , uB / E˛ .uB n n

K˛ ˛ ˛ kf k21 E˛ .eB , eB /. n n ˛2 XnB

˛ This implies that limn!1 uB D 0 and hence limn!1 R˛ n f D R˛ f in F . Since n 1 2 ¹R˛ f : f 2 L .X ; m/\L .X ; m/º is dense in F , [n F XnBn is also dense in F . For XnB any n  k and w 2 F XnBk , since E˛ .U˛   U˛ n , w/ D 0, by equation (2.4.17), XnBn the function v D limn!1 U˛  satisfies E˛ .U˛   v, w/ D 0. Since [k F XnBk is dense in F , this implies that v D U˛ . Q D h, ui, it is enough to show that Hence, for the proof of the relation E .A.1// XnB Q B // D h, U˛ i for any finely open set B such that kU b ˛XnB k1 < 1. E .A. Since Mk .B / D AQk .B / converges in L2 .P /, so does the martingale Mk .t ^ B /.

123

Section 4.1 Positive continuous additive functionals

Hence, noting that limk!1 R˛ fk D u q.e. and increasingly, we have     E AQ t^B C E e ˛t^B U˛XnB .X t^B / D lim E .Mk .t ^ B // D lim E .Mk .B // k!1

k!1

D lim E .AQk .B // D lim h, R˛XnB fk i k!1

D h, u 

HB˛ ui

k!1 h, U˛XnB i.

D

XnB

By letting t ! 1 and noting lim t!1 E .e ˛.t^B / U˛ .X t^B // D 0, we obtain the result. For the proof of the uniqueness, suppose that A.1/ and A.2/ are PCAFs satisfying the condition of the theorem. Then Z 1  Z 1  .1/ .2/ e ˛t dA t e ˛t dA t D Ex D u.x/ q.e. Ex 0

0

Z

Let vij .x/ D Ex Then vij .x/ D 2Ex

R 1 0

1

e ˛t dA t

.i/

0 .k/

e 2˛t u.X t /dA t Z

h, vij i D 2E



1

e

Z

1

e ˛t dA t

.j /

 .

0

for k D i , j . Hence for any  2 S00 ,

2˛t

.k/ u.X t /dA t



0

is independent of i , j . Therefore we have Z 1 2  Z 1 .1/ .2/ ˛t ˛t e dA t  e dA t D h, v11  2v12 C v22 i D 0. E 0

0

Lemma 4.1.11. Let  and A be as in the preceding theorem. Then, for any f 2 BbC .X / and ˛ > 0, UA˛ f is a q.c. version of U˛ .f  /. Proof. First assume that ˛ > ˛0 . It is sufficient to consider the case f D 1G for an open set G with .@G/ D 0. Since UA˛ 1G and UA˛ 1XnG are ˛-excessive functions satisfying UA˛ 1G C UA˛ 1XnG D U˛  2 F q.e., by virtue of Lemma 1.4.2, UA˛ 1G and UA˛ 1XnG belong to F . Hence there exist positive Radon measures ,  2 S0 such that UA˛ 1G D U˛  and UA˛ 1XnG D U˛  q.e. Since UA˛ 1G D HG˛ UA˛ 1G and ˛ UA˛ 1XnG , Lemma 2.4.3 and Theorem 3.5.1 imply that the measures UA˛ 1XnG D HXnG  and  are supported by GN and X nG, respectively. Since  D C, this implies that .@G/ D .@G/ D 0 and hence  D 1G . Therefore UA˛ 1G D U˛  D U˛ .1G  /. This implies the assertion of the lemma. ˇ If ˛  ˛0 , then by taking ˇ > ˛0 , UA˛ satisfies UA˛ f .x/ D UA f .x/ C .ˇ  ˛/ ˇ R˛ UA f .x/. Similarly, since U˛ .f  / satisfies U˛ .f  / D Uˇ .f  / C .ˇ  ˛/ R˛ Uˇ .f  /, the assertion also holds.

124

Chapter 4 Additive functionals and smooth measures

Corollary 4.1.12. Let  and A be as in Theorem 4.1.10. Then equation (4.1.7) holds. h be an ˛-coexcessive function. Then by Lemma 4.1.11, Proof. Let f 2 C0C .X / and b ˇ C˛ bˇ C˛b h/ h, Uˇ C˛ .f  // D ˇEˇ C˛ .Uˇ C˛ .f  /, R ˇ.b h, UA f / D ˇ.b Z bˇ C˛b Dˇ h.x/.dx/. f .x/R X

bˇ C˛b Since limˇ !1 ˇ R h Db h increasingly, ˇ C˛ lim ˇhb h  m, UA f i D

ˇ !1 ˇ

ˇ C˛

Z

f .x/b h.x/.dx/.

X

ˇ

Noting that UA f D UA f C ˛Rˇ C˛ UA f and   ˇ bˇ R b˛Cˇ b hi h, ˛Rˇ C˛ UA f D lim hf  , ˛ˇ R lim ˇ b ˇ !1

ˇ !1

˛ hf  , b hi D 0, ˇ !1 ˇ  ˛

 lim equation (4.1.7) follows.

Theorem 4.1.13. For  2 S0 and a PCAF A, the following are equivalent to each other: UA˛ 1 is a q.c. version of U˛  for ˛ > ˛0 ; b˛ h/ for f , h 2 B C .X / and ˛ > 0; (ii) .UA˛ f , h/ D .f , R Rt p s hids for f , h 2 B C .X /; (iii) Ehm ..fA/ t / D 0 hf , b ..fA/ t / D h, b hf i for any f 2 B C .X / and q.c. ˛-coexcessive (iv) lim t!0 1t Eb hm function b h for some ˛ > 0.; (i)

(v)

ˇ

limˇ !1 ˇ.u, UA 1/ D h, ui for any q.c. function u 2 F \ B C .X / such that juj  b h for some -integrable ˛-coexcessive function b h 2 F for some ˛ > 0.

Proof. (i) , (ii) follows from Lemma 4.1.11 for ˛ > ˛0 . If (ii) holds for ˛ > ˛0 , as in the proof of Theorem 3.3.1, it also holds for ˛ > 0 by the resolvent equation. (ii) , (iii) is clear. (i) , (iv) is a consequence of Theorem 4.1.4 and Corollary 4.1.12. For bˇ C˛ juj  b h and any sequence any function u satisfying the condition of (v), since ˇ R bˇ C˛ uº contain a subsequence converging to u q.e., using Lebesgue theorem, ¹ˇn R n we obtain (v) from (ii). Therefore, Theorem 4.1.4 implies (v) ) (i). Thus we have shown the converse result of Theorem 4.1.10 when  2 S0 . To show it for  2 S , we shall first show the following lemma.

125

Section 4.1 Positive continuous additive functionals

Lemma 4.1.14.  2 S if and only if there exists a nest ¹En º satisfying equation (4.1.12) and 1En   2 S0 . Proof. The if part is clear. For the proof of the converse assertion, we may assume that .X / < 1. In fact, if .X / D 1, for the nest ¹Fn º in (S.2) in the definition of a smooth measure, it is enough to show the assertion for 1Fk   because, for any n  1, if there exists a nest ¹Enk º such that 1Enk \Fk   2 S0 and .Fk n .Enk \ Fk // ! 0 as S n ! 1, then En D nlD1 .Fl \ Enl / satisfies the desired property. Hence we assume that .X / < 1. We shall first show that there exists a decreasing sequence of open sets such that Cap.˛/ .Gn / ! 0,

.Gn / ! 0,

.A/  2n Cap.˛/ .A/,

for all A  Fn D X n Gn

(4.1.17)

Once equation (4.1.17) has been shown, then we obtain that IFn   2 S0 because for any non-negative v 2 F \ C0 .X /, by putting a D E˛ .v, v/1=2 , we have Z Fn

v.x/.dx/  a.X / C  a.X / C

1 X kD0 1 X

a2kC1 .¹x : a2k < v.x/  a2kC1 º \ Fn / a2kC1 2n Cap.˛/ .¹x : a2k < v.x/º/

kD0

 a.X / C 2n 

D .X / C

1 X

2kC1 .a22k /1 K˛2 E˛ .v, v/

kD0  2 nC2 E˛ .v, v/1=2 . K˛ 2

This implies that IFn   2 S0 . Now, we shall prove equation (4.1.17). Let ı D inf¹2n Cap.˛/ .A/  .A/ : A is a Borel subset of X º. Clearly ı  .X /. If ı  0, then it is enough to put Gn D ;. If ı < 0, choose an open set B1 such that 2n Cap.˛/ .B1 /  .B1 /  ı2 and set ı1 D inf¹2n Cap.˛/ .A/  .A/ : A  X n B1 º. Then ı  2n Cap.˛/ .A [ B1 /  .A [ B1 /  2n Cap.˛/ .A/  .A/ C 2n Cap.˛/ .B1 /  .B1 / ı  2n Cap.˛/ .A/  .A/ C 2

126

Chapter 4 Additive functionals and smooth measures

for A  X n B1 , that is ı=2  ı1 . If ı1  0, then put Gn D B1 . If ı1 < 0, take an open set B2  X n B1 satisfying 2n Cap.˛/ .B2 / .B2 / < ı1 =2. Continuing this argument, we can find open sets B1 , B2 , : : : such that 2n Cap.˛/ .B1 [    [ Bk /  .B1 [    [ Bk /  0 2k ı  2n Cap.˛/ .A/  .A/

for all A  X n .B1 [    [ Bk /.

S1 n .˛/ .A/  .A/ for all A  X n G 0 and Put Gn0 D n kD1 Bk . Then 2 Cap S n .˛/ 0 0 0 is a decreasing sequence G 2 Cap .Gn /  .Gn /. Then the set Gn D 1 mDn m  2nC1 .X / ! 0. Since  does not charge the set of open sets satisfying Cap.Gn / T T n Gn of zero capacity, 0 D . n Gn / D limn!1 .Gn /. Lemma 4.1.15. Let  2 S0 and A be its corresponding PCAF. Then, for any nearly Borel set F , ˛-coexcessive function b h with ˛ > 0 and v 2 B C .X /, Z F  Z .ˇ C˛/t b lim ˇEb h.x/v.x/.dx/, (4.1.18) e v.X /dA t t D hm ˇ !1

F0

0

increasingly, where F 0 is the quasi-interior of F , that is, the maximal quasi-open set contained in F . Proof. We first assume that the ˛-coexcessive function b h belongs to F . For any v 2 ˇ C˛ L2 .X ; m C /, since UA v is a q.c. version of Uˇ C˛ .v  / and, by Theorem 3.5.1, ˇ C˛ ˇ C˛ ˇ C˛ b˛ b h are q.c. modifications of vXnF and b hXnF , respectively, we have HXnF v and H XnF from Theorem 2.4.2 that Z F    ˇ C˛ ˇ C˛ ˇ C˛ b .ˇ C˛/t ˇEb e v.X /dA v  H U v, h D ˇ U t t A XnF A hm 0   ˇ C˛ ˇ C˛ ˇ C˛ b b D ˇEˇ C˛ UA v  HXnF UA v, R ˇ C˛ h   ˇ C˛ ˇ C˛ ˇ C˛ b b b ˇ C˛ b b D ˇEˇ C˛ UA v  HXnF UA v, R ˇ C˛ h  H XnF Rˇ C˛ h   ˇ C˛ b b b ˇ C˛ b b D ˇEˇ C˛ UA v, R ˇ C˛ h  H XnF Rˇ C˛ h Z   ˇ C˛ bF b bF b v.x/R D ˇEˇ C˛ UA v, Rˇ C˛ h D ˇ ˇ C˛ h.x/.dx/. X

R By letting ˇ " 1, the right-hand side increases to F 0 f .x/v.x/b h.x/.dx/. b b b b˛ g/ 2 F for a For a general q.c. ˛-coexcessive function h, since hn D h ^ .nR 1 1 strictly positive function g 2 L .X ; m/ \ L .X ; m/ and ˛ > ˛0 , it is enough to apply equation (4.1.18) to b hn and then let n " 1. Theorem 4.1.16. For any  2 S , there exists a unique PCAF A satisfying equation (4.1.7).

Section 4.1 Positive continuous additive functionals

127

Proof. Let  2 S and ¹En º be the sequence of closed sets in Lemma 4.1.14. Since 1En  2 S0 , by Theorem 4.1.13, there is a corresponding PCAF A.n/ related to 1En  by equation (4.1.7). If ` > k, then A.k/ and 1Ek A.`/ are equivalent. Choose an ex.k/ ceptional set N and a defining set ƒ commonly for all A.k/ and satisfying A t .!/ D .1Ek  A.kC1/ / t .!/, limk!1 Ek .!/  .!/ for all ! 2 ƒ. For ! 2 ƒ, let ´ .k/ Ek1 .!/  t < Ek .!/, A t .!/ A t .!/ D A .!/ .!/ t  limk!1 Ek .!/. Then A is a PCAF and, for any ˛-coexcessive function b h and f 2 B C .X /, since the limits of the second term of the following equalities are increasing relative to ˇ and k, we have Z E  k ˇ C˛ .ˇ C˛/t b e f .X t /dA t lim ˇ.UA f , h/ D lim lim ˇEb hm ˇ !1 ˇ !1 k!1 0 Z E  k .k/ e .ˇ C˛/t f .X t /dA t D lim lim ˇEb hm k!1 ˇ !1 0 Z b h.x/f .x/.dx/ D lim k!1 E 0 k

Z D

b h.x/f .x/.dx/

X

S .k/ because X n k Ek0 is exceptional. Since A t .!/ D .1Ek  A/ t is the unique PCAF associated with 1En   by Theorem 4.1.10, the uniqueness assertion holds. Example 4.1.17. Let M be the Brownian motion on Rd . Then the measure .dx/ D g.x/dx is smooth if g is locally integrable.R In this case, as we have seen in Example t 4.1.3, the associated PCAF is A t .!/ D 0 g.Xs .!//ds. The present definition of PCAFs admits a not necessarily locally integrable function g. For example, let g.x/ D jxj˛ , then .dx/ D g.x/dx is a smooth measure when d  2 because Fn D ¹x 2 Rd : 1=n  jxj  nº satisfies the conditions of the definition of a smooth measure. The corresponding PCAF A t given above satisfies Px .A t < 1/ D 1, t > 0, if x ¤ 0 but, when x D 0, and ˛  2,   Z t Z t jXs j˛ ds < 1  P0 jXs.1/ j˛ ds < 1 D 0 P0 0

since

0

 P0 lim t#0

 .1/ Xt D 1 D 1. .2t log log 1=t /1=2

But the one point set ¹0º is an exceptional set and hence A t is a PCAF in the sense of this section.

128

Chapter 4 Additive functionals and smooth measures

4.2 Dual PCAFs and duality relations As in the preceding section, let M be a Hunt process associated with a regular Dirichlet form .E, F / on L2 .X ; m/. We assume that the path space ., F t / is canonical, that is  is the family of functions ! from Œ0, 1/ to X [ ¹º such that the coordinate function X t .!/ D !.t / is right continuous with left limit and F t is a right continuous complete  -field generated by  .¹Xs ; s  t º/. Furthermore, we assume that .E.5/ b holds. For a fixed ı > 0 and a strictly positive q.c. ı-coexcessive  functionhı , we have .ı/ .ı/ b D X bt, b seen in Theorem 3.3.6 that there exists a Hunt process M P x on ., F t / which possesses the resolvent  Z 1 1 b .ı/ ˛t b b b.ı/ R˛Cı .b e f .X t /dt D R hı f /.x/. Ex ˛ f .x/ D b 0 hı .x/ hı belongs to F and is bounded from below by a posiIf ı > ˛0 , then we assume that b tive constant on each compact set. This is possible as we have seen in Theorem 2.4.8. As introduced at the last part of Section 3.3, we can define a dual pseudo Hunt process b D .X bt, b P x / on ., F t / possessing the transition function b p t and in duality with M M relative to m. .ı/ b .ı/ D .X bt, b P x / on ., F t / similarly by Define an additive functional b A t of M b .ı/ define the associated Revuz b .ı/ . For a PCAF b A t of M (A.1) and (A.2) relative to M .ı/ measure b  similarly to equation (4.1.7) by b A Z  1 .ı/  .ı/ .f  b A/ t f .x/h.x/b  .dx/ D lim b E h b b m t!0 t A X   b .ı/,ˇ f D lim ˇ h, U (4.2.1) b m b A ˇ !1 for any q.c. ı-excessive function h, where Z 1  .ı/,ˇ .ı/ ˇ t b b b b f .x/ D Ex e f .X t /d A t . U b A 0 defined by b b .dx/ D Then equation (4.2.1) yields that the measure b b A A .ı/ .1=b hı .x//b  .dx/ satisfies b A Z  1b  b b ˇ f /, .f  A/ D lim ˇ.h, U f .x/h.x/b b .dx/ D lim (4.2.2) E t h m P A b t!0 A t ˇ !1 X where b ˇ f .x/ D b Ex U b A

Z

1

e 0

ˇ t

b t /d b f .X At



b .ı/,ˇ ı f .x/. Db hı .x/U b A

129

Section 4.2 Dual PCAFs and duality relations

As in the proof of Theorem 4.1.4, b b is a smooth measure determined independently A b as the smooth Revuz measure of the choice of h and b hı . In this sense, we can regard b A b b associated with the PCAF A t of M. Conversely, for any smooth measure, there exists b associated with it, that is, we have the following theorem. a PCAF of M b such that Theorem 4.2.1. For any measure  2 S , there exists a PCAF b A t of M C holds for any f 2 B .X / and ˛ > 0. equation (4.2.2) with  in place of b b b A Proof. Since the proof is similar to Theorem 4.1.10 and Theorem 4.1.16, we shall only mention the outline of it. By considering 1Fn   for a nest ¹Fn º satisfying 1Fn  b ˛  2 F and assume that it is u D U  2 S0 , we may assume that  2 S0 . Put b b˛ gn " u and weakly in .E˛ , F / for b g n .x/ D n.u.x/  quasi-continuous. Then R b b b ˛b b gn º nRnC˛ u.x//. By using the CesJaro means ¹R˛ f k º of the subsequence of ¹R R t ˛s .1/ bk D b bk .X b s /ds. Then, for b˛ f f such that limk!1 R u in .E, F /, put b Ak .t / D 0 e any  2 S00 , similarly to equation (4.1.14), we have    1=2 .1/ .1/ bk  f b` /, R bk  f b` / b b˛ .f b˛ .f E .b Ak .1/  b A` .1//2  C E˛ R for C D 2K˛ E˛ .U˛ , U˛ /1=2 . Then, by taking a suitable subsequence ¹b Ak º of i .1/ .1/ A.1/ .t , !/ exists as a uniformly convergent limit on A .t , !/ D b ¹b A º, limi!1 b .1/

k

ki

b ˛ i. E .b A.1/ .1// D h, U each compact interval a.s. b P  . Furthermore, it satisfies b R t ˛s .1/ As . Hence, it is enough to put b A t .!/ D 0 e d b b we have Using the strong Markov property of M, b˛ U b ˇ g/. b˛ g  U b ˇ g D 1 .U R b b b A A A ˇ˛ Hence, for any non-negative measurable functions f and g, using equation (4.2.2), we have by a similar method to Lemma 4.1.5 that b g/ D lim ˇ.f , R b˛ U b g/ D lim ˇ.R˛ f , R .R˛ f , g/b b b A A ˇ !1 ˇ !1 b A   ˇ bˇ g b˛ g  U f ,U D lim b b A A ˇ  ˛ ˇ !1 ˇ

ˇ

b ˛ g/. D .f , U b A If f 2 L2 .X ; m/ and g  b b 2 S0 , then the left-hand side can be written as A     b ˛ .g  b b ˛ .g  b R D E f , U  / D f , U  / .R˛ f , g/b ˛ ˛ b b A A b A

(4.2.3)

130

Chapter 4 Additive functionals and smooth measures

for ˛ > ˛0 . Hence it follows that b ˛ .g  b b˛ g D U b / q.e. U A b A

(4.2.4)

Let  2 S . By virtue of Lemma 4.1.14, there exists a nest ¹Fn º such that 1Fn   2 S0 . If ˛ > ˛0 , then equation (2.3.4) implies that there exists the ˛-copotential b ˛ .gb b ˛ .1Fn / 2 F . We may assume that it is quasi-continuous. Furthermore, U b / U A b ˛ .g  b b ˇ .g  b b˛ U b ˇ .g  b can be defined for any ˛ > 0 by U b /DU b / C .ˇ  ˛/R b / A A A ˇ ˇ b ˛ g similarly by U b˛ g D U b g C .ˇ  ˛/R b˛ U b g, for any fixed ˇ > ˛0 . If we define U b b b b A A A A then equation (4.2.4) can be extended to ˛ > 0. pA

˛ and R For a PCAF A t of M and ˛ > 0, define the kernels Up,A ˛ by

Z ˛ f .x/ Up,A

D Ex

R˛pA f .x/ D Ex

1

e Z0 1



˛tpA t

f .X t /dA t  ˛tpA t e f .X t /dt .

0

b define the kernels U b˛ Similarly, for a PCAF b A t of M,

A b˛pb and R by

 b b e f .X t /d A t  Z0 1 pb A ˛tpb At b b b e f .X t /dt . R˛ f .x/ D Ex

b ˛ f .x/ D b Ex U p,b A

Z

A p,b

1

˛tpb At

0

b˛ g D U b ˛ g. Furthermore, define the corresponding kernels U b .ı/,˛ f and Then U b 0,b p,b A A A b .ı/,p A .ı/ b instead of M. b Then b˛ R f similarly for M b .ı/,˛ .f =b b ˛Cı f .x/ D b hı .x/U hı /.x/ U p,b A A p,b A A b b˛.ı/,pb bpb f .x/ D b hı .x/R .f =b hı /.x/. If A t and b A t are PCAFs of M and M and R ˛Cı associated with the measure  2 S by the relations in equations (4.1.7) and (4.2.2) b ˛ g are q.c. modifications of U˛ .f  / and U b ˛ .g  /, respectively, then UA˛ f and U b A respectively.

b associated with A t are PCAFs of M and M Theorem 4.2.2. Suppose that A t and b  2 S by the relations in equations (4.1.7) and (4.2.2), respectively. Then, for any ˛, p > 0 and non-negative measurable functions f and g,         ˛ A b˛ g , U ˛ f , g D f , R b˛pb f ,g D f ,U g (4.2.5) Up,A p,A A p,b

131

Section 4.2 Dual PCAFs and duality relations

where .u, v/ D

R X

u.x/v.x/.dx/. Furthermore, we have

        A b ˛ g , R˛pA f , g D f , R b˛pb g . R˛pA f , g D f , U A p,b

(4.2.6)

Proof. We shall first assume that  2 S00 . Suppose that p D 0, ˛ > ˛0 and f , g 2 b ˛ g are q.c. modifications of U˛ .f  / and L1 .X ; m/ \ L2 .X ; m/. Since UA˛ f and U b A b ˛ .g  / respectively, it holds that U b ˛ g/ . .UA˛ f , g/ D .f , U b A

(4.2.7)

In fact, b ˛ .g  // .UA˛ f , g/ D E˛ .UA˛ f , U b ˛ g/ . b ˛ .g  // D .f , U D .f , U b A This gives the first relation of equation (4.2.5). Similarly, the second relation follows from b˛ g/ D hf  , R b˛ gi. .UA˛ f , g/ D E˛ .UA˛ f , R If ˛  ˛0 , then by taking ˇ > ˛0 , it holds that ˇ

UA˛ f  UA f C .˛  ˇ/Rˇ UA˛ f D 0. Hence we have UA˛ f D

1 X

ˇ

.ˇ  ˛/n Rˇn UA f .

nD0

Applying the result of ˇ > ˛0 shown above, we have 

UA˛ f , g



D D D

1 X

  ˇ .ˇ  ˛/n Rˇn UA f , g

nD0 1 X



  bn U b ˇ .g  / .ˇ  ˛/n f , R ˇ



nD0 1 X

  bn U bˇ g . .ˇ  ˛/n f , R ˇ A

nD0



  b ˛ g by the relation The right-hand side is equal to f , U b A b ˇ g C .ˇ  ˛/R bˇ U b ˛ g D 0. b˛ g  U U b b b A A A

132

Chapter 4 Additive functionals and smooth measures

Hence the first relation of equation (4.2.5) also holds for ˛  ˛0 . By choosing ˇ > ˛0 , the second relation of equation (4.2.5) for p D 0 and ˛  ˛0 follows from ˇ

ˇ

.UA˛ f , g/ D .UA f C .ˇ  ˛/R˛ UA f , g/ bˇ R bˇ g/ C .ˇ  ˛/.f , R b˛ g/ D .f , R b˛ g/ . D .f , R To show equation (4.2.5) for p > 0, note that kUA˛ 1k1 < 1 for any ˛ > 0 and  2 S00 . If p < 1=kUA˛ 1k1 , then the relation ˛ ˛ UA˛ f  Up,A f  pUA˛ Up,A f D0 ˛ f D implies for any bounded Borel function f that Up,A Hence, by the result given above for p D 0, it follows that ˛ .Up,A f

, g/ D D

1 X nD0 1 X nD0

P1

˛ n ˛ nD0 .pUA / UA f

.

  .p/n .UA˛ /nC1 f , g   b ˛ /nC1 g .p/n f , .U b A

b ˛ g/ , D .f , U A p,b b ˛ f  U ˛ f  pU b ,˛ U ˛ f D 0. Since ¹U ˛ ºp>0 and ¹U b ˛ ºp>0 satbecause U p,A b b A A A p,b A P A p,b p,b 1 ˛ ˛ nC1 n isfy the resolvent equation, Up,A f D f for jp  qj < nD0 .p  q/ .Uq,A / .˛

1=kUA 1k1 . Hence the first relation of equation (4.2.5) holds for all p > 0. The proof of the second relation is similar. If  2 S , then by applying the result shown above to 1Fn   for a nest ¹Fn º such that 1Fn  2 S00 and letting n ! 1, we get equation (4.2.5). The proof of equation (4.2.6) is similar. b ˛ f is a q.c. modification of U b ˛ .f  /, applying the argument of LemSince U b A b we have the following lemma. ma 4.1.2 to the PCAF b A t of M, b ˛ .1Fn  k1 < Lemma 4.2.3. For any  2 S , there exists a nest ¹Fn º satisfying kU 1 for all ˛ > 0 and n  1. Let Y be the support of  and YQ be the support of the PCAF A t , that is, Y is the smallest closed set outside of which  vanishes and YQ D ¹x 2 X n N : Px .A t > 0, for all t > 0/ D 1º,

(4.2.8)

where N is the exceptional set appearing in (A.2). Also let  .t / be the right continuous inverse function of A t given by  .t , !/ D ¹s > 0 : As .!/ > t º and .!/ D  .0, !/.

Section 4.2 Dual PCAFs and duality relations

133

Since YQ D ¹x 2 X : Ex .e ˛ / D 0º and Ex .e ˛ / is an ˛-excessive function, YQ is a q.e. finely closed set. Lemma 4.2.4. Let YQ be the hitting time of YQ . Then YQ D  a.s. Px for q.e. x. Proof. Since YQ is q.e. finely closed, XYQ 2 YQ on ¹YQ < º a.s. Px for q.e. x. Hence, Px .YQ <  / D Px .YQ <  , 0 <  ı YQ /   D Ex PX Q .0 <  / : YQ <  D 0 q.e. Y

On the other hand, for any t > 0, we have Px . < YQ / D Px .ACt > 0,  < YQ /   D Ex PX .A t > 0/ :  < YQ . If x 2 YQ r , then the last term vanishes because Px .YQ D 0/ D 1. If x 2 X n YQ , then X 2 X n YQ a.s. Px on ¹ < YQ º and hence lim PX .A t > 0/  lim PX . < t / D PX . D 0/ D 0.

t!0

t!0

Since YQ n YQ r is semipolar and hence exceptional by Theorems 3.2.6 and 3.4.4, we get the result. Lemma 4.2.5. Px .I  Z  J / D 1 q.e. x, where I.!/ D ¹t  0 : A tC .!/  A t .!/ > 0 for any  > 0º, J.!/ D ¹t  0 : A tC .!/  A t .!/ > 0 for any  > 0º, Z.!/ D ¹t  0 : X t .!/ 2 YQ º. Proof. Since ¹! : Z.!/  RC n J.!/º is contained in the set [ ® ¯ ! : A t .!/  Ar .!/ D 0, r C YQ ı .r !/ < t , r 0, r C YQ ı .r !/ > qº. ¹! : I.!/  RC n Z.!/º  r,q2Q,r0 of MG is given by R˛G f D R˛ f  HF˛ R˛ f . In Chapter 5, we use the part .E G , F G / of the Dirichlet form on the not necessarily open set G. In particular, the denseness of ¹R˛G f : f 2 L2 .G; m/º in F G is used. To ensure such a property, as in Theorem 1.1.4, it is enough to show for u 2 F G that E.u, u/ D lim˛!1 ˛.u  ˛R˛G , u/. To show this for any nearly Borel set G, let us temporary introduce the notation .EN G , FN G / defined by FN G D ¹u 2 L2 .G; m/ : lim˛!1 ˛.uR ˛R˛G u, u/mG < 1º, EN G .u, v/ D lim˛!1 ˛.u  ˛R˛G u, u/mG , where .u, v/mG D G u.x/v.x/m.dx/. We shall identify the function u 2 FN G with a function on X by putting u D 0 on X n G. Theorem 4.2.9. For any nearly Borel set G, FN G D F G and EN G D E on FN G  FN G . In particular, the family ¹R˛G f : f 2 L2 .G; m/º is dense in F G and the union of F G for all relatively compact open set G is E˛ -dense in F for ˛ > ˛0 . Proof. As we have stated above, if G is open, the first part of the theorem has already shown in Theorem 3.5.7. To show the theorem for a nearly Borel set G, we shall show that R˛G u 2 F G for any u 2 L2 .G; m/. We shall first show this for any closed

136

Chapter 4 Additive functionals and smooth measures

Rt set G. Let A t be the PCAF given by A t D 0 IF .Xs /ds for F D X n G. Since F pA is open, F D  D inf¹t > 0 : A t > 0º and hence R˛G f .x/ D limp!1 R˛ f .x/ for q.e. x 2 X and in L2 .X ; m/ for any bounded non-negative quasi-continuous integrable function f on G. For any bounded q.c. integrable function u, by equations (4.2.6) and (4.2.12),       b˛ u D f , H b˛ u D lim p f , U (4.2.15) lim p R˛pA f , u F p!1 p!1 A p,b mF where mF .B/ D m.F \ B/. Furthermore, since u is quasi-continuous, it is cofine continuous q.e. by the dual version of Theorem 3.5.6. In particular, if u D 0 q.e. on F , // D 0 for q.e. x 2 r F , where r F is the set of coregular points then u.x/ D b Ex .u.Xb F b ˛ .x, / is supported by F [ r F for q.e. x by Lemma 3.1.3, it holds that of F . Since H F b ˛ u D 0 q.e. Hence, noting that R˛G f D 0 q.e. on F , by applying equation (4.2.15) H F

to u D R˛G f , we have   lim ˇ .I  ˇRˇ /R˛G f , R˛G f ˇ !1   D lim lim ˇ .I  ˇRˇ /R˛pA f , R˛G f ˇ !1 p!1   ˇ D lim lim ˇ.Rˇ  pUA R˛pA  ˛Rˇ R˛pA /f , R˛G f ˇ !1 p!1

bˇ R˛G f /mF  ˛.R˛G f , R˛G f / D .f , R˛G f /  lim lim ˇ.pR˛pA f , R ˇ !1 p!1   G G b˛ R b G D .f , R˛G f /  lim ˇ f , H F ˇ R˛ f  ˛.R˛ h, R˛ f / ˇ !1   b ˛ R˛G f  ˛.R˛G f , R˛G f / D .f , R˛G f /  f , H F D .f , R˛G f /  ˛.R˛G f , R˛G f /. This implies that R˛G f 2 F and E˛ .R˛G f , R˛G f / D .f , R˛G f /. Similarly, for any f 2 L2 .G; m/ and v 2 F G , E˛ .R˛G f , v/ D .f , v/mG .

(4.2.16)

Assume next that G is a Borel set. Take an increasing sequence of closed subsets ¹Gn º of G such that limn!1 XnGn D XnG a.s. Px for a.e. x. The existence of such a sequence is a consequence of equation (3.1.10). Noting that E˛ .R˛Gn f , R˛Gn f / D .f , R˛Gn f /  .f , R˛ f / < 1, ¹R˛Gn f º is a uniformly bounded sequence of .E, F /. Then, by considering its Cesàro means if necessary, we may consider that limn!1 R˛Gn f D R˛G f in F and q.e. uniformly. Hence R˛G f is a function of F G satisfying equation (4.2.16) for any f 2 L2 .G; mG /. Clearly this also holds for any nearly Borel set G.

137

Section 4.2 Dual PCAFs and duality relations

To show the assertion of the theorem, let EN G,ˇ be the approximating form of EN G defined by EN G,ˇ .u, v/ D ˇ.u  ˇRˇG u, v/mG for u, v 2 L2 .G; mG /. By virtue of Theorem 1.1.4., since limˇ !1 ˇRˇ u D u in F and, by choosing a subsequence, q.e. uniformly for q.c. function u 2 F , we have by equation (3.5.5) that ˇ ¹HF .ˇRˇ u/º is an E˛0 -bounded sequence and limˇ !1 HF˛ .ˇRˇ u/ D HF˛ u in F . In particular, if u 2 FCG , then u D 0 q.e. on F and hence HF˛ u D 0. Therefore ˇ

limˇ !1 HF .ˇRˇ u/  limˇ !1 HF˛ .ˇRˇ u/  HF˛ u D 0 which implies that ˇ

lim jˇRˇG u  uj  lim jˇRˇ u  uj C lim HF .ˇRˇ u/ D 0

ˇ !1

ˇ !1

ˇ !1

ˇ

in F . More generally, if v is a q.c. function of FC , then limˇ !1 HF .ˇRˇ v/  HF˛ v. Since lim˛!1 HF˛ v D 0 a.e. on G and decreasingly, it also holds that lim ˇRˇG v D lim ˇRˇ v D v

ˇ !1

ˇ !1

in L2 .G; m/. For any v 2 L2 .G; m/, approximating v by functions ¹vn º  F in L2 .X ; m/ and noting that k.ˇRˇG jvn  vj/k  k.ˇRˇ jvn  vj/k  ˇ=.ˇ  ˛0 /kvn  vk, we obtain the strong continuity of ¹R˛G º in L2 .G; m/. In particular, for any v 2 L2 .G; mG / and ˛ > ˛0 ,   lim EN G,ˇ .R˛G v, w/ D lim ˇ R˛G v  ˇRˇG R˛G v, w mG ˇ !1 ˇ !1   D lim ˇ RˇG v  ˛RˇG R˛G v, w mG ˇ !1   G D v  ˛R˛ v, w mG

for any w 2 L2 .G; mG /. By putting w D R˛G v, it follows that R˛G v 2 FN G . As we have already seen, R˛G v 2 F G and satisfies   EN G .R˛G v, w/ D w  ˛R˛G v, w D E.R˛G v, w/ for all v 2 F G . Similarly to equation (1.1.8), for any u 2 L2 .G; mG /, EN G .ˇRˇG u, ˇRˇG u/ D ˇ.u  ˇRˇG u, ˇRˇG u/  EN G,ˇ .u, u/. In particular, if u 2 FN G , then the right-hand side remains bounded as ˇ increases to infinity. On the other hand, since the left-hand side is equal to E.ˇRˇG u, ˇRˇG u/, a subsequence of the Cesàro means of ¹ˇRˇG uº converges to u in F and hence u 2 F . Furthermore, since u can be chosen as a q.e. uniformly convergent limit of q.c. functions ˇRˇG u vanishing q.e. on X n G, u 2 F G . Hence FN G  F G .

138

Chapter 4 Additive functionals and smooth measures

For any v 2 L2 .G; mG / and w 2 F G , similarly to equation (1.1.9), ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ N G,˛ ˇE .v, w/ˇ D ˇ˛.v  ˛R˛G v, w/ˇ D ˇE.˛R˛G v, w/ˇ  KE˛0 .˛R˛G v, ˛R˛G v/1=2 E˛0 .w, w/1=2 D K EN ˛ .˛RG v, ˛RG v/1=2 E˛ .w, w/1=2 . ˛

0

˛

0

(4.2.17)

In particular, if u 2 F G then, combining these inequalities as in the proof of equation (1.1.10), we get that p ˛ ˛0 EN G,˛ .u, u/  K 2 E˛0 .u, u/ C K E˛ .u, u/1=2 . ˛  ˛0 0 This implies that EN G,˛ .ˇRˇG u, ˇRˇG u/ K

2

E˛0 .ˇRˇG u, ˇRˇG u/

p ˛ ˛0 CK E˛ .ˇRˇG u, ˇRˇG u/1=2 . ˛  ˛0 0

Since, limˇ !1 ˇRˇG u D u in .E, F G /, the right-hand side remains bounded as ˇ ! 1. Hence equation (4.2.17) with v D w D u yields that ¹EN G,˛ .u, u/º is uniformly bounded relative to ˛. This implies that F G  FN G and EN G .u, u/ D lim EN G .ˇRˇG u, u/ ˇ !1

D lim E.ˇRˇG u, u/ D E.u, u/ ˇ !1

for all u 2 FN G D F G . The last assertion has been already shown.

4.3 Time changes and killings In this section, we assume that .E, F / is a regular Dirichlet form on L2 .X ; m/ satisfying .E.5/. Let A t be a PCAF with Revuz measure  and M t be a multiplicative functional given by M t .!/ D exp.A t .!//. Let M D .X t , Px / be the subprocess of M corresponding to M t . By the definition of the subprocess introduced in Theorem 3.1.4, M is obtained from M by making X t jump to the extra point  at a random time which is independent of M and distributed by the exponential distribution with parameter A t .!/. In particular, since such time is inaccessible, M is also a Hunt process on .X n N , B  .X //. We denote the transition function and the resolvent of M by p t and R˛ , respectively.

139

Section 4.3 Time changes and killings

Let .E , F / be the Dirichlet form corresponding to M , that is ° ±   F D u 2 L2 .X ; m/ : lim ˛ u  ˛R˛ u, u < 1 ˛!1   E .u, v/ D lim ˛ u  ˛R˛ u, v . ˛!1





Let ¹G˛ º be the resolvent associated with .E , F / and R˛ f be a q.c. modification of G˛ f for  2 F . Theorem 4.3.1. F D FQ \ L2 .X ; / and E .u, v/ D E.u, v/ C .u, v/

for u, v 2 F .

(4.3.1)

In particular .E , F / is a regular Dirichlet form on L2 .X ; m/.

Proof. Suppose that  2 S0 , .X / < 1 and let u D Rˇ f be a q.c. version of the ˇ-resolvent of non-negative function f 2 C0 .X /. Then  Z 1 e ˇ tA t f .X t /dt q.e. u.x/ D RˇA f .x/ D Ex 0

Hence kuk1  .1=ˇ/kf k1 and satisfies u D R˛ f C .˛  ˇ/R˛ u  UA˛ u. In particular u 2 F \ L1 .X ; / and quasi-continuous. By virtue of Theorem 1.1.4 and Theorem 2.3.5, for any fixed ˛ > ˛0 , since Z b˛ u.x/  u.x/j.dx/  K˛ ., U˛ /1=2 E˛ .˛ R b˛ u  u, ˛ R b˛ u  u/1=2 ! 0 j˛ R X

b˛ u D u in L1 .X ; /. Hence as ˛ ! 1, lim˛!1 ˛ R   lim ˛ .u  ˛R˛ u, u/ D lim ˛ R˛ f  ˇR˛ u  UA˛ u, u ˛!1 ˛!1   b˛ u D .f  ˇu, u/  lim u, ˛ R ˛!1



D .f  ˇu, u/  .u, u/ . This implies that u 2 F \L2 .X ; / and satisfies E.u, u/ D .f , u/ˇ.u, u/.u, u/ . On the other hand, since   E .u, u/ D lim ˛.u˛R˛ u, u/ D lim ˛R˛ f  ˛ˇR˛ Rˇ f , u D .f ˇu, u/, ˛!1

˛!1

it holds that Eˇ .u, u/ C .u, u/ D Eˇ .u, u/. Since the family ¹RˇA f : f 2 C0 .X /º is dense in .E , F /, we get that F  F \ L2 .X ; / and E .u, v/ D E.u, v/ C .u, v/ for any v 2 F .

140

Chapter 4 Additive functionals and smooth measures

For general  2 S , take a nest ¹Fn º such that .Fn / < 1 and 1Fn   2 S0 . For any f 2 C0 .X /, by putting n D 1Fn  , we have Eˇ .Rˇ n f , Rˇ n f / C .Rˇ n f , Rˇ n f / n D .f , Rˇ n f /  1=.ˇ ˛0 /.f , f /. In particular, since ¹Rˇ n f º is uniformly bounded relative to Eˇ and in L2 .X ; /, the Banach–Saks theorem implies that a subsequence of Cesàro means of ¹Rˇ n f º converges to Rˇ f relative to Eˇ and in L2 .X ; /. Hence F  F \ L2 .X ; / and E .u, v/ D E.u, v/ C .u, v/ for u, v 2 F . Suppose conversely that u is a q.c. function of F \ L2 .X ; /. By truncating if necessary, we may assume that u is bounded. Then lim˛!1 ˛R˛ u D u q.e. and b˛ u D boundedly. If  2 S0 is a bounded measure, as we have seen above, lim˛!1 ˛ R 1 u in L .X ; /. Hence, by     lim ˛ u  ˛R˛ u, u D lim ˛ .u  ˛R˛ u, u/ C lim ˛ 2 UA˛ R˛A u, u ˛!1 ˛!1 ˛!1  A  2 b˛ u D E.u, u/ C lim ˛ R˛ u, R ˛!1

D E.u, u/ C .u, u/ , it follows that u 2 F and E .u, u/ D E.u, u/ C .u, u/ . For general  2 S , take a relatively compact open set G and consider the part process MG of M on G. For any u 2 F G , since Z u.x/.dx/  C E˛ .u, u/1=2 D C E˛G .u, u/1=2 G

for C D K˛ E˛ .U˛ , U˛ /1=2 , G D jG is a measure of a finite energy integral relative to .E G , F G / with G .G/ < 1. Let .E G, , F G, / be the Dirichlet form corresponding to the subprocess MG, of MG by the multiplicative functional G G e A t given by the PCAF AG t of M with Revuz measure G . Then, by the result G shown above, we obtain that F \ L2 .G; G /  F G,  F and E G, .u, u/ D E G .u, u/ C .u, u/ G D E.u, u/ C .u, u/ G for u 2 F G \ L2 .G; G /. By virtue of the proof of Theorem 3.5.7, since the union of F G over a relatively compact open set G contains C0 .X / \ F which is dense in F , we obtain that F \ L2 .X ; /  F and E .u, u/ D E.u, u/C.u, u/ for u 2 F \L2 .X ; /. Furthermore, by Theorem 3.5.7, for each relatively compact open set G, C0 .G/ \ F G is uniformly dense in C0 .G/ and E˛ -dense in F G . Since .G/ < 1, it also dense in L2 .G; /. Noting that the family [G F G, is dense in F , we obtain the regularity of .E , F /. The Dirichlet form .E , F / given in Theorem 4.3.1 is called a perturbed Dirichlet form of .E, F / by the smooth measure . b .ı/ D As in the last part of Section 3.3, by using the m b -dual Hunt process M .ı/ b b b b b .X t , P x / of M, the pseudo Hunt process M D .X t , P x / in duality with M relative

141

Section 4.3 Time changes and killings

to m is defined by

  b b bx ./ D b P hı .x/b E.ı/ x .1=hı /.X t / : 

(4.3.2)

b t . In particular, for the dual CAF b A t of A t associated with a smooth measure for  2 F b .ı/,ˇ .f =b b ˇ f .x/ D b hı .x/U hı /.x/ is independent of the choice of b hı , the function U b b A A and satisfies Z  b ˇ f .x/ D b Ex U b A

and

1

b t /d b e ˇ t f .X At

0

  bˇ f . h, f i D lim ˇ m, U b A ˇ !1

(4.3.3)

In the rest of this section, we assume that ˛0 D 0. We fix a smooth measure  and its associated PCAF A t with an exceptional set N . Define the right continuous inverse function  t .!/ of A t by  t .!/ D inf¹s > 0 : As .!/ > t º.

(4.3.4)

Then it satisfies sCt D s C  t ı s . We occasionally use the notation  .t / instead of  t . Let FLt D F .t/ D ¹ƒ 2 F : ƒ \ ¹ .t /  sº 2 Fs , for all s  0º. Denote by Y the support of , that is, the smallest closed set outside of which  vanishes. As defined by equation (4.2.8), let YQ be the support of A: YQ D ¹x 2 X n N : Px .A t .!/ > 0, for all t > 0/ D 1º.   Let N D , FQt , Y t , Px ,  be the time changed process of M by the PCAF A t defined by Y t D X .t/ and L D A . Then N is a right continuous strong Markov process on .Y , B  .Y //. Let q t and Vp be the transition function and the resolvent of N, that is q t '.x/ D Ex .'.Y t //  Z 1 pt e '.Y t /dt Vp '.x/ D Ex 0 Z 1  e pA t '.X t /dA t D Up,A '.x/. D Ex 0

b By virtue of Theorem 4.2.1, there  is a corresponding  PCAF A t associated with  .ı/ b .ı/ D , F t , Y bt , b by equation (4.2.1). Let N P x ,  be the time changed process of b .ı/ by b M A t , that is b , bt D X Y .t / b

b  .t / D ¹s > 0 : b As > t º.

142

Chapter 4 Additive functionals and smooth measures

b .ı/ given by b .ı/ Let .V ˛ /˛>0 be the resolvent of N  Z 1 pt bp.ı/ '.x/ D b b V E.ı/ e '. X /dt x b .t / Z0 1  pb At b b b .ı/ '.x/. Db E.ı/ e '. X /d A t t DU x A p,b 0 b As we have seen in Theorem 4.2.1, the PCAF b A t can be considered a PCAF of M b b b associated with . Let us denote by N D ., F t , Y t , P x , / the time changed process b Then V b relative to b b ˛ º˛>0 be the resolvent of N. bp g.x/ D U b g.x/ of M A t . Let ¹V p,b A and satisfies     bp g Vp f , g D f , V (4.3.5) To give a characterization of the time changed process N D .Y t , Px /, we first assume that .E, F / is transient. Then we may assume that ı D 0. Let b h Db h0 be a strictly positive 0-coexcessive function such that b h is bounded from below by a positive constant on every compact set. b .0/,˛b u are In Theorem 3.5.1, for any open set B, we have seen that HB˛ u and H B c ˛ ˛ ub h/B =b h for any q.c. functions u,b u 2 F, respectively q.c. modifications of uB and .b .0/,˛ ˛ ˛ b b b u.x/ D b b b˛b ˛ > 0. Let us define H h.x/H .b u=b h/.x/. Then H B u by H B b B u is a B ˛ q.c. modification of b uB . As we have seen in Theorem 1.3.9, .E, Fe / is complete. Furthermore, by equation (2.4.33), any function u 2 Fe is decomposed orthogonally into the functions .0/ .0/ XnB b Be . Similarly to Theorem 3.5.1, HBe u and H u uB 2 HB and F XnB .u/ 2 Fe .0/

.0/

are q.c. modifications of uB and b uB respectively and hence satisfy b Be E.HBe u, v/ D E.v, H u/ D 0

(4.3.6)

XnB

for any v 2 Fe . L FL / on L2 .Y ; / of the time changed process Let us define the Dirichlet form .E, by ³ ² 2 L F D ' 2 L .Y ; / : lim q.'  qVq ', '/ < 1 q!1 (4.3.7) L E.', / D lim q.'  qVq ', / . q!1

Put  uQ D uj Q Y and let us temporarily introduce the bilinear form .EL 0 , FL 0 / defined by D ¹' 2 L2 .Y ; / : ' D e u for some u 2 Fe º and EL 0 .', / D E.HYQ e u, HYQ e v/ for ' D e u and D e v . Since .E.1/ and .E.2/ with ˛0 D 0 are extended to .E, Fe /, .EL 0 , FL 0 / satisfies the corresponding properties, that is, FL 0

EL 0 .', '/  0 jEL 0 .', /j  K EL 0 .', '/1=2 EL 0 . , /1=2

(4.3.8) (4.3.9)

143

Section 4.3 Time changes and killings

for any ', 2 FL 0 . Furthermore, since Fe is complete relative to the norm E., / and E.HYQ u, HYQ u/  E.u, u/ for u 2 Fe , FL 0 is complete relative to EL ˛0 D EL 0 C ˛., / . Therefore, .EL 0 , FL 0 / on L2 .Y ; / is a bilinear form satisfying .E.1/, .E.2/ and .E.3/ with ˛0 D 0. Hence, if we can show that ¹Vp º is the resolvent associated with .EL 0 , FL 0 /, L then Theorem 1.1.4 shows that FL 0 D FL and EL 0 D E. Theorem 4.3.2. Assume that .E, F / is transient and ˛0 D 0. Then FL D ¹e u:u2 Fe º \ L2 .Y ; / and L u, e u, HYQ e u/ D E.e u/ (4.3.10) E.HYQ e for all u 2 Fe \ L2 .Y ; /. Proof. As we stated above, it is enough to show that ¹Vp º is the resolvent associated with .EL 0 , FL 0 /. For any ˛, p > 0 and bounded measurable function f 2 L2 .X ; / such that jf j   2 S0 , since ˛ ˛ f D UA˛ f  pUA˛ Up,A f 2F, Up,A

it satisfies ˛ ˛ ˛ ˛ f , Up,A f / D .f  pUp,A f , Up,A f / . E˛ .Up,A

(4.3.11)

In particular, ˛ ˛ ˛ f , Up,A f /˛mCp  .f , Up,A f / .Up,A ˛ ˛ ˛ which implies that kpUp,A f k  kf k . Hence E.Up,A f , Up,A f / is bounded rel0 ˛ 0 ative to ˛. Since Up,A f D lim˛!0 Up,A f and Vp f D .Up,A f /, we obtain that 0 f 2 Fe and kpVp f k  kf k . Furthermore, since Up,A Z 1  ˛ ˛tpA t Up,A f .x/ D Ex e f .X t /dA t  Z0 1  ˛tpA t e f .X t /dA t D Ex Q

 Y  ˛ f .XYQ / : YQ < 1 , D Ex e ˛YQ Up,A it holds that 0 0 f .x/ D HYQ Up,A f .x/. Up,A

Therefore, for any v 2 Fe and its approximating sequence ¹vn º  F , by replacing ˛ f of equation (4.3.11) by H ˛ v and letting n ! 1 and ˛ # 0, the second factor Up,A n YQ we have 0 f , HYQ e v / D .f ,e v /  p.Vp f , e v / . E.HYQ Up,A On the other hand, since limq!1 qVq f D f in L2 .Y ; /, v / D lim q.Vp f  qVqCp Vp f , e v / ELp .Vp f , e q!1

D lim q.VqCp f , e v / D .f , e v / . q!1

144

Chapter 4 Additive functionals and smooth measures

0 By putting v D Vp f , this implies that Vp f D .Up,A f / 2 FL and satisfies

L p f , e E.V v / D ELp0 .Vp f , e v / D .f , e v / for any v 2 Fe and hence ¹Vp º is the resolvent associated with .EL 0 , FL 0 /. To consider the time changed process of not necessarily transient Dirichlet forms, we prepare some preliminary results. The following theorem shows that the correspondence of a smooth measure and its associated PCAF is invariant under the time change. Theorem 4.3.3. Let B be a PCAF of M with associated smooth measure , then B.A1 .t // is a PCAF of N associated with jY . Proof. It is easy to see that the right continuous inverse function  t of A t satisfies  tCs .!/ D s .!/ C  t ı  s .!/. To show the assertion of the theorem, for any fixed integrable positive coexcessive function b h, it is enough to show that Z 1  Z b h.x/g.x/ .dx/ (4.3.12) e qt g.Y t /dB t D lim qEb q!1

h

Y

0

for all non-negative bounded b h  -integrable functions g on Y . By virtue of Lemma 4.1.2, we may suppose that UB˛ g is bounded for any ˛ > 0. Then by the resolvent in equation (4.1.3), we have for any ˛ > 0, Z 1  Z 1  e qt g.Y t /dB t D Ex e qA t g.X t /dB t Ex 0

0

q

D UB˛ g.x/  qUA UB˛ g.x/ C ˛RqA UB˛ g.x/. Since lim˛!1 UB˛ g D 0 decreasingly we have Z 1  lim qEb e qt g.Y t /dB t q!1

h

0

  q h D lim lim q UB˛ g  qUA UB˛ g C ˛RqA UB˛ g, b q!1 ˛!1   b˛ U bq b hb D lim lim q˛ g, R b q!1 ˛!1 A h b b b Q h/ D hh  , gi. D .g, H Y

Lemma 4.3.4. Suppose that .E, F / is recurrent. If B is a non-exceptional nearly Borel set, then Px .B < 1/ D 1 for q.e. x 2 X . Proof. Let u.x/ D Px .B < 1/ and F D ¹x : u.x/ D 1º. Since B is nonexceptional by assumption and B n B r is exceptional by Theorem 3.2.6 and Theorem

145

Section 4.3 Time changes and killings

3.4.5, B r is non-exceptional. Since B r  F , F is non-exceptional. Since u is bounded and excessive, u  p t u  0 a.e. and Z t Z T Cs Z T ps .u  p t u/ds D ps uds  ps uds 0

0

T

remains bounded as T " 1. By the recurrence and the right continuity of p t u relative to t , it then implies p t u D u m-a.e. and hence q.e. for all t . In particular, for q.e. x 2 X , u.X t / is a bounded Px -martingale. For any  > 0, put F D ¹x 2 X : u.x/  1  º and  D F . Then, for q.e. x 2 F , by the optimal sampling theorem,   1 D u.x/ D Ex u.X ^T / D Ex .u.XT / : T <  / C Ex .u.X / :   T /  Px .T <  / C .1  /Px .  T / D 1  Px .  T /. Thus Px .  T / D 0 and hence Px .XnF < 1/ D 0 for q.e. x 2 F . This implies that F is a T t -invariant set. By irreducibility, this yields that m.X n F / D 0 and hence X n F is exceptional. Corollary 4.3.5. Suppose that .E, F / is recurrent. If h is a bounded excessive function, then h is equal to a constant q.e. Proof. Suppose that h is not q.e. constant. Then there exist numbers a < b such that both F D ¹x : h.x/ < aº and E D ¹x : h.x/ > bº are non-exceptional. Hence, by Lemma 4.3.4, Px .F < 1/ D 1 and Px .E < 1/ D 1 for q.e. x. Define n by 1 D F , 2 D 1 CE ı1 , 2nC1 D 2n CF ı2n 2nC2 D 2nC1 CE ı2nC1 . Then P .n < 1 for all n  1/ D 1 q.e. Since h.X2nC1 /  a and h.X2n /  b, we see that Px . lim h.Xn / does not exist/ D 1 q.e. n!1

This contradicts the martingale property of h.Xn / relative to Px . Corollary 4.3.6. Suppose that M is recurrent. If A t is a PCAF of M associated with a non-trivial smooth measure, then A1 D 1 a.s. Px for q.e. x 2 X . Proof. Since A t is non-trivial, there exist positive constants a, c and a non-exceptional finely closed set B such that Px .a < 1/  c > 0 for any x 2 B, where a D inf¹t > 0 : A t > aº. Using the continuity of A t , it holds for any stopping time S  a that a ı S D inf¹t > 0 : AS Ct  AS > aº  inf¹t : AS Ct > aº D a  S . In particular, Px .a < 1/ is excessive and hence, by the supermartingale convergence theorem, Z D lim t!1 PX t .a < 1/ exists. Since M visits B infinitely often, Z  c.

146

Chapter 4 Additive functionals and smooth measures

Furthermore, since a ı  t C t D a on ¹a > t º, Px .a < 1 j F a ^t /  Px .a ı  t < 1, a > t j F a ^t / C Px .a  t j F a ^t /  1¹ a >tº PX t .a < 1/ C 1¹ a tº . By letting t ! 1, this yields that 1¹ a 0. This is obvious by Corollary 4.3.6 because f   is the smooth measure associated with .f  A/ t . Now we shall be concerned with the time changed process N of the irreducible recurrent Markov process M. To apply the results of transient Dirichlet forms, taking a non-trivial non-negative function g such that 0  g  1 on X , consider the Dirichlet form .E  , F / defined by equation (2.4.21) for D g  m. Let Mg be the transient Hunt process determined as a subprocess of M relative to the multiplicative functional ¹exp.B t /º with Z t

Bt D

g.X t /dt . 0

Denote by R˛g and p gt the resolvent and the transition function of Mg . Let N g D .Y tg , Pxg / be the time changed process of Mg by A t . We use the superfix g to represents the notions related to N g , for example the resolvent and its associated transition function of N g are denoted by .Vpg / and q tg , respectively. Let .EL g , FL g / be the Dirichlet form on L2 .Y ; / corresponding to the resolvent .V˛g / by equation g (4.3.7) and .EL g , FLe / be its extended Dirichlet form. By virtue of Theorem 4.3.2, the Dirichlet form .EL g , FL g / on L2 .Y ; / of the Hunt processes N g is given by FL g D ¹e u : u 2 Feg º \ L2 .Y ; / g g EL g . u,  u/ D E g .H e u, H e u/, YQ

where

H gQ e u.x/ Y

D

Egx .e u.YYQ //.

YQ

147

Section 4.3 Time changes and killings

Lemma 4.3.8. If  is a measure associated with a PCAF A t of M, then  is the measure associated with the PCAF A t of Mg . Proof. Since the definition of the measure associated with a PCAF is independent of b the choice of the ˛-coexcessive Z function h by Theorem  Z 4.1.4, it is enough to show that lim ˇEg b hm ˇ !1

1

e ˇ t f .X t /dA t

b h.x/f .x/.dx/

D

Y

0

for any non-negative b h  -integrable function f on Y . By considering f  1F instead hi < 1 if necessary, we may assume of f for any measurable set F satisfying h, 1F b Rt thath, b hi < 1. Using the CAF B D g.X /ds associated with g m, put ˇ .x/ D t s 0  R 1 ˇ tB ˇ ˇ t f .X /dA Ex 0 e t t . Then it satisfies ˇ D UA f  UB ˇ . Since ˇ h/ . ˇ , b ˇ˛ ˇ ˇ ˇ bˇ b h/ h/  .UA f , b .f , R  ˇ˛ ˇ˛ ˇ  .f , b h/ , .ˇ  ˛/2

ˇ bˇ b h/gm  ˇ.UB ˇ , b h/ D ˇ. ˇ , R

it follows that ˇ ˇ h/ D lim ˇ.UA f , b h/  lim ˇ.UB ˇ , b h/ D .f , b h/ . lim ˇ. ˇ , b

ˇ !1

ˇ !1

ˇ !1

Theorem 4.3.9. Assume that .E, F / is an irreducible, recurrent Dirichlet form satisL FL / on L2 .Y ; / corresponding to the time fying ˛0 D 0. Then, the Dirichlet form .E, L changed process N is given by F D ¹e u : u 2 Fe º \ L2 .Y ; / and L u, e E.e u/ D E.H Q e u, H Q e u/ (4.3.13) Y

Y

for any u 2 Fe such that e u 2 L2 .Y ; /. u : u 2 Feg º \ L2 .Y ; / and Proof. As we have noted before Lemma 4.3.8, FLeg D ¹e g g g g g u, H Q e u/ for all u 2 FL . Since 0  E.u, u/  E g .u, u/  EL . u,  u/ D E .H Q e Y Y E˛ .u, u/ for ˛  1, it holds that F D F g and hence Fe D Feg . Similarly FLe D FLeg . R   Q For any q.c. function ' on Y , by putting Rg,XnY '.x/ D Eg 0 YQ '.X t /dt and Rt B t D 0 g.Xs /ds, we have Z  Q  Y g,XnYQ Bs R .gHYQ '/.x/ D Ex e HYQ '.Xs /dBs Z0  Q  Y Bs e EXs .'.XYQ //dBs D Ex Z0  Q  Y Bs e '.XYQ /dBs D Ex  0  B D Ex '.XYQ /.1  e YQ / D HYQ '.x/  H gQ '.x/. Y

148

Chapter 4 Additive functionals and smooth measures

Since Q Q b g w/ D .v, w  H b g w/ E g .Rg,XnY v, w/ D E g .Rg,XnY v, w  H Q Q Y

Y

for any v, w 2 F g , we have   Q g g g E g .H Q e u, H Q e u/ D E g HYQ e u  Rg,XnY .gHYQ e u/, H Q e u Y Y Y   Q g g g u, H Q e u/ C .HYQ e u, H Q e u/gm  E Rg,XnY .gHYQ e u/, H gQ e u D E.HYQ e Y

Y

Y

bge u, HYQ e u/ C .HYQ e u, H u/gm , D E.HYQ e Q Y

u, HYQ e u  H gQ / D 0 following from HYQ e u where we used the the property that E.HYQ e Y

Q

u 2 F XnY . H gQ e Y On the other hand, for ' D e u, g '.x/ D Ex Up,A

Z

1

e pA t B t '.X t /dA t



0

satisfies

g ' D Up,A '  Rg,pA .gUp,A '/, Up,A R 1  where Rg,pA v.x/ D Egx 0 e pA t v.X t /dt . Define the dual kernels similarly by R b t D t g.X b s /ds of A t and B t , respectively. Since using the dual CAFs b A t and B 0 g g Vp ' D .Up,A '/ and Vp ' D .Up,A '/, using the convergences

b g '.x/ D lim b Ex lim p U p!1 p!1 A p,b

Z

1

e

 b b g '.x/ b  .s=p/ '.X /ds D H .s=p/ b YQ

sb B

0

and limp!1 pUp,A ' D HYQ ', it holds that   EL g .', '/ D lim p '  pVpg ', ' p!1   D lim p '  pVp ' C pRg,pA .gUp,A '/, ' p!1   g L b D E.', '/ C lim pUp,A ', p U p,A ' p!1

gm

L b g '/gm . D E.', '/ C .HYQ ', H Q Y

L Since E g .H Q e u, H Q e u/ D EL g .', '/, we obtain E.HYQ e u, HYQ e u/ D E.', '/. g Y

g Y

Chapter 5

Martingale AFs and AFs of zero energy

This chapter concerns stochastic calculus related to the semi-Dirichlet forms. In Section 5.1, Fukushima’s decomposition of an additive functional into a martingale additive functional and an additive functional of zero energy is given. But, due to the lack of the dual Markov process, we use a weak sense energy instead of the energy being used in the stochastic calculus of symmetric Dirichlet forms. In Section 5.2, the Beurling–Deny type decomposition of the Dirichlet forms is given under an additional condition. In the present settings, the drift term appears in the decomposition. A characterization of the zero energy part of Fukushima’s decomposition is given in Section 5.3. For any local Dirichlet form, associated martingale part and stochastic integrals related to them are studied in Section 5.4. In Section 5.5, transformations by multiplicative functionals and their applications to the absolute continuity of diffusion processes are given. In Section 5.6, general criteria for conservativeness and recurrence are studied. Finally, the criteria are applied to the diffusion processes corresponding to the differential generators on Euclidean space.

5.1

Fukushima’s decomposition of AFs

Throughout this section, we assume that .E, F / is a regular Dirichlet form on L2 .X ; m/ satisfying .E.5/. For ı > ˛0 , fix a strictly positive ı-coexcessive funchı 2 F and b hı tion b hı 2 L1 .X ; m/ given by Theorem 2.4.8. We may assume that b is bounded from below by a positive constant on every compact set. Let .A, G .ı/ / be the bilinear form defined in Section 1.4 using b hı , that is, A.u, v/ D limˇ !1 ˇ.u  b for m b D h  m. If .E, F / is transient, then we may assume that ı D 0 and ˇGˇ u, v/b ı m b h ı 2 Fe . By choosing a suitable resolvent ¹R˛ º corresponding to .E, F / such that R˛ f is quasi-continuous for all ˛ > 0 and f 2 L1 .X ; m/, there exists a Hunt process .ı/ b.ı/ M with resolvent ¹R˛ º. As in Section 3.3, put R˛ D R˛Cı and R ˛ .x, dy/ D .ı/ .ı/ b b b b b b b b .1=hı .x//R˛Cı .x, dy/hı .y/. Let M D .X t , P x / and M D .X t , b P x / be the .ı/ b b Hunt process and the pseudo Hunt process with resolvents R˛ and R˛ respectively, introduced in Section 3.3. Let D.L/ D ¹R˛ f : f 2 L2 .X ; m/º be the domain of the generator L defined by Lu D ˛u  f for u D R˛ f with ˛ > ˛0 and f 2 L2 .X ; m/. Similarly, define the generator L.ı/ with domain D.L.ı/ / by L.ı/ u D ˛u  f for .ı/ .ı/ b /º. Then D.L/  F and u D R˛ f 2 D.L.ı/ / D ¹R˛ f : ˛ > ˛0 , f 2 L2 .X ; m

150

Chapter 5 Martingale AFs and AFs of zero energy

E.u, v/ D .Lu, v/ for u 2 D.L/ and v 2 F . Furthermore, D.L.ı/ /  G .ı/ and D .f  ˛u, v/b A.ı/ .u, v/ D .L.ı/ u, v/b m m .ı/

for u 2 D.L.ı/ / given by u D R˛ f and v 2 G .ı/ , where A.ı/ .u, v/ D A.u, v/ C . ı.u, v/b m In this section, we are concerned with a decomposition of AFs which are not nechı essarily positive. For an AF A of M, define the energy e .ı/ .A/ of A relative to b by  Z 1 1 2 .ı/ ˛t 2 e .A/ D lim e A t dt . (5.1.1) ˛ Eb m ˛!1 2 0 Since e

.ı/

1 .A/ D lim ˛!1 2

Z

1

t e t

0

  1 2 A Eb t=˛ dt , t =˛ m

it is easy to see that

   2 1 1 At e ıt A2t D lim Eb Eb m m t!0 2t t!0 2t if the right-hand side converges boundedly. Let „ be the family of sequences of relatively compact quasi-open sets given by ³ ² Bn is a relatively compact quasi-open set . „ D ¹Bn º : Bn  BnC1 , [n Bn D X q.e. e .ı/ .A/ D lim

Furthermore, put „.ı/ D ¹¹B` º 2 „ :

1 b < hı < ` on B` for any `  1º. `

Let us define the family Floc of functions u which belongs to F locally in F by ² ³ there exist ¹Bn º 2 „ and ¹un º  F Floc D u : . such that u D un ma.e. on Bn .ı/

Similarly, define the family Gloc of functions which belongs to G .ı/ locally. Also define the weak sense energy of .A t / relative to v 2 F C by  Z 1 1 2 ˛t 2 ev .A/ D lim ˛ Evm e A t dt . (5.1.2) ˛!1 2 0 If v has a compact support and e .ı/ .A/ < 1, then hı k1 e .ı/ .A/. ev .A/  kv=b For a fixed nearly Borel set B, put evB .A/

1 D lim ˛ 2 Evm ˛!1 2

Z

1

e 0

˛t

(5.1.3) 

A2t^ B dt

.

Section 5.1 Fukushima’s decomposition of AFs

151

Then evB .A/ D ev .AB / for AB t D A t^ B . In particular, for any increasing sequence ` B ` ¹B` º 2 „, put ev .A/ D e .A/. If Evm .A2t / is increasing relative to t , then ev` .A/  ev .A/. We are mainly concerned with the following three kinds of additive functionals.

5.1.1

AFs generated by functions of F

Suppose that a function u has a version e u such that e u is q.e. finely continuous and finite. Then Œu u.X t /  e u.X0 / (5.1.4) At D e is an AF of M. Since any function u 2 G .ı/ satisfies limˇ !1 ˇ.u  ˇGˇ u, u/b D m A.u, u/, by writing as  Z 1 1 Œu 2 2 ˇ t e .A t / dt D ˇ 2 hb m, Rˇ u2  2uRˇ u C u2 i ˇ Eb m ˇ 0 b.ı/ 1/ D 2ˇ.u  ˇRˇ u, u/b  ˇ.u2 , 1  .ˇ  ı/R ˇ ı b m m 2 b.ı/ C ˇı.u , Rˇ ı 1/b , m we have from Theorem 3.5.14 that, for any u 2 F \ C0 .X /, Z 1 .ı/ Œu e u2 .x/.b k .ı/ .dx/ C ıb m.dx//. e .A / D Aı .u, u/  2 X

(5.1.5)

In particular, e .ı/ .AŒu / and .1=2/hb k .ı/ ,e u2 i are dominated by Aı .u, u/. For any u, v 2 G .ı/ put res

A

1 .u, v/ D A.u, v/  2

Z X

  e u.x/e v .x/ b hı .x/k.dx/ C b k .ı/ .dx/ .

Since k .ı/ .dx/ D b hı .x/.k.dx/ C ım.dx// by Theorem 3.5.14, Z 1 e u2 .x/k .ı/ .dx/. e .ı/ .AŒu / D Ares .u, u/ C 2 X

(5.1.6)

For any u 2 Fb and v 2 FbC , ev .A

Œu

 Z 1 1 2 2 ˇ t /D e .u.X t /  u.X0 // dt lim ˇ Evm 2 ˇ !1 0   1 D lim 2ˇ.u  ˇRˇ u, uv/  ˇ.u2  ˇRˇ u2 , v/ 2 ˇ !1 1 D E.u, uv/  E.u2 , v/. 2

(5.1.7)

152

Chapter 5 Martingale AFs and AFs of zero energy

If u, v are bounded, noting that juv.x/  uv.y/j  .kuk1 _ kvk1 /.ju.x/  u.y/j C jv.x/  v.y/j/ and juv.x/j  .kuk1 C kvk1 /.ju.x/j C jv.x/j/, we obtain from equations (1.4.9), .E.5/ and (5.1.7) that  Z 1 e ˇ t .u.X t /  u.X0 //2 dt ˇ 2 Evm 0    KE˛0 .u, u/1=2 E˛0 .u, u/1=2 C E˛0 .v, v/1=2 (5.1.8) for some constant K depending on kuk1 C kvk1 C kuk C kvk but independent of ˇ. In particular, we have the following result. Lemma 5.1.1. If ¹un º  Fb is a uniformly bounded sequence converging to u relative to E˛ for some ˛ > ˛0 , then lim ev .AŒun u / D 0

(5.1.9)

n!1

for all v 2 FbC . Proof. Put un  um instead of u in equation (5.1.8). Then  Z 1 ˇ 2 Evm e ˇ t ..un  um /.X t /  .un  um /.X0 //2 dt 0    KE˛0 .un  um , un  um /1=2 E˛0 .un  um , un  um /1=2 C E˛0 .v, v/1=2 . Since ¹E˛ .un  um , un  um /º is uniformly bounded relative to m, n, by letting m ! 1 and then ˇ ! 1, it holds that ev .un  u/  K1 E˛0 .un  u, un  u/1=2 for some constant K1 independent of n. This implies equation (5.1.9).

5.1.2 Martingale additive functionals of finite energy Let M be the family of martingale additive functionals (MAFs in abbreviation) of M, that is, ¯ ® (5.1.10) M D M : M is an AF, Ex .M t2 / < 1, Ex .M t / D 0 q.e. x . Since Ex .M tCs j F t / D Ex .M t C Ms ı  t j F t / D M t C EX t .Ms / D M t

a.e. Px for q.e. x, .ı/

M 2 M is a square integrable martingale. Since, e ıs b p s 1  1,   .ı/   2 ı.sCt/ 2 2 E e ı.sCt/ Eb .M / D e .M / C b p 1, E .M /  sCt s s t b b m m m  e ıs Eb .Ms2 / C e ıt Eb .M t2 /. m m

153

Section 5.1 Fukushima’s decomposition of AFs

Hence e ıt Eb .M t2 / is subadditive relative to t and hence m 1 ıt 1 e Eb .M t2 / D sup e ıt Eb .M t2 / m m t>0 2t 2t 1 D sup e ıt Eb .hM i t /, m t>0 2t

e .ı/ .M / D lim

t!0

(5.1.11)

where hM i t is the sharp bracket of ¹M t º determined as the unique PCAF such that ı

M t2  hM i t is a Px -martingale for q.e. x. Let M be the family of martingale additive functionals of finite energy relative to b hı ı

.ı/ M D ¹M 2 M : e .M / < 1º.

(5.1.12)

Since hM i t is a PCAF, there is a corresponding smooth measure hM i by equation (4.1.7). Then, 1 hı i. (5.1.13) e .ı/ .M / D hhM i , b 2 We shall call hM i the energy measure of M . ı

Define the family Mloc of MAFs locally of finite energy by ´ μ ı ı there exists ¹Dn º 2 „ and M .n/ 2M Mloc D M : . .n/ such that M t^ Dn D M t^ Dn for all n  1

(5.1.14)

For any sequence ¹B` º 2 „.ı/ and a non-negative bounded function v 2 F , we defined by equation (5.1.2) the weak sense energy ev .M / of the MAF M t relative to 

` D ¹v 2 F ` : v  0º. Define the family v. Put F ` D F B` and Fb,C M of MAF of b finite energy in the weak sense by μ ´ 2  / < 1 for any ¹B` º 2 „.ı/ , .M t^ ` / 2 M, Evm .M t^ ` . (5.1.15) MD M : ` and ev` .M / < 1 for all v 2 Fb,C for all `  1 

For any M 2M put M t` D M t^ ` and ev` .M / D evB` .M /. Then ev` .M / D ev .M ` /. By virtue of Lemma 4.1.15, since hM ` i D 1B`  hM i , it holds that Z 1 1 vdhM i . (5.1.16) ev` .M / D hhM ` i , vi D 2 2 B` 

Let M loc be the family of MAFs locally of finite energy in the weak sense defined by ´ μ  .n/ º   there exists ¹D º 2 „ and ¹M M . n (5.1.17) M loc D M : .n/ such that M t D M t for all t < Dn

154

Chapter 5 Martingale AFs and AFs of zero energy .n/

In particular, if M t is continuous, then M t^ Dn D M t^ Dn . For any ¹B`1 º 2 „.ı/ , the .n/

sequence ¹B` º defined by B` D B`1 \ D` belongs to „.ı/ and M t^ B` D M t^ B is a `



MAF with ev` .M / < 1. This implies that, for any M 2 M loc , there exists a sequence ` and `  1. ¹B` º such that ev` .M / < 1 for all v 2 Fb,C

5.1.3 CAFs of zero energy Let N be the family of continuous additive functionals of zero energy, that is N D ¹N : N is a CAF, e .ı/ .N / D 0, Ex .jN t j/ < 1 q.e.º. Also, define the family Nloc of CAFs locally of zero energy by μ ´ there exists ¹Dn º 2 „ and ¹N .n/ º  N . Nloc D N : .n/ such that N t^ Dn D N t^ Dn for all n  1

(5.1.18)

(5.1.19)



Furthermore, let N be the family of CAFs of zero energy in the weak sense defined by μ ´  N is a CAF, for any ¹B` º 2 „.ı/ , ev` .N / D 0 . (5.1.20) ND N : ` and `  1 and Ex .jN t^ ` j/ < 1 for all v 2 Fb,C 

Denote by N loc the family of CAFs locally of zero energy in the weak sense, that is ´ μ   there exists ¹Dn º 2 „ and ¹N .n/ º N . (5.1.21) N loc D N : .n/ such that N t^ Dn D N t^ Dn for all n  1 



Since N t 2N loc is continuous, similarly to the remark on M 2M loc at the end of 

the last subsection, for any N 2 N loc , there exists a sequence ¹B` º 2 „.ı/ such that ` and `  1. ev` .N / D 0 for all v 2 Fb,C The purpose of this section is to show the Fukushima’s decomposition of additive functional AŒu of the type in equation (5.1.2) for u 2 F into the sum of additive   functionals M 2M and N 2N . Lemma 5.1.2. For any PCAF A associated with a smooth measure  such that S 00 , ˇ > ı > ˛0 and t > 0, h, b hı i < 1,  2 b b .ı/ k1 h, b Eb  .A t /  .1 C ˇt e ıt /kU hı i. ˇ ı hı

(5.1.22)

Furthermore, for any bounded function v 2 F C with compact support, Evm .A t /  t e ıt kv=b hı k1 h, b hı i.

(5.1.23)

155

Section 5.1 Fukushima’s decomposition of AFs

Proof. By virtue of Lemma 4.1.14, there exists a nest ¹Fn º of closed sets such that hı   2 S0 . 1Fn   2 S0 and 1Fn hı   2 S0 . Hence we may assume that  2 S0 and b .ı/ b p t .vb hı /i for all Put c t .x/ D Ex .A t /. Then c t 2 F \ G and A.c t , v/ D h, v hı  b v 2 F . In fact, noting that b ˛ .b hı  , UA˛ 1i D e ˛t h, U hı  /i hb hı  , c t i  e ˛t hb   b ˛ .b D e ˛t E˛ U˛ , U hı  / < 1, and

Z

Z X

c t2 d m b



Eb .A2t / m Z

t

D2 Z

D 2Eb m

EXs .A ts /dAs

hı ids hc ts  , b p sb

t

e ıs hb hı  , c ts ids < 1,

0

c t belongs to Theorem 4.1.13

0

0

 2 L2 .X ; m b /.



t

Since ˛R˛ c t .x/ D ˛

R1 0

e ˛s ps E .A t /ds, we have from

.c t  ˛R˛ c t , c t /b Z 1 m    D˛ e ˛s .E .A t /, b hı c t /  E .A t /, b p s .b hı c t / ds 0  Z t Z t Z 1 e ˛s h, b p .b hı c t /id   h, b p sC .b hı c t /id  ds D˛ 0 0 0 ! Z 1 Z tCs=˛ Z s=˛ s e h, b p .b hı c t /id   h, b p .b hı c t /id  ds. D 0

t

0

Since c tb hı is finely continuous, it follows that Z 1   s b b D e s h, h c i  h, b p . h c i ds lim ˛ .c t  ˛R˛ c t , c t /b t ı t ı t m ˛!1

0

D h, b hı c t  b p t .b hı c t /i. hı c t  b p t .b hı c t /i. Similarly, for any This shows that c t 2 G .ı/ and A.c t , c t / D h, b .ı/ b hı v  b p t .b hı v/i. q.c. hı  -integrable function v 2 G , we have A.c t , v/ D h, b Furthermore, by Theorem 4.2.2, Z t hı idt .A / D h, b p sb hb m, c t i D Eb t hı m 0 Z t hı ids  t e ıt h, b D e ıt h, e ıs b p sb hı i. 0

156

Chapter 5 Martingale AFs and AFs of zero energy

Let  be a measure such that b hı   2 S00 . Then for any ˇ > ı, equation (5.1.22) follows from   b .ı/  hı  , c t i D Aˇ c t , U Eb  .A t / D hb ˇ ı hı   .ı/ b b .ı/  b b .ı/ /i C ˇ c t , U D h, b hı U   b p . h U t ı ˇ ı ˇ ı ˇ ı m  b .ı/ .ı/ .ı/ .ı/ b b b D hb hı  , U  b pt U i C ˇ c t , U  ˇ ı ˇ ı ˇ ı m b   .ı/ b b  kU ˇ ı k1 hı  .X / C ˇhb m, c t i . Similarly, equation (5.1.23) follows from hı k1 hb m, c t i  t e ıt kv=b hı k1 h, b hı i. Evm .A t / D .c t , v/  kv=b .1/

For two MAFs M t and M .2/ given by e

.ı/

.M

.1/

,M

.2/

.2/

and M t

ı

of M , let us introduce the mutual energy of M .1/

˛2 / D lim Eb m ˛!1 2

Z

1

e

˛t

.1/ .2/ M t M t dt

 .

(5.1.24)

0 ı

This gives an inner product among the elements of M . Similarly, define the mutual energy in the weak sense by  Z 1 ˛2 .1/ .2/ e ˛t M t M t dt (5.1.25) Evm ev .M .1/ , M .2/ / D lim ˛!1 2 0 for v 2 F C if the limit exists. Theorem 5.1.3. (i)

ı

ı

.M , e .ı/ / is a Hilbert space. If ¹M .n/ º M is an e .ı/ -Cauchy sequence, ı

then there exists a unique M 2 M and a subsequence ¹nk º such that .n / limn!1 e .ı/ .M .n/  M / D 0 and limk!1 M t k D M t uniformly on any compact interval a.s. Px for q.e. x. 

(ii) Suppose that ¹M .n/ º M is an ev -Cauchy sequence for all bounded nonnegative functions v 2 F . Then there exists a unique local martingale M such .n / that, for any ¹B` º 2 „.ı/ , limn!1 ev` .M .n/  M / D 0 and limi!1 M t^ i ` D M t^ ` uniformly on any compact interval a.s. Pv for all bounded non-negative functions v 2 F B` .

157

Section 5.1 Fukushima’s decomposition of AFs ı

Proof. (i) For all M 2M , since e .ı/ .M / D 12 hhM i , b hı i, Lemma 5.1.2 implies for any ˇ > ı that b .ı/ k1 e .ı/ .M /, Eb  .M t2 /  2.1 C ˇt e ıt /kU ˇ ı hı

for all  2 b S 00 .

Hence .M t , Pb / is a square integrable martingale and by equation (3.2.3), hı 

Pb hı 



 2 b .ı/ k1 e .ı/ .M /. sup jMs j >   2 .1 C ˇT e ıT /kU ˇ ı 0sT 

If ¹M .n/ º is an e .ı/ -Cauchy sequence, then we can choose a subsequence ¹nk º so that e .ı/ .M .nkC1 /  M .nk / / < 1=23k . For such subsequence  Pb  hı

sup

0sT

.n / jMs kC1



Ms.nk / j

1 > k 2



b k1  2k .  2.1 C ˇT e ıT /kU ˇ ˛ .ı/

Hence, by the Borel–Cantelli lemma, ¹M .nk / º converges uniformly on any compact t S 00 and hence a.s. Px for q.e. x. Put Ms .!/ D interval on Œ0, 1/ a.s. Pb  for all  2 b .nk /

limk!1 Ms

hı

.n/

.!/. Since ¹M t º converges in L2 .Pb /, Eb .M t / D 0 for all hı  hı  ı

.ı/ .n/   2b S 00 which implies that M t 2M . For any  > 0,  choose n0 so that e .M .n/ .m/ .M t  M t /2   because M .m/ / <  for any n, m  n0 . Then .e ıt =2t /Eb m

the left-hand side increases to e .ı/ .M .n/  M .m/ / as t decreases to 0. By letting m ! 1 and then t ! 0, it follows that e .ı/ .M .n/  M /   by Fatou’s lemma. Thus ı

M 2M and M .n/ converges to M relative to e .ı/ . (ii) Suppose that ¹M .n/ º is an ev -Cauchy sequence for any non-negative bounded function v 2 F . Since ev .M /  ev` .M / for any M 2 M and `  1, ¹M .n/ º is a Cauchy sequence relative to ev` for all `  1. Hence, by taking a bounded non-negative function v 2 F such that v  1 on B` , it holds that     Eb m hM .n/  M .m/ i t^ `  ` Evm hM .n/  M .m/ i t . hı .n0 /

.n/

Hence ¹M t^ ` º is an e .ı/ -Cauchy sequence and hence a subsequence ¹M t^ ` º conı

verges to some M`,t 2M relative to e .ı/ and uniformly on every compact interval .n0 / a.s. Pz for q.e. z by (i). By the diagonal argument, we may assume that ¹M t^ ` º conhı  ` on verges to M`,t for all `  1. Since M`C1,t D M`,t for t  ` and 1=`  b  B` , 0/ .n  M i t^ ` D 0 M t defined by M t D M`,t for t  ` satisfies limn!1 Evm hM for any `  1.

158

Chapter 5 Martingale AFs and AFs of zero energy

Furthermore, for any `  1 and bounded non-negative function v 2 F ,  Z 1 e .ˇ Cı/t .M .n/  M .m/ /2t^ ` dt ˇ 2 Evm 0  Z 1  `kvk1 ˇ 2 Eb m e .ˇ Cı/t .M .n/  M .m/ /2t^ ` dt hı

 `kvk1 e

.ı/

0

.M .n/,`  M .m/,` /, .i/,`

where M .i/,` is a MAF defined by M t 

.i/

D M t^ ` . By letting m ! 1 and then

ˇ ! 1, this implies that M 2M and limn!1 ev` .M .n/  M / D 0. If M t0 and M t00 satisfy the condition of (ii), then ev` .M 0  M 00 / D 0 and hence 0 00 D M t^ . This gives the uniqueness of M . M t^ ` ` For ¹B` º 2 „.ı/ , let R˛` D R˛B` be the resolvent of the part process MB` . By virtue of Theorem 4.2.9, for any u 2 F B` , limˇ !1 ˇRˇ` u D u relative to E˛ for ˛ > ˛0 . Since b hı 2 F is bounded on B` , lim E˛ .ˇRˇ` u  u, b hı .ˇRˇ` u  u// D 0

ˇ !1

for any u 2 FbB` . Therefore limˇ !1 ˇRˇ` u D u relative to E˛ and A˛ . We call the decomposition given by the following theorem Fukushima’s decomposition of AŒu . For the symmetric and non-symmetric Dirichlet forms, since we can take b hı D 1 and G .ı/ D F , conditions .E.5/ and .E.6/ hold. Hence, it generalizes the decomposition given for the symmetric Dirichlet forms. Theorem 5.1.4. If u 2 FbB` for some `  1 for ¹B` º 2 „.ı/ , then there exist uniquely ı

M Œu ,` 2M and N Œu ,` 2 N satisfying Œu

Œu ,`

A t^ ` D M t

Œu ,`

C Nt

(5.1.26)

for ` D B` . In particular, if .E, F / satisfies .E.6/, then for any u 2 F \ G .ı/ , there ı

exist unique M Œu 2M , N Œu 2 N satisfying Œu

At

Œu

D Mt

Œu

C Nt .

(5.1.27) ı

Proof. Uniqueness: It is enough to show that M t` D 0 for any M ` 2M \N . For such  .M t` /2  t e ıt hhM ` i , b hı i D 2t e ıt e .ı/ .M ` / D 0, it follows that M ` , since Eb m

M t` D 0.

159

Section 5.1 Fukushima’s decomposition of AFs

Existence: Suppose first that u D R˛` f for f 2 L2 .B` ; m/ and ˛ > ˛0 . Define Œu ,` and M t by Z t^ ` Œu ,` Œu Œu ,` .˛u.Xs /  f .Xs // ds, Mt D A t^ `  N t . N Œu ,` D

Œu ,` Nt

0

Then N Œu ,` 2 Nc , because

Z   Œu ,` 2 .N t / D 2Eb Eb m m

t^ `

.˛u.Xs /  f .Xs //

0

Z

 EXs Z tZ

ts

D2 0

0

Z tZ D2 Z

0

0

t

2 0

ts

.ts/^ `

  .˛u.Xv /  f .Xv //dv ds

0

hb m, ps` ..˛u

 f /p ` .˛u  f //id  ds

e ıs .b p .ı/,` 1, .˛u  f /p ` .˛u  f //b d  ds s m

ˇ  ˇ ˇ ˇ t e ıt ˇ .˛u  f /, p ` .˛u  f / ˇ d  , m b

.ı/,` b .ı/,B` , pt are the transition functions of the part processes MB` and M where p `t and b .ı/ Œu ,` / D 0. respectively. Since m b  `m on B R `t , this implies that e .N If u D R˛` f , then p `t uu D 0 p ` .˛uf /d  , in fact the Laplace transformations of both sides are equal to R˛` u  ˛1 u. From this we can obtain the martingale property of M Œu ,` , that is Œu ,`

Ex .M tCs j Fs^ ` /   Z .sCt/^ ` .˛u  f /.X /d  j Fs^ ` D Ex u.X.tCs/^ ` /  u.X0 /  0 Z s^ ` D p `t u.Xs^ ` /  u.X0 /  .˛u  f /.X /d  0 Z t p ` .˛u  f /.Xs^ ` /d   0 Z s^ ` D u.Xs^ ` /  u.X0 /  .˛u  f /.X /d  D MsŒu ,` . 0

Œu ,`

Furthermore, since M t

is square integrable, and e .ı/ .M Œu ,` / D e .ı/ .AŒu /  ı

Aı .u, u/ < 1 by equation (5.1.5), it belongs to M . For any u 2 FbB` , by Theorem 4.2.9, there exists an approximating sequence ¹un º of u of the form un D nRn` u D R˛` fn with fn D n.u  .n  ˛/Rn` u/ 2 b/ \ L1 .B` ; m b /. By the uniqueness of the decomposition, M Œun um ,` D L2 .B` ; m

160

Chapter 5 Martingale AFs and AFs of zero energy

M Œun ,` M Œum ,` and e .ı/ .M Œun ,` M Œum ,` /  Aı .un um , un um / by equation (5.1.5). Since limn!1 un D u relative to A˛ , M Œun ,` is an e .ı/ -Cauchy sequence and ı

hence converges to M Œu ,` 2M relative to e .ı/ and satisfies e .ı/ .M Œu ,` /  A˛ .u, u/. Œu ,` Œu Œu ,` It remains to show that N Œu ,` defined by N t D A t^ `  M t belongs to N . Œu

,` Œu ,` n According to Lemma 4.1.9 and Theorem 5.1.3, N converges to N uniformly on every finite t -interval a.s. PxB` for q.e. x. Hence N Œu ,` is a CAF. Furthermore, 1 Em ..N Œu ,` /2 / 2t b   1 Œuun Œu ,` Œun ,` Œun ,` 2 D lim sup Eb .A  .M  M / C N / t^ ` t t t t!0 2t m  3 .e .ı/ .AŒuun / C e .ı/ .M Œu ,`  M Œun ,` //

lim sup t!0

 6 Aı .u  un , u  un / for all n. Letting n ! 1, we have N Œu ,` 2 N . Suppose that .E, F / satisfies .E.6/. Then, for any u 2 F \ G .ı/ , limn!1 nGn u D u in F and in G .ı/ by Theorem 1.1.4 and Theorem 1.4.8. Hence the above proof works for any u 2 Fb \ G .ı/ and B` D X . As in the first part of the proof of the existence of the decomposition, if u D UA˛ f for a PCAF A t associated with a smooth measure  such that f   2 S00 , then Rt Œu u.X t /u.X0 / can be decomposed as in equation (5.1.26) for N t D ˛ 0 u.Xs /ds  Rt 0 f .Xs /dAs . Œu We call the decomposition of A t given by the next theorem Fukushima’s decomposition (in the weak sense). 



Theorem 5.1.5. For any u 2 Fb , there exist uniquely M Œu 2M and N Œu 2N such that Œu Œu Œu (5.1.28) At D Mt C Nt . Proof. According to the denseness of the domain D.L/ of the generator L in .E, F /, there exists a sequence ¹un º of uniformly bounded functions un D nRn u D R˛ fn with ˛ > ˛0 and fn D n.u C .n  ˛/Rn u/ 2 L2 .X ; m/ satisfying limn!1 E˛ .un  Rt Œu u, un  u/ D 0. As in the proof of Theorem 5.1.4, N t n D 0 .˛un  fn /.Xs /ds 

satisfies ev` .N Œun / D ev .N Œun / D 0 for all v 2 FbC and `  1. Define M t n 2M by Œu Œu Œu At n D Mt n C Nt n . (5.1.29) Œu

Then, by equation (5.1.7), ev .AŒun / D ev .M Œun / D E.un , vun / 

1 E.u2n , v/ 2

(5.1.30)

161

Section 5.1 Fukushima’s decomposition of AFs

and limn!1 AŒun D AŒu relative to ev for all v 2 FbC . Hence ¹M Œun º is an ev 

Cauchy sequence. By virtue of Theorem 5.1.3, this implies that there exists M Œu 2M Œu Œu such that, for any ¹B` º 2 „.ı/ , limn!1 M t n D M t relative to ev` for any `  1 C and v 2 Fb . Furthermore, it also converges uniformly on every compact t -interval of Œ0, `  a.s Pvm . Since ¹un º converges to u relative to E˛ , by choosing a subsequence Œu Œu if necessary, limn!1 A t n D A t uniformly on every compact t -interval a.s. Px for q.e. x by Lemma 4.1.9. We shall show that it also converges relative to ev` for all `  1 and v 2 FbB` . ˛ u and To show this, fixing `  1, put B D B` and  D B` . Let u.2/ D HXnB .1/

.2/

u.1/ D u  u.2/ . Also decompose un E˛ -orthogonally as un D un C un for .1/ .1/ .2/ .1/ ˛ ˛ R˛ fn 2 HXnB , where fn D 1B fn . un D R˛B fn 2 F B and un D HXnB .i/

.i/

.i/

.i/

.i/

Since E˛ .un  um , un  um /  K˛2 E˛ .un  um , un  um /, limn!1 un D u.i/ in F and q.e. uniformly by choosing a subsequence for i D 1, 2. We shall show the convergence of AŒun to AŒu relative to ev` by showing separately the convergences Œu

.i /

Œu.i /

of A t n to A t for i D 1, 2, respectively. .1/ For any ˇ > 0, since .u.1/  un /.X / D 0, it holds that  Z 1 .1/ Œu.1/ un 2 2 ˇ t ˇ Evm e .A t^ / dt 0 Z   2 e ˇ t .u.1/  u.1/ D ˇ 2 Evm n / .X t / 0     .1/ .1/ .1/ .1/ 2 /.X /.u  u /.X / dt C ˇ .u  u / , v  2.u.1/  u.1/ 0 t n n n   B .1/ .1/  u.1/  u.1/ D 2ˇ u.1/  u.1/ n  ˇRˇ .u n /, .u n /v   2 B .1/ 2  u.1/ ˇ .u.1/  u.1/ n /  ˇRˇ .u n / ,v . Similarly to equation (1.1.9), we can see that   B .1/ .1/ .1/ .1/  ˇR .u  u /, .u  u /v ˇ u.1/  u.1/ n n n ˇ   .1/  u.1/ D E ˇRˇB .u.1/  u.1/ n /, .u n /v  1=2 B .1/ .1/ /, ˇR .u  u /  K˛ E˛ ˇRˇB .u.1/  u.1/ n n ˇ  1=2 .1/ E˛ .u.1/  u.1/  u.1/ . n /v, .u n /v .1/

(5.1.31)

.1/

Note that ¹E˛ .u.1/  un , u.1/  un /º is uniformly bounded relative to n and, as in equations (1.1.8) and (1.1.10), E˛ .ˇRˇB w, ˇRˇB w/ is bounded by a constant multiple of E˛ .w, w/ C E˛ .w, w/1=2 uniformly relative to ˇ. Hence there exists a constant

162

Chapter 5 Martingale AFs and AFs of zero energy

K1 independent of ˇ and n such that equation (5.1.31) is bounded by K1 E˛ .u.1/  .1/ .1/ un , u.1/  un /1=2 . Therefore, by letting ˇ ! 1 and then n ! 1 in equation .1/ .1/ (5.1.31), we obtain that limn!1 AŒun D AŒu relative to evB . Furthermore, for v 2 F B, .1/ evB .AŒu /

Z 1  2  1 2 ˇ t .1/ .1/ u .X t^ /  u .X0 / dt D e lim ˇ Evm 2 ˇ !1 0  ˇ D lim ˇ.u.1/  ˇRˇB u.1/ , u.1/ v/  ..u.1/ /2  ˇRˇB .u.1/ /2 , v/ 2 ˇ !1  1    (5.1.32) D E u.1/ , u.1/ v  E .u.1/ /2 , v . 2 .2/

.2/

˛ R˛ fn is the difference of the ˛Let us next consider AŒu . Since un D HXnB C reduced functions of ˛-excessive functions R˛ fn and R˛ fn on X n B, it can be .2/ e ˛ n for a signed Radon measure n on X n B by Lemma 2.4.3. written as un D U For each ˇ > 0,

 .2/ Œun 2 e ˇ t .A t^ / dt 0  Z 2 2 .2/ .2/ D ˇ Evm e ˇ t ..u.2/ / .X /  2u .X /u .X //dt t 0 n t n n 0      2 ˇ .2/ 2 C ˇEvm e ˇ .u.2/ un .X / C ˇ .u.2/ n / .X /  2ˇEu.2/ vm e n / ,v     .2/ .2/ 2 .2/ 2  ˇ .u.2/ D 2ˇ u.2/ n  ˇRˇ un , vun n /  ˇRˇ .un / , v     ˇ 2 .2/ 2 ,v C ˇ HXnB .u.2/ /  ˇR .u / ˇ n n     ˇ .2/  2ˇ HXnB u.2/ , vu.2/ n  ˇRˇ un n Z

ˇ 2 Evm

1

.2/ .2/ .2/ D 2 Iˇ .u.2/ n /  IIˇ .un / C IIIˇ .un /  2 IVˇ .un /.

.2/

.2/

.2/

(5.1.33)

Since ¹un º and ¹un  um º are uniformly E˛ -bounded, there exists a constant K2 .2/ .2/ .2/ .2/ .2/ .2/ such that E˛ .un , un /  K2 and E˛ .un  um , un  um /  K2 for all m, n  1. Furthermore, we may consider that it also satisfies E˛ .v, v/ C kunk1 C kun k  .2/ .2/ .2/ .2/ .2/ .2/ K2 , E˛ .vun , vun /  K2 and E˛ v.un  um /, v.un  um /  K2 for all m, n  1. .2/ .2/ .2/ .2/ By virtue of equation (5.1.8), putting un  um instead of un in Iˇ .un /, we have .2/ .2/ .2/ .2/ .2/ .2/ .2/ .2/ Iˇ .un  um /  K2 E˛ .un  um , un  um /1=2 . Similarly, noting that ¹un  um º is uniformly bounded relative to m, n  1, equations (1.1.8), (1.1.10) and .E.5/ imply

163

Section 5.1 Fukushima’s decomposition of AFs

that there exists a constant K3 such that  1=2 .2/ .2/ .2/ 2 .2/ .2/ 2 .u  u /  K E  u / , .u  u / v IIˇ .u.2/ 2 ˛ n m n m n m .2/ .2/ .2/ 1=2  K3 E˛ .u.2/ . n  um , un  um / .2/

Since un D U˛ n with a measure n supported by X n B,   ˇ ˇ .2/ IVˇ .u.2/ n / D ˇ HXnB Uˇ n  ˛HXnB Rˇ U˛ n , vun     ˇ D ˇ Uˇ n , vu.2/  ˛ˇ HXnB Rˇ U˛ n , vu.2/ . By using the approximating form E ˇ of E, the first term of the right-hand side can be written as .2/ .2/ ˇ.Uˇ n , vu.2/ n / D ˇE˛ .R˛ Uˇ n , vun / D ˇE˛ .Rˇ U˛ n , vun /   D ˇ U˛ n  .ˇ  ˛/Rˇ U˛ n , vu.2/ n   .2/ .2/ D ˇ u.2/ n  .ˇ  ˛/Rˇ un , vun .2/ .2/ .2/  E ˇ .u.2/ n , vun / C ˛kun k  kvun k. .2/

.2/

By equation (1.1.9), the right-hand side is dominated by K4 E˛ .un , un /1=2 for some constant K4 independent of ˇ  ˛0 . Similarly, by equation (1.1.3), the second term can be written as     ˇ ˇ .2/ b D ˛ˇE H R U  , R .vu / ˛ˇ HXnB Rˇ U˛ n , vu.2/ ˇ ˇ n n XnB ˇ ˛ n  1=2 ˇ ˇ  ˛ˇKˇ Eˇ HXnB Rˇ U˛ n , HXnB Rˇ U˛ n  1=2 .2/ bˇ .vu.2/ b  Eˇ R /, R .vu / ˇ n n 1=2  1=2  .2/ bˇ .vu.2/ vun , ˇ R  ˛Kˇ Eˇ Rˇ U˛ n , ˇRˇ U˛ n / n .2/  ˛Kˇ ku.2/ n k  kvun k. .2/

.2/

Since Kˇ in equation (1.1.3) is bounded relative to ˇ  ˛, by putting un  um .2/ instead of un , we have  1=2 .2/ .2/ .2/ .2/ .2/ u IVˇ .u.2/  u /  K E  u , u  u 5 ˛ n m n m n m

for some K5 independent of m, n and ˇ  ˛. .2/ .2/ To estimate IIIˇ .un /, we note that .un /2 is an ˛-potential of a signed smooth .2/

Œun

measure. In fact, in the decomposition A t

.2/

Œun

D Mt

.2/

Œun

C Nt

.2/

Œun

, since N t

is a

164

Chapter 5 Martingale AFs and AFs of zero energy .2/

.2/

.2/

CAF of bounded variation, ev .N Œun / D 0. Put A2,n D AŒun , M 2,n D M Œun and .2/ N 2,n D N Œun , respectively. Then, by equations (5.1.7) and (5.1.19), the associated measure hM 2,n i of hM 2,n i satisfies Z vdhM 2,n i D 2ev .M 2,n / D 2ev .A2,n / X

.2/ 2 D 2E.u.2/ , vu.2/ n /  E..un / , v/ Z n .2/ 2 .2/ 2 D2 vu.2/ n dn  ˛..un / , v/  E˛ ..un / , v/, X

.2/

R

.2/

.2/

that is E˛ ..un /2 , v/ D X vd n and hence .un /2 D U˛ n for n D 2un dn  .2/ ˛.un /2 d m  dhM 2,n i . Using this, we have  ˇ  IIIˇ .u.2/ n / D ˇ HXnB .U˛ n  ˇRˇ U˛ n /, v   ˇ D ˇ HXnB .Uˇ n  ˛Rˇ U˛ n /, v . .2/

In the expression of n , since 2un dn is supported by X n B,  ˇ    .2/ ˇ HXnB Uˇ .2u.2/ n  n /, v D ˇ Uˇ .2un  n /, v .2/ b b D 2ˇhu.2/ n  n , Rˇ vi D 2E˛ .U˛ n , un ˇ Rˇ v/  1=2 .2/ 1=2 .2/ b b E˛ u.2/  2KE˛ .u.2/ n , un / n .ˇ Rˇ v/, un .ˇ Rˇ v/ .2/ 1=2  K6 E˛ .u.2/ , n , un / .2/ b .2/ b 1=2 : n  1, ˇ  ˛º < 1. for K6 D 2K sup¹E˛ .un .ˇ R ˇ v/, un .ˇ Rˇ v// .2/ 2 .2/ 2 Since Uˇ .˛.un /  m/ D ˛Rˇ .un / ,  ˇ    2 .2/ 2 b ˇ b ˇ HXnB Uˇ .˛.u.2/ n /  m/, v D ˛ˇEˇ Rˇ .un / , H XnB Rˇ v     2 .2/ 2 1=2 bˇ v 1=2 bˇ v, R Eˇ R  ˛ˇKˇ Eˇ Rˇ .u.2/ n / , Rˇ .un / 2 .2/ .2/ 1=2  ˛Kˇ k.u.2/ n / k  kvk  K7 E˛ .un , un /

for some constant K7  ˛Kˇ kvk=.˛  ˛0 / which can be taken independently on ˇ. .2/ Since N 2,n is of bounded variation, equation (5.1.30) holds for un and v 2 FbC . Hence, by equations (5.1.7) and (5.1.16), we have  ˇ    ˇ bˇ v ˇ HXnB Uˇ hM 2,n i , v D ˇEˇ HXnB Uˇ hM 2,n i , R   bˇ R bˇ R bˇ v D ˇhhM 2,n i , H b vi D ˇEˇ Uˇ hM 2,n i , H XnB XnB ˇ bˇ vi D 2e .M 2,n / D 2eˇb .A2,n /  hhM 2,n i , ˇ R ˇb Rˇ v Rˇ v   .2/ 2   .2/ b b D 2E u.2/ n , un .ˇ Rˇ v/  E .un / , ˇ Rˇ v .

165

Section 5.1 Fukushima’s decomposition of AFs

.2/ 2 1=2 2 bˇ vº is uniformly E˛ -bounded relative to ˇ and E˛ ..u.2/ Since ¹ˇ R  n / , .un / / .2/ .2/ 1=2 K2 E˛ .un , un / , there exists a constant K8 independent of ˇ and n such that   ˇ .2/ 1=2 . ˇ HXnB Rˇ hM 2,n i , v  K8 E˛ .u.2/ n , un /

Combining these estimates, we can see that there exists a constant K9 satisfying ˇ .2/ .2/ ˇ.HXnB Uˇ n , v/  K9 E˛ .un , un /1=2 for all ˇ  ˛. Since Eˇ .Rˇ w, ˇRˇ w/ D .2/

.w, ˇRˇ w/  kwk2 for any w 2 F , the remaining term of IIIˇ .un / is estimated as  ˇ    ˇ bˇ v ˛ˇ HXnB Rˇ U˛ n , v D ˛ˇEˇ HXnB Rˇ U˛ n , R  ˇ 1=2 ˇ bˇ v, R bˇ v/1=2  Eˇ .R  ˛ˇKˇ Eˇ HXnB Rˇ U˛ n , HXnB Rˇ U˛ n  ˛Kˇ Eˇ .Rˇ U˛ n , ˇRˇ U˛ n /1=2 kvk   2 .2/ 2 1=2  ˛Kˇ kvk .u.2/ n / , ˇRˇ .un / .2/  ˛Kˇ kvk  ku.2/ n k  sup¹kun k1 : n  1º

 ˛Kˇ kvk  kuk1  ku.2/ n k. .2/

.2/

.2/

.2/

.2/

By replacing un with un  um we get that IIIˇ .un  um / is also dominated by .2/ .2/ .2/ .2/ a constant multiple of E˛ .un  um , un  um /1=2 . Finally we obtained that there exists a constant K10 such that  Z 1  1=2 .2/ .2/ Œun um 2 .2/ .2/ .2/ e ˇ t .A t^ / dt  K10 E˛ u.2/ . ˇEvm n  um , un  um 0

.1/

.2/

.2/

As in the case of AŒun , this implies that limn!1 AŒun D AŒu relative to ev` . Therefore, ev` .AŒu  AŒun /  K11 E˛ .u  un , u  un /1=2 for some constant K11 which implies the convergence of AŒun to AŒu relative to ev` for all v 2 F B` . The Œu

Œu.1/

Œu.2/

C Nt is a CAF. Furthermore, since E˛ .u.i/  functional given by N t D N t .i/ .i/ .i/ un , u  un /  E˛ .u  un , u  un /, we have  Z 1 Œu.i / 2 2 ˇ t e .N t^ / dt ˇ Evm 0   Z 1 Z 1 .i / .i / Œun 2 Œu.i / un 2 2 ˇ t 2 ˇ t e .N t^ / dt C 3ˇ Evm e .A t^ / dt  3ˇ Evm 0 0  Z 1 .i / Œu.i / u e ˇ t .M t^ n /2 dt C3ˇ 2 Evm  Z 10 .i / Œun 2 2 ˇ t  3ˇ Evm e .N t^ / dt C .K2 C K10 /E˛ .u  un , u  un /1=2 . 0 .i /

By letting ˇ ! 1 and then n ! 1, this implies that ev` .N Œu / D 0 for i D 1, 2 and hence ev` .N Œu / D 0. Therefore, we obtained the decomposition in equation (5.1.28).

166

Chapter 5 Martingale AFs and AFs of zero energy

  R If M t 2M \ N , then for M t` D M t^ ` , 0 D ev` .M / D X vdhM ` i for all `  1 and hence M D 0. This implies the uniqueness of the decomposition.

Lemma 5.1.6. For any u 2 Fb and quasi-open set B,  Z 1 2 ˇ t lim ˇ Evm e .u.X t^ B /  u.X0 //dt D E.u, v/ ˇ !1

(5.1.34)

0

for all v 2 FbB . Furthermore, for any functions u, v 2 FbB , 1 evB .AŒu / D E.u, uv/  E.u2 , v/. 2

(5.1.35)

˛ e u and u1 D u  u2 be Proof. For any fixed ˛ > ˛0 , let u D u1 C u2 with u2 D HXnB the E˛ -orthogonal decomposition given by Theorem 2.4.2. Since u1 .X B / D 0 a.s., for any v 2 F B ,  Z 1 2 ˇ t e .u1 .X t^ B /  u1 .X0 //dt lim ˇ Evm ˇ !1 0  Z B 1 D lim ˇ 2 Evm e ˇ t u1 .X t /dt  u1 .X0 / ˇ ˇ !1 0   B D  lim ˇ u1  ˇRˇ u1 , v D E.u1 , v/. ˇ !1

Since B ı  t D B  t for t < B and PXB .B D 0/ D 1 a.s. Pvm ,  Z 1 ˇ 2 Evm e ˇ t .u2 .X t^ B /  u2 .X0 //dt 0  Z B 2 D ˇ Evm e ˇ t EX t .e ˛ B e u.X B //dt 0    C ˇEvm e ˇ B e u.X B /  ˇEvm e ˛ B e u.X B /  Z B e .ˇ ˛/t e ˛ B e u.X B /dt D ˇ 2 Evm 0     C ˇEvm e ˇ B e u.X B /  ˇEvm e ˛ B e u.X B /     ˇ˛  u.X B /  Evm e ˇ B e u.X B / . Evm e ˛ B e D ˇ˛ Therefore, for any v 2 F B ,

Z

lim ˇ Evm

ˇ !1

1

ˇ t

.u.X t^ B /  u.X0 //dt   D E.u1 , v/ C ˛Evm e ˛ B e u.X B / 2

e

0

D E.u, v/ C E˛ .u2 , v/ D E.u, v/.



167

Section 5.1 Fukushima’s decomposition of AFs

Similarly to the equality before equation (5.1.33),  Z 1 1 Œu B Œu 2 ˇ t 2 e .A t^ B / dt lim ˇ Evm ev .A / D 2 ˇ !1 0   1 1 ˇ lim ˇ HXnB .u2  ˇRˇ u2 /, v D E.u, uv/  E.u2 , v/ C 2 2 ˇ !1  ˇ   lim ˇ HXnB .u  ˇRˇ u/, uv . ˇ !1

b ˇ .ˇ R bˇ v/ D ˇ R bˇ v  ˇ R bB v is E˛ -bounded relative to ˇ and converges Since H XnB ˇ to zero as ˇ ! 1 for v 2 F B , we have  ˇ  lim ˇ HXnB .u2  ˇRˇ u2 /, v ˇ !1

 ˇ  bˇ v D lim ˇEˇ HXnB .u2  ˇRˇ u2 /, R ˇ !1

  bˇ R b v D lim ˇEˇ u2  ˇRˇ u2 , H XnB ˇ ˇ !1

  bˇ R b v D 0. D lim E u2 , ˇ H XnB ˇ ˇ !1

ˇ

Similarly, limˇ !1 ˇ.HXnB .uˇRˇ u/, uv/ D 0, which yields equation (5.1.35). Theorem 5.1.7. For any u 2 Fb , the Revuz measure hui D hM Œu i associated with M Œu is a  -finite smooth measure satisfying Z e v .x/hui .dx/ D 2E.u, uv/  E.u2 , v/ (5.1.36) X

for all v 2 Fb . Œu ,`

Œu

D M t^ ` . By Lemma 4.1.15, Proof. For any sequence ¹B` º 2 „.ı/ , put M t Œu ,` the associated measure hM Œu,` i of M is related to the associated measure hui of M Œu by hM Œu,` i D 1B`  hui . For any `, since ev` .M Œu / D ev` .AŒu / for any

v 2 FbB` by equation (5.1.28), using equation (5.1.35) we obtain equation (5.1.36). Since [`1 FbB` is dense in .E, Fb /, for any v 2 Fb , there exists a uniformly bounded sequence ¹vn º  [1 F B` such that limn!1 E˛ .vn  v, vn  v/ D 0. `D1 b For any m, n  1, since Z je vm  e v n j.x/hui .dx/ D 2E.u, ujvm  vn j/  E.u2 , jvm  vn j/ X

vn D e v in L1 and the right-hand side converges to zero as n, m ! 1, limn!1 e .X ; hui / and q.e. Therefore, equation (5.1.36) holds for any v 2 Fb .

168

Chapter 5 Martingale AFs and AFs of zero energy

Corollary 5.1.8. Suppose that .E, F / is strongly local, then for any u 2 Floc , there 



exist uniquely M Œu 2 M loc and N Œu 2 N loc satisfying equation (5.1.28). Proof. Let u 2 Floc . Then there exist ¹un º  Fb and ¹Dn º 2 „ such that un D u on 

Dn . For each n  1, since Theorem 5.1.5 holds for any ¹un º, there exist M Œun 2M 

Œu

and N Œun 2N such that AŒun D M Œun C N Œun . By the strong locality, A t n is Œu Œu continuous relative to t 2 Œ0, 1/. Since N t n is continuous already, M t n is also ŒunC1 Œu D Mt n continuous. Furthermore, by the uniqueness of the decomposition, M t Œu

Œu and N t nC1 D N t n for t < Dn and hence for t  Dn . Therefore, M Œu and N Œu Œu

defined by M t

Œun

D Mt

Œu

and N t 

Œun

D Nt





for t < Dn belong to M loc and N loc .

If there exists a function u 2F such that M Œu D N Œu , then there exist ¹Dn º 2 „ Œu Œu Œu Œu and ¹un º  Fb such that M t D M t n and N t D N t n for t < Dn . For Œu Œu any sequence ¹B` º 2 „.ı/ , this implies that M t^ nDn \B D N t^ nDn \B and hence `

`

evDn \B` .M Œun / D 0 for any v 2 FbDn \B` . Therefore, M t^ Dn \B D 0 for any ` n, `  1. This yields the uniqueness of the decomposition. Œu

Example 5.1.9. For a domain D of Rd and the Lebesgue measure m.dx/ D dx, consider the Dirichlet form .E, F / on L2 .D; m/ determined by the smallest closed extension of d Z X @u @v aij .x/ dx E.u, v/ D @x i @xj D i,j D1

C

d Z X

bi .x/

iD1 D

@u v.x/dx, u, v 2 C01 .D/. @xi

We assume that .aij / satisfies the conditions of Section 1.5.1. Furthermore, assume that .bi / satisfies the condition (i) given there. Then, by Theorem 1.5.2 and Theorem 1.5.5, .E, F / satisfies .E.5/ and .E.6/. Hence, for u, v 2 C01 .X /, equation (5.1.36) yields that   Z d Z X @u @v @ v.x/dhui .x/ D 2 aij .x/ .uv/  u.x/ dx @xi @xj @xj D D i,j D1

D2

d Z X

aij .x/

i,j D1 D

Hence we get that hui .dx/ D 2

d X i,j D1

@u @u v.x/dx. @xi @xj

aij .x/

@u @u dx. @xi @xj

169

Section 5.2 Beurling–Deny type decomposition

If u 2 Fb , this implies that hui is the measure associated with the PCAF Z hM

Œu

t

it D 2

d X

0 i,j D1

aij .Xs /

@u @u .Xs /ds. @xi @xj

(5.1.37)

Example 5.1.10. Let .E, F / be the jump type Dirichlet form on L2 .X ; m/ given in Section 1.5.2, that is, E is defined by equation (1.5.11): “ E.u, v/ D .u.x/  u.y// v.x/j.x, y/m.dx/m.dy/. XXnd

Assume that j.x, y/ satisfies the conditions given in Section 1.5.2. Then, by Theorem 1.5.7 and Theorem 1.5.9, .E, F / satisfies .E.5/ and .E.6/. For any u, v 2 lip C0 .D/, equation (5.1.36) can be written as “ Z v.x/dhM Œu i.x/ D .u.x/  u.y//2 v.x/j.x, y/m.dx/m.dy/. D

DDnd

Hence

Z hM Œu i .dx/ D

 .u.x/  u.y// j.x, y/m.dy/ m.dx/. 2

¹y¤xº

This implies hM

Œu

Z tZ it D 0

5.2

D

.e u.Xs /  e u.y//2 j.Xs , y/m.dy/ds.

(5.1.38)

Beurling–Deny type decomposition

In this section, we consider the Beurling–Deny type decomposition of a regular Dirichlet form .E, F / on L2 .X ; m/ satisfying .E.5/. Let k and J be the killing measure and the jumping measure given by equations (3.5.21) and (3.5.25), respectively. By using the results obtained in the preceding sections, we first give the probabilistic descrip  tions of them. Let M c and M d be the families of continuous and purely discontinuous 

martingale AFs of M respectively, that is 







M c D ¹M 2M : Px .M t is continuous in t / D 1, q.e.xº



` ` M d D ¹M 2M : hM , L i D 0 Px -a.s. for q.e. x for any L 2M c and `  1º, 

where M t` D M t^ ` and ` D B` for any fixed ¹B` º 2 „.ı/ . For any M 2M , since 

M ` 2 M, the results holding for M 2 M are applicable to M ` . In particular, M 2M

170

Chapter 5 Martingale AFs and AFs of zero energy

can be decomposed uniquely as 

M D Mc C Md,



M c 2M c , M d 2M d .1 



In particular, for any u 2 Fb , there exist uniquely M Œu ,c 2M c and M Œu ,d 2M d such that Œu Œu ,c Œu ,d (5.2.1) M t^ ` D M t^ ` C M t^ ` Œu for all `  1. We may assume  that u is quasi-continuous already. Since N is continuous and u.X /  u.X / 1¹tº D u.X /1¹tº , equations (5.1.28) and (5.2.1) Œu

Œu

Œu

Œu

imply that u.X /1¹  t º D M 1¹tº , where M t D M t  M t . Usp  ing the dual predictable projection u.X /1¹tº of u.X /1¹tº , define the purely discontinuous MAF M Œu ,k by p  Œu ,k Mt D u.X /1¹tº  u.X /1¹tº . Furthermore, put M Œu ,j D M Œu ,d  M Œu ,k . Then equation (5.2.1) can be written as Œu

Œu ,c

Œu ,j

Œu ,k

M t^ ` D M t^ ` C M t^ ` C M t^ `

(5.2.2)

j

for all `  1. Put chui D hM Œu,c i , hui D hM Œu,j i and khui D hM Œu,k i . Then the Œu

restriction of hui to B` is the associated measure of M t^ ` . Similar correspondence also holds for each term on the right-hand side of equation (5.2.2). Since equation (5.2.2) is an orthogonal decomposition relative to h, i, this implies that their associated measures are related by j

hui D chui C hui C khui .

(5.2.3)

Let .N.x, dy/, H / be the Lévy system of the Hunt process M. Then p   p Œu hM Œu ,k i t D .M /2 1¹tº D u2 .X /1¹tº  X p D 1¹ º .Xs /u2 .Xs /1¹Xs ¤Xs º Z D

0º

Theorem 5.2.1. Assume that assumption (J) holds. Then, for any u, v 2 F \ C0 .X /, E.u, v/ can be written as “ 1 1 c .u.y/  u.x// .v.y/  v.x// J.dxdy/ E.u, v/ D E .u, v/ C 2 2 d.x,y/>0 “   1  .u.y/  u.x// v.x/ J.dxdy/  b J .dxdy/ 2 d.x,y/>0 Z 1 b u.x/v.x/k.dx/, (5.2.10)  E .u, v/ C 2 X where J is a positive Radon measure on X  X n d, b J .dxdy/ D J.dydx/, k is a positive Radon measure on X , E c .u, v/ is a symmetric bilinear form satisfying E c .u, v/ D 0 for any functions u, v such that one of them is equal to a constant on a neighborhood of the support of the other, and E b is a bilinear form satisfying E b .u, v/ D 0 for any functions u, v such that u is a constant on a neighborhood of the support of v and E b .uv, w/ D E b .u, v/ C E b .v, u/

(5.2.11)

for any u, v, w 2 F \C0 .X / such that w D 1 on a neighborhood of suppŒu[suppŒv. The measures J.dxdy/, k.dx/ and the bilinear forms E c .u, v/ and E b .u, v/ possessing the stated properties are unique. In particular, J and k are given by equations (5.2.6) and (5.2.5), respectively, in terms of the Lévy system .N , H / of the associated Hunt process M. c Proof. Uniqueness: If the supports of u, v 2 F \ C R R 0 .X / are disjoint, then E .u, v/ D b E .u, v/ D X uvd k D 0. Hence E.u, v/ D  d.x,y/>0 u.y/v.x/J.dxdy/. As in

173

Section 5.2 Beurling–Deny type decomposition

the proof of Theorem 3.5.15, this implies that J is uniquely determined by E. For any v 2 F \ C0 .X /, take a function u 2 F \ C0 .X / such that u D 1 on an neighborhood B .v/ of the support B.v/ of v. Since E c .u, v/ D E b .u, v/ D 0 and .u.x/  u.y//.v.x/  v.y// D 0 for x, y 2 B .v/, “ .u.y/  u.x// .v.y/  v.x// J.dxdy/ d.x,y/>0

“

  .u.y/  u.x// v.x/ J.dxdy/  b J .dxdy/

 d.x,y/>0

“ D

¹.x,y/;x…B .v/,y2B.v/º

.1  u.x// v.y/J.dxdy/

“



¹.x,y/;x2B.v/,y…B .v/º

“  “ D 2 “

¹.x,y/;x2B.v/,y…B .v/º

¹.x,y/;x2B.v/,y…B .v/º

D 2

.u.y/  1// v.x/J.dxdy/   .u.y/  1/ v.x/ J.dxdy/  b J .dxdy/

.u.y/  1// v.x/J.dxdy/

.u.y/  u.x// v.x/J.dxdy/, d.x,y/>0

we have Z X

“ v.x/k.dx/ D E.u, v/ C

.u.y/  u.x// v.x/J.dxdy/. d.x,y/>0

This implies that k is determined by E.u, v/ and J.dxdy/ and hence k is unique. Therefore E c .u, v/E b .u, v/ is uniquely determined by E.u, v/. To show the uniqueness of E b .u, v/ and E c .u, v/, assume that E c .u, v/E b .u, v/ D 0, that is, E c .u, v/ D E b .u, v/ for all u, v 2 F \ C0 .X /. Then E b is symmetric and hence, by equation (5.2.11), E b .uv, w/ D E b .u, vw/ C E b .v, uw/ D 2E b .u, v/ for any w 2 F \ C0 .X / such that w D 1 on a neighborhood of the supports of u and v. Since E b .uv, w/ D E c .uv, w/ D 0, this implies E b .u, v/ D 0 and E c .u, v/ D 0, that is E c D E b D 0. Existence: Consider the measures J and k constructed in Section 3.5 so that they satisfy the formulae (3.5.25) and (3.5.21), respectively. They can be then specified by equations (5.2.6) and (5.2.5), respectively, in term of the Lévy system .N , H / of the bˇ .dxdy/ D associated Hunt process M. Put Rˇ .dxdy/ D Rˇ .x, dy/m.dx/ and R bˇ .x, dy/m.dx/. Let u, v 2 C0 .X / \ F and take a non-negative function w 2 R C0 .X / \ Fb such that w D 1 on an -neighborhood of the support of v. Then by

174

Chapter 5 Martingale AFs and AFs of zero energy

writing

Z Z

ˇ.u  ˇRˇ u, v/ D ˇ 2

.u.y/  u.x// v.x/w.x/Rˇ .dxdy/

Z uv.x/.1  ˇRˇ 1.x//m.dx/ (5.2.12) Cˇ X Z Z .u.y/  u.x// .v.y/  v.x// w.x/Rˇ .dxdy/ D ˇ2 X X Z Z .u.y/  u.x// v.y/w.x/Rˇ .dxdy/  ˇ2 ZX X uv.x/.1  ˇRˇ 1.x//m.dx/, (5.2.13) Cˇ X

X

X

and taking the mean of equations (5.2.12) and (5.2.13), we have Z Z ˇ2 .u.y/  u.x// .v.y/  v.x// w.x/Rˇ .dxdy/ ˇ.u  ˇRˇ u, v/ D 2 X X Z Z ˇ2 .u.y/  u.x// v.x/Rˇ .dxdy/  2 X X Z Z ˇ2 .u.y/  u.x// v.y/w.x/Rˇ .dxdy/  2 X X Z uv.x/.1  ˇRˇ 1.x//m.dx/ Cˇ X Z Z ˇ2 D .u.y/  u.x// .v.y/  v.x// w.x/Rˇ .dxdy/ 2 X X Z Z   ˇ2 bˇ .dxdy/ .u.y/  u.x// v.x/ Rˇ .dxdy/  w.y/R  2 X X Z uv.x/.1  ˇRˇ 1.x//m.dx/. (5.2.14) Cˇ X

Since Z Z ˇ2 .u.y/  u.x//2 w.x/Rˇ .dxdy/ X

Z

X

 ˇ Ewm 2

1

e

ˇ t

 .u.X t /  u.X0 // dt 2

0

is bounded relative to ˇ by equation (5.1.7) and a similar result holds for v, we obtain by the Schwarz inequality that ˇZ Z ˇ ˇ ˇ 2ˇ .u.y/  u.x// .v.y/  v.x// w.x/Rˇ .x, dy/m.dx/ˇˇ ˇ ˇ X X 1=2  Z Z 2 2 .u.y/  u.x// w.x/Rˇ .x, dy/m.dx/  ˇ   ˇ

X

X

Z Z

1=2 .v.y/  v.x// w.x/Rˇ .x, dy/m.dx/ 2

2 X

X

175

Section 5.2 Beurling–Deny type decomposition

is bounded relative to ˇ. Hence, there exists a sequence ˇn " 1 such that E j , .u, v/ D

“

lim ˇ 2 n!1 n

E c, .u, v/ D

d.x,y/

.u.y/  u.x// .v.y/  v.x// w.x/Rˇn .x, dy/m.dx/

“

lim ˇ 2 n!1 n

d.x,y/ 0. For any  > 0 which does not belong to such an exceptional set, since “ “ F .x, y/J.dxdy/ D lim F .x, y/'n .d.x, y//J.dxdy/ n!1

d.x,y/>

XX

for any continuous function F .x, y/ with compact support and a sequence of continuous functions 'n .t / on R1 such that limn!1 'n .t / D 1.,1/ .t /, equation (3.5.25) implies that, “ j , .u.y/  u.x// .v.y/  v.x// w.x/ E .u, v/ D lim n!1

 “

d.x,y/ 2 ˇn Rˇn .x, dy/m.dx/

D

.u.y/  u.x// .v.y/  v.x// w.x/J.dxdy/ d.x,y/

exists. By letting  # 0 through  satisfying the above condition, we obtain the existence of “ E j ,s .u, v/ D .u.y/  u.x// .v.y/  v.x// w.x/J.dxdy/. d.x,y/>0

As we noted after equation (5.2.5), we may consider that Z Z lim ˇn uv.x/.1  ˇRˇn 1.x//m.dx/ D u.x/v.x/k.dx/. n!1

X

X

Using equations (3.5.25) and (3.5.27) we have “   2 bˇ .dxdy/ .u.y/  u.x// v.x/ Rˇn .dxdy/  w.y/R lim ˇn n n!1

d.x,y/m

“

D d.x,y/m

  .u.y/  u.x// v.x/ J.dxdy/  w.y/b J .dxdy/ .

176

Chapter 5 Martingale AFs and AFs of zero energy

Note that v.x/w.y/b J .dxdy/ does not depend on w if d.x, y/ is small. Hence assumption .J / implies that the right-hand side of the above equality converges as m increases to infinity. We denote the limit by “   E j ,a .u, v/ D .u.y/  u.x// v.x/ J.dxdy/  w.y/b J .dxdy/ . d.x,y/>0

Therefore, equation (5.2.14) implies the existence of “     b 2 bˇ .dxdy/ . E .u, v/ D lim lim ˇn u.y/u.x/ v.x/ Rˇ .dxdy/ R m!1 n!1

d.x,y/º

Therefore, J satisfies the condition .J /. A comparison of equation (5.2.10) with equation (1.5.12) and the uniqueness statement of Theorem 5.2.1 imply the last assertion.

5.3

CAFs of locally zero energy in the weak sense

In this section, we also assume that .E, F / is a regular Dirichlet form on L2 .X ; m/ satisfying .E.5/. Under this condition, we have seen in the preceding section that, for Œu



any function u 2 Fb , there is a corresponding CAF N t 2N . We shall be concerned here with more concrete correspondence between them. For any sequence ¹B` º 2 „.ı/ , put ` D B` . Theorem 5.3.1. Suppose that N t is an AF. Then N D N Œu for some u 2 Fb if and 

only if N 2N and satisfies  Z 1 2 ˛t e N t^ ` dt D E.u, v/ f or al l v 2 FbB` . lim ˛ Evm ˛!1

0

(5.3.1)

180

Chapter 5 Martingale AFs and AFs of zero energy 

Furthermore, if .E, F / is strongly local, then a CAF N 2N loc satisfies N D N Œu for some u 2 Floc if and only if there exists a sequence ¹Dn º 2 „ such that equation (5.3.1) holds for Dn \ B` instead of B` for each n. Proof. If N D N Œu for u 2 FbB` , then by Lemma 5.1.6,  Z 1 2 ˛t e N t^ ` dt lim ˛ Evm ˛!1 0  Z 1 D lim ˛ 2 Evm e ˛t .u.X t^ ` /  u.X0 //dt D E.u, v/ ˛!1

for any v 2

0

FbB` .



Conversely,R suppose that N 2N satisfies equation (5.3.1). Then the function 1 C .x/ D Ex . 0 e  t N t^ ` dt / belongs to L2 .X ; v  m/ for any v 2 FbC . We shall temporary introduce the resolvent R˛ ` defined by  Z 1 R˛ ` f .x/ D Ex e ˛t f .X t^ ` /dt . 0

Since f .X t^ ` ıs^ ` / D f .X.sCt/^ ` /, ¹R˛ ` º satisfies the resolvent equation. Define ` ` ` b bb bb the resolvent ¹R ˛ º similarly relative to M. Then .R˛ f , g/ D .f , R˛ g/. Since N.tCs/^ ` D Ns^ ` C N t^ ` ı s^ ` ,  Z 1 Z 1 ` ˇs  t e ds e .N.tCs/^ `  Ns^ ` /dt Rˇ C .x/ D Ex 0 0  Z 1 Z 1 Z 1 1 ˇ s e .ˇ /s ds e  t N t^ ` dt  e Ns^ ` ds D Ex  0 0 s 1 1 D .C  Cˇ /.x/  Cˇ .x/ ˇ  1 . C  ˇCˇ /.x/, D .ˇ   / 1 . C .x/  ˇCˇ .x// D R ` .ˇCˇ /.x/. Hence, noting that is, Rˇ ` . C /.x/ D ˇ  that lim!1 . C , w/ D 0 for any w 2 F and using equation (5.3.1) we have ` ` ` bb bb .ˇCˇ , v  ˛ R ˛ v/ D lim . C  .ˇ   /Rˇ . C /, v  ˛ R˛ v/ !1

` ` bb bb D lim . C , .I  .ˇ   /R /.I  ˛ R ˛ /v/ ˇ !1

` ` bb bb D lim . 2 C , R .I  ˛ R ˛ /v/ ˇ !1   ` ` bb bb D E u, R .I  ˛ R ˛ /v ˇ     ` ` ` bb bb bb D  u, .I  ˛ R ˛ /v C ˇ u, Rˇ .I  ˛ R˛ /v   ` bb D  u  ˇRˇ ` u, .I  ˛ R ˛ /v ,

181

Section 5.3 CAFs of locally zero energy in the weak sense ` bb that is, .ˇCˇ C u  ˇRˇ ` u, v  ˛ R ˛ v/ D 0. We shall show that this implies

ˇCˇ .x/ D ˇRˇ ` u.x/  u.x/

m-a.e.

(5.3.2)

To see this, let v 2 F and set `ˇ D .ˇCˇ C u  ˇRˇ ` u, v/. Then   ` bb `ˇ D ˇCˇ C .I  ˇRˇ ` /u, ˛ R ˛ v   D ˛ R˛ ` .ˇCˇ / C R˛ ` .I  ˇRˇ ` /u, v   D ˛ Rˇ ` .˛C˛ / C Rˇ ` .I  ˛R˛ ` /u, v ˛ ` bb D ˛.˛C˛ C .I  ˛R˛ ` /u, R v/ D `˛ , ˇ ˇ that is ˇ`ˇ D ˛`˛ . Since lim˛!1 ˛ 2 .C˛ , v/ D E.u, v/ by equation (5.3.1), using equation (5.1.34) we obtain that lim ˛`˛ D lim ˛.˛C˛ C u  ˛R˛ ` u, v/ ˛!1   D E.u, v/ C lim ˛ u  ˛R˛ ` u, v D 0,

˛!1

˛!1

which gives equation (5.3.2). Œu Œu Finally, put M t D N t N t D e u.X t /e u.X0 /M t N t . By the inverse Laplace u.X t^ ` /  e u.X0 //, transformation of equation (5.3.2), since Ex .N t^ ` / D Ex .e u.X t^ ` /  e u.X0 /  N t^ ` / D 0 m-a.e. x. Hence .M t^ ` / is Ex .M t^ ` / D Ex .e 

a MAF. Furthermore, since N Œu and N belong to N , M t` D M t^ ` satisfies  Z 1 Z 2 ˛t 2 v.x/hM ` i .dx/ D lim ˛ Evm e M t^ ` dt D 0, ˛!1

X

0

2 Therefore, it follows from Lemma 5.1.2 that Evm .M t^ / D 0 and for any v 2 F ` hence M t D 0 for all t . B` .



The assertion concerning N 2N loc follows similarly. We have seen in Theorem 4.1.16 that, for each  2 S , there is a corresponding PCAF A t satisfying equation (4.1.7). In particular, if  2 S0 , then U˛  2 F and UA˛ 1 is a q.c. modification of U˛ . Lemma 5.3.2. If  2 S0 , then ŒU˛

Nt

Z D 0

t

˛UA˛ 1.Xs /ds  A t .

(5.3.3)

Proof. By equation (5.3.1) it is enough to prove that Z 1 Z t^ `   e ˇ t ˛UA˛ 1.Xs /ds  A t^ ` dt D E.U˛ , v/ lim ˇ 2 Evm ˇ !1

0

0

182

Chapter 5 Martingale AFs and AFs of zero energy

for all v 2 FbB` . For any PCAF C t with associated measure , Theorem 4.1.13 (v) implies that  Z 1 ˇ 2 ˇ t lim ˇ Evm e C t dt D lim ˇ.v, UC 1/ D h, vi ˇ !1

ˇ !1

0

for any q.c. function v 2 F By virtue of Lemma 4.1.15, since C t^ ` is a PCAF of MB` associated with 1B`  , similarly to the above equation, it holds that  Z ` 2 ˇ t e C t dt D h, vi lim ˇ Evm B` .

ˇ !1

for any v 2 we have

FbB` .

0

Applying this result to C t D Z

1

ˇ t

Z

t^ `

Rt 0

˛UA˛ 1.Xs /ds as well as C t D A t , 

˛UA˛ 1.Xs /ds

e lim ˇ Evm 0 0     ˛ D ˛UA 1, v  h, vi D E UA˛ 1, v 2

ˇ !1



 A t^ ` dt

which yields the desired result. Œu

By virtue of Lemma 5.3.2, if u D U˛  for  2 S0 and ˛ > ˛0 , then N t bounded variation. Generally, we have the following result.

is of

Theorem 5.3.3. The following are equivalent to each other for u 2 Fb . (i)

N Œu is a CAF of locally bounded variation.

(ii)

There exist smooth measures  .1/ and  .2/ and ¹B` º 2 „.ı/ such that 1B`  .i/ 2 S0 for any k, i D 1, 2 and E˛ .u, v/ D h .1/   .2/ ,e v i,

(5.3.4)

for all v 2 F B` and ˛ > ˛0 . Proof. Suppose that N Œu is locally of bounded variation. Then it can be written as the .1/ .2/ Œu difference of two PCAFs, that is, there exist PCAFs A t and A t such that N t D .1/ .2/ .i/ A t C A t . Let  .i/ be the smooth measure corresponding to A t and let ¹Fk º be a nest such that 1Fk .1 C b hı /   .i/ 2 S0 for i D 1, 2. Let B` D ¹x : Ex .e XnF` / > 1 0º \ ¹x 2 B` : 1=` < b hı < `º for an increasing sequence of relatively compact 1 open sets ¹B` º increasing to X , then ¹B` º 2 „.ı/ . For any bounded q.c. non-negative hı , using Theorem 4.1.13 and Theorem 5.3.1 function v 2 F B` , since jvj  `kvk1b we have  Z 1 .1/ .2/ .1/ .2/ 2 ˇ t h   , vi D lim ˇ Evm e .A t^ `  A t^ ` /dt ˇ !1 0  Z 1 Œu D  lim ˇ 2 Evm e ˇ t N t^ ` dt ˇ !1

D E.u, v/.

0

183

Section 5.3 CAFs of locally zero energy in the weak sense

Conversely, suppose that (ii) holds. Let A.i/ be the PCAF associated with  .i/ . Put ` D 1B`  . .1/   .2/ /, then ˛ E˛ .u, v/ D h` , vi D E˛ .U˛ ` , v/ D E˛ .U˛ `  HXnB U˛ ` , v/. `

for all v 2 F B` . On the other hand, the left-hand side can be written as E˛ .u  ˛ ˛ ˛ u, v/. Since this holds for all v 2 F ` , u  HXnB u D U˛ `  HXnB U˛ ` . HXnB ` ` ` Rt .1/ .2/ Noting that C t D 0 IB` .Xs /.dAs  dAs / is the CAF associated with ` , we have from Lemma 5.3.2 that Z t Z t ŒU .2/ U˛ ` .Xs /ds  1B` .Xs /.dA.1/ Nt ˛ ` D ˛ s  dAs /. 0

0

˛ Since HXnB U˛ ` is the difference of two ˛-excessive functions belonging to F , it `

can be written as U˛ ` for some ` D `C  ` with `C , ` 2 S0 . For any v 2 F B` , since ˛ U˛ ` , v/ D 0, h` , vi D E˛ .U˛ ` , v/ D E˛ .HXnB ` .`/

k is supported by X n B` . Using the PCAF C t ˛ ŒuHXnB u `

Nt

associated with ` , it holds that

˛ ŒU˛ k HXnB U˛ k

` D Nt Z t ˛ D˛ .U˛ `  HXnB U˛ ` /.Xs /ds ` 0 Z t .`/ .2/  1B` .Xs /.dA.1/ s  dAs / C C t .

0

Rt

.i/

.`/

1XnB` .Xs /dAs D C t D 0 for t < B` , we have Z t ˛ ŒuHXnB u   .1/ .2/ ˛ ` u.Xs /  HXnB D˛ u.Xs /ds  A t C A t Nt `

Noting that

0

0

for t < B` .

˛ uº is E˛ bounded relative to `  1 and converges as ` tends to infinity to Since ¹HXnB `

˛ ˛ u, HXnB u/ D 0. Hence Ex .e ˛ u.X // D 0 q.e., it follows that lim`!1 E˛ .HXnB ` ` ˛ lim`!1 HXnB u.X t / D 0 uniformly on every finite interval a.s. Px for q.e. x. Fur`

˛ ŒHXnB u

˛ thermore, E˛ convergence of HXnB u to 0 implies that lim`!1 M t^ k ` D 0 uni` formly on every compact ˛t -interval for all k by Lemma 5.1.1 and Theorem 5.1.3 (ii). ŒH u Therefore, limk!1 N XnB` .X t / D 0 uniformly on any compact t -interval and R t Œu .1/ .u/ hence N t D ˛ 0 u.Xs /ds  A t C A t which implies (i).

If .E, F / is local, then the assertion of Theorem 5.3.3 holds for u 2 Floc . As we 

have seen in Theorem 5.3.1, a CAF N 2N coincides with N Œu for some u 2 F if  it satisfies equation (5.3.1). For general N 2N , we cannot expect the expression as

184

Chapter 5 Martingale AFs and AFs of zero energy

N D N Œu by using a single function u 2 F , but if we restrict it to a suitable compact set, it is possible to represent it by a single function. To show this, we prepare the following two lemmas. 

Lemma 5.3.4. Let N 2N and G be a relatively compact quasi-open set. Suppose that (5.3.5) Px .G < 1/ > 0 q.e. x 2 G. Then there exist an increasing sequence ¹Bn º of quasi-open subsets of G and an exceptional set Z such that (i)

limn!1 Bn D G a.s. Px for x 2 G n Z.

(ii)

supx2Bn nZ Ex .NN 2n / < 1, where n D Bn and NN t D max0st jNs j.

N N Proof. Let .!/ D max0t G jN G  N t j. Then  it holds  that  2N G and N G  N 2 . Put d .x/ D Ex .e  / and dN .x/ D Ex e NG for  > 0. Then they satisfy d2 .x/  dN .x/  d2= .x/. Since there exists an exceptional set Z such that   Ex .1  d .Xs / : s < G / D Ex 1  e  maxsuG jNG Nu j : s < G increases to 1  d .x/ as s # 0 for any x 2 G n Z, 1  d is excessive relative to the part process MG . Define a quasi-open set Bn by Bn D ¹x : x 2 G, dN .x/ > 1=nº. Then ¹Bn º is an increasing sequence of quasi-open subsets of G. If x 2 Bn n Z, then 1  dN=2 .x/  Px .NN G  T / C e T =2 n D 1  Px .NN G > T / C e T =2 . Hence

1 C e T =2 . n By taking T as large, we can choose a number C such that Px .NN n > T /  Px .NN G > T /  1  Px .NN n > T /  C < 1

for q.e. x 2 Bn n Z.

Put k D inf¹t > 0 : NN t D kT º. Noting that k  k1 C 1 ı k1 we have Px .NN n > kT / D Px . k < n /    Ex PXk1 . 1 < n / : k1 < n  C Px . k1 < n /  C k . By this, we get that Px .NN n > t /  C1 e C2 t , x 2 Bn n Z for some positive constants C1 and C2 . In particular, it follows that supx2Bn nZ Ex .NN 2n / < 1. By the hypothesis, d .x/ > 0 and inf0t n d .X t / > 0 a.s. Px , for q.e. x 2 Bn n Z. Then n .!/ D

Section 5.3 CAFs of locally zero energy in the weak sense

185

G .!/ for sufficiently large n depending on !. Thus we get Px .limn!1 n D G / D 1 for q.e. x 2 G n Z. 

Lemma 5.3.5. For N 2N and a relatively compact quasi-open set G, assume that 1 b  < hı <  on G for some  > 0 and q.e. sup Ex .NN 2G / < 1. x2X

(5.3.6)

Then, for v.x/ D Ex .N G / there exists an increasing sequence of functions ¹u˛n º  F G \G .ı/ such that [n ¹x : u˛n .x/ D 1º q.e. on G and the function vu˛n 2 FbG \G .ı/ . Proof. Let M t D v.X t /v.X0 /N t . We shall show that M t^ G is a square integrable martingale. In the right-hand side of  2   2  2  2     3E v v N t^ G , Ex M t^ .X / C 3E .X / C 3E x t^ x 0 x G G  2  2 it is clear that Ex .v .X0 // and Ex N t^ G are q.e. bounded. Also since       Ex v 2 .X t^ G /  Ex EX t ^G .N 2G / D Ex .N G  N G ^t /2 ® ¯  2 Ex .N 2G / C Ex .N 2G ^t / is bounded, .M t^ G / is square integrable. The martingale property follows from   Ex .M t^ G / D Ex EX t ^G .N G / C Ex .N G /  Ex .N t^ G / D 0 q.e. x 2 G. Thus, M t is a locally square integrable martingale. Let  be the associated smooth measure of hM i t . By virtue of Lemma 4.1.14 and Lemma 4.1.15, there exº of closed sets such that Fn " 1 a.s. and ists an increasing R sequence ¹Fn R Fn ˇ t limˇ !1 ˇEb hı d < 1. By replacing Fn with a fine e d hM i t D Fn b 0 m ˛ XnF n / > 0º, we may assume that F is a quasi-open set. Put open set ¹x : Ex .e n Bn D G \ Fn . For a strictly positive integrable function such that f  1 and positive integer n, let u˛n D .kR˛n f / ^ 1 for R˛n f D R˛Bn f . Then u˛n 2 F \ G .ı/ by Lemma 1.4.5 and  Z B   n 1 ˇ t ˛ ˛ 2 D ˇ 2 Eb e .vu .X /  vu .X // dt ˇ .I  ˇRˇn /vu˛n , vu˛n t 0 n n m m b 2 0  ˇ C .vu˛n /2 , 1  ˇRˇn 1 m b 2  ˇ b.ı/,n 1 , C .vu˛n /2 , 1  ˇ R ˇ ı m b 2 .ı/,n 1 n b ˛ b b where Rˇ ı f .x/ D R .hı f /.x/. Since un is ˛-excessive, b hı .x/ ˇ   ˇ ˇ b.ı/,n 1 .vu˛n /2 , 1  ˇRˇn 1 C .vu˛n /2 , 1  ˇ R ˇ ı m m b b 2 2    ˇ ˛ 2 ˇ b.ı/,n 1  kvk1 .un / , 1  ˇRˇn 1 C kvk1 .u˛n /2 , 1  ˇ R ˇ ı m m b b 2 2 ˛ ˛ ˛ ˛  kvk1 A˛ .un , un /   kvk1 E˛ .un , un /.

186

Chapter 5 Martingale AFs and AFs of zero energy

By the definition of v, Z 2 lim ˇ Eb m ˇ !1

n

e

0

ˇ t

 .vu˛n .X t /

Z

 lim ˇ Eb m ˇ !1 2

n

e

ˇ t

0



vu˛n .X0 //2 dt 

2 .M t^ n

C

2 N t^ /dt n

 Z n  ˛ 2 ˇ t ˛ u C kvk1 lim ˇ 2 Eb e .X /  u .X / dt t 0 n n m ˇ !1 0 Z b hı d C kvk1 A.u˛n , u˛n / < 1.  X

This implies that vu˛n 2 FbBn \ G .ı/ . As defined after equation (2.4.9), a function u 2 F is said to be harmonic on G if  E.u, v/ D 0 for all v 2 F G . If v.X t /  v.X0 / D M t , t  G for some M 2M , then v is harmonic on G. Hence, the above two lemmas show the following theorem. 

Theorem 5.3.6. Let N 2N and G be a relatively compact quasi-open set of X on hı <  for some  > 0. Suppose that equation (5.3.5) holds. Then there which 1 < b exists a sequence ¹Bn º of quasi-open subsets of G with limn!1 Bn D G a.s. Px Œu for q.e. x 2 G and a sequence ¹un º  F \ G .ı/ such that N t D N t n for t < Bn . If ¹Bn0 º and ¹u0n º are other sequences satisfying the same conditions as .Bn , un /, then un  u0n is harmonic on Bn \ Bn0 . For a measurable set C with m.C / > 0, consider the Dirichlet form .E  , F / defined by (5.3.7) E  .u, v/ D E.u, v/ C .u, v/ ., with .dx/ D 1C .x/m.dx/. By virtue of Theorem 4.3.1, .E  , F / is a regular Dirichlet form on L2 .X ; m/. For an additive functional A t , define the weak sense energy ` .A/ of an AF A relative to the e,v .A/ and, for a fixed sequence ¹B` º 2 „.ı/ , e,v t   Hunt process M associated with .E , F /. Then they coincides with ev .A/ and ev` .A/, respectively. In fact,  Z 1 ˇ ˇ Rt ˇ 2 2 ˇ t ˇ  0 1C .Xs /ds e  1ˇ A t dt jev .A/  e,v .A/j  lim ˇ Evm ˇe ˇ !1 0  Z 1  lim ˇ 2 Evm e ˇ t tA2t dt D 0 ˇ !1

0







` .A/. Define if ev .A/ < 1. Similarly, ev` .A/ D e,v M  and N  similarly to M and 





N . Then N  DN . For any u 2 F , let Œu ,

u.X t /  u.X0 / D M t

Œu ,

C Nt

Œu

D Mt

Œu

C Nt

(5.3.8)

187

Section 5.3 CAFs of locally zero energy in the weak sense

be the Fukushima’s decompositions relative to the processes M and M, respectively. Lemma 5.3.7. For u 2 F , N Œu , and N Œu are related by Z t  Z t Œu , Œu D Nt  eAt NsŒu e As dAs  u.Xs /e As dAs Nt for A t D

Rt 0

0

(5.3.9)

0

1C .Xs /ds. 

Proof. We shall denote by NQ t the right-hand side of equation (5.3.9). Since NQ 2N , for the proof of equation (5.3.9), it is enough to show  Z 1 2 ˇ t A t Q e e (5.3.10) N t^ ` dt D E  .u, v/ lim ˇ Evm ˇ !1

0

for all non-negative functions v 2 Fb` corresponding to ¹B` º 2 „.ı/ by Theorem 5.3.1. In the expression  Z 1 Œu e ˇ t e A t N t^ ` dt D ˇ 2 Evm 0   Z 1 Z 1 Œu Œu 2 e ˇ t N t^ ` dt  ˇ 2 Evm e ˇ t N t^ ` .1  e A t /dt , ˇ Evm 0

0

the first term of the right-hand side tends to E.u, v/ by Theorem 5.3.1. Since   Rt Œu N t 2N and 1  e A t D 0 e As dAs 2N , the second term of the right-hand side converges to zero as ˇ ! 1, because ˇ ˇ Z 1 ˇ ˇ Œu e ˇ t N t^ ` .1  e A t /dt ˇˇ lim ˇˇˇ 2 Evm ˇ !1 0 1=2  Z 1 Œu e ˇ t .N t^ ` /2 dt  lim ˇ 2 Evm ˇ !1

 Z  ˇ 2 Evm

0

1

e ˇ t .1  e A t /2 dt

1=2 D 0.

0

Similarly, ˇ Z ˇ ˇ lim ˇ 2 Evm ˇˇ !1

1

0

ˇ t

e Z

1

 lim ˇEvm ˇ !1

0

 Z  lim ˇ Evm ˇ !1

0

 Z  Evm

1 0

Z 0

t^ `

NsŒu e As dAs

e ˇsAs jNs^ ` jdAs Œu

1





e ˇsAs .Ns^ ` /2 dAs

e ˇsAs dAs

Œu

1=2

ˇ ˇ dt ˇˇ

1=2

188

Chapter 5 Martingale AFs and AFs of zero energy

 Z  lim ˇ Evm ˇ !1

1

0

Rt

 1=2 Œu e ˇs .Ns^ ` /2 ds  hm, vi D 0.

Furthermore, noting that 0 u.Xs /e As dAs is a CAF of M with associated Revuz measure 1C u  m, as in the proof of Lemma 5.3.2, we have Z 1 Z t^ `   e ˇ t u.Xs /e As dAs dt lim ˇ 2 Evm ˇ !1 0 0 Z 1 Z t^ `   2 ˇ t D lim ˇ Evm e u.Xs /dAs dt ˇ !1

0

0

D .u, v/1C m . Hence the assertion follows. Corollary 5.3.8. For any u 2 F and a measurable set C with 0 < m.C / < 1, Z t  Z t Œu , Œu At Œu As As D Mt C e Ns e dAs  e u.Xs /e dAs , Mt (5.3.11) for D 1C  m and A t D

Rt 0

0

0

1C .Xs /ds.

Theorem 5.3.9. Suppose that ˛0 D 0. Then Fukushima’s decomposition in the weak sense given by Theorem 5.1.5 holds for any bounded function u 2 Fe . Proof. If .E, F / is transient, then any bounded function u 2 Fe is approximated by a sequence of uniformly bounded sequences ¹un º of the form un D Rfn with fn 2 L1 \ L1 .X ; m/. Since each function un can be decomposed as in equation (5.1.28) 

and ev .M Œun / D 2E.un , un v/  E.u2n , v/, there exists a MAF M Œu 2M such that Œu ,` Œu ,` relative to ev` for v 2 F B` and uniformly on every filimn!1 M t n D M t Œu ,` Œu Œun Œu nite interval, where M t n D M t^ n` . Also A t^ converges to A t^ ` uniformly on ` every compact interval by Lemma 4.1.9. Hence the assertion holds similarly to Theorem 5.1.5. Suppose that .E, F / is recurrent. Note that the extended Dirichlet space Fe coin cides with the extended Dirichlet space Fe of a transient Dirichlet form .E  , F / for D 1C  m with a measurable set C such that 0 < m.C / < 1. For u 2 Fe , take a uniformly bounded sequence ¹un º  F such that limn!1 E  .un  u, un  u/ D 0 and limn!1 un D u a.e. Then, by the result of the transient case, there exists a MAF 



M Œu , 2M  and a CAF N Œu , 2N  satisfying equation (5.3.8). Since ¹un º conŒu , verges to u relative to E  and E, as in the proof of Theorem 5.1.5, limn!1 N t^ n` D Œu ,

Œu

Œu

N t^ ` and lim!1 N t^ n` D N t^ ` uniformly on any finite t -interval. By virtue of ,Œu

Œu

Lemma 5.3.7, since N t n and N t n are related by equation (5.3.9) for any n  1, ,Œu Œu ,Œu Œu and N t have the same relation. In particular, N t  N t is locally of Nt

189

Section 5.3 CAFs of locally zero energy in the weak sense

bounded variation. Furthermore, since the difference M Œun ,  M Œun is a CAF of ` .M Œun , / D e ` .M Œun / for all `  1. Thus, the convergence bounded variation, e,v v Œun

` implies the convergence of M of M Œun , to M Œu , relative to e,v t 

Œu

to M t

relative

In particular, 2M . The relation in equation (5.1.28) holds from equation to (5.3.8), Lemma 5.3.7 and Corollary 5.3.8. ev` .

M Œu

Example 5.3.10. Let .E, F / be a regular Dirichlet form on L2 .R1 ; m/ with m.dx/ D dx determined by Z 1 Z 1 1 0 0 u .x/v .x/dx  b.x/u0 .x/v.x/dx (5.3.12) E.u, v/ D 2 1 1 for a bounded measurable function b.x/. Let .X t , Px / be its associated diffusion process. Then F D H 1 .R1 / and the function u1 .x/ D x C belong to Floc with ¹Dn º 2 „ of a family of increasing relatively compact open sets such that [Dn D R1 . For any v 2 C01 .R1 /, since Z 1 Z 1 1 0 v .x/dx  b.x/v.x/dx E.u1 , v/ D 2 0 Z 1 0 1 b.x/v.x/dx, (5.3.13) D  v.0/  2 0 Œu1

is the PCAF associated with the measure 12 ı0 .dx/ C 1Œ0,1/ b.x/dx, and hence Z t 1 Œu N t 1 D L0 .t / C 1Œ0,1/ .X t /b.X t /dt 2 0 for the local time L0 .t / of X t at the origin. 2 D For the same process, consider the function u2 .x/ D x log jxj  x. Then du dx 2 1 1 1 log jxj 2 Lloc .R ; m/ and hence u2 2 Floc . For any function v 2 C0 .R /, noting that lim!0 log .v./  v.// D 0, we have Z Z 1 0 log jxjv .x/dx  b.x/ log jxjv.x/dx E.u2 , v/ D 2 R1 R1  Z  Z 1 1 0 0 v .x/ log.x/dx C v .x/ log xdx D lim 2 !0 1  Z  bv.x/ log jxjdx R1  Z Z  Z 1 1 v.x/ v.x/ b.x/v.x/ log jxjdx dx  dx   lim 2 !0 1 x x  R1 Z 1 b.x/v.x/ log jxjdx, D v.p. .v/  x R1 where v.p. x1 is the Cauchy principal value which is not of a signed measure. Therefore, Nt

Œu2

Nt

is not of bounded variation.

190

Chapter 5 Martingale AFs and AFs of zero energy

Example 5.3.11. Let D be a domain of Rd and .E, F / be the regular local Dirichlet form given by equation (1.5.3) with c D 0 and kbi k1 < 1 for i D 1, 2, : : : , d . If aij .x/ 2 C 1 .D/ and u 2 F , since u  with the mollifier  converges to u relative Œu to E˛ , N t is given by  Z t X d d X @ @.u  / @.u  / Œu N t D lim .aij /C bi .Xs /ds. (5.3.14) !0 0 @xi @xj @xi i,j D1

iD1

5.4 Martingale AFs of strongly local Dirichlet forms In this section, assuming that .E, F / is a regular strongly local Dirichlet form satisfy

ing the condition .E.5/, we give some structures of the elements belonging to M and stochastic calculus related to them. As we have seen in Corollary 5.1.8, Fukushima’s decomposition in equation (5.1.28) holds for any u 2 Floc . Let hui be the smooth measure associated with the PCAF hM Œu i t for u 2 Floc . As in Theorem 5.1.7, for all u 2 Fb and v 2 [`1 FbB` with ¹B` º 2 „.ı/ , Z e v .x/dhui .dx/ D 2E.u, uv/  E.u2 , v/. (5.4.1) X

By the strong locality, this can be extended to all u 2 Floc and v 2 [`1 FbB` such that u coincides with a function belonging to Fb on the support of v. Define a signed measure hu,vi for u, v 2 Floc by  1 (5.4.2) hu,vi D huCvi  hui  hvi . 2 For any u, v 2 Floc , there exist sequences ¹un º, ¹vn º 2 Fb and common ¹Dn º 2 „ such that u D un , v D vn on Dn . Furthermore, for any ¹B` º 2 „.ı/ , since ¹D` \ B` º 2 „.ı/ , replacing D` by D` \ B` , we may assume that D` D B` for some fixed sequence ¹B` º 2 „.ı/ . Put ` D B` and F ` D F B` as before. As in the proof of Lemma 5.3.2, hu,vi is the measure associated with the CAF hM Œu , M Œv i of bounded variation by the relation Z `  Z ˇ t Œu Œv b w e.x/hı .x/hu,vi .dx/ D lim ˇEwm e d hM , M i t ˇ !1 X 0  Z 1 Œu Œv 2 ˇ t e M t^ ` M t^ ` dt D lim ˇ Ewm ˇ !1 0  Z 1 Œu Œv D lim ˇ 2 Ewm e ˇ t M t M t dt (5.4.3) ˇ !1

0

for any w 2 F By virtue of equation (5.4.1) it satisfies Z w e.x/dhu,vi .x/ D E.u, vw/ C E.v, uw/  E.uv, w/ `.

X

(5.4.4)

191

Section 5.4 Martingale AFs of strongly local Dirichlet forms

for all u, v 2 Fb and w 2 [`1 FbB` . This also holds for u, v 2 Floc such that u and v respectively coincide with some functions of Fb on some B` . To show the derivation property of hu,vi , we need the following Burkholder, Davis and Gundy inequality. Lemma 5.4.1. For any ˛ > 0, there exist constants C1  C2 such that, for any continuous square integrable martingale .M t , F t , P /,       ˛  E .M1 , (5.4.5) C1 E hM i˛=2 /  C2 E hM i˛=2 1 1 where M t D sup0st jMs j. Proof. For any  > 0, put  D inf¹t > 0 : jM t j > º and S t D .M tC  M /2  .hM i tC  hM i /. Then .S t , F tC / is a continuous martingale. Define a stopping time a by a D inf¹t > 0 : S t D aº. For any a < 0 < b, since 0 D E.Sa ^b : a ^ b < 1/ D aP .a < b / C bP .b < b /  a .1  P .b < a // C bP .b < a /,  .!/ > it holds that P .b < a /  a=.b a/. For any ! satisfying  .!/ < 1, if M1 2 2 ˇ for ˇ > 1 and hM i1  ı  , then there exists t > 0 such that M tC .!/ > ˇ. Furthermore, since hM is .!/  ı 2 2 for all s > 0, it holds that

S t .!/ > .ˇ  1/2 2  ı 2 2 . Hence b .!/  t for b D .ˇ  1/2 2  ı 2 2 . By noting that inf Ss .!/   sup .hM isC .!/  hM i .!// > a

st

for a D

st

ı 2 2

it holds that a .!/ > t . Therefore    P M1 > ˇ, hM i1  ı 2 2 j F  P .b < a / 

 > /, this yields that Since P . < 1/ D P .M1 !    hM i1 1=2 M1 < , < 1 P ı ˇ

   D P M1 > ˇ, hM i1  ı 2 2 ,  < 1 

ı2 . .ˇ  1/2

ı2  P .M1 > /. .ˇ  1/2

Integrating both sides by d ˛ , we have Z 1   ˛ ı2 ı2  D E .M / P .M1 > /d ˛ 1 .ˇ  1/2 .ˇ  1/2 0   Z 1  hM i1 1=2 M  P   < 1 ,  < 1 d ˛ ı ˇ 0     Z 1   hM i1 1=2 M1  >  d ˛ > P P ˇ ı 0   1 1   ˛ . D ˛ E .M1 /  ˛=2 E hM i˛=2 1 ˇ ı

(5.4.6)

192

Chapter 5 Martingale AFs and AFs of zero energy



Hence

     ˛ 1 ı2 1 ˛=2  .  / E hM i E .M 1 1 ˇ ˛ .ˇ  1/2 ı ˛=2

 and This gives the second inequality of equation (5.4.5). By changing the roles of M1 hM i1 in equation (5.4.6), the first inequality of equation (5.4.5) follows similarly.

Theorem 5.4.2. For any u, v, w 2 Floc , huv,wi .dx/ D e u.x/hv,wi .dx/ C e v .x/hu,wi .dx/.

(5.4.7)

Proof. We may assume that ¹B` º 2 „.ı/ is a sequence such that there exist functions u` , v` and w` of Fb which coincide with u, v and w on B` , respectively. Hence, we may assume that u, v, w 2 Fb already. Since  1 h.uCv/2 ,wi  h.uv/2 ,wi , huv,wi D 4 it is enough to prove the relation Z Z f .x/hu2 ,vi .dx/ D 2 f .x/e u.x/hu,vi .dx/ (5.4.8) X

X B` .

for any bounded q.c. function f 2 [`1 F For f 2 F B` , the left-hand side of equation (5.4.8) is equal to  Z 1  2   2 ˇ t 2 u .X t^ ` /  u .X0 / v.X t^ ` /  v.X0 / dt e lim ˇ Ef m ˇ !1 0  Z 1    D lim 2ˇ 2 Ef um e ˇ t u.X t^ ` /  u.X0 / v.X t^ ` /  v.X0 / dt ˇ !1 0  Z 1  2   2 ˇ t u.X t^ ` /  u.X0 / v.X t^ ` /  v.X0 / dt C lim ˇ Ef m e ˇ !1

0

D lim I.ˇ/ C lim II.ˇ/. ˇ !1

ˇ !1

By equation (5.4.3) the first term is equal to Z lim I.ˇ/ D lim 2ˇ 2 Ef um ˇ !1

ˇ !1

1

0

e ˇ t .M t^ ` M t^ ` / dt Œu

Œv



D 2hhu,vi , f e ui. The second term vanishes because Z lim II.ˇ/ D lim ˇ 2 Ef m ˇ !1

ˇ !1

 lim ˇ 2 Ef m ˇ !1

Z

1 0

1 0

e ˇ t .M t^ ` /2 M t^ ` / dt Œu

e ˇ t .M t^ ` /4 dt Œu

1=2

 1=2 1 Œu D lim Ef m .M t^ ` /4 hhvi , f i1=2 t!0 t 

Œv

Z Ef m



1 0

Œv

.M t^ ` /2 dt

1=2

Section 5.4 Martingale AFs of strongly local Dirichlet forms

193

and, by Lemma 5.4.1,     1 1 Œu lim Ef m .M t^ ` /4  C2 lim Ef m hM Œu i2t^ ` t!0 t t!0 t Z t^ `  2 Œu Œu D C2 lim Ef m hM.ts/^ ı s id hM is ` t!0 t 0 Z t^ `  2 Œu Œu EXs .hM i.ts/^ ` /d hM is D C2 lim Ef m t!0 t 0 Z t 2 hE .hM Œu i t^ ` /  , b p s f ids  C2 lim t!0 t 0 Œu /  , b hı i D 0. D 2`kf k1 C2 lim hE .hM t^ `

t!0

This yields the assertion of the theorem. Theorem 5.4.3. For any ˆ 2 C 1 .Rm / such that ˆ.0/ D 0 and u1 , u2 , : : : , um 2 Floc , ˆ.u/ D ˆ.u1 , u2 ,    , um / 2 Floc and satisfies hˆ.u/,vi .dx/ D

m X

ˆui .u.x//hui ,vi .dx/

(5.4.9)

iD1

for all v 2 Floc . Proof. Let G be a relatively compact quasi-open set on which b hı is bounded from below and above by positive constants on G and w1 , w2 , : : : , wm 2 Fb be suitable functions satisfying wi D ui , i D 1, 2, : : : , m a.e. on G. Put K1,G .ˆ/ D sup¹jˆxi .w.x//j : x 2 Gº, K0,G .ˆ/ D sup¹jˆ.w.x//j : x 2 Gº and KG .ˆ/ D K1,G .ˆ/ _ K0,G .ˆ/, where w D .w1 .x/, w2 .x/, : : : , wd .x//. Then for any f 2 FbG , jf .x/ˆ.w.x//  f .y/ˆ.w.y//j   d X  KG .ˆ/ kf k1 jwi .x/  wi .y/j C jf .x/  f .y/j , iD1

X  m kf k1 jwi .x/j C jf .x/j . jf .x/ˆ.w.x/j  KG .ˆ/ iD1

By virtue of .E.5/, this implies that f ˆ.w/ 2 Fb and satisfies   m X E˛ .wi , wi / jE.f ˆ.w/, f ˆ.w//  KG E˛ .f , f / C

(5.4.10)

iD1

for ˛ > ˛0 and KG D 2K0 KG .ˆ/.kf k1 C 1/ with the constant K0 appearing in the definition of .E.5/. In particular, this yields that ˆ.u/ 2 Floc . To prove equation (5.4.9), let D be the family of functions of ˆ 2 C 1 .Rm / such that equation (5.4.9) holds. If ˆ, ‰ 2 D, then Theorem 5.4.2 and equation (5.4.9)

194

Chapter 5 Martingale AFs and AFs of zero energy

yield the relation dhˆ.w/‰.w/,vi D ˆ.w/.x/dh‰.w/,vi C ‰.w/dhˆ.w/,vi m X @.ˆ.w/‰.w// D dhwi ,vi , @wi iD1

which implies that ˆ‰ 2 D. Hence D contains all polynomials. If f is a q.c. function of F such that f D 0 outside of G and R f D 1 on a compact subset F Rof G, then by the local property and equation (5.4.4), X gwdhf ,vi D 0 and hence X gdhf w,vi D R X gdhw,vi for any non-negative q.c. function g 2 F such that g D 0 outside of F . Therefore, for the proof of equation (5.4.9), it is enough to prove Z m Z X g.x/dhf ˆ.w/,vi D g.x/ˆwi .u/dhwi ,vi (5.4.11) X

iD1 X

for ˆ 2 C 1 .Rm / and v 2 F . Take a sequence ¹ˆ.k/ º of polynomials such that .k/ @ˆ limk!1 ˆ.k/ D ˆ and limk!1 @ˆ D @w uniformly on any compact subset @wi i m of R including the range of ¹.w1R.x/, w2 .x/, : : : , wm .x// R : x 2 Gº. Then equation (5.4.11) holds for all ˆ.k/ . Since X gdhwi i < 1 and X gdhvi < 1, as will be shown in the next lemma, g  jhwi ,vi j is a bounded measure. Hence, by Lebesgue theorem, m Z m Z X X .k/ g.x/ˆwi .w/dhwi ,vi D g.x/ˆwi .w/dhwi ,vi . lim k!1

iD1 X

iD1 X

On the other hand, ˇZ ˇ ˇ ˇ ˇ g.x/d ˇ hf ˆ.k/ .w/f ˆ.w/,vi ˇ  ˇ X Z 1=2 Z 1=2 gdhf ˆ.k/ .w/f ˆ.w/i  gdhvi . X

X

In the right-hand side, by equation (5.1.36), we have Z gdhf ˆ.k/ .w/f ˆ.w/ D X     2E f .ˆ.k/ .w/  ˆ.w//, gf .ˆ.k/ .w/  ˆ.w//  E f 2 .ˆ.k/ .w/  ˆ.w//2 , g . Applying the argument used to show equation (5.4.10) to the function ˆ.k/ ˆ instead of ˆ, and noting that the constant KG .ˆ.k/  ˆ/ appearing there converges to zero as k tends to infinity, we obtain that the right-hand side of the above relation tends to zero as k increases to infinity. This yields equation (5.4.11). Therefore, equation (5.4.9) holds on G. Since G is arbitrary, we arrive at the result. For general L, M 2 M, let hL,M i be the smooth measure associated with hL, M i.

195

Section 5.4 Martingale AFs of strongly local Dirichlet forms

Lemma 5.4.4. If f 2 L2 .X ; hLi / and g 2 L2 .X ; hM i /, then fg is integrable relative to the total variation measure jhL,M i j and satisfies 2 Z Z Z 2 jfgjd jhL,M i j  jf j dhLi jgj2 dhM i . (5.4.12) X

X

X

Proof. Let  D jhL,M i j C hLi C hM i and k1 , k2 , k3 be the density function of hL,M i , hLi , hM i relative to , respectively. Then, for any constants a, b,   dhaLCbM i D a2 k2 C 2abk1 C b 2 k3 d. This implies that the set B0 D [a,b2Q ¹x : a2 k2 .x/ C 2abk1 .x/ C b 2 k3 .x/ < 0º is -negligible. Hence, for all x 2 X nB0 , a2 k2 .x/C2abk1 .x/Cb 2 k3 .x/  0 for all real numbers a, b. Therefore, by putting a D ˛jf .x/j and b D ˇ sgn.k1 .x//jg.x/j, it holds that ˛ 2 f 2 .x/k2 .x/ C 2˛ˇjf .x/g.x/k1 .x/j C ˇ 2 g 2 .x/k3 .x/  0 for all x 2 X n B0 . By integrating relative to , we obtain that Z Z Z 2 2 2 ˛ f dhLi C 2˛ˇ jfgjd jhL,M i j C ˇ g 2 dhM i  0 X

for all a, b 2

X

R1 .

X

From this equation (5.4.12) follows. 

Theorem 5.4.5. For any M 2M and f 2 L2 .X ; hM i /, there exists a unique ele ment f  M 2M such that, for any relatively compact quasi-open set B on which b hı is bounded from below and above by positive constants, Z 1 B ew .f  M , L/ D f .x/w.x/hM ,Li .dx/ (5.4.13) 2 X 

for all L 2M and w 2 F B . Proof. Let B be a relatively compact open set on which b hı is bounded from below 



and above by positive constants. For M 2M , put M tB D M t^ B and let MB be the 

family of MAFs M B for all M 2M . Then hM i D hM B i on B by Lemma 4.1.15. 

By virtue of Lemma 5.4.4, for any L 2M and non-negative function w 2 FbB , ˇZ ˇ ˇ 1 ˇˇ ˇ f .x/w.x/ .dx/ B B hM ,L i ˇ ˇ 2 X 1=2 Z 1 f 2 .x/w.x/hM B i .dx/ ew .LB /1=2 . p 2 X R This implies that LB 7! 12 X f wdhM B ,LB i is a continuous linear functional on the 



Hilbert space .MB , ew / and hence, by Riesz theorem, there exists f  M B 2 MB

196

Chapter 5 Martingale AFs and AFs of zero energy

satisfying ew .f  M B , LB / D

1 2

Z X

f .x/w.x/hM B ,LB i .dx/.

(5.4.14)

The function f  M B is determined independently of w. To show this, let  be a partition of Œ0, t  given by  : 0 D t0 < t1 <    < tn D t and put jj D max¹ti  ti1 : 1  i  nº. For such a partition , put .f 

. / M B /t

D

n X

f .X ti 1 /.M tBi  M tBi 1 /

iD1

  . / for any f 2 C0 .X /. Then ¹.f  M B / t º t0 , Px is a square integrable martingale for q.e. x satisfying  X n  2  B . / 2 B B Ex .f  M / t D Ex f .Xi1 /.hM i ti  hM i ti 1 / . iD1 .0/,B

Hence there exists a MAF M t such that  2  . / .0/,B D 0, lim Ex .f  M B / t  M t j j!0 Z t    .0/,B 2 Ex .M t / D Ex f 2 .Xs /d hM is .

(5.4.15)

0

In particular, M .0/,B 2 M and Z   Ex hM .0/,B , LB i t D Ex

t^ B

 f .Xs /d hM , Lis

(5.4.16)

0

for all L 2 M. This yields the relation Z Z wdhM .0/,B ,LB i D f wdhM B ,LB i X

(5.4.17)

X



for any w 2 F B and hence M .0/,B 2 MB and Z 1 f wdhM B ,LB i , ew .M .0/,B , LB / D 2 X that is, M .0/,B D f  M B . This implies that f  M B is defined independently of the choice of w. By virtue of Theorem 4.1.4 and equation (5.1.25), f  M B is a MAF such that the associated measure of hf  M B , LB i is f dhM B ,LB i . If B1  B2 are relatively compact quasi-open sets satisfying the condition assumed on B, since hM B1 ,LB1 i D     hM B2 ,LB2 i on B1 , it holds that ew .f  M B2 /B1 , LB1 D ew f  M B1 , LB1 for any

197

Section 5.4 Martingale AFs of strongly local Dirichlet forms 

w 2 F B1 and hence .f  M B2 / t^ B1 D .f  M B1 / t . Therefore, f  M 2M is well defined by .f  M / t D .f  M B / t for t < B and satisfies equation (5.4.13). The MAF f  M defined by TheoremR 5.4.5 is called the stochastic integral of f t relative to M and denoted by .f M / t D 0 f .Xs /dMs . The following lemma follows from Lemma 5.4.4 and Theorem 5.4.5. 

Lemma 5.4.6. Suppose that M , L 2M , f 2 L2 .X ; hM i / and g 2 L2 .X ; hLi /. Then the following results hold: 

(i) hf M ,Li D f  hM ,Li for any L 2M ; (ii) g  .f  M / D .gf /  M ; R 1

B .f  M , g  L/ D (iii) ew

2 X

fgw dhM ,Li for any non-negative function w 2 FbB .

Theorem 5.4.7. Let C1 and D1 be a uniformly dense sub-family of C0 .X / and an E˛ 

dense sub-family of F , respectively. Then, for all `  1, the family M 1 D ¹f  M Œu : 

` -dense in f 2 C1 , u 2 D1 º is ew M for any non-negative function w 2 Fb` . 



` -orthogonal to the family Proof. Assume that a MAF M 2M is ew M 1 , that is, R Œu ,` ` D X f wdhM Œu,` ,M ` i D 0 for all f 2 C1 , u 2 D1 and w 2 Fb , where M t Œu

M t^ ` and M ` is defined similarly. Then hM Œu ,` , M ` i D 0 for all u 2 D1 and hence for all u 2 F . In particular, by taking a bounded quasi-continuous function ' on B` R t^ Œu ,` and ˛ > ˛0 , put u D R˛` '. Then M t D u.X t^ ` /u.X0 / 0 ` .˛u'/.Xs /ds and satisfies   Œu ,` Ex .M t  MsŒu ,` /.M t`  Ms` / j Fs^ `   D Ex hM Œu ,` , M ` i t  hM Œu ,` , M ` is j Fs^ ` D 0 for q.e. x. Similarly to equations (5.4.15) and (5.4.16),  we can define  the R t ˛s .1/ Œu .1/,` ` stochastic integral M t D D dMs satisfying Ex hM , L it 0 e R     t^ Ex 0 ` e ˛s d hM Œu , Lis for all L 2M . Then Ex hM .1/,` , M ` i t D 0 and ` ew .M .1/ /

1 D lim Ewm t!0 2t

Z

t^ `

e 0

2˛s

 d hM

Œu

is

` .M Œu / D ew

Rt D e ˛t u.X t /  u.X0 / C 0 e ˛s '.Xs /ds 2 M. Since   1 .2/ ` .M .2/ / D lim Ewm .M t^ ` /2 ew t!0 2t   1 ` .M Œu /, D lim Ewm .e ˛t u.X t^ ` /  u.X0 //2 D ew t!0 2t .2/

for all w 2 Fb` . Put M t

198

Chapter 5 Martingale AFs and AFs of zero energy 

M .2/ belongs to M . Furthermore, since

  1 .1/,` Ewm .u.X t^ ` /  u.X0 //M t t!0 2t   1 Œu ,` .1/,` D lim Ewm M t M t t!0 2t Z t  1 ` D lim Ewm e ˛s d hM Œu ,` is D ew .M Œu /, t!0 2t 0

` .M .1/ , M .2/ / D lim ew

` .M .1/  M .2/ / D 0 for all `  1 and hence M .1/,` D M .2/,` . Therefore , noting ew R ` ˛ .2/,` .2/,` that M1  Ms D e ˛s^ ` u.Xs^ ` / C s^ e '.X /d , we have `

  .1/,` 0 D Ex .M t  Ms.1/,` /.M t`  Ms` / j Fs^ `   .2/ D Ex .M1  Ms.2/ /.M t`  Ms` / j Fs^ ` Z `  e ˛ '.X /d .M t`  Ms` / j Fs^ ` D Ex s^ Z 1`  ˛.Cs/ e '.X.Cs/^ ` /.M.Cs/^ `  Ms^ ` /d j Fs^ ` . D Ex 0

Since this holds for any ˛ > ˛0 , it follows that Ex .'.X /.M^ `  Ms^ ` / j Fs^ ` / D 0 for q.e. x. This can be extended to all ' 2 L2 .B` ; m/. Using the Markov property, it yields for 0 D t0 < t1 < t2 <    < tn D t and '1 , '2 , : : : , 'n 2 L2 .B` ; m/ that   Ex '1 .X t1 ^ ` /'2 .X t2 ^ ` /    'n .X tn ^ ` /M tn ^ ` D D

n1 X j D0 n1 X j D1

  Ex '1 .X t1 ^ ` /'.X t2 ^ ` /    'n .X tn ^ ` /.M tj C1 ^ `  M tj ^ ` /  Ex '1 .X t1 ^ ` /    'j .X tj ^ ` /EX tj C1 ^` .'j C1 .X0 /    'n .X.tn tj C1 /^ ` //

  .M tj C1 ^ `  M tj ^ ` / D 0 

` -denseness of for q.e. x. Hence, we get that M t^ ` D 0 which implies the ew M1 

in M . By virtue of Corollary 5.1.8, for u1 , u2 2 Floc , there exist sequences ¹B` º 2 „.ı/ and ¹u`i º  Fb such that ui D u`i on B` for i D 1, 2. In particular, if u1 D u2 on B` , Œu Œu then M t 1 D M t 2 for t  ` and hence hu1 u2 i .B` / D 0. More generally, this also holds if u1  u2 is equal to a constant c on B` . To show this, it is enough to show

199

Section 5.4 Martingale AFs of strongly local Dirichlet forms

Z

that

X

f .x/hu1 u2 i .dx/ D 0

(5.4.18)

for any q.c. function f 2 F ` . By applying equation (5.4.7) to u D f and v D w D u1  u2 , it follows that dhf .u1 u2 /,u1 u2 i D f dhu1 u2 ,u1 u2 i C .u1  u2 /dhf ,u1 u2 i . Since u1  u2 D c on B` , the integral of the left-hand side of the above equation is equal to Z Z Z dhf .u1 u2 /,u1 u2 i D dhcf ,u1 u2 i D c dhf ,u1 u2 i . B`

B`

B`

The right-hand side can be written as Z Z f dhu1 u2 ,u1 u2 i C c B`

B`

dhf ,u1 u2 i .

This implies equation (5.4.18). Lemma 5.4.8. If u1 u2 is equal to a constant on a quasi-open set B, for u1 , u2 2 Fb , then Œu Œu for all t < B . (5.4.19) Mt 1 D Mt 2 Proof. If u1  u2 is equal to a constant on B, then hu1 u2 i .B \ B` / D 0 for ¹B` º 2 „.ı/ . Since 1B\B`  hu1 u2 i is the measure associated with the PCAF hM Œu1  M Œu2 i t^ B\B` of MB by Lemma 4.1.15, hM Œu1  M Œu2 i t^ B\B` D 0. Since lim`!1 B` D 1, equation (5.4.19) holds. 

We have seen in Theorem 5.4.3 that, for u1 , u2 , : : : , um 2F loc,b and ˆ 2 C 1 .Rm / such that ˆ.0/ D 0, the composite function ˆ.u1 , u2 ,    , um / 2 Floc satisfies equation (5.4.9). Furthermore, by Lemma 5.4.6 (iii), for any v 2 Fb , w 2 Fb` with ¹B` º 2 „.ı/ and f , g 2 C0 .X /, X  m   ` ` f  M Œˆ.u/ ,` , g  M Œv ,` D ew f ˆui .u/  M Œui ,` , g  M Œv ,` . ew iD1

Hence, by Theorem 5.4.7 and Lemma 5.4.8, the following result holds. Theorem 5.4.9. For any locally bounded functions u1 , u2 , : : : , um 2 Floc and ˆ 2 C 1 .Rm / such that ˆ.0/ D 0, ˆ.u1 , u2 , : : : , um / 2 Floc and satisfies M Œˆ.u/ D

m X iD1

ˆui .u/  M Œui .

(5.4.20)

200

Chapter 5 Martingale AFs and AFs of zero energy

Let D be a domain of Rd and .E, F / be a strongly local Dirichlet form satisfying .E.5/. Assume that the coordinate function xi belongs to Floc for i D 1, 2, : : : d . Let ij D hxi ,xj i be the measure associated with hM Œxi , M Œxj i. Then jij .F /j  i i .F /1=2 jj .F /1=2

(5.4.21)

for any compact subset F of B` . Since we are assuming that .E, F / is strongly local, assumption (J) in Theorem 5.2.1 holds. In particular, if .E, F / is the Dirichlet form given in Section 1.5.1 with b satisfying condition (i) given there, then C01 .D/  F \ G .ı/ , and equation (5.4.21) holds for any compact set F . Hence, Theorem 5.4.9 implies the following result stated in Corollary 5.2.2. Corollary 5.4.10. Let D be a domain of Rd and .E, F / be a strongly local regular Dirichlet form with core C01 .D/ satisfying .E.5/. Then the following results hold. (i) C 1 .D/  Floc and, for any u 2 C 1 .D/, M Œu D

d X

uxi  M Œxi .

(5.4.22)

iD1

(ii) For any u, v 2 C01 .D/, E.u, v/ D

d Z 1 @u @v 1 X dij  E b .u, v/. 2 2 D @xi @xj

(5.4.23)

i,j D1

In the rest of this section, we assume that .E, F / is a Dirichlet form satisfying the conditions of Corollary 5.4.10. Denote M Œxi by M Œi for i D 1, 2, : : : , d . Lemma 5.4.11. Under the conditions of Corollary 5.4.10, it holds that P  ²X ³ d d ` Œi < 1  f  M e i Œi w iD1 fi  M : , MD for all w 2 F ` for any ¹B` º 2 „.ı/ iD1 X  d d Z 1 X ` fi  M Œi D fi .x/fj .x/w.x/ij .dx/ ew 2 B` iD1

(5.4.24)

(5.4.25)

i,j D1

for any w 2 Fb` . Proof. Let M1 D ¹f  M Œu : f , u 2 C01 .Rd /º and M2 be the family of local MAFs given by the right-hand side of equation (5.4.24). Since f  M Œu can Pd Œi be written as f  M Œu D iD1 .f uxi /  M , M1 is contained in the family M2 ` -dense given by the right-hand side of equation (5.4.24). Furthermore, since M1 is ew

Section 5.4 Martingale AFs of strongly local Dirichlet forms

201



in M by Lemma 5.4.7 for all w 2 F ` , it is enough to show that M2 is closed in  Pd .n/ ` /. Suppose that lim ` .M , ew  M Œi n!1 ew .Mn  Mm / D 0 for Mn D iD1 fi R Pd P .n/ .n/ such that i,j D1 D fi fj w.x/dij < 1. Put  D di,j D1 jij j and let aij D dij =d be the density. Then A D .aij / is a non-negative definite symmetric matrix. Let S.x/ D .ij .x// D A1=2 .x/ and T .x/ D lim!0 S.x/.A.x/ C I /1 . Then S.x/T .x/ D T .x/S.x/ D ER .x/ is the projection operator on the range A.x/.Rd /. For a vector b, denote by b its transposed vector. Then, by putting .n/ .n/ .n/ f .n/ D .f1 , f2 , : : : , fd / , we have Z ` .n/ .m/ 2ew .M  M / D .f .n/  f .m/ / A.f .n/  f .m/ /wd D Z     ER .f .n/  f .m/ / A ER .f .n/  f .m/ / wd D ZD     S T .f .n/  f .m/ / A S T .f .n/  f .m/ / wd D ZD     T .f .n/  f .m/ / A2 T .f .n/  f .m/ / wd D ZD ˇ ˇ2 ˇ ˇ D ˇAT .f .n/  f .m/ /ˇ wd. D

P P P .n/ .n/ Hence jdD1 .AT /ij fj D jdD1 .SER /ij fj D jdD1 .ER /ij .S f .n/ /j converges to some function hi .x/ in L2 .D; w  / for i D 1, 2, : : : , d . Clearly, ER h D h for P h D .h1 , h2 , : : : , hd / . Put M D diD1 fi  M Œi for fi D .T h/i . Then Z ˇ ˇ2 1 ˇ ˇ ` .M .n/  M / D ew ˇAT .f .n/  f /ˇ wd 2 B` Z ˇ ˇ2 1 ˇ ˇ D ˇER S f .n/  ER S f ˇ wd 2 B` Z ˇ ˇ2 1 ˇ 2 ˇ hˇ wd. D ˇER S f .n/  ER 2 B` 2 ` .M  M / D 0 and h D h, it follows that limn!1 ew Since ER n

` .M / ew

Z d Z 1 X 1 D fi fj wdij D .T h/ A.T h/wd 2 2 B B ` ` i,j D1 Z Z 1 1 kER hk2 wd D khk2 wd < 1. D 2 B` 2 B`

`. Hence limn!1 Mn D M 2 M2 relative to ew

202

Chapter 5 Martingale AFs and AFs of zero energy

  Example 5.4.12. Let aij .x/ 1i,j d be a uniformly elliptic family of locally inte-

grable functions on a domain D  Rd satisfying aij D aj i and b 2 L1 .D; m/ for m.dx/ D dx. Let .E, F / be the Dirichlet form on L2 .D; dx/ given by d Z d Z X 1 X @u @v @u aij dx  bi vdx E.u, v/ D 2 @xi @xj @xi D D i,j D1

(5.4.26)

iD1

for u, v 2 F D H01 .D/ the Sobolev space on D. Then equation (5.4.4) yields that, Z w.x/dhu,vi D E.u, vw/ C E.v, uw/  E.uv, w/ D

D

d Z X

aij

i,j D1 D

Hence hu,vi .dx/ D

d X i,j D1

and hM

Œu

,M

Œv

it D

d Z X i,j D1 0

@u @v wdx. @xi @xj

aij

@u @v dx @xi @xj

t

aij .Xs /

@u @v .Xs /ds. @xi @xj

Let ij D .a1=2 /ij and put B .i/ D

d X

. 1 /ij .X t /  M Œj .

j D1

Then, for any  2 S00 , B D .B .1/ , B .2/ ,    , B .d / / is a family of martingales on ., F t , P / satisfying hB .i/ , M .j / i t D ıij t . Hence, B is a d -dimensional Brownian motion. Using this, M Œi is expresses as M Œi D

d X

ij .X t /  B .j / .

(5.4.27)

j D1

In the next section, we shall apply these results to the transformation of local Dirichlet forms by multiplicative functionals.

203

Section 5.5 Transformations by multiplicative functionals

5.5

Transformations by multiplicative functionals

In this section, we are concerned with a transformation by a multiplicative functional of a diffusion process associated with a strongly local regular Dirichlet form. Let .E, F / be a strongly local and regular Dirichlet form on L2 .X ; m/ satisfying .E.5/ and let M D ., F t , X t , Px , / be the associated diffusion process. We assume that the path space  is canonical, that is,  is the family of continuous functions ! from Œ0, 1/ to X [¹º such that !.t / D  for all t  . Define X t .!/ by X t .!/ D w.t / for ! 2 . As we defined before equation (3.1.12), a right continuous F t -adapted process L t is called a multiplicative functional if there exists a defining set ƒ and an exceptional set N such that Px .ƒ/ D 1 for all x 2 X n N and, for ! 2 ƒ, L t .!/ is right continuous with left limit relative to t , L0 .!/ D 1, 0  L t .!/  1, L t .!/ D L .!/ for t   and L tCs .!/ D Ls .!/L t .s !/ for all s, t  0. Let 2 Floc be a q.c. strictly positive function. Then there exists an increasing sequence ¹Dn º of quasi-open sets such that [n Dn D X q.e. and 0 < 1=n  .x/  n q.e. on Dn and .x/ D n .x/ on Dn for some n 2 Fb . For a smooth function n with compact support contained in .0, 1/ such that n .t / D log t for 1=n  t  n, kn k1 < 1, kn0 k1 < 1 and a function n 2 F such that D n on B` , it holds that jn . n .x//  n . n .y//j  kn0 k1 j n .x/  n .y/j,

kn . n .x//k1 < 1.

Therefore n . / 2 F and hence log 2 Floc . Similarly, for D Let Œlog  Œlog  C Nt log .X t /  log . X0 / D M t 

p

, log 2 Floc . (5.5.1) 

be Fukushima’s decomposition into M Œlog  2M loc and N Œlog  2N loc . Define the  multiplicative functional L t by   1  Œlog  (5.5.2) L t D exp M t  hM Œlog  i t . 2 



Then L is a local martingale and hence Ex .L t /  1.2 Let M D ., F t , Px ,   /   be the transformed process of M relative to L , that is, Px ./ D Ex .L t : / for  2 Ft . As in the preceding section, we fix a positive ı-coexcessive function b hı 2 F for 



ı > ˛0 . Let us denote by M and N  the families of MAFs of finite energy and CAFs of zero energy in the weak sense relative to M , respectively. For u 2 Floc define an ,Œu by AF M t ,Œu Œu D M t  hM Œu , M Œlog  i t . (5.5.3) Mt 2

See Theorem 5.2 in [72].

204

Chapter 5 Martingale AFs and AFs of zero energy

Lemma 5.5.1. For any u, v 2 Floc , M ,Œu and M ,Œv are local martingales relative to M satisfying hM ,Œu , M ,Œv i t D hM Œu , M Œv i t . (5.5.4) Proof. We may assume that u, v 2 F . By virtue of Ito’s formula, since Z t   Z t  ,Œu D Ls Ms,Œu dMsŒlog  C dMs,Œu C Ls d hM Œlog  , M Œu is Lt Mt 0 0 Z t   D Ls Ms,Œu dMsŒlog  C dMsŒu , 0

M ,Œu is a local martingale relative to M . To show equation (5.5.4), it is enough to  ,Œu 2  /  L t hM Œu i t is a local martingale relative to M. This also show that L t .M t follows from Ito’s formula, in fact    ,Œu 2 /  hM Œu i t L t .M t Z t   D Ls .Ms,Œu /2  hM Œu is dMsŒlog  0 Z t   C Ls 2Ms,Œu dMs,Œu C d hM ,Œu , M ,Œu is  d hM Œu is 0 Z t C2 Ls Ms,Œu d hM Œlog  , M ,Œu is 0 Z t Z t   ,Œu 2 D Ls .M t /  hM Œu i t dMsŒlog  C 2 Ls Ms,Œu dMsŒu . 0

0



To show that M ,Œu 2M , we prepare the following lemma, where we consider that  m is the basic measure of M . Lemma 5.5.2. If A t is a PCAF of M associated with a smooth measure , then it is a PCAF of M associated with  . Rt Proof. Since A t is approximated by a sequence of PCAFs of the form 0 f .Xs /ds as Rt in the proof of Theorem 4.1.10, it is enough to show the assertion for A t D 0 f .Xs /ds for a non-negative function f 2 L2 .X ; m/ \ L2 .X ;  m/. In this case, the associated for the basic measures of A t relative to M and M are f  m and f  m respectively Rt measure m D  m of M . Hence the result holds for A t D 0 f .Xs /ds. For ¹B` º 2 „.ı/ , by considering B` \ D` , we may assume that B`  D` . Let ,` M` D .X t , Px` / and M,` D .X t , Px / be the part processes of M and M on B` ,

Section 5.5 Transformations by multiplicative functionals

205

respectively. Let us denote by .E  , F  / the Dirichlet form corresponding to M on L2 .X ;  m/, that is,  1  E  .u, u/ D lim u  p t u, u m t!0 t F  D ¹u 2 L2 .X ;  m/ : E  .u, u/ existsº, 

where p t is the transition function of M . 

Theorem 5.5.3. For any `  1, Fb`  Fb and E  .u, v/ D E.u , v /  E. , uv /

(5.5.5)

for any u, v 2 [1 F `. `D1 b Proof. Let u 2 F ` . Then u 2 F ` and hu,i is a bounded signed measure on B` . Hence by equation (5.4.4) we have for any v 2 Fb` , lim

t!0

    1 1 ,` u  p t u, v m D  lim Evm L t^ ` u.X t^ ` /  u.X0 / t!0 t t   1  D  lim Evm u.X t^ ` /  u.X0 / C .u.X t^ ` /  u.X0 //.L t^ `  1/ t!0 t   Z t^ ` 1 Œu Œu Ls dM Œlog  D E.u, v /  lim Evm .M t^ ` C N t^ ` / t!0 t 0 Z t^ `  1  Œu Œlog  Ls d hM , M is D E.u, v /  lim Evm t!0 t 0 Z v dhu,i D E.u, v /  X

D E.u , v /  E. , uv /. From this we get that u 2 F  and that equation (5.5.5) holds. As an application of this theorem, we consider a domain D of Rd and two diffu.k/ sion processes M.k/ D .X t , Px /, k D 1, 2, associated with the local regular semiDirichlet forms .E .k/ , F .k/ / with core C01 .D/ on L2 .D; m.k/ / given by E

.k/

.u, v/ D

d Z X i,j D1 D



d Z X

iD1 D

@u @v .k/  .dx/ @xi @xj ij @u .k/ v.x/ i .dx/ @xi .k/

(5.5.6)

for u, v 2 C01 .D/ and signed smooth measures i . Taking the maximum of the constants ˛0 appearing in .E.1/ and .E.2/ for E .k/ , we can assume that ˛0 is common

206

Chapter 5 Martingale AFs and AFs of zero energy

for both forms. Furthermore, by considering a part process if necessary, we assume that the M.k/ , k D 1, 2 are transient. We also assume that .E.5/ is satisfied for both semi-Dirichlet forms. Assume further that the following conditions are satisfied: .1/

.2/

(i) Px and Px are mutually absolutely continuous on each F t for M.1/ -q.e. x; .1/

(ii) there exists an M.1/ -q.e. positive and q.c. function 2 Floc such that m.2/ .dx/ D .x/m.1/ .dx/. Lemma 5.5.4. If a Borel subset of D is M.1/ -exceptional, then it is M.2/ -exceptional. .1/

Proof. Suppose that B is an M.1/ -exceptional Borel subset of D. Then Px .B < .2/ 1/ D 0 for M.1/ -q.e. x. By assumption (i), this implies that Px .B < 1/ D 0 for M.1/ -q.e. x and hence m.2/ -a.e. x by (ii). This yields that B is exceptional relative to M.2/ . Under the present assumption, for q.e. x there exists a .t , x/-measurable multiplica.2/ .1/ tive functional Lxt of M.1/ such that Px ./ D Ex .Lxt : / for M.1/ -q.e. x for 0 any  2 F t . Since M.1/ is a diffusion process, by taking a modification L t of LX t , .2/ there exists a continuous multiplicative functional L t such that Px ., t <  .1/ / D .1/ Ex .L t : , t <  .2/ / for all  2 F t .3 By the absolute continuity condition (i), L t > 0 .1/ .1/ .2/ for any t > 0 a.s. Px for q.e. x and Ex .L t : t <  .1/ / D Px .t <  .2/ /  1. Fur.1/ thermore, there exists a local MAF M t of M such that log L t D M t  12 hM i t , that is,   1 (5.5.7) L t D exp M t  hM i t . 2 .1/

.2/

For any u 2 Floc \ Floc , let k,Œu

u.X t /  u.X0 / D M t

k,Œu

C Nt

Œu

be the weak sense Fukushima’s decompositions of A t 





into M .k/,Œu 2 M.k/ and

N .k/,Œu 2 N .k/ for k D 1, 2, where M.k/ is the family of MAFs locally of finite  energy in the weak sense and N .k/ is the family of CAFs of locally zero energy in the weak sense relative to M.k/ . .1/

.2/

Lemma 5.5.5. If u 2 Floc \ Floc , then .2/,Œu

Mt

3

.1/,Œu

D Mt

See e.g. [88].

 hM .1/,Œu , M i t ,

.2/,Œu

Nt

.1/,Œu

D Nt

C hM .1/,Œu , M i t . (5.5.8)

207

Section 5.5 Transformations by multiplicative functionals .1/

Proof. By restricting to a suitable B` for ¹B` º 2 „.ı/ , there exist u.1/ 2 Fb u.2/

2

.2/ Fb

such that u D

u.1/

D

and

Œu A t^ `

u.2/

on B` . In particular, is a CAF of finite .1/,` .2/ and M . Furthermore, we may assume that ew .M / < .1/,` FbB` , where ew is the weak sense energy relative to M.1/ . Denote by

energy relative to both M.1/

1 for any w 2 e t the right-hand sides of the two equalities of equation (5.5.8), respectively. f t and N M R f t D t .Ls dMs.1/,Œu C Ls M fs dMs / is a local martingale Then, by Ito’s formula, L t M 0 f t is a local martingale CAF of M.2/ . Therefore, it is relative to M.1/ and hence M  e 2 N .2/ . enough to prove that N 

Since N .1/,Œu 2N .1/ , by taking a smaller set B` and a suitable subsequence t # 0 .1/,Œu if necessary, we may assume that lim t!0 .1=t /Eb ..N t^ ` /2 / D 0. Hence m lim

k!1

nX k 1

E.1/ wm

iD1 nX k 1

 lim

k!1

iD1

 .1/,Œu N.j C1/t=n

k ^ `

.1/,Œu k ^ `

 Njt=n

2 

  .1/ .1/,Œu .N1=n ^ /2 D 0, kw=b h ı k1 E k ` b m

which implies that lim

k!1 .1/

nX k 1 

.1/,Œu

N.j C1/t=n

iD1

k ^ `

.1/,Œu k ^ `

 Njt=n

2 

D0

.2/

a.s. Pwm.1/ and Pwm.2/ by choosing a further subsequence if necessary. Furthermore, since hM .1/,Œu , M i is locally of bounded variation, it is locally of zero energy in the e t^ ` is of zero energy in the weak sense weak sense relative to M.2/ . Therefore, N .2/ relative to M . .1/

.2/

As in Lemma 5.5.1, for any u, v 2 Floc \ Floc , hM .2/,Œu , M .2/,Œv i t D hM .1/,Œu , M .1/,Œv i t .

(5.5.9)

.k/ b .k/ D .X bt, b P x / be the pseudo Hunt process which is an m.k/ -dual process Let M .k/ of M for k D 1, 2. .2/

.1/

Theorem 5.5.6. For any 1  i , j  d , ij .dx/ D .x/ij .dx/ and .2/

.1/

i .dx/ D .x/ i .dx/  .x/ .k/

Proof. Since ij

.1/ .dx/ hM ,M Œxi  i



d X @ .1/  .dx/. @xj ij

(5.5.10)

j D1

is the measure associated with hM .k/,Œxi , M .k/,Œxj i of M.k/ , .2/

.1/

Lemma 5.5.2 implies that ij .dx/ D .x/ij .dx/. For any u 2 C01 .D/ and

208

Chapter 5 Martingale AFs and AFs of zero energy

v 2 FbB` for ¹B` º 2 „.ı/ , Lemma 5.1.6 implies that  Z 1 .2/ 2 .2/ ˇ t e .u.X t^ ` /  u.X0 //dt E .u, v/ D lim ˇ Evm ˇ !1 0  Z 1 .1/ ˇ t D lim Evm e .L t^ `  L0 /.u.X t^ ` /  u.X0 //dt ˇ !1 0  Z 1 ˇ t C lim E.1/ e .u.X /  u.X //dt t^ ` 0 vm ˇ !1 0  Z 1 Z t .1/ ˇ t Œu D lim Evm e Ls^ ` d hM , M is dt C E .1/ .u, v / ˇ !1 0 0 Z .1/ D v .x/ hM ,M Œu i .dx/ C E .1/ .u, v /. D

This relation also holds for u 2 Floc \ C .X /. In this relation, by putting u D xi and removing the term Z X Z X d d @v @v .2/ .1/ dij D dij @x @x j j D D j D1

j D1

from both sides, we obtain that Z Z .2/ .1/  v.x/ i .dx/ D v .x/  D

hM ,M

D

Z



D

Œxi 

i

.dx/ C

d Z X

v.x/

j D1 D

@ .1/  .dx/ @xj ij

.1/

v .x/ i .dx/

which yields equation (5.5.10).

5.6 Conservativeness and recurrence of Dirichlet forms In this section we consider a regular Dirichlet form .E, F / on L2 .X ; m/. Let M D At .X t , Px / be the associated Hunt process. For a non-negative function g, let A t and b be the PCAF and its dual PCAF given by Z t Z t b b t /dt . At D g.X t /dt and A t D g.X 0

0

By equation (4.2.5), the function

Z

A b bb R ˛ f2 .x/ D Ex

satisfies

1

At b t /dt e ˛tb f2 .X

0

  A bb f1 , R f 2 ˛

gm

  D R˛A .gf1 /, f2 .



Section 5.6 Conservativeness and recurrence of Dirichlet forms

209

A f is a q.c. modification of G A f by bb b g f . Hence we shall denote R bb If ˛ D 0, then R 0 0 bg f . R

Theorem 5.6.1. (i)

If there exists a sequence ¹un º  F \ L1 .X ; m/ such that 0  un  1, limn!1 un D 1 a.e and b˛ f / D 0, lim E.un , R

n!1

(5.6.1)

for any non-negative non-trivial function f 2 L1 .X ; m/ with compact support and ˛ > ˛0 , then .E, F / is conservative, that is p t 1 D 1 q.e. (ii)

Suppose that .E, F / is irreducible and ˛0 D 0. If there exists a sequence ¹un º  F \ L1 .X ; m/ such that 0  un  1, limn!1 un D 1 a.e. and bg f / D 0, lim E.un , R

n!1

(5.6.2)

1 1 for some non-trivial functions g 2 L1 C .X , m/ and f 2 LC .X ; m/\L .X ; m/, then .E, F / is recurrent.

Proof. Suppose that the condition in (i) holds. Then   b˛ f / D lim E˛ .un , R b˛ f /  .un , ˛ R b˛ f / 0 D lim E.un , R n!1

n!1

D lim .un  ˛R˛ un , f / n!1

D .1  ˛R˛ 1, f /, which yields that 1  ˛R˛ 1 D 0 a.e. Hence p t 1 D 1 a.e. Suppose that the condition in (ii) holds. For any non-trivial non-negative functions b g f 2 Fe g 2 L1 .X ; m/, since .E g , F / is a transient Dirichlet form with ˛0 D 0, R 1 g g 1 b f / D .u, f / for any f 2 L .X ; m/ \ L .X ; m/ by Theoand satisfies E .u, R C rem 1.3.9. Then, similarly to the proof of (i), we have   bg f / D lim E g .un , R bg f /  .un , R bg f /gm 0 D lim E.un , R n!1 n!1   g D lim un  R .un g/, f n!1

D .1  Rg g, f /. Hence 0 D 1  Rg g.x/ D Ex .e A1 / on the support of f which implies that A1 D 1 a.s. Px on ¹x : f .x/ > 0º. By irreducibility, this yields that A1 D 1 a.s. Px for q.e. x and hence .E, F / is recurrent. Example 5.6.2. For a strictly positive locally bounded function m, let .E, F / be the Dirichlet form on L2 .Rd ; m/ with m.dx/ D m.x/dx determined by the smallest

210

Chapter 5 Martingale AFs and AFs of zero energy

closed extension of E.u, v/ D

d Z X d i,j D1 R n Z X



iD1

Rd

aij .x/

@u @v dx @xi @xj

bi .x/

@u v.x/dx @xi

(5.6.3)

for locally bounded functions ¹bi º and ¹aij .x/º satisfying .r/kk2 

d X

aij .x/i j  ƒ.r/kk2

i,j D1

on ¹x : jxj  rº for some positive constants .r/  ƒ.r/ and  D .1 , 2 , : : : , d /. d For any function u.x/, by using the unit R sphere S1 D @B1R in R and the surface element  .d / on S1 , put u.r/ D S1 u.r / . /. Then Rd u.x/m.x/dx D R1 d 1 dr. Note that, for any non-negative non-trivial function g 2 0 um.r/r 1 L .X ; m/ and any non-negative function f 2 L1 .X ; m/ \ L1 .X ; m/ with combg f 2 Fe . For such functions f and g, put v˛ D R b˛ f b˛ f 2 F and R pact support, R g b f . Define ˛ and g by and vg D R ˛ .r/ D .An, rv˛ /.r/r d 1  .b, n/v˛ .r/r d 1 , and

g .r/ D .An, rvg /.r/r d 1  .b, n/vg .r/r d 1 ,

where n D x=jxj, A D .aij / and b D .bi /. Lemma 5.6.3. (i)

b˛ f corresponding to f 2 For some ˛ > ˛0 and any function v˛ D R 1 1 L .X ; m/ \ L .X ; m/ with compact support, if limr!1 ˛ .r/ D 0, then M is conservative.

(ii)

Suppose that .E, F / is irreducible and ˛0 D 0. If limr!1 g .r/ D 0 for bg f corresponding to some non-trivial functions g 2 L1 .X ; m/ and vg D R C 1 f 2 L1 C .X ; m/ \ L .X ; m/, then M is recurrent.

Proof. For any 2 C01 .Œ0, 1//, put u.x/ D .jxj/ 2 F . Since E.u, v˛ / D .u, f  ˛v˛ /, Z Z .ru, Arv˛ /dx  .b, ru/v˛ dx E.u, v˛ / D d Rd Z 1 ZR1 0 d 1

.r/.An, rv˛ /.r/r dr 

0 .r/.b, n/v˛ .r/r d 1 dr D 0 0 Z 1

0 .r/ ˛ .r/dr. D 0

Section 5.6 Conservativeness and recurrence of Dirichlet forms

211

This implies that d ˛ D .˛v˛  f /m.r/r d 1 , dr where the derivative is taken in the distribution sense. In particular, for r  r0 , since d˛ Rf .r/ D 0, it holds that dr  0 and hence ˛ .r/ is increasing. Furthermore, ˛ .r/ D jxjr .˛v˛  f /d m  0 for r  r0 . Assume that limr!1 ˛ .r/ D 0. Then, for a sequence of smooth functions n .r/ on Œ0, 1/ such that n .r/ D 1 for r  n, n .r/ is decreasing relative to r and n .r/ D 0 for r  n C 1, put un .x/ D n .jxj/. Then Z E.un , v˛ / D

1

0

n0 .r/ ˛ .r/dr   ˛ .n/

which converges to zero. Since r0 is arbitrary, we obtain the result by Theorem 5.6.1 (i). The proof of (ii) is similar by replacing ˛, v˛ and ˛ by g, vg and g , respectively. For the function v˛ , vg and u.x/ D .jx// given above, Z Z .r.uv˛ /, Arv˛ / dx  .b, r.uv˛ // v˛ dx E˛ .uv˛ , v˛ / D Rd Rd Z C˛ uv˛2 d m d Z 1 Z 1 R

.r/.rv˛ , Arv˛ /r d 1 dr C

0 .r/.n, Arv˛ /v˛ r d 1 dr D 0 Z 01 Z 1 d 1

.r/.b, rv˛ /v˛ r dr 

0 .r/.b, n/v˛2 r d 1 dr  0 0 Z 1 d 1

.r/v˛2 mr dr. C˛ 0

On the other hand, the left-hand side is equal to Z 1

.r/f v˛ m.r/r d 1 dr. E˛ .uv˛ , v˛ / D 0

Hence the function V˛ .r/ D .n, Arv˛ /v˛ r d 1  .b, n/v˛2 r d 1 satisfies d V˛ D .rv˛ , Arv˛ /r d 1  .b, rv˛ /v˛ r d 1 C .˛v˛2 m  f v˛ m/r d 1 dr and Z 1 d V˛ lim V˛ .r/ D dr D E˛ .v˛ , v˛ /  .f , v˛ / D 0. r!1 dr 0

(5.6.4)

212

Chapter 5 Martingale AFs and AFs of zero energy

For any r > 0, .rv˛ , Arv˛ /  .b, rv˛ /v˛  .rv˛ , Arv˛ /  .b, A1 b/1=2 v˛ .rv˛ , Arv˛ /1=2 2 1  1 D .rv˛ , Arv˛ /1=2  .b, A1 b/1=2 v˛  .b, A1 b/v˛2 2 4 1 1 2   .b, A b/v˛ . 4 Hence, equation (5.6.4) implies that d V˛ =dr  0 for r  r0 if ² ³ .b, A1 b/.x/ ˛  sup : jxj  r . 4m.x/

(5.6.5)

In particular, if this holds, then V˛ .r/ increases to zero as r increases to infinity. Let a.r/ N D .n, An/.r/ and 

2˛m.x/  .b, A1 b/ inf ˛ .r/ D 2.n, An/ jxjDr

1=2 .

(5.6.6)

dr D 1

(5.6.7)

Theorem 5.6.4. (i)

Suppose that k.b, A1 b/=mk1 < 1 and Z r Z 1 1 exp d 1 N r0 a.r/r r0

 ˛ .t /dt

for ˛ satisfying equation (5.6.5). Then M is conservative. (ii)

Suppose that .E, F / is an irreducible Dirichlet form with ˛0 D 0. If .b, n/  0, r  b  0 and Z 1 1 dr D 1, (5.6.8) d 1 N r0 a.r/r then .E, F / is recurrent.

Proof. Using the definition of ˛ .r/ and equation (5.6.4), for any r  r0 , it holds that  2 r 22d 2˛ .r/ D .n, Arv˛  bv˛ C Anv˛ /  .n, An/v˛  2 2  2 A1=2 n, A1=2 rv˛  A1=2 bv˛ C A1=2 nv˛ C 2.n, An/v˛    2a.r/ N .rv˛ , Arv˛ / C .b, A1 b/v˛2 C 2.n, An/v˛2 2   C 4a.r/ N .b, rv˛ /v˛ C .rv˛ , An/v˛  .b, n/v˛2

213

Section 5.6 Conservativeness and recurrence of Dirichlet forms

   4a.r/ N .rv˛ , Arv˛ /  .b, rv˛ /v˛ C ˛v˛2 m   rv˛ , An/v˛  .b, n/v˛2 C 4a.r/ N   C 2a.r/ N .b, A1 b/v˛2 C 2.n, An/v˛2 2  2˛mv˛2   1d d V˛  4a.r/r N C V˛ dr   C 2a.r/ N .b, A1 b/ C 2.n, An/ 2 /v˛2  2˛mv˛2   1d d V˛  4a.r/r N C V˛ . dr Therefore, 1 4

Z

R

r0

1 2˛ .r/ d 1 r a.r/ N Z



R r0

d dr

Z



r

.t /dt

exp r0

Z  V˛ .r/ exp Z

 V˛ .R/ exp

r0 R

.t /dt r0



r

.t /dt

dr

  V˛ .r0 /  V˛ .r0 / < 1.

Since 2 .r/ is decreasing, this implies that Z lim 2˛ .r/

r!1

1 r0

1 exp d 1 r a.r/ N

Z



r

.t /dt r0

˛0 and fn 2 H . Therefore, the proof of lim˛!1 ˛G˛ u D u in F is reduced to the case of u D Gˇ f for ˇ > ˛0 and f 2 F . In this case, lim Bˇ .˛G˛ u  u, ˛G˛ u  u/ D lim .˛G˛ f  f , ˛G˛ u  u/ D 0.

˛!1

˛!1

(iv) Put u. , x/ D ˛G˛ f . , x/ for 0  f  1. As in the proof of Theorem 1.1.5 .E.4b/ ) .E.4c/, it holds that ˛0 ku  uC ^ 1k2 ¯ 1®   B.u C uC ^ 1, u  uC ^ 1/ C B.u  uC ^ 1, u  uC ^ 1/ 2 D B.u, u  uC ^ 1/   @u C C ,u  u ^ 1 D E.u, u  u ^ 1/  @   @u C C D ˛.u  f , u  u ^ 1/ C ,u ^ 1 . @ Let wˇ 2 W be the solution of equation (6.1.21) corresponding to uC ^ 1, that is, @wˇ =@ C ˇwˇ D ˇ.uC ^ 1/. Since limˇ !1 wˇ D uC ^ 1 in F and 0  wˇ  1,   @u C ,u ^ 1 @     @wˇ @u D lim , wˇ D  lim ,u @ ˇ !1 @ ˇ !1   ³ ² @wˇ @wˇ C ,u ^ 1 C , u  uC ^ 1 D  lim @ @ ˇ !1  ² @wˇ 1 @wˇ D  lim , . C ˇwˇ / @ ˇ @ ˇ !1  ¯ C ˇ.wˇ  uC ^ 1/, u  uC ^ 1 ®    ¯   lim  ˇwˇ , u C ˇ.wˇ  1/, .u  1/C  0.

ˇ !1

222

Chapter 6 Time dependent Dirichlet forms

Furthermore, the second term of the right-hand side of ˛.u  f , u  uC ^ 1/

  D ˛ku  uC ^ 1kH C ˛ uC ^ 1  f , u  uC ^ 1

is non-negative. Therefore we have ˛0 ku  uC ^ 1k2  ˛ku  uC ^ 1k2 which yields that ˛G˛ f D .˛G˛ f /C ^ 1. Lemma 6.1.3. There exists a strongly continuous semigroup T t on H such that Z 1 G˛ f . , x/ D e ˛t T t f . , x/dt , f or al l f 2 H . (6.1.22) 0

Moreover, for any non-negative measurable functions  on R1 and f 2 H , T t .f /. , x/ D . C t /T t f . , x/ a.e.

(6.1.23)

b t on H such that Similarly there exists a dual strongly continuous semigroup T Z 1 b ˛ f . , x/ D b t f . , x/dt , G (6.1.24) T 0

b t f . , x/ a.e. b t .f /. , x/ D .  t /T T

(6.1.25)

Proof. The existence of the semigroup T t satisfying equation (6.1.22) is a consequence of Hille–Yoshida’s theorem. To prove equation (6.1.23), put Z 1 e ˛t . C t /T t f . , x/dt . u. , x/ D 0

It then suffices to show that u D G˛ .f / for all  2 C02 .R1 /. Write u as Z 1 u. , x/ D ./G˛ f . , x/ C .. C t /  .// e ˛t T t f . , x/dt 0 Z 1Z t D ./G˛ f . , x/ C  0 . C s/e ˛t T t f . , x/dsdt Z0 1 Z0 1  0 . C s/e ˛t T t f . , x/dsdt D ./G˛ f . , x/ C Z0 1 s  0 . C s/e ˛s G˛ Ts f . , x/ds. (6.1.26) D ./G˛ f . , x/ C 0

Then, for any ' 2 F , we have     @G˛ f @u , ' D  0 . /.G˛ f . , /, '/ C ./ . , /, ' @ @ Z 1 C  00 . C s/e ˛s .G˛ Ts f . , /, '/ds   Z0 1 @ 0 ˛s  . C s/e .G˛ Ts f /. , /, ' ds. C @ 0

223

Section 6.1 Time dependent Dirichlet forms and associated resolvents

Using equations (6.1.16) and (6.1.26), this can be rewritten as     @u , ' D  0 . /.G˛ f . , /, '/ C ./ E˛. / .G˛ f . , /, '/  .f . , /, '/ @ Z 1  00 . C s/e ˛s .G˛ Ts f . , /, '/ds C Z0 1    0 . C s/e ˛s E˛. / .G˛ Ts f . , /, '/  .Ts f . , /, '/ ds C 0

D  0 . /.G˛ f . , /, '/  .. /f . , /, '/ C E˛. / .u. , /, '/ Z 1  00 . C s/e ˛s .G˛ Ts f . , /, '/ds C 0 Z 1  0 . C s/e ˛s .Ts f . , /, '/ ds.  0

This combined with the relation obtained from equation (6.1.26) by replacing  with  0 yields   @u , ' D  .. /f . , /, '/ C E˛. / .u, '/. @ Since u 2 W , we get that u D G˛ .f / by the uniqueness of the solution. Lemma 6.1.4. Let u 2 H . If limˇ !1 ˇ.u  ˇGˇ u, u/ < 1, then u 2 F and limˇ !1 ˇ.u  ˇGˇ u, v/ D E.u, v/

(6.1.27)

for any v 2 W . b ˇ u/, it holds Proof. Since ˇ.u  ˇGˇ u, u/ D ˇ.u  ˇGˇ C˛0 u, u/  ˛0 ˇ 2 .Gˇ C˛0 u, G that lim ˇ.u  ˇGˇ u, u/ D lim ˇ.u  ˇGˇ C˛0 u, u/  ˛0 .u, u/. ˇ !1

ˇ !1

Therefore, by the relation B.ˇGˇ C˛0 u, ˇGˇ C˛0 u/ D E.ˇGˇ C˛0 u, ˇGˇ C˛0 u/ D ˇ.u  ˇGˇ C˛0 u, ˇGˇ C˛0 u/  ˛0 .ˇGˇ C˛0 u, ˇGˇ C˛0 u/ D ˇ.u  ˇGˇ C˛0 u, u/  ˇku  ˇGˇ C˛0 uk2H  ˛0 kˇGˇ C˛0 uk2  ˇ.u  ˇGˇ C˛0 u, u/,

(6.1.28)

it follows that ¹ˇGˇ uº is B1 -bounded. Hence a subsequence ¹ˇn Gˇn uº converges to some u0 2 F weakly. Since for any f 2 H .f , u0 / D E1 .G1 f , u0 / D lim E1 .G1 f , ˇn Gˇn u/ n!1

D lim .f , ˇn Gˇn u/ D .f , u/, n!1

224

Chapter 6 Time dependent Dirichlet forms

it follows that u D u0 2 F . Therefore lim ˇ.u  ˇGˇ u, v/ D lim E.ˇGˇ u, v/ D E.u, v/, for all v 2 W .

ˇ !1

ˇ !1

In the rest of this chapter, we use the notations Z D R1  X and d. , x/ D d  d m.x/. Then any u 2 H belongs to L2 .Z; / and kukH D kukL2 .Z;/ . Since ¹G˛C˛0 º is sub-Markov, as in Section 1.1, ¹G˛ º can be extended to a resolvent on b˛ º L1 .Z; / satisfying k˛G˛ uk1  kuk1 for any u 2 L1 .X ; /. Analogously, ¹G 1 b is defined as a resolvent on L .Z; / for ˛ > 0 satisfying k˛ G ˛ vkL1 ./  kvkL1 ./ .

6.2 A parabolic potential theory As in Section 1.4, a non-negative measurable function u is called an ˛-excessive function if ˇGˇ C˛ u  u -a.e. Let us denote by P˛ the family of all ˛-excessive functions. We put P D P1 . Lemma 6.2.1. (i)

A function u 2 F is ˛-excessive if and only if it satisfies E˛ .u, w/  0, f or al l w 2 W C .

(ii)

(6.2.1)

If u 2 P˛ for ˛ > ˛0 and u  w for some w 2 W , then u 2 F .

b ˇ u 2 W C . Then Proof. Suppose that u 2 F satisfies equation (6.2.1). Let vˇ D ˇ K it satisfies

@vˇ @t

C ˇvˇ D ˇu . Hence   @vˇ C 0  E˛ .u, vˇ / D , u  u C B˛ .u, vˇ / @t   1 @vˇ @vˇ  C , C ˇvˇ C B˛ .u, vˇ / D .ˇu  ˇvˇ , u /  ˇ @t @t  B˛ .u, vˇ /.

Since limˇ !1 vˇ D u in F , this implies that B˛ .u, u /  0. Hence, using the Markov property, B˛ .u , u /  B˛ .uC , u / D B ..u/ ^ 0, .u/  .u/ ^ 0/  0 which yields that u D 0. Thus u  0. For any v 2 H C , since the dual resolvent b ˛Cˇ v is non-negative, we have G   b ˛ .v  ˇ G b ˇ C˛ v/ D E˛ u, G b ˇ C˛ /v .u  ˇGˇ C˛ u, v/ D .u, v  ˇ G b ˇ C˛ v/  0. D E˛ .u, G This implies that ˇGˇ C˛ u  u a.e., that is, u is ˛-excessive.

225

Section 6.2 A parabolic potential theory

Suppose conversely that u 2 F C satisfies ˇGˇ C˛ u  u. Then, equation (6.1.19) implies that     E˛ .u, w/ D lim E˛ ˇGˇ C˛ u, w D lim ˇ u  ˇG˛Cˇ u, w  0 ˇ !1

ˇ !1

for all w 2 W C which proves equation (6.2.1). (ii) For any ˇ > 0, since ˇGˇ C˛ u  u  w, we have B˛ .ˇGˇ C˛ u  w, ˇGˇ C˛ u  w/ D E˛ .ˇGˇ C˛ u  w, ˇGˇ C˛ u  w/ D ˇ.u  ˇGˇ C˛ u, ˇGˇ C˛ u  w/  E˛ .w, ˇGˇ C˛ u  w/  E˛ .w, ˇGˇ C˛ u  w/  KkwkW kˇGˇ C˛ u  wkF for a constant K depending on ˛. Hence ¹kˇGˇ ˛ u  wkF º and consequently ¹kˇGˇ C˛ ukF º is uniformly bounded relative to ˇ > 0. Hence there exists a sequence ˇn " 1 such that limn!1 ˇn Gˇn C˛ u converges weakly in F . Since it converges to u in H , u belongs to F . If u 2 P˛ , then for any w 2 W C , . @w , u/  B˛ .u, w/. Hence @ E˛ .w, u/  B˛ .u, w/ C B˛ .w, u/.

(6.2.2)

Similarly to Theorem 2.3.1, the following lemma holds. Lemma 6.2.2. For any u 2 P˛ with ˛ > ˛0 , there exists a positive Radon measure u on Z such that Z E˛ .u, v/ D v.z/du .z/, (6.2.3) Z

for all v 2 W \ C0 .Z/. If u 2 P˛ and u are related by equation (6.2.3), then we write u D U˛ u and call it the ˛-potential of u . For any h 2 H C , put Lh D ¹u 2 F : u  h -a.e.º.

(6.2.4)

b the family of all ˛-coexcessive functions and 1-coexcesb ˛ and P Let us denote by P sive functions, respectively. To define a capacity related to .E, W /, let us prepare the following lemma. Lemma 6.2.3. Suppose that Lh \ W is non-empty. Then, for any " > 0 and ˛ > ˛0 , b ˛ satisfying u" 2 W \ P there exist unique functions u" 2 W \ P˛ and b 1 (6.2.5) E˛ .u" , v/ D ..u"  h/ , v/ , " 1 E˛ .v,b u" / D .v, .b (6.2.6) u"  h/ / , " for all v 2 F , respectively.

226

Chapter 6 Time dependent Dirichlet forms

Proof. First we note the inequality ..u  h/  .v  h/ , u  v/  0.

(6.2.7)

holding for any u, v 2 H . .i/ Uniqueness: Suppose that u" , i D 1, 2 are solutions of equation (6.2.5). Then  1  .1/ .2/ .1/ .2/  .1/ .2/ B˛ .u.1/ , .u"  h/  .u.2/ "  u" , u"  u" / D "  h/ , u"  u" " .1/

.2/

which is non-positive by equation (6.2.7). Hence u" D u" . Existence: For ˛ > ˛0 , define the sequence ¹u.n/ º inductively by u.0/ D 0, u.n/ D  1 .n1/ G˛ " .u  h/ . Then it satisfies 0  u.n/  1" G˛ h and !   1  @u.n/ , ' C E˛. / u.n/ . , /, ' D .u.n1/  h/ . , /, ' a.e.  .  @ " By putting ' D u.n/  u.n1/ and integrating over   t  T , we have   1  1  .n/ 2 2  k u.n/  u.n1/ .T , /kH k u  u.n1/ . , /kH 2 2 Z T   C E˛.t/ .u.n/  u.n1/ /.t , /, .u.n/  u.n1/ /.t , / dt

Z  1 T  .n1/ .u  h/  .u.n2/  h/ .t , /, .u.n/  u.n1/ /.t , / dt D " Z 1 T k.u.n1/  u.n2/ /.t , /kH k.u.n/  u.n1/ /.t , /kH dt  " 1=2 Z T 1 2  ku.n1/  u.n2/ /.t , /kH dt " Z T 1=2 .n/ .n1/ 2  k.u  u /.t , /kH dt .

In particular, Z T 2 k.u.n/  u.n1/ /.t , /kH dt 

1 .˛  ˛0 /2 "2

Z

T

2 k.u.n1/  u.n2/ /.t , /kH dt .

Combining these inequalities, we have     2 2  k u.n/  u.n1/ .T , /kH k u.n/  u.n1/ . , /kH Z T 2 2  ku.n1/  u.n2/ .t , /kH dt .˛  ˛0 /"2  n1 .T   /n1 2 khk .  .˛  ˛0 /"2 ".˛  ˛0 / .n  1/Š

Section 6.2 A parabolic potential theory

227

2 For any ı > 0, since there exists T such that k.u.n/  u.n1/ /.T , /kH  .n/ .2="/kG˛ h.T , /kH < ı, this implies that ¹u º converges to a function u" 2 C.R1 ; H / uniformly on any compact interval. Since ¹un º is dominated by .1="/G˛ h, it also converges to u" in H . Furthermore, since

ku.n/  u.n1 k2F  .1 _ .1=˛//B˛^1 .u.n/  u.n1/ , u.n/  u.n1/ / 1  2 ku.n/  u.n1/ kH ku.n1/  u.n2/ kH , " we obtain that u.n/ also converges to u" in F . For any  2 C01 .R1 /, by letting n ! 1 in the relation Z R1

Z   u.n/ . , /, '  0 . /d  C

  E˛. / u.n/ . , /, ' ./d  R1 Z   1 .u.n1/  h/ . , /, ' d  , D " R1

we obtain the same relation for u" instead of u.n/ . This implies equation (6.2.5). The proof of equation (6.2.6) is similar. Theorem 6.2.4. Suppose that Lh \ W is non-empty. Then lim"!0 u" D eh˛ and u" D b e ˛h exist increasingly a.e., strongly in H and weakly in F . Furthermore, lim!0 b they satisfy   @v (6.2.8) , v  eh˛ C B˛ .eh˛ , v  eh˛ /  0,  @   @v e ˛h ,b e ˛h /  0, (6.2.9) , v b e ˛h C B˛ .v b @ for all v 2 Lh \ W , respectively. Proof. We divide the proof into several steps. .1/ u" is increasing as " # 0. Let " < ı. Then  1 1 .u"  h/  .uı  h/ , .u"  uı / . (6.2.10) E˛ .u"  uı , .u"  uı / / D " ı R dv Since the function vˇ D ˇ 1 e ˇ. t/ .u"  uı / .t , /dt 2 W satisfies d ˇ C   ˇvˇ D ˇ.u"  uı / and limˇ !1 vˇ D .u"  uı / in F , we have   @  .u"  uı /, .u"  uı / @   @ .u"  uı /, vˇ D lim ˇ !1 @

228

Chapter 6 Time dependent Dirichlet forms



 @vˇ C  D  lim , .u"  uı /  .u"  uı / @ ˇ !1  ² ³   @vˇ 1 @vˇ  C , C vˇ D lim  ˇvˇ C ˇ.u"  uı / , .u"  uı / C @ ˇ @ ˇ !1  0. Therefore, by using the Markov property we obtain that the left-hand side of equation (6.2.10) is dominated by B˛ .u"  uı , .u"  uı / /  B˛ ..u"  uı / , .u"  uı / / . On the other hand, the right-hand side of equation (6.2.10) is non-negative. In fact, 1  1 .u"  h/  .uı  h/ , .u"  uı / " ı ı" 1 ..uı  h/ , .u"  uı / / D ..u"  h/  .uı  h/ , .u"  uı / / C " "ı  0. Now we have B˛ ..u"  uı / , .u"  uı / / D 0 and hence u"  uı . .2/ ¹B˛ .u" , u" /º is uniformly bounded relative to " > 0. Take a function u0 2 Lh \ W . Since .u0  h/ D 0, we have from equation (6.2.7) B˛ .u"  u0 , u"  u0 / D E˛ .u"  u0 , u"  u0 / 1 D ..u"  h/  .u0  h/ , u"  u0 /  E˛ .u0 , u"  u0 / "  E˛ .u0 , u"  u0 /  .1 C K˛ /ku0 kW B˛ .u"  u0 , u"  u0 /1=2 . Hence B˛ .u"  u0 , u"  u0 /  .1 C K˛ /2 2 ku0 k2W . .3/ lim"!0 ..u"  h/ , u" / D 0 and lim"!0 ..u"  h/ , v/ D 0 for all v 2 W . Since ..u"  h/ , v/ D "E˛ .u" , v/ for all v 2 W , the assertion of this step is an easy consequence of .2/. .4/ lim"!0 u" D eh˛ 2 Lh exists strongly in H , weakly in F . By virtue of .2/, a subsequence of ¹u" º converges weakly in F to some eh˛ 2 F . Further, since u" is increasing and uniformly bounded in H , eh˛ is independent of the choice of the subsequence and converges in H . Hence ¹u" º converges to eh weakly in F . By equation (6.2.7), we have ..u"  h/  .v  h/ , u"   v/  0for˛any v 2 F and hence, by letting " ! 0 and noting .3/, it follows that  .v  h/ , eh  v  0.   Put v D eh˛ C ıw for w 2 F C to get that .eh˛ C ıw  h/ , w  0. Finally, by  ˛  letting ı ! 0, we have .eh  h/ , w  0 and so eh˛  h. .5/ eh satisfies equation (6.2.8). Let v 2 Lh \ W . Since .v  h/ D 0, we have from equations (6.2.5) and (6.2.7),   1 @u" , v  u" C B˛ .u" , v  u" / D ..u"  h/  .v  h/ , v  u" /  0.  @ "

Section 6.2 A parabolic potential theory

Furthermore, since  lim

"!0



229

     @u" @v @v ˛ , v  u" D lim , u" D , eh "!0 @ @ @

and lim"!0 B˛ .e" , e" /  B˛ .eh˛ , eh˛ /, it follows that   @v ˛ , e C B˛ .eh˛ , v  eh˛ /  0. @ h This implies equation (6.2.8). In the above proof, we only used the Markov property to show that B˛ .w C , w  /  0. But, since B˛ .w  , w C / D B˛ ..w/C , .w/ /  0, equation (6.2.9) follows similarly. Lemma 6.2.5. Suppose that Lh \ W ¤ ;. If u 2 Lh satisfies   @w , w  u C B˛ .u, w  u/  0 f or al l w 2 Lh \ W ,  @t

(6.2.11)

for ˛ > ˛0 , then u  eh˛ . Proof. Suppose that u 2 Lh satisfies equation (6.2.11). It suffices to prove u  u" for all " > 0. Put g D u  u" . For any function w 2 Lh \ W ,   1 @u" , w  u C B˛ .u" , w  u/ D ..u"  h/ , w  u/ ,  @ " and hence   1 @.w  u" / , w  u C B˛ .g, w  u/   ..u"  h/ , w  u/ .  @ "

(6.2.12)

Taking w0 2 Lh \ W and f 2 W C put w D w0 C ıf for ı > 0. Then   @ .w0  u" C ıf /, w0  u" C ıf  g C B˛ .g, w0  u" C ıf  g/  @ 1   ..u"  h/ , w0  u" C ıf  g/ . " Therefore,   @ .w0  u" C ıf /, g C B˛ .g, w0  u"  g/ C ıB˛ .g, f /  @ ı 1  ..u"  h/ , w0  u"  g/  ..u"  h/ , f / . " " Dividing both sides by ı and letting ı ! 1, we have   1 @f , g C B˛ .g, f /   ..u"  h/ , f / . @ "

230

Chapter 6 Time dependent Dirichlet forms

 b  In particular,   let f D fˇ for fˇ D ˇ K ˇ g . Since kfˇ kH  kg kH , it holds that  @fˇ , g D ˇ.g   fˇ /, g  0. Hence @

 1 .u"  h/ , fˇ . "   Since limˇ !1 fˇ D g in F and .u  h/ D 0, B˛ .g C , fˇ /  B˛ .g  , fˇ /  

1 B˛ .g C , g  /  B˛ .g  , g  /   ..u"  h/  .u  h/ , .u  u" / /  0. " This implies 0  B˛ .g C , g  /  B˛ .g  , g  / and hence u  u" D g  0. Theorem 6.2.6. Suppose that W \ Lh \ P˛ ¤ ; for ˛ > ˛0 . Then eh˛ is an ˛excessive function satisfying eh˛ D min¹u 2 P˛ \ F : u  h a.e.º.

(6.2.13)

Proof. Let w0 2 W \ Lh \ P˛ . Applying equation (6.2.8) to w D w0 C ˇf , ˇ > 0, f 2 W C we have   @ ˛ .w0 C ˇf /, w0 C ˇf  eh C B˛ .eh˛ , w0 C ˇf  eh˛ / 0   @ ² ³    @f ˛ @w0 ˛ ˛ ˛ ˛ D , e C B˛ .eh , w0  eh / C ˇ , e C B˛ .eh , f / . @ h @ h   ˛ Hence @f C B˛ .eh˛ , f /  0. This implies that eh˛ is ˛-excessive and conse, e @ h quently eh˛  min¹u 2 P˛ : u  h a.e.º. For the proof of the  converse inequality, it suffices to prove that u"  u for any u 2 P˛ \ Lh . Since .u"  h/ , .u"  u/C D 0 for u 2 Lh , we have      @u" 1 C  , .u"  u/ / C B˛ u" , .u"  u/C D .u"  h/ , .u"  u/C D 0. @ "   b ˇ .u"  u/C . Then limˇ !1 vˇ D .u"  u/C in F and @vˇ , u"  u  Let vˇ D ˇ K @ 0. Since u 2 P˛ , we then have     @u" C  , .u"  u/ C B˛ u, .u"  u/C @        @vˇ @u" D lim  , vˇ C B˛ .u, vˇ / D lim , u" C B˛ .u, vˇ / @ @ ˇ !1 ˇ !1    @vˇ D lim E˛ .u, vˇ / C , u"  u  0. @ ˇ !1 Now we get that       @u" C C B˛ u"  u, .u"  u/ D , .u"  u/  B˛ u, .u"  u/C  0. @

231

Section 6.2 A parabolic potential theory

    Hence, B˛ .u"  u/C , .u"  u/C  B˛ .u"  u/ , .u"  u/C  0, that is, u"  u. For any relatively compact open set A of Z put eA˛ D e1˛A and eA D eA1 . The function is called the ˛-equilibrium potential (equilibrium potential if ˛ D 1) of A. Since 1 is 1-excessive, similarly to Lemma 1.4.2, eA ^ 1 is also 1-excessive and belongs to F . Hence eA D eA ^ 1 and 0  eA  1. By virtue of Theorem 6.2.2, there exists a positive Radon measure A such that Z E1 .eA , w/ D wdA , for all w 2 W \ C0 .Z/. (6.2.14) eA˛

Z

The measure A is called the equilibrium measure of A. Lemma 6.2.7. For any relatively compact open set A, 0  eA  1, eA D 1 a.e. on A N and A is supported by A. Proof. It suffices to prove the last assertion. To this end, take a function w 2 W C \ N Let u" 2 W be the solution of equation (6.2.5) for C0 .Z / such that suppŒw  Z n A. ˛ D 1 and h D IA . Then, noting that .u"  IA / D 0 on Z n A, we have   Z @w wdA D , eA C B1 .eA , w/ @ Z    @w 1 D lim , u" C B1 .u" , w/ D lim ..u"  IA / , w/ "!0 "!0 @ " D 0. N D 0. This implies that A .Z n A/ ˛ ˛ Define the ˛-coequilibrium potential b eA and the ˛-coequilibrium measure b A similarly. They enjoy analogous properties to the above. In particular,

b ˛ : u  h a.e.º b e ˛h D min¹u 2 P Z eA / D w.z/b A .dz/ for all w 2 W \ C0 .Z/, E1 .w,b

(6.2.15) (6.2.16)

Z

1 where b eA D b e 11A and b A D b A .

Since b eA  1 on A, it holds for the function u" specified by equation (6.2.5) that 1 eA  u" / D ..u"  1A / ,b eA  u" /  0. E1 .u" ,b " Noting that 0  u"  eA  1, this implies B1 .eA , eA /  lim B1 .u" , u" / D lim E1 .u" , u" / !0 "!0 Z eA / D lim u" db A  Cap.A/.  lim E1 .u" ,b "!0

"!0 X

(6.2.17)

232

Chapter 6 Time dependent Dirichlet forms

Let O be the family of all relatively compact open sets of Z . For any A 2 O, define N Then, for any w 2 W \ C0 .Z/ such the capacity Cap.A/ of A by Cap.A/ D b A .A/. that w D 1 on A, Z w.z/b A .dz/ D E1 .w,b eA /. (6.2.18) Cap.A/ D Z

As in Lemma 2.1.2, we have the following lemma. Lemma 6.2.8. Cap./ satisfies the following conditions: (i)

A1 , A2 2 O, A1  A2 ) Cap.A1 /  Cap.A2 /;

(ii)

A1 , A2 2 O ) Cap.A1 [ A2 /  Cap.A1 / C Cap.A2 /;

(iii) An , A 2 O, An " A ) Cap.A/ D supn Cap.An /. .i/

eA1  b eA2 by Theorem 6.2.6. Letb u" be the approximating Proof. (i) If A1  A2 , thenb sequence of b eAi determined by Lemma 6.2.3 for h D 1Ai . By noting that eA1 D 1 a.e. on Ai , we have for any function w 2 W \ C0 .Z/ with w D 1 on A2 , we have Z Cap.A1 / D wdb A1 D E1 .w,b eA1 / Z

 1  .1/ D lim E1 .w,b u.1/ / D lim .b u"  1A1 / , w " "!0 "!0 "  1  .1/ u"  1A1 / , eA2 D lim .b "!0 " u.1/ u.1/ D lim E1 .eA2 ,b " / D lim lim E1 .ˇGˇ C1 eA2 ,b " /. "!0

"!0 ˇ !1

.1/

u" / is increasing relative to " # 0 and ˇ " 1, the last term is Since E1 .ˇGˇ C1 eA2 ,b equal to u.1/ eA1 / lim lim E1 .ˇGˇ C1 eA2 ,b " / D lim E1 .ˇGˇ C1 eA2 ,b

ˇ !1 "!0

ˇ !1

 lim E1 .ˇGˇ C1 eA2 ,b eA2 /, ˇ !1

where, in the last inequality, we used the 1-excessiveness of ˇGˇ C1 eA2 and the relation eA2 . Then, by tracing back the above argument, we see that the last term is equal b eA1R  b A2 D Cap.A2 /. to Z wdb .i/ (ii) Let b u" ,b u" be the approximating sequences of b eA1 [A2 and b eAi , respectively. Since 1  eA1 _ eA2  eA1 C eA2 on A1 [ A2 , similarly to the proof of (i), we have  1 Cap.A1 [ A2 / D lim .b u"  1A1 [A2 / , eA1 _ eA2 "!0 " D lim lim E1 .ˇGˇ C1 .eA1 _ eA2 /,b u" / "!0 ˇ !1

eA1 [A2 /  lim E1 .ˇGˇ C1 .eA1 _ eA2 /,b ˇ !1

233

Section 6.2 A parabolic potential theory

 lim E1 .ˇGˇ C1 .eA1 _ eA2 /,b eA1 Cb eA2 / ˇ !1

 1 1 e .2/  1A2 / , eA1 _ eA2 .b e "  1A1 / C .b " "!0 "   .2/  1  1 e "  1A2 / , eA2 .b e "  1A1 / , eA1 C .b D lim "!0 " D Cap.A1 / C Cap.A2 /.

D lim

(iii) We may assume that supn B1 .eAn , eAn / < 1. Then, by choosing a subsequence if necessary, limn!1 eAn D v 2 F exists weakly. Since v is 1-excessive and v  1 a.e. on A D [n An , v  eA . Hence b ˛C1b b ˛C1b eA /  lim E1 .v, ˛ G eA / Cap.A/ D lim E1 .eA , ˛ G ˛!1

˛!1

b ˛C1b eA / D lim lim E1 .eAn , ˛ G ˛!1 n!1

b ˛C1b eA /. D lim lim E1 .eAn , ˛ G n!1 ˛!1

As in the proof of (i) and (ii), by using the approximating sequence of eAn , we can show that the last term is equal to limn!1 Cap.An /. Since the converse inequality is clear, we obtain (iii). For any open set A, put Cap.A/ D sup¹Cap.B/ : B is a relatively compact open subset of Aº, and for any set B  Z, let Cap.B/ D

inf

B A;open

Cap.A/.

(6.2.19)

By virtue of equations (6.1.4) and (6.2.8), there exist constants K1 and K2 such that, for any relatively compact open set A and w 2 W \ C0 .Z/ such that w  1 on A, eA k2F  B1 .b eA ,b eA /  E1 .w,b eA /  K2 kb eA kF kwkW . K1 kb Hence, for K3 D K2 =K1 , This combined with

kb eA kF  K3 kwkW .

(6.2.20)

Z

Cap.A/ D

Z

wdb A D E1 .w,b eA /  K2 kb eA kF kwkW

implies that Cap.A/  K2 K3 kwk2W .

(6.2.21)

For any Borel set F of Z, put dF D inf¹u 2 P \ F : u  1 a.e. on some neighborhood of F º.

(6.2.22)

b instead of P . For a 1-excessive function u 2 F and Define similarly b d F by using P a 1-coexcessive function b u 2 F , put b ˛C1b u/ D lim E1 .˛G˛C1 u,b u/ D lim E1 .u, ˛ G u/. E1 .u,b ˛!1

˛!1

234

Chapter 6 Time dependent Dirichlet forms

Theorem 6.2.9. Let F be a compact set and ¹An º be a decreasing sequence of relatively compact open sets such that F  ANnC1  An and \An D F . Then eAn D b d F decreasingly and weakly in F . The limn!1 eAn D dF and limn!1 b sequence ¹b An º of coequilibrium measures of ¹An º converges vaguely to a positive Radon measure b  on F satisfying Cap.F / D b .F /. The sequence ¹An º of the equilibrium measures of ¹An º also converges to a positive Radon measure F vaguely and satisfies lim E1 .eAm ,b eAn / D lim

Cap.F / D lim

m!1 n!1

lim E1 .eAm ,b eAn /  F .F /. (6.2.23)

n!1 m!1

eAn given by Lemma Proof. For each n, letb un," 2 W be an approximating sequence ofb un," , .b un,"  6.2.3. For any function w 2 W \C0 .X / such that w D 1 on A1 , since .wb 1An / /  0, Cap.An / D E1 .w,b eAn / D lim E1 .w,b un," / "!0   1 w b un," , .b un,"  1An / C lim E1 .b un," ,b un," / D lim "!0 " "!0 un," ,b un," / D B1 .b eAn ,b eAn /.  lim B1 .b ı!0

eAn º is decreasing as Therefore, ¹b eAn º is a uniformly bounded sequence of F . Since ¹b well, it follows that limn!1 b eAn D b d F exists as a decreasing and weak limit of F . Hence, for any v 2 W \ C0 .X /, Z v.z/b An .dz/ D lim E1 .v,b eAn / D E1 .v, b d F /. lim n!1 Z

n!1

Since the measures ¹b An º are uniformly R bounded and W \ C0 .Z/ is dense in C0 .Z/, An .dz/ for all v 2 C0 .Z/. Hence ¹b An º this implies the existence of limn!1 Z v.z/b converges vaguely to a measure b F on F . In particular, if v satisfies v D 1 on A1 , then Z Z vdb An D vdb F D b F .F /. lim Cap.An / D lim n!1

n!1 Z

Z

On the other hand, since limn!1 Cap.An / D Cap.F /, it follows that Cap.F / D b F .F /. The existence of F is similar. If m  n, noting that eAm D 1 a.e. on Am , it follows that b 1b E1 .eAm ,b eAn / D lim E1 .˛R˛C1 eAm , U An / ˛!1 Z ˛R˛C1 eAm db An D b An .ANn /. D lim ˛!1 X

Similarly, since b eAm  1 a.e. on Am , E1 .eAn ,b eAm /  An .ANn /. Since E1 .eAm ,b eAn / is decreasing relative to m and n, we have Cap.F / D lim b An .ANn / D lim lim E1 .eAm ,b eAn / n!1

D lim

m!1 n!1

lim E1 .eAm ,b eAn /  F .F /.

n!1 m!1

235

Section 6.2 A parabolic potential theory

Example 6.2.10. If we consider that .B, F / is a Dirichlet form on L2 .Z; / in the sense of Chapter 1, then its associated resolvent ¹G ˛ º is given by G ˛ f . , x/ D . / . / G˛ f . , /.x/ by using the resolvent ¹G˛ º associated with .E . / , F /, because for 2 any f 2 L .Z; /, Z   E˛. / G˛. / . , /, v. , / d  D .f , v/. B˛ .G ˛ f , v/ D R1

Let A D ¹0 º  K for a compact sets K  X and let An be a decreasing sequence such that An D .sn , tn /  Un for sn " 0 , tn # 0 and a relatively compact open set eAn . , x/ of An relative to .B, F / is given Un # K. Then the coequilibrium potential b . / . / b e Un .x/ by means of the coequilibrium potential b e Un of Un by eAn . , x/ D 1.sn ,tn / . /b relative to .E . / , F /. Hence, the capacity Cap.A/ of A relative to .B, F / is given by Z tn . / b Cap.A/ D lim B1 .w, eAn / D lim ./E . / .',b e Un /d  D 0 n!1

n!1 s n

for any function ./ 2 C0 .R1 / such that ./ D 1 on .s1 , t1 / and ' 2 F \ C0 .X / such that ' D 1 on U1 . In general, this does not coincide with the capacity Cap.A/ defined by equation (6.2.19). For example, assume that E . / is independent of  . Then for the set A given above, Cap.A/ D m.K/. (6.2.24) To show this, let ¹An º be the sequence given above. As we have seen in Theorem 6.2.9,  such that limn!1 b eAn D b d K weakly in F and limn!1 b An D there exist b d K and b b  vaguely. If f 2 C0 .Z/ satisfies f . , / D 0 for   sn , then G1 f . , x/ D 0 for all An is supported by ANn ,   sn . Since b Z G1 f . , x/db An D E1 .G1 f ,b e An / D .f ,b e An /, 0D Z

 is supported by ¹0 º  K, it which implies that b eAn . , / D 0 for a.e.  < sn . Since b can be written as b  D ı¹ 0 º  K .dx/ for a positive Radon measure K on K. Hence, for any  2 C01 .R1 / and ' 2 F \ C0 .X /, Z Z     d K . , / d  dK/ D   0 . / ', b d K . , / d  C ./E1 ', b E1 . ˝ ', b 1 R1 Z R  ˝ 'db  D .0 /h K , 'i. D X

This implies that     d K . , / d  D h K , 'iı¹ 0 º .d  /. d ', b d K . , / C E1 ', b

236

Chapter 6 Time dependent Dirichlet forms

In particular d.', b d K . , // is a signed measure and h K , 'i D .', b d K .0 C, //  .', b d K .0 , //. Noting that .', b d K .0 , // D 0, we have h K , 'i D .', b d K .0 C, //. Since b d K .0 , x/ D 1 for x 2 K and ' is arbitrary, we finally have K D 1K  m and Cap.A/ D b .A/ D m.K/. By using the capacity defined in the paragraph preceding Lemma 6.2.8, we define and use the same quasi-notions as in Section 2.2. Theorem 6.2.11. Any function w 2 W has a quasi-continuous modification w e. Further, for any u 2 P \ F and w 2 W ,   Z @w , u C B1 .u, w/. w edu D (6.2.25) @t Z Proof. By the denseness of C0 .Z/ \ W in W , there exists a sequence ¹wn º  W \ C0 .X / such that kwnC1 wn kW  2n . Hence using equation (6.2.21), we can obtain a q.c. modification w e of w 2 W as in Theorem 2.2.3. To prove equation (6.2.25), take a function u 2 P . In the proof of Lemma 6.2.2 we saw that, for any " > 0 and v 2 C0 .Z/ with K D SuppŒv, there exists a function v" 2 W \ C0 .Z/ such that SuppŒv"   K and kv  v" k1 < ". Then Z Z vdu  "u .K/ C v" du D "u .K/ C E1 .u, v" / Z

Z

D "u .K/ C lim E1 .˛G˛C1 u, v" / ˛!1

 "u .K/ C lim E1 .˛G˛C1 u, ev" /. ˛!1

Clearly ev"  ev C "e1K . Hence, by noting that ev C "e1K is excessive, we have from equation (6.2.2) that Z vdu  "u .K/ C lim E1 .˛G˛C1 u, ev C "e1K / ˛!1

Z

 "u .K/ C lim ¹B1 .˛G˛C1 u, ev C "e1K / C B1 .ev C "e1K , ˛G˛C1 u/º ˛!1

D "u .K/ C B1 .u, ev C "e1K / C B1 .ev C "e1K , u/. By letting " ! 0 we obtain Z vdu  B1 .u, ev / C B1 .ev , u/  2KB1 .u, u/1=2 B1 .ev , ev /1=2

(6.2.26)

Z

for a suitable constant K. For any f 2 H , if we denote by ¹jf j" º  W the approximating sequence of ejf j in Theorem 6.2.4, then by ejf j  ef C ef and equation

237

Section 6.2 A parabolic potential theory

(6.2.2), B1 .jf j" , jf j" / D E1 .jf j" , jf j" /  E1 .jf j" , ef C ef /  B1 .jf j" , ef C ef / C B1 .ef C ef , jf j" /  2KB1 .jf j" , jf j" /1=2  B1 .ef C ef , ef C ef /1=2 . This implies that B1 .ejf j , ejf j /  lim B1 .jf j" , jf j" /  4K 2 B1 .ef C ef , ef C ef /. "!0

For f 2 W , equation (6.2.20) with f and f in place of 1A yields kef kF  K3 kf kW and kef kF  K3 kf kW , respectively. Hence by putting v D jwnC1 wn j in equation (6.2.26) for the sequence ¹wn º  F \ C0 .Z/ stated in the beginning of the proof, we have Z jwnC1  wn jdu  2KB1 .u, u/1=2 B1 .ejwnC1 wn j , ejwnC1 wn j /1=2 Z

 K4 B1 .u, u/1=2  kwnC1  wn kW for some constant K4 depending on K3 . Consequently, limn!1 wn D w e in L1 .Z; u /. Equation (6.2.25) follows from the corresponding equality by applying it to wn and then letting n ! 1. A positive Radon measure  on Z is called a measure of finite energy integral if it does not charge any set of zero capacity and satisfies Z jw.z/jd.z/  C kwkW , for all w 2 W \ C0 .Z/ (6.2.27) Z

for some constant C . In this case, equation (6.2.27) also holds for any q.c. function w 2 W . We denote by S0 the family of all measures of the finite energy integral. In view of Theorem 6.1.2 (ii), there exists a constant C˛ such that ˇ  ˇ ˇ ˇ ˇ d G˛ f , v ˇ D j.f , v/  B˛ .G˛ f , v/j ˇ ˇ d  kf kF 0 kvkF C K˛ B˛ .G˛ f , G˛ f /1=2 B˛ .v, v/1=2  C˛ kf kF 0 kvkF . Hence there exists a constant C1 such that kG1 f kW  C1 kf kF 0 for all f 2 F 0 .

(6.2.28)

238

Chapter 6 Time dependent Dirichlet forms

Theorem 6.2.12. The following conditions are equivalent to each other: (i)

 2 S0 ;

(ii)

for any w 2 C0 .Z/, there exists a constant C such that Z jw.z/jd.z/  C B1 .b e jwj ,b e jwj /1=2 ;

(6.2.29)

Z

b \ F such that  D b (iii) There exists v 2 P v . Proof. (ii) ) (i) Since kejwj kF  2K3 kwkW as in the proof of Theorem 6.2.11, equation (6.2.29) implies equation (6.2.27). Hence, it is enough to show that  in (ii) does not charge any set of zero capacity. Suppose that F is a compact set of zero capacity. Then there exists a decreasing sequence of open sets ¹An º such that F  ANnC1  An and Cap.An / # 0. Take a function wn 2 W \ C0 .Z/ such that wn D 1 on AnC1 and wn D 0 on Z n An . Then, by equation (6.2.2), Z wn d D E1 .U1 , wn / D lim E1 .˛G˛C1 U1 , wn / .F /  ˛!1

Z

 lim .B1 .˛G˛C1 U1 , ewn / C B1 .ewn , ˛G˛C1 U1 // ˛!1

D B1 .U1 , ewn / C B1 .ewn , U1 //  K1 B1 .U1 , U1 /1=2 B1 .ewn , ewn /1=2 . Furthermore, by using the approximating sequence wn, of ewn determined by Lemma 6.2.3, we have B1 .ewn , ewn /  lim B1 .wn, , wn, /  lim E1 .wn, ,b e wn / !0 !0 Z eAn / D lim wn, db An  lim E1 .wn, ,b !0

!0 X

 Cap.An /. Therefore, .F / D 0. (i) ) (iii) Suppose that  2 S0 . Let R˛ f be a q.c. modification of G˛ f and L.f / be the linear functional defined by Z R1 f .z/d.z/, f 2 H . L.f / D Z

Since  does not charge any set of zero capacity, L.f / is a positive linear functional on H determined independently of the choice of the Borel measurable version of f . By virtue of equations (6.2.27) and (6.2.28), for C2 D C C1 , Z R1 jf jd  C kR1 jf jkW  C2 kf kF 0  C2 kf kH . (6.2.30) L.f /  Z

239

Section 6.3 Associated space-time processes

Hence there exists a function v 2 H C such that Z R1 f d D .f , v/, L.f / D

for all f 2 H .

Z

For any f 2 H C , since .f  ˛R˛C1 f , v/ D

Z Z

(6.2.31)

Z R1 .f  ˛R˛C1 f / d D

Z

R˛C1 f d  0,

b ˛C1 v  v. From equation (6.2.30), L.f / D .f , v/ can be extended to v satisfies ˛ G a continuous linear functional on F 0 . Since F is a Hilbert space, F is reflexive and b \ F . By virtue of Lemma 6.2.2 and equation (6.2.31), for the measure hence v 2 P b˛b v , b v such that v D U Z Z R1 f d D .f , v/ D E1 .R1 f , v/ D R1 f db v , for all f 2 H . Z

Z

This proves the implication (i) ) (iii). b \ F . For any w 2 C0 .Z/, let b (iii) ) (ii) Assume that  D b v for v 2 P e jwj be the minimal coexcessive function determined by Theorem 6.2.6 for h D jwj. Since bnC1 jwj D jwj q.e. and jwj  b e jwj , by using the dual version of equalimn!1 nR tion (6.2.2), we have Z Z bnC1 jwjd D lim E1 .nR bnC1 jwj, v/ jwjd D lim nR Z

n!1 Z

n!1

bnC1b e jwj , v/  lim E1 .nR n!1 °    ± bnC1b bnC1b  lim B1 v, nR e jwj C B1 nR e jwj , v n!1

e jwj / C B1 .b e jwj , u/ D B1 .u,b  2K1 B1 .b e jwj ,b e jwj /1=2 B1 .v, v/1=2 . Hence equation (6.2.29) holds for C D 2K1 B1 .v, v/1=2 .

6.3

Associated space-time processes

In this section, we study some properties of the Hunt process M D .Z t , Pz / associated with the time dependent Dirichlet form .E, F / on L2 .Z; /. As in Chapter 3, by choosing a suitable resolvent ¹R˛ º such that R˛ f is a q.c. modification of G˛ f for any ˛ > 0, f 2 L1 .X ; m/ and its transition function p t f , we can show the following result. with the Theorem 6.3.1. There exists a Hunt process M D .Z t , Pz/ on Z associated  time dependent Dirichlet form .E, F /. Moreover, if Z t D  .t /, X .t/ is a decompo-

240

Chapter 6 Time dependent Dirichlet forms

sition of Z t into the processes on R1 and X , then  .t / is the uniform motion to the right, that is,  .t / D  .0/ C t . For any Borel subset B of Z, let B be the first hitting time of B relative to M and let   (6.3.1) HB˛ h.z/ D Ez e ˛B h.ZB / . For any h 2 P˛ \ F , there exists a set N of zero capacity such that ˇRˇ C˛ h.z/ is quasi-continuous and increasing for all z 2 X n N . The function defined by e hD limn!1 nRnC˛ h is called an ˛-excessive modification of h 2 P˛ \ F . Furthermore, for any h 2 P˛ with ˛ > 0, by approximating h by an increasing sequence of functions h given by e h D ¹hk º  P˛ \ F , h 2 P˛ also has an ˛-excessive modification e e limn!1 limk!1 nRnC˛ hk D limk!1 hk . As an approximating sequence hk , we can take hk D h ^ .kRˇ g/ for ˇ > ˛0 and a strictly positive function g 2 L2 .Z; / by Lemma 6.2.1 (ii). h De e ˛h1A .z/ Lemma 6.3.2. For any h 2 P˛ \ F with ˛ > ˛0 and an open set A, HA˛e q.e., where e h and e e ˛ are ˛-excessive regularizations of h and e ˛ , respectively. h1A

h1A

Proof. Put D Clearly ^  ^ pA˛  e1˛A a.e. Since ˛ ˛ ˛ ˇGˇ C˛ .e1A ^ pA / and e1A belong to P˛ \ F , using equation (6.2.2), we have   B˛ ˇGˇ C˛ .e1˛A ^ pA˛ /, ˇGˇ C˛ .e1˛A ^ pA˛ /   D E˛ ˇGˇ C˛ .e1˛A ^ pA˛ /, ˇGˇ C˛ .e1˛A ^ pA˛ /    E˛ ˇGˇ C˛ .e1˛A ^ pA˛ /, e1˛A      B˛ ˇGˇ C˛ .e1˛A ^ pA˛ /, e1˛A C B˛ e1˛A , ˇGˇ C˛ .e1˛A ^ pA˛ /   1=2 1=2 B˛ e1˛A , e1˛A  2K˛ B˛ ˇGˇ C˛ .e1˛A ^ pA˛ /, ˇGˇ C˛ .e1˛A ^ pA˛ / HA˛ 1

pA˛ .

ˇGˇ C˛ .e1˛A

pA˛ /

e1˛A

for some constant K˛ , and consequently   B˛ ˇGˇ C˛ .e1˛A ^ pA˛ /, ˇGˇ C˛ .e1˛A ^ pA˛ /  4K˛2 B˛ .e1˛A , e1˛A /. This implies that limˇ !1 ˇGˇ C˛ .e1˛A ^ pA˛ / D e1˛A ^ pA˛ weakly in F and hence e1˛A ^ pA˛ 2 L1A \ P˛ . By virtue of Theorem 6.2.6, this yields e1˛A ^ pA˛  e1˛A , that e ˛1A q.e. is, pA˛  e Conversely, since B D ¹z 2 A : e e ˛1A .z/ ¤ 1º is of -measure zero, we have for any fixed s > 0,   Z 1 Z 1 e .ˇ C˛/t PZ t .Zs 2 B/dt D Ez e .ˇ C˛/t 1B .ZsCt /dt Ez 0

0

 e .ˇ C˛/s Rˇ C˛ 1B .z/ D 0,

q.e. z.

For any finite subset D D ¹t1 , t2 , : : : , tk º of .0, 1/, put  .D/ D min ¹t 2 D; Z t 2 Aº, where  .D/ is considered as infinity if the set in curly brackets is empty. Then the

241

Section 6.3 Associated space-time processes

above inequality implies that, for Pz -a.s. with q.e. z, PZ t

k  ˛  X e e 1A .Z.D/ / ¤ 1,  .D/ < 1  PZ t .Z tj 2 B,  .D/ D tj / D 0, a.e. t . j D1





e ˛1A .Z t /, Pz is a right continuous supermartingale for q.e. z, Also, noting that e ˛te Z 1    .nC˛/t ˛.D/ dt nEz e EZ t e 0 Z 1    .nC˛/t ˛.D/ ˛ D nEz e EZ t e e e 1A .Z.D/ / dt  Z 01  nEz e .nC˛/t e e ˛1A .Z t /dt 0

D nRnC˛ e e ˛1A .z/

q.e. z.

e ˛1A Letting D increase to a dense subset of .0, 1/, it follows that nRnC˛ pA˛  nRnC˛e ˛ ˛ q.e. which implies that pA  e e 1A q.e. The proof of the above step implies that the process Z t does not hit any set B of h does not zero capacity for a q.e. starting point. In particular, for any h 2 P \ F , HA˛e e depend upon the choice of the ˛-excessive regularization h of h. By equation (6.1.8), we defined a bilinear form E on W  F and F  W . But for later use, we need to extend the domain of definition of E. By virtue of Lemma 6.1.4, if u 2 W and v 2 F , then     bˇ v . (6.3.2) E.u, v/ D lim E ˇGˇ u, v D lim E u, ˇ G ˇ !1

ˇ !1

We shall extend the domain by using this relation. Let J D ¹u D u1  u2 C w : ui 2 P˛ \ F for some ˛ > ˛0 , w 2 W º, b b ˛ \ F for some ˛ > ˛0 , w 2 W º. J D ¹b u Db u1  b u2 C w : b ui 2 P

(6.3.3) (6.3.4)

Lemma 6.3.3. J D b J and E.u, v/ is well defined by equation (6.3.2) for any u, v 2 J. Proof. Suppose that u 2 P˛ \ F for some ˛ > ˛0 . By virtue of Lemma 6.2.2, there exists a positive Radon measure u such that E˛ .u, w/ D hu , wi for all w 2 W \ C0 .Z/. Then E˛ .w, u/ D hu , wi C B˛ .w, u/ C B˛ .u, w/. Since the linear functional L./ on F defined by L.v/ D B˛ .v, u/ C B˛ .u, v/

(6.3.5)

242

Chapter 6 Time dependent Dirichlet forms

b ˛ L 2 W such that E˛ .v, G b ˛ L/ D L.v/, for belongs to F 0 , there exists a function G all v 2 F . Equation (6.3.5) then implies that b ˛ L/ D hu , wi for all w 2 W \ C0 .Z/, E˛ .w, u  G b ˛ u a.e. Therefore, u D G b˛ L  U b ˛ u 2 b b ˛ L D U J and hence that is, u  G J. The converse inclusion follows similarly. P˛ \ F  b If u, v 2 P˛ \F , then E˛ .ˇGˇ C˛ u, v/ is bounded relative to ˇ by equation (6.2.2). Further, the last two terms of the right-hand side of E˛ .ˇGˇ C˛ u, v/ D E˛ .v, ˇGˇ C˛ u/ C B˛ .ˇGˇ C˛ u, v/ C B˛ .v, ˇGˇ C˛ u/ converge as ˇ increases to infinity by Theorem 6.1.2. Since the left-hand side is nonnegative and E˛ .v, ˇGˇ C˛ u/ is decreasing relative to ˇ, limˇ !1 E˛ .ˇGˇ C˛ u, v/ converges. Hence limˇ !1 E˛ .ˇGˇ C˛ u, v/ E.u, v/ exists. Similarly, E˛ .u, v/ b˛ \ F . b ˇ C˛ v/ exists for u, v 2 P limˇ !1 E˛ .u, ˇ G In the above proof, we have seen that any function u 2 P˛ \ F can be written as b ˛ \ F . If u and v are ˛-excessive and ˛u D w  v for some w 2 W and v 2 P coexcessive regularizations of u and v, respectively, then w D u C v a.e., but it does not necessarily means that u C v is quasi-continuous already. For example, suppose that E . / D 0 for all  and let u. / D E .e ˛F / for the one point set F D ¹aº. F / D E .e ˛b Then u. / D e ˛. a/ . < a/, D 0 .  a/. In this case, let v. / D b ˛.a / ˛j aj . > a/, D 0 .  a/. Then u C v D e . ¤ a/, D 0 . D a/ e which coincides a.e. with a continuous function e ˛j aj 2 W . If we replace v by its ˛-excessive regularization of v, then u C v is continuous. b ˛ ), then e u (resp.b u) represents an ˛-excessive (resp. ˛If u 2 P˛ (resp. u 2 P coexcessive) regularization of u. In particular, if u has a q.c. modification, then it coincides with ˛-excessive and ˛-coexcessive regularizations q.e. As we have seen in Lemma 6.3.3, any function u 2 J has both ˛-excessive and ˛-coexcessive regularizations e u and b u, respectively. In the rest of this chapter, we consider that E is defined also on J  J by equation (6.3.2). As we have seen in Lemma 6.3.2, for any h 2 P˛ \ F ˛ H ˛e ˛ is an increasing and weak limit e ˛hIA q.e. Hence hA and an open set A, hA Ah De ˛ of the solution hA," 2 W of equation (6.2.5). b ˛ \ F ) and A be an open subset of Z. Lemma 6.3.4. Let h 2 P˛ \ F (resp. h 2 P ˛ ˛ b b e h 2 P˛ \ F .resp. H Then HA e A h 2 P ˛ \ F / and h, w/ D 0 E˛ .HA˛e

b ˛b .resp. E˛ .w, H A h/ D 0/

for any w 2 J, such that w D 0 a.e. on A.

(6.3.6)

243

Section 6.3 Associated space-time processes

˛ Proof. We consider that ˛ D 1 and omit the superfix ˛ of hA," . If w 2 W vanishes a.e. on A, then

1 h, w/ D lim E1 .hA," , w/ D lim ..hA,"  hIA / , w/ D 0. E1 .HA1e "!0 "!0 " Suppose that w1 , w2 2 P \ F satisfies w1 D w2 a.e. on A. Since ˇGˇ C1 hA," is increasing relative to " # 0 and ˇ " 1, h, wi / D lim E1 .ˇGˇ C1 HA1e h, wi / E1 .HA1e ˇ !1  h, wi / C B1 .wi , ˇGˇ C1 HA1e h/ D lim B1 .ˇGˇ C1 HA1e ˇ !1  h/  E1 .wi , ˇGˇ C1 HA1e D B1 .HA1e h, wi / C B1 .wi , HA1e h/  lim lim E1 .wi , ˇGˇ C1 hA," / ˇ !1 "!0

D

h, wi / B1 .HA1e

C

h/ B1 .wi , HA1e

 lim lim E1 .wi , ˇGˇ C1 hA," /, "!0 ˇ !1

which, by a similar argument, is equal to b ˇ C1 wi / lim lim E1 .ˇGˇ hA," , wi / D lim lim E1 .hA," , ˇ G

"!0 ˇ !1

"!0 ˇ !1

D lim E1 .hA," , wi / "!0

h, w1  w2 / D 0. Combining these for i D 1, 2 by equation (6.2.5). Hence E1 .HA1e cases, equation (6.3.6) also holds for h 2 P \ F and w 2 J such that w D 0 a.e. on A. The proof of the dual assertion is similar. b˛ º be a resolvent such that R b˛ f is a q.c. modification As in Chapter 3, let ¹R 2 b ˛ f for any ˛ > ˛0 and f 2 L .X ; m/. Furthermore, for ı > 0, take of G a strictly positive ı-coexcessive function b hı which is regularized already, that is, bnCıb hı D b hı q.e. Define the measure b  by b  .dz/ D b hı .z/.dz/ and a limn!1 nR .ı/ .ı/ b˛ º by R b˛ f .z/ D .1=b b˛Cı .b resolvent ¹R hı .z//R hı f /.z/. Then, there exists a Hunt .ı/ b .ı/ D .Z bt , b b.ı/ process M P z / on Z with resolvent ¹R ˛ º. Furthermore, there exists a b D .Z b˛ º which can be treated as if bt , b P z / with resolvent ¹R pseudo Hunt process M b .ı/ . We denote by H b ˛ .z, dw/ the ˛-order hitting it were a Hunt process through M A b probability of A relative to M. ZnA Let R˛ be the resolvent of the part process on Z n A given by  Z A ZnA ˛t R˛ f .z/ D Ez e f .Z t /dt . (6.3.7) 0

244

Chapter 6 Time dependent Dirichlet forms

According to the strong Markov property, the following Dynkin formula holds. R˛ f .z/ D R˛ZnA f .z/ C HA˛ R˛ f .z/.

(6.3.8)

Define the dual notions similarly. Theorem 6.3.5. For any nearly Borel set A, f , g 2 H and ˛ > ˛0 , b˛ g/ D E˛ .R˛ f , H b ˛R b E˛ .HA˛ R˛ f , R A ˛ g/.

(6.3.9)

Proof. We may assume that f and g are non-negative. If A is an open set, then the assertion is clear from equation (6.3.8) and Lemma 6.3.4. Assume that A is a compact set.Then there exists a decreasing sequence of open sets ¹A    n º containing A such that P z limn!1 OAn D OA D 1 for a.e. z. Since Pz limn!1 An D A D 1 and b B˛ .h, h/  lim E˛ .ˇGˇ C˛ h, ˇGˇ C˛ h/  lim E˛ .ˇGˇ C˛ h, h/ D E˛ .h, h/ ˇ !1

ˇ !1

for any h 2 P˛ \ F , by equation (6.2.2),   B˛ HA˛n R˛ f , HA˛n R˛ f      E˛ HA˛n R˛ f , HA˛n R˛ f D E˛ HA˛n R˛ f , R˛ f      B˛ HA˛n R˛ f , R˛ f C B˛ R˛ f , HA˛n R˛ f  1=2 B˛ .R˛ f , R˛ f /1=2 .  2K˛ B˛ HA˛n R˛ f , HA˛n R˛ f Hence

  B˛ HA˛n R˛ f , HA˛n R˛ f  4K˛2 B˛ .R˛ f , R˛ f / .

(6.3.10)

In particular, a subsequence ¹un º of Cesàro means of ¹HA˛n R˛ f º converges to P n H ˛ R f for some HA˛ R˛ f in F . Each un can be written as un D .1=mn / m kD1 Ak ˛ Pmn b˛ R b v n D .1=mn / kD1 H increasing sequence ¹mn º. Put b Ak ˛ g. Similarly to equab˛R b tion (6.3.10), ¹b v n º is a bounded sequence of F converging to H A ˛ g a.e. Hence ˛ b b R H A ˛ g 2 F . For each n, by equation (6.3.9), we have       b˛ g D E˛ H ˛ R˛ f , H b˛ R b˛ b b E˛ HA˛n R˛ f , R An An ˛ g D E˛ R˛ f , H An R˛ g . b˛ R b v n are the convex combinations of HA˛n R˛ f and H Since un andb An ˛ g, respectively, it follows that     b˛ g D lim E˛ un , R b˛ g E˛ HA˛ R˛ f , R n!1

D lim E˛ .R˛ f , vO n / D lim .f , vO n / n!1 n!1     ˛b b ˛ R˛ g . b D f , H A R˛ g D E˛ R˛ f , H A This implies equation (6.3.9) for any compact set.

Section 6.3 Associated space-time processes

245

If A is a Borel set, take an increasing sequence of compact subsets ¹Fn º of A such bz for a.e. z. By letting n ! 1 in equation that Fn # A a.s. Pz and O Fn # OA a.s. P (6.3.10), it also holds for any compact set. Hence it holds for Fn instead of An . Then we can use the same argument as above to get the assertion for any Borel set A. b ˛ . Then, for any f 2 H and nearly Corollary 6.3.6. Let h 2 F \ P˛ and g 2 F \ P Borel set A,         b˛R b˛ f D E˛ h, H b ˛ g D E˛ H ˛ R˛ f , g . b h, R E˛ HA˛e A ˛ f , E˛ R˛ f , H Ab A (6.3.11) Proof. Let e h be an ˛-excessive regularization of a function of h 2 P˛ \ F and put hn D nRnC˛ h. By virtue of equation (6.3.9), since HA˛ hn D HA˛ R˛ .n.hnRnC˛ h//,       b b˛ f D E˛ hn , H b ˛R b b˛b E˛ HA˛ hn , R A ˛ f D E˛ h, nRnC˛ H A R˛ f Z bnC˛ H b˛R b nR D A ˛ f .z/h .dz/ Z

  b˛R b˛R b˛ f i D E˛ h, H b˛ f . On the other hand, by equawhich converges to hh , H A A tion (6.3.10),   B˛ HA˛ .hn  hm /, HA˛ .hn  hm /  4K˛2 B˛ .hn  hm , hn  hm / . h weakly in F , which gives equation (6.3.11). The proof Hence limn!1 HA˛ hn D HA˛e of the second equality is similar. Corollary 6.3.7. Suppose that  is a measure of a finite energy integral with SuppŒ  F for some compact set F . Then, for any open set A such that F  A, e ˛  q.e. In particular, if  does not charge any semipolar set, then e˛  D U HA˛ U ˛e e HF U ˛  D U ˛ . e ˛  is an ˛-excessive modification of e ˛ Proof. Since A is open, HA˛ U 1A U˛ . Since ZnA b˛ g D 0 a.e. on A, we have from Lemma 6.3.4 R       b˛ g D E˛ U˛ , H b ˛R b e ˛ , R b b˛b E˛ HA˛ U A ˛ g D lim E˛ U˛ , ˇ G ˇ C˛ H A R˛ g ˇ !1 Z Z ˛b b b b˛R b ˇ Rˇ C˛ H A R˛ g d D D lim H A ˛ g d ˇ !1 Z Z Z   b˛ g . b˛ g d D E˛ U˛ , R D R Z

e ˛  D U˛  a.e. and hence q.e. by taking ˛-excessive regularization. This implies HA˛ U

246

Chapter 6 Time dependent Dirichlet forms

b ˛ D . Hence, by virtue of If  on F does not charge a semipolar set, then H F Corollary 6.3.6,   ˛ b˛ R b e ˛ , g D h, H b HF U F ˛ gi D h, R˛ gi   e ˛ , g . D U e ˛  q.e. e˛  D U Therefore, HF˛ U For any q.c. function u 2 W which is bounded q.e. on a nearly Borel set A, define E.HA˛ u, w/ similarly by bn w/ E.HA˛ u, w/ D lim E.HA˛ u, nR n!1

for w 2 b J. Theorem 6.3.8. For any nearly Borel set A and q.c. function u 2 W such that u is bounded on A, E.HA˛ u, w/ is well defined and satisfies b ˛w b/ D E˛ .u, H E˛ .HA˛ u, w A e/

(6.3.12)

for any w 2 J. In particular, if w D 0 a.e. on A, then E˛ .HA˛ u, w b/ D 0. Proof. First we assume that A is an open set. For un D nRnC˛ u, since HA˛ u D limn!1 HA˛ un and un belongs to the range R˛ H , un  HA˛ un belongs to J and vanishes on A. Hence, by using equation (6.1.28) and Lemma 6.3.4, we have B˛ .HA˛ un , HA˛ un /  lim B˛ .ˇRˇ C˛ HA˛ un , ˇRˇ C˛ HA˛ un / ˇ !1

D lim E˛ .ˇRˇ C˛ HA˛ un , ˇRˇ C˛ HA˛ un / ˇ !1

 lim E˛ .ˇRˇ C˛ HA˛ un , HA˛ un / ˇ !1

D E˛ .HA˛ un , HA˛ un / D E˛ .HA˛ un , un / D B˛ .HA˛ un , un / C B˛ .un , HA˛ un /  E˛ .un , HA˛ un / bnC˛ H ˛ un kF .  2K˛ kHA˛ un kF kun kF C kukW knR A bnC˛ H ˛ un kF  KkH ˛ un kF for a suitable conSince kun kF  KkukF and knR A A stant K by Theorem 6.1.2, it follows that ¹kHA˛ un kF º is uniformly bounded relative to n  1. Hence it converges weakly in F . Since lim!1 HA˛ un D HA˛ u q.e. by Lebesgue theorem, we see that HA˛ u 2 F . Hence, for any function w 2 J such that

Section 6.3 Associated space-time processes

247

w D 0 a.e. on A, we have by Theorem 6.3.5, bkC˛ w bkC˛ w E˛ .HA˛ u, w b/ D lim E˛ .HA˛ u, k R b/ D lim lim E˛ .HA˛ un , k R b/

k!1 n!1 ˛ b R b˛R b b lim lim E˛ .un , k H b/ D lim E˛ .u, k H b/ A kC˛ w A kC˛ w k!1 n!1 k!1 b ˛w (6.3.13) E˛ .u, H A b/, k!1

D D

b˛R b˛w b bDH where the last equality follows from limk!1 k H A kC˛ w A b weakly in F . In particular, if w D 0 a.e. on A, then w b D 0 on the coregular points of A and hence b˛w e D 0 which implies that equation (6.3.12) vanishes. H A For any nearly Borel set A, take a sequence of open sets ¹Ak º such that Ak A, P z .limk!1 b Ak D b A / D 1 for a.e. z. As in the Pz .limk!1 Ak D A / D 1 and b inequality shown above, we have B˛ .HA˛k u, HA˛k u/  B˛ .HA˛k u, u/ C B˛ .u, HA˛k u/  E˛ .u, HA˛k u/  .2K˛ kukF C kukW / kHA˛k ukF .

Therefore ¹HA˛k uº is bounded in F and converges weakly to HA˛ u in F . Hence, by equation (6.3.13), bnC˛ w b/ D lim E˛ .HA˛ u, nR b/ E˛ .HA˛ u, w n!1

bnC˛ w b/ D lim lim E˛ .HA˛k u, nR n!1 k!1

b˛ R b b/ D lim lim E˛ .u, nH Ak nC˛ w n!1 k!1

b˛R b b/ D lim E˛ .u, nH A nC˛ w n!1

b˛w D E˛ .u, H A b/. This implies that E˛ .HA˛ u, w b/ is well defined for any w 2 J and satisfies equation (6.3.12). Similarly to the case of open set, if w D 0 a.e. on A, then the last term vanishes. In accordance with the definition given in the paragraph preceding Lemma 3.5.11, we say that .E . / , F / possesses the local property if, for any ', 2 F with disjoint compact supports, E . / .', / D 0. If .E . / , F / possesses the local property for any  , we say that .E, F / has the local property. Then the similar results to Lemma 3.5.11 and Theorem 3.5.12 hold in the time dependent case. Lemma 6.3.9. .E, F / has the local property if and only if, for any relatively compact N open set A of Z, HA1 .z, / is concentrated on the boundary @A for q.e. z 2 Z n A.

248

Chapter 6 Time dependent Dirichlet forms

Proof. Suppose that .E, F / possesses the local property. Let A be a relatively compact 2 H . Put open set and u a function of W \ C0 .Z/ such that SuppŒu  A and @u @ b1 f  H b1 R N Since un H 1 un D 0 b1 f for f 2 C0C .Zn A/. un D nRn u 2 W and v D R A A a.e. on A, we have from Lemmas 6.3.4 b1 f / lim E1 .un  HA1 un , v/ D lim E1 .un  HA1 un , R

n!1

n!1

D .u  HA1 u, f /. On the other hand, lim E1 .un  HA1 un , v/ D lim E1 .un , v/ D E1 .u, v/

n!1

n!1

  , v D 0. which vanishes because B1 .u, v/ D 0 from the local property and @u @ 1 1 N Therefore, .u  HA u, f / D 0 and hence HA u D u D 0 q.e. on Z n A. 1 N Let Conversely, assume that HA .z, / is concentrated on @A for q.e. z 2 Z n A. ', 2 F have disjoint compact supports. Then there exists a relatively compact open N For any interval .a, b/, let set  of X such that suppŒ'   and suppŒ   X  . 1 A D .a, b/  . Taking a function  2 C0 .a, b/, put u D  ˝ '. Then u is supported by a compact subset of A and hence HA1 u D 0 q.e. on Z n AN by the assumption. N r , uH 1 u D Furthermore, since HA1 u D u on AN except at most the semipolar set AnA A 0 a.e. on Z. Therefore, 0 D E1 .u  HA1 u,  ˝ / D E1 . ˝ ',  ˝ Z . / E1 .', / 2 . /d  . D

/

R1

This implies that E . / .', / D 0 for a.e.  and hence .E, F / has the local property. Using this lemma, the following theorem holds similarly to Theorem 3.5.12. Theorem 6.3.10. .E, F / has the local property if and only if M is a diffusion process, that is, the paths t 7! Z t .w/ are continuous a.s. Pz for q.e. z.

6.4 Additive functionals and associated measures In this section, we consider the correspondence between a PCAF and a smooth measure. In Chapter 4, we saw that, for an ˛-potential U˛  2 F of measure , there exists a PCAF A t such that UA˛ 1 is a q.c. modification of U˛ . But, in the present case, due to the presence of non-exceptional semipolar sets, this correspondence does not hold in general. For example, assume that m.X / < 1 and let u.z/ D Ez .e ˛B / with B D ¹bºX for fixed b > 0. Since B D b   a.s. P. ,x/ for any  < b and x 2 X , B is not an

Section 6.4 Additive functionals and associated measures

249

exceptional set but it is a semipolar set because Z t 2R B only the instant t D b   . In 1 this case, we cannot express u as u. , x/ D E. ,x/ . 0 e ˛t dA t / using a PCAF A t , because if this were possible, then for any increasing sequence of stopping times n with limit  , Z 1    lim Ez e ˛n u.Zn / D lim Ez e ˛t dA t n!1 n!1  Z 1 n    ˛t e dA t D Ez e ˛ u.Z / . D Ez 

In particular, if  D B and n D Bn for Bn D .b  1=n, b  X , then  D b   if  < b,  D 1 if   b and n D .b    1=n/ _ 0 if  < b, n D 1 if   b. Hence limn!1 n D  but, for  < b, limn!1 E. ,x/ .e ˛n u.Zn // D e ˛.b / > 0 and E. ,x/ .e ˛ u.Z // D 0 because u.Z / D 0. An ˛-excessive function u of M is called a regular potential if, for any increasing sequence ¹n º of stopping times such that limn!1 n D  a.s. Pz for q.e. z,     u.Zn / D Ez e ˛ e u.Z / . (6.4.1) lim Ez e ˛ne n!1

If u is a regular potential belonging to F , then it can be written as u D U˛  for some  2 S0 . Then, for any Borel set B, u1 D U˛ B with B D 1B   is also a regular potential. In fact, put u2 D U˛ XnB . Since u1 and u2 are ˛-excessive, ui .Z //  Ez .e ˛ e ui .Z // q.e. for any stopping times    and i D 1, 2. Ez .e ˛ e Hence, for any increasing sequence of stopping times ¹n º with limn!1 n D  ,     u.Z / D Ez e ˛ .e u1 C e u2 /.Z / Ez e ˛ e   u1 C e u2 /.Zn /  lim Ez e ˛n .e n!1   u.Zn / D lim Ez e ˛ne n!1     u.Z / D Ez e ˛ .e u1 C e u2 /.Z / q.e. D Ez e ˛ e u1 .Zn // D Ez .e ˛ e u1 .Z //, that is, u1 is a regThis yields that limn!1 Ez .e ˛ne ular potential. Assume that P˛ \ Lg ¤ ; for a quasi-continuous function g 2 H . As in Theorem 6.2.4, denote by eg˛ the minimal ˛-excessive function in Lg . We shall show that e e ˛g is a regular potential. To show this, we need the following two lemmas. Lemma 6.4.1. Suppose that h 2 P˛ \ F and ¹Bn º is a decreasing sequence of quasi-open sets such that limn!1 Bn   a.s. Pz for q.e. z. Then, for q.e. z 2 Z, h.z/ D 0. limn!1 HB˛ne

250

Chapter 6 Time dependent Dirichlet forms

Proof. Since h 2 P˛ \ F , there exists h 2 S0 such that h D U˛ h . Take f 2 H C , then we have from equation (6.3.11),       b˛ f h, f D lim HB˛n ˇRˇ C˛e h, f D lim E˛ HB˛n ˇRˇ C˛e h, R HB˛ne ˇ !1 ˇ !1     b˛ R bˇ C˛ H b˛ f D lim E˛ e b˛ R b˛ f h, H h, ˇ R D lim E˛ ˇRˇ C˛e Bn Bn ˇ !1 ˇ !1 Z Z bˇ C˛ H b˛ R b b˛ R b D lim ˇR H Bn ˛ f .z/dh .z/ D Bn ˛ f .z/dh .z/ ˇ !1 Z

Db E h

Z

O

O Bn

Z

 b t /dt . e ˛t f .Z

By the duality relation, since     P lim Bn < T <  D P [m \n [rT 1=m ¹Zr 2 Bn , T < º n!1   O b [m \n [rT 1=m ¹Z b r 2 Bn , T < º DP   b lim O B < T < O DP n!1

n

for any T < 1, where r moves among positiverational numbers,the assumption bz limn!1 OB <  D 0 for a.e. z. that Pz limn!1 Bn <  D 0 a.e. z implies P n b˛ R b˛ f D 0 a.e. Since b h is a limit of a decreasing sequence Hence b h0 limn!1 H 0 Bn b of ˛-coexcessive functions, it follows that h0 D 0 q.e. by Lemma 3.4.6. Therefore, h.z/ D 0 a.e. and hence q.e. z. limn!1 HB˛ne Lemma 6.4.2. For any q.c. function g 2 H , e e ˛g .z/ D Ez .e ˛B g.ZB // q.e., where ˛ B D ¹z : e e g .z/ D g.z/º. Proof. Let g" be the solution of g" D .1="/G˛ .g"  g/ given by Lemma 6.2.3 and put gn D g"n for "n # 0. Let Bn D ¹z : gn .z/  g.z/º and n D Bn . Then  Z 1 1 e g n .z/ D Ez e ˛t .gn  g/ .Z t /dt "n  Z0 1   1 D Ez e ˛t .gn  g/ .Z t /dt D Ez e ˛ne g n .Zn / "n n     e ˛g .Zn / q.e.  Ez e ˛n g.Zn /  Ez e ˛ne For any a > 0, let Aa D ¹z : e e ˛g .z/ > aº. Since e e ˛g D limn!1 nRnC˛ eg˛ is an increasing limit of q.c. functions, Aa is quasi-open, that is, there exists a nest ¹Fn º such that Aa is open in Fn for all n. Furthermore, from     aEz e ˛Aa : Aa <   Ez e ˛Aae e ˛g .ZAa /  e e ˛g .z/,

Section 6.4 Additive functionals and associated measures

251

it follows that lima!1 Aa   a.s. Pz for q.e. z. By virtue of Lemma 6.4.1, it then e ˛g D 0 q.e. For any non-negative integrable function f let holds that lima!1 HA˛ae   Ef  e ˛n g.Zn /     D Ef  e ˛n g.Zn / : n < Aa C Ef  e ˛n g.Zn / : n  Aa D In,a C IIn,a . e ˛g .Zn /  a on ¹n < Aa º and Put P D limn!1 n . Since jg.Zn /j D g.Zn /  e limn!1 g.Zn / D g.ZP / by the quasi-continuity of g and quasi-left continuity of Z t , we have by Lebesgue theorem that   lim In,a D Ef  e ˛P g.ZP / : P  Aa . n!1

On the other hand, since g.Zn /  0,     e ˛g .ZAa / D f , HA˛ae e ˛g IIn,a  Ef  e ˛Aae which converges to zero as a ! 1. Therefore, limn!1 Ef  .e ˛n g.Zn // D Ef  .e ˛P g.ZP //. By the inequality given in the first paragraph of the proof,     f ,e e ˛g D lim .f , gn / D lim Ef  e ˛ne g n .Zn / n!1 n!1       e ˛g .ZP /  f ,e e ˛g .  Ef  e ˛P g.ZP /  Ef  e ˛P e Let DB be the first entry time of B defined by DB D inf¹t  0 : Z t 2 Bº. Since e ˛g g/.ZP // D 0 by the above relation, it holds that DB  P g e e ˛g and Ef  .e ˛P .e a.s. Pz for a.e. z. Hence DB  B D lim t!0 .t CDB ı t /  lim t!0 .t C P ı t / P C a.s. Pz for q.e. z. Since k  DB , P  DB  P C a.s. Pz for a.e. z. In particular, if z 2 B \ B r , then P  DB D 0 a.s. Pz . If z … B, then g.z/ < gk .z/ for C a.s. Pz and hence large k and hence  ˛Dk > 0 a.s.  Pz . In particular, P D P D DB ˛ B g.ZDB except at most a semipolar set B nB r . Then the assertion e e g .z/ D Ez e of the lemma follows by taking an ˛-excessive regularization of both sides. This result can be applied to the optimal stopping problem. For a q.e. finely continuous function g on Z and a stopping time  , put   (6.4.2) Jz . / D Ez e ˛ g.Z / . Then the problem is to find the stopping time  which gives the maximum of Jz . /. Theorem 6.4.3. Suppose that g 2 F is a q.e. finely continuous function such that e ˛g .z/ D g.z/º and q.e. z, Lg \ W is non-empty. Then, for B D ¹z : e   max¹Jz . / :  i s a st oppi ng t i meº D Ez e ˛B g.ZB / D e e ˛g .z/. (6.4.3)

252

Chapter 6 Time dependent Dirichlet forms

Proof. By virtue of Theorem 6.2.6, eg˛ is the smallest ˛-excessive function belonging to Lg . Hence, for any stopping time  ,     Ez e ˛ g.Z /  Ez e ˛e e ˛g .Z /  e e ˛g .z/ q.e. Hence it is enough to show that   Ez e ˛B g.ZB / D e e ˛g .z/

q.e.

This has been already shown by Lemma 6.4.2. Theorem 6.4.4. Suppose that g 2 H is a q.c. function such that P˛ \ Lg ¤ ;. Then e e ˛g is a regular potential. Proof. In the proof of Lemma 6.4.2, we have seen that P D DB and hence P C D B . e ˛g .z/ D HB˛ g.z/. For any increasing sequence of Also we have seen thate e ˛g .z/ D HB˛e stopping times ¹n º with limn!1 n D  , put n D n CB ı n and limn!1 n D . e ˛g .ZB / D g.ZB /, Then   C B ı  . Since e      e ˛g .Zn / D Ez e ˛ n EZn e ˛Be e ˛g .ZB / Ez e ˛ne    D Ez e ˛ n EZn e ˛B g.ZB /   D Ez e ˛n g.Zn / . Letting n ! 1 we have by quasi-left continuity,     lim Ez e ˛ ne e ˛g .Z n / D lim Ez e ˛ n HB˛e e ˛g .Z n / n!1 n!1  ˛ ˛    n e e g .Zn / D lim Ez e ˛n g.Zn / D lim Ez e n!1 n!1     e ˛g .Z / . D Ez e ˛ g.Z /  Ez e ˛e Since e e ˛g is ˛-excessive and n    , it holds that       e ˛g .Z /  Ez e ˛ e e ˛g .Z /  Ez e ˛ ne e ˛g .Z n / . Ez e ˛e Therefore

    e ˛g .Z n / D Ez e ˛ e e ˛g .Z / , lim Ez e ˛ ne

n!1

and

hence e e ˛g

is a regular potential.

The following two theorems can be shown in a similar manner to Theorem IV.3.13 and Theorem VI.3.5 in [14]. Theorem 6.4.5. An excessive function u is a regular potential if and only if there exists a positive continuous additive functional A t of M such that Z 1  e u.z/ D Ez e ˛t dA t q.e. (6.4.4) 0

253

Section 6.4 Additive functionals and associated measures

Proof. We shall give only the outline of the proof. Assume first that u is bounded and limn!1 un D u uniformly on Z q.e., where un D nRnC˛ u D R˛ gn for gn D ˇ n.u  nRnC˛ u/  0. Let e A t be the PCAF defined by e Ant D

Z

t

e ˛s gn .Zs /ds

0

for ˛ > 0. For a bounded measure  charging no exceptional set, let M tn be the martingale defined by An1 j F t / D e Ant C e ˛t R˛ gn .Z t /. M tn D E .e Then

   n  2 A1  e Am P sup jM tn  M tm j > ı  ı 2 E .e 1/ . t0

As in the proof of Theorem 4.1.10, for any m  n,  n  2 A1  e Am E .e 1/  Z 1 2˛t D 2E e .gn  gm /.un  um /.Z t /dt  Z 01 2˛t e gn .un  um /.Z t /dt  2E 0

Z

 2h, 1i ku  p  2 2kuk1 h, 1iku  um k1 , um k21 E

1=2

1

e

2˛t

2 1=2 gn .Z t /dt

0

because Z E

1

e 0 Z

2˛t

1

 2E

e

2  gn .Z t /dt

2˛t

 gn u.Z t /dt

0

 2kuk21 h, 1i.

Further   P sup jR˛ .gn  gm /.Z t /j > ı t0     D P sup jun  um j.Z t / > ı  ı 2 E sup ju  um j2 .Z t / t0

ı

2

h, 1iku 

t0

um k21

ı

2

h, 1ikuk1 ku  um k1 .

254

Chapter 6 Time dependent Dirichlet forms

Therefore, for any k  1, by choosing a sequence ¹nk º such that kuunk k1 < 24k , we get that   n P sup je A t kC1  e Ant k j > 2k t0   n nk j > 2k1  P sup jM1kC1  M1 t0   C P sup jR˛ .gnkC1  gnk /.Z t /j > 2k1 t0  p  2kC2 2 2 C 1 kuk1 h, 1iku  unk k1 2  p   22kC2 2 2 C 1 kuk21 h, 1i. Rt At Therefore, e A t D limk!1 e Ant k converges uniformly a.e. P . Then A t D 0 e ˛t d e satisfies  Z 1 ˛ ˛t e gnk .Z t /dt D lim nk R˛Cnk u.z/ D u.z/, q.e. UA 1.z/ D lim Ez k!1

k!1

0

For a general bounded regular potential u, let u be the measure R associated with u by equation (6.2.3). Take a strictly positive function f such that Z uf d < 1. Since R b˛ f i D limn!1 nRn u D u q.e. and hu , R Z uf d < 1, using Egorov theorem, we can find an increasing sequence of compact sets ¹Fk º such that limn!1 nRn u D R b˛ f du D 0. Put k D u jF and u uniformly on each Fk and limk!1 ZnFk R k e ˛ k . Then, for any non-negative function g 2 H , u.k/ D U   bnC˛ g, U˛ u .g, u  nRnC˛ u/ D g  nR b˛ g  nR b˛ R bnC˛ gi b gi D hu , R D hu , R   nC˛ bnC˛ gi D g, u.k/  nRnC˛ u.k/ .  hk , R Hence u  nRnC˛ u  u.k/  nRnC˛ u.k/ . In particular, limn!1 nRnC˛ u.k/ D u.k/ uniformly on Fk q.e. Since u is a regular potential, as we noted before Lemma 6.4.1, u.k/ is also a regular potential. By u.k/ D HA˛e u.k/ . virtue of Corollary 6.3.7, for any open set A such that A Fk , e Hence, by approximating Fk by a sequence of open sets ¹An º such that An Fk and limn!1 An D Fk a.s. Pz for a.e. z, we have by the regularity of the potential u.k/ D limn!1 HA˛ne u.k/ D HF˛k e u.k/ a.e. Since both sides are considered to be u.k/ , e excessive regularizations, they coincide q.e. Therefore, if we use the notation q.e.supF to represent the supremum on F except on a set of zero capacity, q.e. sup .u.k/  nRnC˛ u.k/ /.z/  q.e. sup HF˛k .u.k/  nRnC˛ u.k/ /.z/ z2Z

z2Z

 q.e. sup .u.k/  nRnC˛ u.k/ /.z/. z2Fk

255

Section 6.4 Additive functionals and associated measures

Hence u.k/ satisfies the condition assumed on u in the previous step. Then, there ex.k/ e ˛ k q.e. Furthermore, A t D limk!1 A.k/ ists a PCAF A t such that UA˛.k/ 1 D U t converges uniformly on every finite interval a.s. Finally, in the general case, there exists an increasing sequence of compact sets ¹Kn º such that u is bounded on Kn q.e. and limn!1 Ez .e ˛ Kn / D 0 q.e. Similarly e e e ˛ Kn D H ˛ U to the preceding proof, U Kn ˛ Kn and hence U ˛ Kn is bounded q.e. for

n such each n, where Kn D u jKn . Therefore, there is a corresponding PCAF AK t Kn ˛ e that U ˛ Kn D UAKn 1. Then A t D limn!1 A t gives the desired PCAF.

Theorem 6.4.6. For any measure  of a finite energy integral, U˛  is a regular potential if and only if  does not charge any semipolar set. e Proof. ˛  is aregular potential, then there exists a PCAF A t such that U ˛ .z/ D R 1If U ˛t Ez 0 e dA t . Similarly to Lemma 4.1.11, it then holds for any bounded measurable function f , Z 1  e ˛ .f  /.z/ D Ez U e ˛t f .Z t /dA t q.e. 0

If B is a semipolar set, then e ˛ .IB  /.z/ D Ez U

Z

1

e ˛t IB .Zs /dAs

 D0

0

for any ˛ > 0. Hence   e ˛ .IB  / D lim hIB  , ˛R˛ f i D hIB  , f i 0 D lim ˛ f , U ˛!1

˛!1

for any f 2 C0C .Z/. Hence .B/ D 0. Conversely, if  does not charge a semipolar set, then for any compact set F , ˛e e ˛ F for F D jF by Corollary 6.3.7. In the proof of Theorem HF U ˛  F D U 6.4.5, only this property is used to ensure between regular potenRthe correspondence  e ˛  is a regular e ˛ .z/ D Ez 1 e ˛t dA t and hence U tial and PCAF. Hence U 0 potential. If  is the measure associated with a PCAF A t then, as we noted in the proof of Theorem 6.4.6, Z 1  ˛ ˛t e e f .Z t /dA t q.e. U ˛ .f  /.z/ D UA f .z/ D Ez 0

Further,  is the Revuz measure associated with A t , that is  Z 1 ˛ 2 ˛t e A t dt . h, f i D lim hf  , RA 1i D lim ˛ Ef  ˛!1

˛!1

0

(6.4.5)

256

Chapter 6 Time dependent Dirichlet forms

Generally, a 1-excessive function is not necessarily a regular potential. A 1-excessive function u is said to be a natural potential if, for any increasing sequence of stopping times ¹n º with limn!1 n  ,   (6.4.6) lim Ez e n u.Zn / D 0 q.e. z, n!1

where we consider that u is a 1-excessive regularization. An AF A t is called a natural additive functional (NAF in abbreviation), if A t and Z t have no common discontinuities. We only give an outline of the proof of the next theorem.1 A stopping time  is called a predictable stopping time if there exists an increasing sequence of stopping times ¹n º such that n <  on ¹ < 1º and limn!1 n D  a.s. Theorem 6.4.7. If u 2 F \ P˛ is a natural potential, then there exists a unique NAF A t such that u D UA1 1 q.e. Proof. Let W t D u.Z t / and, for a fixed " > 0, put  D inf ¹t : jW t  W t j > ", Z t D Z t º.

(6.4.7)

Then  > 0 and  ı  t D   t on ¹ > t º for all t > 0 a.s. Further,  is a predictable stopping time. In fact it suffices to put n D Bn , Bn D ¹e > 1  1=nº for e .z/ D Ez .e  /. Define a sequence ¹n º of stopping times by 0 D 0 and nC1 D n C ın . Since the supermartingale ¹e t W t º is right continuous, has left limit and jWn  Wn  j  ", limn!1 n D  a.s. Define an AF A"t by X .Wn   Wn / . (6.4.8) A"t D n t

Approximating the excessive function u by functions of the form R1 fk , fk  0 and n by a sequence of stopping times from below, we can see that e n Wn  e n Wn  and consequently A"t  0. Clearly A"t is natural because it jumps only at n at which divide the Z t does not jump. Now Rwe  remaining P1proofinto several steps. 1 " n .W .1/ Put w " .z/ D Ez 0 e t dA"t D Ez n   Wn / and v D nD1 e " " u  w . Then v is 1-excessive. Let n be the sequence given after equation (6.4.7). Put V D limn!1 e  n W n . Then V D e  W on ¹ < º and V D 0 on ¹  º by naturality. Then   (6.4.9) 0  u.z/  lim Ez e  n u.Z n / D u.z/  Ez .V /. n!1

1

See Theorem IV.4.22 in [14] for more details.

257

Section 6.4 Additive functionals and associated measures

Since u is a natural potential, it holds that limn!1 Ez .e n u.Zn // D 0, and hence ®   u.z/  w " .z/ D lim u.z/  Ez e n u.Xn / n!1 n X

 ¯ Ez e k u.Zk /  e k u.Zk /



kD1 n1 X

D lim

n!1 kD0 n1 X

D lim

n!1

  Ez e k u.Zk /  e kC1 u.ZkC1 /   Ez e k u.Zk /  e k EZk .V / .

kD0

Since each term in the last summand is non-negative by equation (6.4.9), u  w " and satisfies lim t!0 e t p t v" D v" . The inequality e t p t v"  v" follows from a direct computation. j .2/ Put w D lim"#0 w " and A t D lim"#0 A"t . Then Aj is a NAF satisfying w D UA1j 1. Since A"t is increasing as " decreases and Z 1  w.z/ D lim w " .z/ D lim Ez e t dA"t "!0

exists, the right-hand side of Z ˇ Z s " e dAs  0 ˛

"!0

ˇ

˛

e

s

0

Z dAıs



1

e 0

s

Z dA"s

 0

1

e s dAıs

j

converges to zero as " < ı # 0. Hence A"t converges to A t uniformly on each finite t -interval. In particular, Aj is a NAF. .3/ Put v D u  w D lim"!0 v " . Then v is a regular potential. By the definition, v is a decreasing limit of 1-excessive functions. Furthermore, since it is a difference of 1-excessive functions, it is q.e. finely continuous. Hence, v is a 1-excessive regularization. Let n be an increasing sequence of stopping times with limn!1 n D   . Since v  u, lim Ez .v.Z n / :  D /  lim Ez .u.Z n / :  D / D 0.

n!1

n!1

Therefore   0  lim Ez e  n v.Z n /  e  v.Z / n!1   D lim Ez e  n v.Z n /  e  v.Z /;  < , ƒ n!1   D Ez e  v.Z /  e  v.Z /;  < , ƒ , for ƒ D ¹n <  , for all nº. Since  is predictable on ƒ \ ¹ < º, Z t does not jump j j at  . Hence w.Z /  w.Z / D A  A  on ƒ \ ¹ < º. In fact, for any  2 F k

258

Chapter 6 Time dependent Dirichlet forms

and k < n, 

Ez e



w.Z /  e

 n



w.Z n / :  D Ez

Z



e n

t

j dA t

 : .

Therefore   Ez e  v.Z /  e  v.Z / :  < , ƒ     D Ez e  u.Z /  u.Z /  Aj C Aj  :  < , ƒ D 0, by the construction of Aj . Now we have shown the theorem from the expression u D v C w by using Theorem 6.4.5. Lemma 6.4.8. Suppose that  is a strictly positive predictable stopping time such that  ı  t D   t on ¹ > t º for any t and u.z/ D Ez .e  / 2 F . Then there exist nearly Borel sets B and ¹Bn º such that Bn <  and limn!1 Bn D B D  a.s. Pz for q.e. z. Proof. Let u.z/ D Ez .e  /, Bn D ¹z : u.z/  1  1=nº and n D Bn . Let ¹n º be an increasing sequence of stopping times such that n <  and limn!1 n D  . Then       (6.4.10) Ez e  n u.Z n / D Ez e  n ın D Ez e  . Since n is increasing, limn!1 u.Z n / exists. Hence by letting n ! 1 in equation (6.4.10), it follows that limn!1 u.Z n / D 1 a.s. on ¹ < 1º. In particular, n   for all n  1. Further, if n D  for some n, then u.Z / D u.Zn / D u.Zm /  1 for all m  n which contradicts the condition that u < 1 following from 1 m the positivity of  . Therefore, n <  for all n  1 a.s. on ¹ < 1º and hence u.z/ D HB1 n u.z/ as in equation (6.4.10). Finally, by the relation   1 Ez e .n / : n < 1 D Ez .u.Zn / : n < 1/  .1  /Pz .n < 1/ n we get that limn!1 n D  a.s. e 1 u . Since u belongs to P \ F , there exists a measure  2 S0 such that u D U Noting that u D HB1 n u, it follows from equation (6.3.11) that b1 f i D E1 .u, R b1 f / D E1 .H 1 u, R b1 f / hu , R Bn

b1 R b1 b b D E1 .u, H Bn 1 f / D hu , H Bn R1 f i,

b 1 is supported by the cofine closure B bn of Bn , b 1 . Since H and hence u D u H Bn Bn T bn . Furthermore, by letting n ! 1 u is supported by the cofine closed set B D B 1 b b b in E1 .u, R1 f / D E1 .u, H Bn R1 f /, the same relation holds for B instead of Bn . Then

259

Section 6.5 Some stochastic calculus

equation (6.3.11) implies that u.z/ D HB1 u.z/. Therefore,     Ez .e  / D Ez e B EZB .e  / D Ez e B ıB . Consequently B  B C  ı B D  a.s. To prove the converse inequality, put n,m D n C Bm ı n , where Bm is the exit time from Bm . Since n <  , we have n,m   C Bm ı  and n,m <  C Bm ı  if and only if Z t 2 Z n Bm for some t 2 Œn ,  /. By noting that n "  and lim t" u.Z t / D 1, we can see that n,m D  C Bm ı  for large n. Now, for any v D R1 f with non-negative function f , we have    1 n,m e v.z/ D lim E v.Z / lim HB1 n HZnB z  n,m m n!1 n!1   ı   Bm v.ZC Bm ı / D Ez e   1 v.Z / . D Ez e  HZnB m 1 v.z/. Also, noting that Since B  Bn , the left-hand side dominates HB1 HZnB m 1 1 limm!1 HZnBm v D v increasingly, we have HB v.z/  Ez .e  v.Z //. By applying this to v D nRn 1 D nR1 .1  .n  1/Rn 1/ and letting n ! 1, we obtain   B a.s.

e 1  is a natural potential. Theorem 6.4.9. If  2 S0 , then u D U Proof. Let ¹n º be an increasing sequence of stopping times such that limn!1 n  . Since u.Z / D 0, to show equation (6.4.6), it is enough to assume that  D limn!1 n   is a predictable stopping time and n D Bn for a decreasing sequence of nearly Borel sets ¹Bn º by Lemma 6.4.8. Then, for any non-negative function f 2 H , by equation (6.3.11),       b1 f D E1 H 1 u, H b1 R b1 f f , HB1 n u D E1 HB1 n u, R Bn Bn   1 b 1 b b b  E1 u, H Bn R1 f D h, H Bn R1 f i  Z  b DE e t f .Z t /dt . O Bn

Hence, HB1 n u is decreasing and limn!1 HB1 n u D 0 q.e. by Lemma 3.4.6.

6.5

Some stochastic calculus

In Chapter 5, we studied some stochastic calculus related to the semi-Dirichlet forms satisfying .E.5/. In this section, assuming .E.5/ for all E . / , we shall show that such calculus is also possible for time dependent Dirichlet forms. Since the results and proofs are similar to the corresponding ones in Chapter 5, we only give Fukushima’s

260

Chapter 6 Time dependent Dirichlet forms Œu

decomposition (in the weak sense) of A t for u 2 W . As in Section 5.1, define the weak sense energy ev .A/ of a CAF A t relative to v 2 Fb by  Z 1 1 2 ˇ t 2 e A t dt . (6.5.1) lim ˇ Ev ev .A/ D 2 ˇ !1 0 As in Section 5.1, for ı > ˛0 , take a strictly positive ı-coexcessive function b hı 2 F such that b hı is bounded from below by a positive constant on every compact set of Z. As in Theorem 2.4.8, such a function exists. By taking an ˛-coexcessive regularization, we may assume that b hı is q.e. cofine continuous. Let „.ı/ be the family of sequences of relatively compact q.e. cofinely open sets ¹B` º such that [B` D Z q.e. and 1=` < b hı < ` on B` . Fixing ¹B` º 2 „.ı/ , put A`t D A t^ ` for ` D B` and ev` .A/ D ev .A` /, that is  Z 1 1 ev` .A/ D lim ˛ 2 Ev e ˛t A2t^ ` dt . 2 ˛!1 0 Note that, for any nearly Borel set B with finite capacity, B n B r is semipolar. Let be the set of coregular points of B and put F D r B n B r . Then F [ r F  r B. e ˛ F q.e. By virtue of Denote by F the equilibrium measure of F , that is HF˛ 1 D U XnF b˛ f , we have equation (6.3.12) with w bDR rB

b˛XnF f / D hF , R b˛XnF f i 0 D E˛ .HF˛ 1, R for any non-negative bounded integrable function f . Since we can take a function f b˛XnF f > 0 q.e. on X n .r F [ F /, F is supported by r F [ F  r B. such that R Hence, b˛ R e ˛ F , f / D hF , H b .HB˛ HF˛ 1, f / D .HB˛ U B ˛f i b˛ f i D .H ˛ 1, f / D hF , R F

which implies that D In particular, since F  X n B r , HF˛ 1 D HB˛ HF˛ 1 < 1 on F . Similarly to the proof of Theorem 3.2.6, this implies that F is a semipolar set. Similarly, B r n r B is a semipolar set. In particular, it holds that 1B r  A D 1r B  A for any PCAF A. For any u 2 W , define an AF AŒu by HB˛ HF˛ 1

HF˛ 1.

Œu

At

De u.Z t /  e u.Z0 /,

Then, for any u 2 Wb and v 2 Fb , it holds similarly to equation (5.1.7) that   1 1 ev AŒu D E.u, uv/  E.u2 , v/ D B.u, uv/  B.u2 , v/. 2 2

(6.5.2)

(6.5.3)

For any u 2 W and ˛ > ˛0 , we defined in Theorem 6.2.4 the smallest ˛-excessive function eu˛ .x/ 2 F such that eu˛  u. For any approximating sequence ¹u" º  W of

261

Section 6.5 Some stochastic calculus

eu˛ and any w 2 Lu \ W , it holds that B˛ .u" , u" /  E˛ .u" , w/  KkwkW ku" kF . Hence, there exists a constant K1 satisfying keu˛ kF  K1 kwkW .

(6.5.4)

Lemma 6.5.1. Let u 2 W . Then, for any  2 S0 , " > 0 and T > 0, there exists a constant K independent of T ,  and  such that   K u.Z t /j >   (6.5.5) P sup je e ˛T B˛ .U˛ , U˛ /1=2 kukW . 0tT  Proof. Let ¹Fn º be a nest such that e u is continuous on each Fn and put A D ¹x : e u.x/  º. For B D X n .A \ Fn / X n A, Lemma 6.3.2 implies that   u.Z t /j >  D P .XnA  T /  P .B  T / P sup je 0tT Z ˛T ˛B ˛T ˛  e E .e /De e eB d Z Z ˛ ˛ .e eXnF Ce eXnA /d.  e ˛T n Z

In the right-hand side, using equations (6.2.2) and (6.2.17), we have Z Z ˛ ˛ ˛ e eXnFn d D lim ˇR˛Cˇ eXnF d D lim E˛ .U˛ , ˇR˛Cˇ eXnF / n n ˇ !1 Z ˇ !1 Z   ˛ ˛ / C B .ˇR e , U /  lim B˛ .U1 , ˇR˛Cˇ eXnF ˛ ˛ ˛Cˇ XnF n n ˇ !1

˛ ˛ / C B˛ .eXnF , U˛ / D B˛ .U˛ , eXnF n n ˛ ˛ , eXnF /1=2  2K˛ B˛ .U˛ , U˛ /1=2 B˛ .eXnF n n

 2K˛ B˛ .U˛ , U˛ /1=2 Cap.X n Fn /1=2 . Similarly, Z Z Z 1 1 ˛ ˛ e eXnA d  e e d  .e e ˛ Ce e ˛u /d  Z juj  Z u Z  1  e ˛u Ce e ˛u / C B˛ .e e ˛u Ce e ˛u , U˛ / B˛ .U˛ ,e  2K˛  e ˛u Ce e ˛u ,e e ˛u Ce e ˛u /1=2 . B˛ .U˛ , U˛ /1=2 B˛ .e  In the last term, by equation (6.5.4), ˛ ˛ B˛ .eu˛ , eu˛ / C B˛ .eu , eu /  K2 kuk2W

262

Chapter 6 Time dependent Dirichlet forms

for some constant K2 depending on ˛. Therefore,   u.Z t /j >  P sup je 0tT   1 ˛T 1=2 1=2 Cap.X n Fn / C kukW  K e B˛ .U˛ , U˛ /  for K D 2K˛ .1 C K2 /. By letting n ! 1, we get equation (6.5.5). Theorem 6.5.2. If ¹un º is a Cauchy sequence of functions of W , then there exists unk .Z t /º converges uniformly on each compact t a subsequence ¹unk º such that ¹e interval a.s. Pz for q.e. z. In particular, for any u 2 W , limk!1 nk Rnk u.Z t / D u.Z Q t / uniformly on each compact t -interval a.s. Pz for q.e. z. Proof. Take a subsequence ¹unk º such that kunkC1 unk kW  22k for each k. Then Lemma 6.5.1 implies that the set ƒk D ¹sup0tT junk .Z t /  unkC1 .Z t /j > 2k º satisfies P .ƒk /  Ke ˛T e k B˛ .U˛ , U˛ /1=2 . Hence, by Borel–Cantelli’s lemma, P .limk!1 ƒk / D 0 for all  2 S0 . Therefore Pz .limk!1 ƒk / D 0 q.e. z which implies the uniform convergence of limk!1 uQ nk .Z t /. If u 2 W , then by equation (6.1.16) uQ D R˛ f for f . , x/ D . / du=d   L˛ u. , x/ 2 F 0 . Then the relation nRn f D u  Rn f C ˛Rn u implies that limn!1 nRn u D u in W . This implies the last assertion of the theorem. As in Chapter 2, we denote by S00 the family of all measures  2 S0 with bounded ˛-potential. Define the dual family b S 00 similarly. Also denote by kuk`1 the essential supremum of u on B` . Lemma 6.5.3. Let A be a PCAF and A be its associated measure. For any 2 S0 b ˛ is bounded on B` and t > 0, such that U b ˛ k`1 A .B` /. E .A`t /  e ˛t kU Proof. Since UA˛` 1 is an excessive modification of U˛ .1B`  A /, Z

 Z 1  e ˛s dA`s  e ˛t E e ˛s dA`s 0  0  ˛t ˛t e b˛ D e h , U ˛ .1B`  A /i D e E˛ U˛ .1B`  A /, U

E .A`t /  e ˛t E

t

b ˛ i  e ˛t kU b ˛ k`1 A .B` /. D e ˛t h1B` A , U

(6.5.6)

263

Section 6.5 Some stochastic calculus

We defined the family J by equation (6.3.3). By virtue of Lemma 6.3.3, any function u 2 J has an ˛-excessive as well as an ˛-coexcessive regularization. These two regularizations differ at most on a semipolar set. For any u 2 J, e u represents an ˛-excessive (resp. ˛-coexcessive) regularization of u if we consider it unb For example, H ˛e b ˛ u) is considered an ˛-excessive der the M (resp. M). B u (resp. H B e (resp. ˛-coexcessive) regularization. In particular, if u 2 W , then e u represents a .ı/ ` e D 0 q.e. on X n B` º q.c. modification. For any ¹B` º 2 „ , put J D ¹w 2 W : w ` be the family of bounded non-negative functions of J. and let Jb,C As in Section 5.1, let Mloc be the family of locally square integrable MAFs and 

M the family of MAFs M such that M ` is of finite energy in the weak sense for all  ¹B` º 2 „.ı/ . Also let N be the family of CAFs N such that N ` is of zero energy in the weak sense for all ¹B` º 2 „.ı/ . 

For M 2M , let hM i be the associated measure of the PCAF hM i t . Then, as in equation (5.1.16), the weak sense energy ev .M / is given by Z 1 e v .z/hM i .dz/ (6.5.7) ev .M / D 2 Z ` for any v 2 Jb,C . Since hM i does not charge semipolar sets by Theorem 6.4.6, v . As in Theorem 5.1.3, if ev .M / is independent of the choice of the regularization e 

` for all ¹M .n/ º M is ev -Cauchy sequence for all non-negative functions v 2 Jb,C 

.n /

`, then there exist M 2M and a subsequence ¹M .nk / º such that limk!1 M t^ k` D M t^ ` uniformly on every finite t -interval a.s. Pz for q.e. z. For the Hunt processes associated with time dependent Dirichlet forms, we can show Fukushima’s decomposition similarly to Theorem 5.1.4 and Theorem 5.1.5. 



Theorem 6.5.4. For any u 2 Wb , there exist uniquely M Œu 2M and N Œu 2 N such that Œu Œu Œu (6.5.8) At D Mt C Nt . Proof. The proof is similar to the proof of Theorem 5.1.5. 



Uniqueness: If M 2M \ N , then for any `  1 and non-negative function v 2 Jb` , M t` D M t^ ` satisfies 0 D ev` .M ` / D 12 hhM ` i , vi. This implies that hM ` i D 0 for all ` and hence M D 0.  Existence: Put un D nRn u D R1 fn for fn D n.u  .1 n/Rn u/. Define N .n/ 2N 

and M .n/ 2M by .n/ Nt

Z D

t

.n/

.un  fn /.Zs /ds, M t 0

Œun

D At

.n/

 Nt .

264

Chapter 6 Time dependent Dirichlet forms

Then by equation (6.5.3), 0  ev .M .n/  M .m/ / D ev .AŒun um /  1  D B .un  um , v.un  um //  B .un  um /2 , v . 2 This implies that ¹M .n/ º is an ev -Cauchy sequence and hence ¹M .n/,` º converges 

relative to ev` and uniformly on each compact interval to M Œu 2M . Further, Theorem 6.5.2 gives us the existence of a subsequence ¹AŒunk º which converges uniformly on any compact intervals. As in the proof of Theorem 5.1.5, it also holds that lim e ` .AŒun n!1 v

 AŒu / D 0

for any `  1 and v 2 J B` . Hence ¹N .nk / º converges to a CAF N Œu satisfying AŒu D M Œu C N Œu . Furthermore, N Œu is of zero energy in the weak sense because, ` , for any non-negative function v 2 Wb,C   ° ±2  Œu Œuun .n/ Œu .n/ Ev .N t /2  Ev A t  N t^ `  .M t^ `  M t^ ` /   Œuu .n/ Œu .n/  3Ev .A t^ ` n /2 C .N t^ ` /2 C .M t^ `  M t^ ` /2 , and hence

  ev` .N Œu /  3 ev` .AŒuun / C ev` .M Œu  Mn /  6B˛ .u  un , u  un / 

for all n  1. Therefore N 2N . Similarly to Theorem 5.1.7, Theorem 6.5.4 and equation (6.5.7) imply the following result. Corollary 6.5.5. For any u 2 Wb , hM Œu i is the unique measure on Z charging no semipolar set satisfying Z w e.z/dhM Œu i .z/ D 2B.u, uw/  B.u2 , w/ (6.5.9) Z

for all w 2 Jb . . /

Example 6.5.6. Let D be a domain of Rd and .ij / be a measurable family of non. /

negative definite symmetric measures and .i / be a family of locally bounded signed measures on D which are measurable relative to  . For , 2 C01 .D/, consider the

265

Section 6.5 Some stochastic calculus

bilinear form defined by E

. /

. , / D

Z

X @ @ . / dij C @xi @xj d

D

Z

iD1 D

@

. / d i . @xi

(6.5.10)

For any  2 R1 , we assume that .E . / , C0 .D// is closable on L2 .D; m/ with d m.x/ D dx. Then the right-hand side of equation (6.5.9) is equal to ³ Z Z X d ² @u @.uw/ @u2 @w . / 2  2B.u, uw/  B.u , w/ D 2 dij @xi @xj @xi @xj R1 D i,j D1

 Z X d  @u2 @u . / C uw  w d i 2 @xi @xi R1 D Z

Z D

iD1

Z

2w

R1

D

d X @u @u . / d . @xi @xj ij

i,j D1

This implies that dhM Œu i . , x/ D 2

d X @u @u . / d .x/d  . @xi @xj ij

(6.5.11)

i,j D1 . /

In particular, if dij D aij . , x/d m.x/ for a family of non-negative definite symmetric measurable functions .aij /, then d X

dhM Œu i D 2

aij . , x/

i,j D1

From this, we get that hM

Z Œu

it D 2

t

d X

0 i,j D1

aij

@u @u d. @xi @xj

@u @u .Zs /ds. @xi @xj

(6.5.12)

We have shown the decomposition in equation (6.5.8) for u 2 Wb . More generally, assume that u 2 J is given by u D w C u1  u2 for w 2 W and u1 , u2 2 P˛ \ F . Since ui 2 P˛ \ F , there exists i 2 S0 such that ui D U˛ i . Then the following result holds. Theorem 6.5.7. For any function u 2 J of the form stated above, decomposition as in equation (6.5.8) is possible if and only if 1  2 does not charge any semipolar set. In this case, Z t Œu Œw .1/ .2/ .e u1  e u2 /.Zs /ds  A t C A t (6.5.13) Nt D Nt C 0

for the PCAFs

.i/ At

associated with i for i D 1, 2.

266

Chapter 6 Time dependent Dirichlet forms

Proof. For u D w C u1  u2 2 J with w 2 W and ui 2 F \ P , if AŒu can be ` decomposed as in equation (6.5.8), then for any v 2 Jb,C  Z 1 Œu lim ˇ 2 Ev e ˇ t N t^ ` dt ˇ !1 0   D  lim ˇ u  ˇRˇ u, v/ D E.w C u1  u2 , v ˇ !1

D E.w, v/ C ˛.u1  u2 , v/  h1  2 , vi.  Rt Œu Œw .1/ .2/ .u/ u1 e u2 /.Xs /ds  A t C A t . Since N t 2N This implies that N t D N t C ˛ 0 .e .1/ .2/ is continuous and N Œw is also continuous by Theorem 6.5.4, A t C A t is a CAF and hence 1  2 does not charge any semipolar set. Conversely, if 1 2 does not charge any semipolar set, then we may consider that both of 1 and 2 do not charge any semipolar set. Then it follows that their associated Œu additive functionals are continuous. Therefore, the AF N t given by equation (6.5.13) 

Œu

Œu

belongs to N . Furthermore, it holds that A t  N t



2M . Œu

As we have seen in the proof of Theorem 6.5.7, N t (5.3.1), that is  Z 1 e ˛t N t^ ` dt Œu

lim Ev

˛!1

0

is characterized by equation

D E.u, v/

(6.5.14)

` . for all v 2 Jb,C

Example 6.5.8. Consider again the regular local Dirichlet forms given by equation . / . / (6.5.10) with dij .x/ D aij . , x/dx and i .dx/ D bi . , x/dx which have the . /

. /

common core C01 .D/. Assume further that .@=@xj /aij and bi belong locally to H for all i , j  1, that is, they coincide with some functions of H on every compact set respectively. Then, for any u 2 C02 .Z/,  Z t d d X X @ @u @u @u Œu Nt D .aij / bi (6.5.15) C .Zs /ds. @ @xi @xj @xi 0 i,j D1

iD1

In fact, the right-hand side of equation (6.5.15) is of zero energy in the weak sense and satisfies  Z 1   d d X X @ @u @u 2 ˇ t @u e .aij / bi C lim ˇ Ev .Z t^ ` /dt @ @xi @xj @xi ˇ !1 0 i,j D1

  d d X X @u @ @u @u D v, .aij / bi C @ @xi @xj @xi i,j D1

D E.u, v/, ` . for all v 2 Jb,C

iD1

iD1

Notes

Chapter 1 The results in Sections 1.1, 1.2 and 1.3 are similar to those presented in [123] and [104] for the non-symmetric Dirichlet forms. For the earlier works related to non-symmetric Dirichlet forms, see also [12, 13, 84, 87, 89, 97, 98, 106, 141, 147, 148]. Throughout this volume, we intended to extend the results of the symmetric Dirichlet forms to the lower bounded semi-Dirichlet forms. Various results on symmetric Dirichlet forms can be found in [20, 55, 58, 81, 145]. Section 1.1. Theorem 1.1.1 is taken from [150]. It is used for construction of resolvents and potentials related to the semi-Dirichlet forms. Basic relationships between semi-Dirichlet forms and their associated resolvents are given in this section. Section 1.2. Since the topology of F uses only the symmetric part of E, definition and arguments concerning the closability of a bilinear form are similar to the symmetric case. We prove in Theorem 1.2.2 that the smallest closed extension of a bilinear form satisfying the Markov property also satisfies the same property. The regularity of semi-Dirichlet forms is similarly defined to the symmetric cases. Section 1.3. The notion of strongly ¹T t º-invariant sets of semi-Dirichlet forms and their properties are shown in Vol. 2 of [81] and [94]. In this book, we also define a b t º and use it to define the irreducibility and notion of invariant sets of ¹T t º and ¹T recurrence. Our definition of irreducibilities coincides with the corresponding ones of the symmetric cases. Further, transience of semi-Dirichlet forms and related properties are studied in this section. If E is non-negative, then the extended Dirichlet form is well defined similarly to the case of a symmetric Dirichlet form. Section 1.4. If .E, F / satisfies the dual Markov property, then any positive constant is ı-coexcessive. If .E, F / is a semi-Dirichlet form, then, for any strictly positive ıb .ı/ b coexcessive function b hı , Doob’s h-transformation ¹G ˛ º of the coresolvent ¹G ˛Cı º .ı/ .ı/ is sub-Markov. In this section, a bilinear form .A , G / corresponding to the dual b .ı/ pair of resolvents ¹G˛Cı º and ¹G ˛ º is introduced. Although the sector condition for .ı/ A is not clear, the bilinear form is used in the stochastic calculus after Chapter 3. The related h-transformations of semigroups and their relations to Markov processes are also investigated by [65]. Section 1.5. In this section, typical examples of semi-Dirichlet forms of a diffusion type as well as a jump type are given. The jump type example is taken from [61]. For other examples, see e.g. [55, 78, 81, 84, 103, 104, 123, 143, 174, 175].

268

Notes

Chapter 2 In this chapter, we are concerned with an analytic potential theory related to semiDirichlet forms. By using Theorem 1.1.1 instead of Riesz theorem, we can define a capacity and quasi-notions related to it as in the symmetric case. For the related notions see e.g. [67, 84, 87, 89, 97, 98, 103, 104]. Section 2.1. We introduce an ˛-capacity of regular semi-Dirichlet forms by using the equilibrium potential. If .E, F / is non-negative, then a 0-capacity is also defined. The capacity without regularity assumption are given in [4,103,104]. For the capacity related to Lp -semigroups, see [81]. Section 2.2. In this section, using the capacity given in Section 2.1, quasi-properties such as existence of quasi-continuous versions of functions of F are shown. In particular, for any ˛ > 0, existence of a q.c. modification of G˛ f for f 2 L1 is proved. If .E, F / is transient, then the result also holds for ˛ D 0. See also [103] for more general results without regularity. Section 2.3. For an excessive function u belonging to the domain F of a regular semi-Dirichlet form, its expression as a potential of a measure is studied. In particular, as in the symmetric case, any set of zero capacity is negligible relative to such measures. Section 2.4. In this section, parts on subsets of the underlying space of semi-Dirichlet forms are considered. For an ˛-excessive function, its decomposition into a reduced function and its orthogonal complement is formulated. We also construct a ı-coexceshı is bounded from below by a positive consive function b hı 2 L1 .X ; m/ such that b stant on each compact set. In the case of ˛0 D 0, the 0-order version of the decomposition is also studied.

Chapter 3 In this chapter, Markov processes associated with regular semi-Dirichlet forms and their properties are studied. Fundamental properties related to Markov processes are stated including their proofs. More detailed results can be found in [14,20,55,81,107]. A construction of a Hunt process associated with a regular semi-Dirichlet form is given. For the correspondence between Markov processes and non-symmetric Dirichlet forms, see [4, 34, 84, 89, 103, 104, 106, 123]. Some characterizations of killing measures and jumping measures are also stated by using the Lévy system given by [90, 112]. Related analytic characterizations in the case of non-symmetric Dirichlet forms are given by [67]. Section 3.1. In this section, some necessary properties of Markov processes to be used in the stochastic calculus related to Dirichlet forms are collected. The results are mainly based on [14] and Appendix of [55].

Notes

269

Section 3.2. This section is also devoted to the study of the general theory of excessive functions of general Hunt processes. In particular, potential theory related to the Markov processes corresponding to a Dirichlet form is stated. Section 3.3. A construction of a Hunt process associated with semi-Dirichlet forms given in this section is essentially taken from [123]. For more general results on a construction of a special standard process corresponding to a quasi-regular Dirichlet form, see [32, 103, 104]. Although the dual Markov process does not exist in general for semi-Dirichlet forms, we introduce the dual pseudo Hunt process which behaves like an ordinary Hunt process. Section 3.4. In this section, an identification of equilibrium potential with hitting probability is shown. By using this, exceptionality of any Borel set and uniform convergence of an E˛ -Cauchy sequence of functions of F along the paths are formulated. Section 3.5. An orthogonal decomposition of F into the part of Dirichlet form F XnA and its orthogonal complement HA˛ is formulated. Regularity of .E, F XnA / for any closed set A is studied. The local property of Dirichlet forms in connection with diffusion processes is also studied. By using the Lévy system, some characterizations of the killing and jumping measures are stated. By an analytic method, existence of such measures is given by [67].

Chapter 4 In this chapter, we give the Revuz type correspondence between PCAFs and smooth measures. Furthermore, some related properties and the time change of Markov processes corresponding to non-symmetric Dirichlet forms by a PCAF are studied. Related results are also given by [34, 104]. In the symmetric and non-symmetric cases, related results can be found in [20, 55, 81, 84, 104, 106, 123, 145, 146]. Section 4.1. Although the underlying measure is not necessarily excessive, the Revuz measure of any PCAF is well determined by making all coexcessive functions participate in the formula. Conversely, for a smooth measure, a corresponding PCAF is constructed. Most of the results parallel to symmetric Dirichlet forms also hold by modifying the proofs. General correspondences between PCAFs and excessive functions can be found in [14, 109]. Section 4.2. For any smooth measure, an associated PCAF of the dual pseudo Hunt process is constructed and its duality relationship to the associated PCAF of the original Hunt process is studied. Section 4.3. In this section, for a fixed PCAF, time change and killing relative to the PCAF of the Markov process associated with a regular Dirichlet form are treated. But, for the time change, we need to assume that the Dirichlet forms satisfies ˛0 D 0. For the symmetric Dirichlet forms, time change, killing and their applications have been well studied. See e.g. [20, 36, 55, 96, 145].

270

Notes

Chapter 5 If .E, F / is a symmetric Dirichlet form, a decomposition of u.X t /  u.X0 / for any q.c. function u 2 F into a MAF of finite energy and a CAF of zero energy is given by [46]. Such decomposition is also possible for non-symmetric Dirichlet forms (see [84, 123]). But, for semi-Dirichlet forms, since any constant function is not coexcessive in general, we modify the notion of energy to get Fukushima’s decomposition. In this chapter, introducing a weak sense energy, we give a similar decomposition. Furthermore, some properties of the zero energy part as well as the martingale part are studied. The Beurling–Deny type decomposition is also given in this chapter. Some related results are also obtained by [67, 105]. Section 5.1. After defining the weak sense energy, Fukushima’s decomposition relative to this energy is given. For the decomposition, we use the weak sense energy of stopped additive functionals at the exist time from any set B` for an increasing hı given in Secsequence ¹B` º of finely open sets on which the coexcessive function b tion 1.4 is bounded. Section 5.2. The killing measure k and the jumping measure J corresponding to the Markov process associated with a semi-Dirichlet form are given in Chapter 3. Under an assumption on J , we give the Beurling–Deny type decomposition of semi-Dirichlet forms. Analytic proof of such decomposition for quasi-regular Dirichlet forms of any symmetric principal value integrable function is given by [67]. Section 5.3. Some characterizations of CAFs locally of zero energy are studied in this section. Most of the results are essentially the same as for the symmetric cases. Furthermore, we show that any CAF of locally zero energy in the weak sense can be represented locally as a weak sense zero energy part of suitable function of F (see [133]). Section 5.4. In this section, for a local semi-Dirichlet form, basic properties such as the derivation property of the martingale part of Fukushima’s decomposition are given. Since the mutual energy of two MAFs is symmetric, most of the corresponding results for symmetric cases are inherited by the present settings. Section 5.5. As an application of the results of Section 5.4, we consider a transformation of a diffusion process by a multiplicative functional. A characterization of Dirichlet forms corresponding to the transformed process is given (see [57, 88, 132]). Furthermore, a relation of the semi-Dirichlet forms associated with mutually absolutely continuous diffusion processes is presented. In this section, we only consider the local case. See [19] for the non-local case. Section 5.6. In this section we give general recurrence and conservativeness criteria for the Hunt processes associated with semi-Dirichlet forms. Such criteria were given in Section 1.6 in the first edition of [55] in the case of symmetric Dirichlet forms. As an example, in the case of a differential generator, the criteria are applied to give sufficient conditions for the conservativeness and recurrence. More exact condition for the conservativeness of the non-symmetric Dirichlet forms case is given by [165].

Notes

271

Chapter 6 In this chapter, a time dependent family of semi-Dirichlet forms and its associated time inhomogeneous Markov process are considered. It is essentially taken from [127, 130]. The parabolic potential theory is based on [101, 102, 110, 134]. In this chapter, we assumed that the motion of the time direction is uniform motion. More general Dirichlet forms and their associated Markov processes are given by [151]. For the related results, see also [140, 152, 171, 172]. Section 6.1. For a family of time dependent semi-Dirichlet forms, existence of the associated resolvent and their mutual relations are studied. The construction is based upon the well studied analytic method. See, for example, [101,102,134] and references therein. Section 6.2. In this section, we introduce the notion of excessive functions and equilibrium potentials by using the result in [110]. Using the equilibrium potential, we can define a capacity of Borel sets of R1  X and the quasi-notions related to it. Existence of a q.c. modification of every function belonging to W is shown in this section. Section 6.3. In this section, existence of a Markov process corresponding to a time dependent family of semi-Dirichlet forms is shown. For a given family of time dependent semi-Dirichlet forms, the domain of definition is extended to a class of functions including W , excessive functions and coexcessive functions. After extending the semiDirichlet forms, an orthogonal decomposition similar to Section 3.5 is given. Section 6.4. For a family of time dependent semi-Dirichlet forms, there exist nonexceptional semipolar sets. Hence the Revuz measures of, not necessarily continuous, additive functionals can charge semipolar sets. But, as in [14], there are one-to-one correspondences between PCAFs and measures charging no semipolar sets. In this section, such Revuz correspondences are stated. Furthermore, an application to the optimal stopping problem is studied; see [59, 113, 115, 131, 186]. Section 6.5. Stochastic calculus similar to Chapter 5 is also possible for the time dependent Dirichlet forms. In this section, only Fukushima’s decomposition in the weak sense and a characterization of the measure corresponding to the quadratic variation of the martingale part are given. For some related results on time dependent Dirichlet forms and their associated time inhomogeneous diffusion processes on moving domains; see [129, 140].

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Index

˛-coprojection, 43 ˛-projection, 43 additive functional continuous —, 112 additive functional, 112 natural, 256 positive —, 112 positive continuous —, 112 approximating form, 5 approximating sequence, 9, 19 branching points, 83 capacity, 45, 46 closable form, 9 closed form, 1 coequilibrium — measure, 58 coequilibrium potential, 44 coexcessive —˛-coexcessive, 20 coharmonic, 61 conservative, 18 continuous additive functionals zero energy, 154 contractive, 7 copotential, 56 copotential operator, 13 core, 11 coreduced function, 60 cosweeping out, 62 defining set, 112 Dirichlet form, 1, 9 —lower bounded, 9 —non-negative, 9 —non-symmetric, 9 —symmetric, 9 regular —, 11

time dependent, 217 Dynkin’s formula, 95 energy, 150 energy measure, 153 equilibrium — measure, 58 — potential, 44 exceptional, 90 excessive, 20, 77 —˛-excessive, 20 excessive function, 224 excessive modification, 240 extended Dirichlet form, 18 fine topology, 80 finely open, 80 finely continuous q.e., 97 finite energy integral, 55 first exit time, 74 Fukushima’s decomposition, 154, 158 —weak sense, 160 harmonic, 61 hitting time, 74 Hunt process, 73 invariant set, 12 irreducible, 13 irregular, 75 jumping measure, 108 killing measure, 107 local property, 247 local property —, 103

284 locally in F , 150 lower bounded, 1 MAF of finite energy weak sense, 153 MAFs of finite energy locally, 153 MAFs of finite energy in the weak sense locally, 153 Markov process, 72 strong —, 73 martingale, 76 martingale additive functional, 152 finite energy, 153 multiplicative functional, 75, 203 mutual energy, 156 weak sense—, 156 natural potential, 256 nearly Borel measurable, 74 nest, 51 regular —, 51 non-negative, 2 optimal sampling theorem, 76 optimal stopping problem, 251 part, 100 part process, 99 perturbed Dirichlet form, 140 polar, 97 polar set, 79 potential, 56 potential operator, 13 predictable stopping time, 256 pseudo Hunt process, 89 q.e. uniformly, 53 quasi-closed, 62 quasi-continuous, 51 — modification, 52

Index quasi-everywhere, 51 quasi-left continuous, 73 quasi-open, 62 recurrent, 13 reduced function, 60 reference form, 26 regular, 75 regular potential, 249 Revuz measure, 117 sector condition, 1 semipolar set, 79 sharp bracket, 153 smallest closed extention, 9 smooth measure, 119 standard process, 73 stochastic integral, 197 stochastic process, 72 stopping time, 73 strong Feller property, 97 sub-Markov resolvent, 7 submartingale, 76 supermartingale, 76 supermedian, 77 support of PCAF, 132 sweeping out, 62 symmetric, 2 thin, 79 time changed process, 141 transient, 13 universally measurable, 72 weak sense energy, 150 zero energy locally, 154 locally in the weak sense, 154 weak sense, 154