Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35) (London Mathematical Society Monographs)

SYMMETRIC MARKOV PROCESSES, TIME CHANGE, AND BOUNDARY THEORY London Mathematical Society Monographs Editors: Martin B...

Author: Zhen-Qing Chen | Masatoshi Fukushima

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SYMMETRIC MARKOV PROCESSES, TIME CHANGE, AND BOUNDARY THEORY

London Mathematical Society Monographs Editors: Martin Bridson, Ben Green, and Peter Sarnak Editorial Advisers: J. H. Coates, W. S. Kendall, and János Kollár The London Mathematical Society Monographs Series was established in 1968. Since that time it has published outstanding volumes that have been critically acclaimed by the mathematics community. The aim of this series is to publish authoritative accounts of current research in mathematics and highquality expository works bringing the reader to the frontiers of research. Of particular interest are topics that have developed rapidly in the last ten years but that have reached a certain level of maturity. Clarity of exposition is important and each book should be accessible to those commencing work in its field. The original series was founded in 1968 by the Society and Academic Press; the second series was launched by the Society and Oxford University Press in 1983. In January 2003, the Society and Princeton University Press united to expand the number of books published annually and to make the series more international in scope. Vol. 35, Symmetric Markov Processes, Time Change, and Boundary Theory, by Zhen-Qing Chen and Masatoshi Fukushima Vol. 34, Log-Gases and Random Matrices, by Peter J. Forrester Vol. 33, Prime-Detecting Sieves, by Glyn Harman Vol. 32, The Geometry and Topology of Coxeter Groups, by Michael W. Davis Vol. 31, Analysis of Heat Equations on Domains, by El Maati Ouhabaz

SYMMETRIC MARKOV PROCESSES, TIME CHANGE, AND BOUNDARY THEORY

Zhen-Qing Chen Masatoshi Fukushima

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

c 2011 by Princeton University Press Copyright Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved ISBN: 978-0-691-13605-9 Library of Congress Control Number: 2011934606 British Library Cataloging-in-Publication Data is available This book has been composed in Times Roman Printed on acid-free paper ∞ press.princeton.edu Typeset by S R Nova Pvt Ltd, Bangalore, India Printed in the United States of America 10

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To Chen, Minda, and Angela Z .- Q . C . To Masako M . F.

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Contents

Notation

ix

Preface

xi

Chapter 1. SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS

1

1.1 1.2 1.3 1.4 1.5

Dirichlet Forms and Extended Dirichlet Spaces Excessive Functions and Capacities Quasi-Regular Dirichlet Forms Quasi-Homeomorphism of Dirichlet Spaces Symmetric Right Processes and Quasi-Regular Dirichlet Forms

Chapter 2. BASIC PROPERTIES AND EXAMPLES OF DIRICHLET FORMS 2.1 2.2 2.3 2.4

Transience, Recurrence, and Irreducibility Basic Examples Analytic Potential Theory for Regular Dirichlet Forms Local Properties

Chapter 3. SYMMETRIC HUNT PROCESSES AND REGULAR DIRICHLET FORMS 3.1 3.2 3.3 3.4 3.5

Relations between Probabilistic and Analytic Concepts Hitting Distributions and Projections I Quasi Properties, Fine Properties, and Part Processes Hitting Distributions and Projections II Transience, Recurrence, and Path Behavior

Chapter 4. ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES 4.1 4.2 4.3

Positive Continuous Additive Functionals and Smooth Measures Decompositions of Additive Functionals of Finite Energy Probabilistic Derivation of Beurling-Deny Formula

1 15 24 30 33

37 37 50 77 88

92 92 103 106 113 118

130 130 143 151

viii

CONTENTS

Chapter 5. TIME CHANGES OF SYMMETRIC MARKOV PROCESSES 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Subprocesses and Perturbed Dirichlet Forms Time Changes and Trace Dirichlet Forms Examples Energy Functionals for Transient Processes Trace Dirichlet Forms and Feller Measures Characterization of Time-Changed Processes Excursions, Exit System, and Feller Measures More Examples

Chapter 6. REFLECTED DIRICHLET SPACES 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Terminal Random Variables and Harmonic Functions Reflected Dirichlet Spaces: Transient Case Recurrent Case Toward Quasi-Regular Cases Examples Silverstein Extensions Equivalent Notions of Harmonicity

Chapter 7. BOUNDARY THEORY FOR SYMMETRIC MARKOV PROCESSES 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Reflected Dirichlet Space for Part Processes Douglas Integrals and Reflecting Extensions Lateral Condition for L2 -Generator Countable Boundary One-Point Extensions Examples of One-Point Extensions Many-Point Extensions Examples of Many-Point Extensions

Appendix A. ESSENTIALS OF MARKOV PROCESSES A.1 A.2 A.3 A.4

Markov Processes Basic Properties of Borel Right Processes Additive Functionals of Right Processes Review of Symmetric Forms

166 168 174 190 202 206 221 225 232 240 241 246 260 262 269 275 283

300 302 310 324 334 340 352 369 377 391 391 413 423 440

Appendix B. SOLUTIONS TO EXERCISES

443

Notes

451

Bibliography

457

Catalogue of Some Useful Theorems

467

Index

473

Notation 1. R: set of all real numbers, R+ : set of all non-negative real numbers, Q: set of all rational numbers, Q+ : set of all non-negative rational numbers, Z: set of all integers, N: set of all natural numbers 2. A [−∞, ∞]-valued function is called a numerical function. An Rvalued function is called a real function. For a family H of numerical functions, bH (resp. H+ ) denotes the family of all bounded (resp. [0, ∞]-valued) functions in H. Sometimes we also denote bH by Hb . 3. For a, b ∈ [−∞, ∞], a ∨ b and a ∧ b denote the larger and smaller one, respectively. We let a+ = a ∨ 0 and a− := (−a) ∨ 0. For numerical functions f , g, we define ( f ∨ g)(x) = f (x) ∨ g(x), f + (x) := ( f (x))+ , and so on. 4. If f is a measurable map from a measurable space (E, B) to another measurable space (F, G), we write f ∈ B/G. If in particular f is a measurable numerical function on a measurable space (E, B(E)), we write f ∈ B(E) by regarding B(E) as the space of all B(E)-measurable functions on E. Further, f ∈ bB(E) ( f ∈ B+ (E)) indicates that f is a bounded (non-negative) measurable function on E. 5. The symmetric difference (A \ B) ∪ (B \ A) of sets A and B is denoted by AB. 6. The characteristic function (indicator function) 1B of a set B is defined by 1 x∈B 1B (x) = 0 x∈ / B. 7. Rn denotes the n-dimensional Euclidean space for n ∈ N. R1 is denoted by R. For x ∈ Rn , A, B ⊂ Rn , we define A − x = {y − x : y ∈ A} and A − B = {y − x : y ∈ A, x ∈ B}. 8. For ∈ N, the contractive real functions ϕ and ϕ are defined by ϕ (t) = ((−) ∨ t) ∧ ,

t ∈ R,

ϕ (t) = t − ((−1/) ∨ t) ∧ (1/), 9. f = g [m] means that f = g m-a.e.

t ∈ R.

x

NOTATION

10. A ⊂ B m-a.e. (respectively, A = B m-a.e.) means that m(A \ B) = 0 (respectively, m(AB) = 0). 11. For the Lp -norm in the space Lp (E; m) is denoted as f p = p ≥ 1, p ( E | f (x)| m(dx))1/p . 12. δa denotes the probability measure concentrated at a point a. 13. For a measure µ and a non-negative function f ∈ B+ , the measure A → A f (x)µ(dx) is denoted by f · µ or by f µ. 14. For a measure µ and a function f on E, the integral E fdµ is denoted by µ, f or by f , µ or by µ( f ). 15. σ { · }: the smallest σ -field making the family of sets or functions { · } measurable. Sometimes we also use σ ( · ) for σ { · }. 16. For a Markov process X, Px -a.s. means that Px -a.e. on . Further, almost surely or a.s. means that Px -a.s. for every x ∈ E. 17. C(E) denotes the space of all continuous real functions on a topological space E. When E is locally compact, Cc (E) (C∞ (E)) denotes the family of all functions in C(E) with compact support (vanishing at infinity). For a metric space (E, d), we denote by Cu (E) the collection of all d-uniformly continuous real functions on E. 18. We use := as a means of definition. For a Markov process X and a subset A on its state space, we use σA := inf{t > 0 : Xt ∈ A} and / A} to denote the first hitting time of A and the τA := inf{t > 0 : Xt ∈ first exit time from A, respectively, by X. The entrance time of A by X is defined to be σ˙ A := inf{t ≥ 0 : Xt ∈ A}. We use the convention that inf ∅ = ∞.

Preface

The seminal paper of A. Beurling and J. Deny on the theory of Dirichlet spaces appeared in 1959, just half a century ago. Since then the theory of Dirichlet spaces has been growing in close relationship with the theory of Markov processes, especially symmetric Markov processes. The scope of the original Dirichlet space theory has since been greatly expanded both in theory and applications. Books bearing the term “Dirichlet forms” or “symmetric Markov processes” in the title continue to be published. For instance, the following volumes, listed in the chronological order, are devoted to this theory: [138, 63, 140, 64, 15, 119, 47, 73, 100, 141]. In 1960’s and 1970’s, the study of the Beurling-Deny theory was motivated by a desire to comprehend and develop the boundary theory for Markov processes as is evident in the papers by M. Fukushima, H. Kunita, M. L. Silverstein, and Y. LeJan and in the two books of Silverstein. Indeed, the concept of the reflected Dirichlet space and the space of functions with finite Douglas integrals involved there were the outgrowth of the idea in the preceding works by W. Feller in 1957 and J. L. Doob in 1962 reinterpreted in terms of Dirichlet forms. But the study in this direction was left halfway until Z.-Q. Chen gave an appropriate reformulation of the reflected Dirichlet space in 1992. Over the last 30 years, time changes of symmetric Markov processes have been extensively studied. It is well understood now that time change of a symmetric Markov process by means of a positive continuous additive functional (PCAF in abbreviation) with full support corresponds precisely to the replacement of the symmetrizing measure while keeping the extended Dirichlet space invariant. The relevant stochastic calculus is also well developed accompanied by basic decomposition theorems of (not necessarily positive) additive functionals. However, the intrinsic Beurling-Deny decomposition of the trace Dirichlet form or characterization of the time-changed process by a non–fully supported PCAF in terms of a (generalized) Douglas integral has been obtained only quite recently; this is the topic of Chapter 5 of this book. The notion of the quasi-regular Dirichlet form due to S. Albeverio, Z.-M. Ma, and M. Röckner and a related result of P. J. Fitzsimmons have

xii

PREFACE

enabled us to reduce the study of a general symmetric (not necessarily Borel) right process to the study of a symmetric Borel special standard process. It is established by Z.-Q. Chen, Z.-M. Ma, and M. Röckner that a Dirichlet form is quasi-regular if and only if it is quasi-homeomorphic to a regular Dirichlet form on a locally compact separable metric space. This quasi-homeomorphism allows one to transfer problems concerning those quasi-regular Dirichlet forms and symmetric right processes on (possibly infinite dimensional) general Hausdorff topological spaces to problems for regular Dirichlet forms and symmetric Hunt processes on locally compact separable metric spaces. The development of the Dirichlet form theory benefits from its interaction with other areas of probability theory such as the theory of general Markov processes, martingale theory, and stochastic analysis, and with analytic potential theory, harmonic analysis, Riemannian geometry, theory of function spaces, partial differential equations and pseudo-differential operators, and mathematical physics. On the other hand, Dirichlet form theory has wide range of applications to these fields. For example, it is an effective tool in studying various probabilistic models as well as analytic problems with nonsmooth data or in non-smooth media, such as reflecting Brownian motion on non-smooth domains, Brownian motion with random obstacles, diffusions and analysis on fractals, Markov processes and analysis on metric measure spaces, and diffusion processes and differential analysis on path spaces or loop spaces over a compact Riemannian manifold. It is a powerful machinery in studying various stochastic differential equations in infinite dimensional spaces, stochastic partial differential equations, various models in statistical physics such as quantum field theory and interacting particle systems. It also provides a probabilistic means to study various problems in partial differential equations with singular coefficients, analytic potential theory, and theory of function spaces. The aim of this book is twofold. First, it gives a systematic introduction to the essential ingredients of both the probabilistic part and the analytic part of the theory of quasi-regular Dirichlet form. This is done in the first four and a half chapters of the book, where the theory of quasi-regular Dirichlet form and that of regular Dirichlet form are developed in a unified way. Second, it presents some recent developments of the theory in the last two and a half chapters. Its aim is, along with the characterization of the trace Dirichlet form by the Douglas integral, to give a comprehensive account of the reflected Dirichlet space and then to show the important role they play in the recent development of the boundary theory of symmetric Markov processes. We strived to make the contents of this book self-contained so that it, especially its first four and a half chapters, can be used as a textbook for advanced graduate students. Chapters 2, 3, 5, 6, and 7 contain many examples illustrating the theory presented. Exercises given throughout the book are an integral part of the book. The solutions to these exercises are given in Appendix B.

PREFACE

xiii

The rest of the book is organized as follows. In Chapter 1, we introduce the concepts of Dirichlet form on L2 (E, B; m) and its extended Dirichlet space, where (E, B) is a measurable space without any topological assumption imposed on E. In the remaining sections of Chapter 1, we give a quick introduction to the basic theory of quasi-regular Dirichlet forms, where E is assumed to be a topological Hausdorff space with the Borel σ -field B(E) being generated by the continuous functions on E. After the concept of a quasi-regular Dirichlet form is introduced, it is shown that every quasi-regular Dirichlet form is quasi-homeomorphic to a regular Dirichlet space on a locally compact separable metric space. In Chapter 2, we investigate the transience and recurrence of the semigroups associated with general Dirichlet forms. Analytic potential theory for regular Dirichlet forms, such as capacity, smooth measures, and their potentials, are studied in Section 2.3. Various equivalent characterizations of the local property for a quasi-regular Dirichlet form are given in Section 2.4. Some basic examples of Dirichlet forms corresponding to symmetric Markov processes are presented in Section 2.2, including symmetric pure jump step processes, symmetric Lévy processes, one-dimensional diffusions, multidimensional Brownian motions, and Brownian motions on manifolds. An example of a quasi-regular but not regular Dirichlet form on Rn is given in Example 5.1.11. Probabilistic potential theory of symmetric Markov processes and its relationship to analytic potential theory of the associated Dirichlet forms are presented in Chapter 3. In Chapter 4, additive functionals of symmetric Markov processes are studied. In particular, the one-to-one correspondence between positive continuous additive functionals and the smooth measures is established. Fukushima decomposition, which serves as a counterpart of Itô’s formula for symmetric Markov processes, is presented in Section 4.2. It plays an important role in analyzing the sample path properties of the processes as well as in other areas of the Dirichlet form theory. Beurling-Deny decomposition of a regular Dirichlet form is derived in Section 4.3 by utilizing martingale additive functionals. The first half of Chapter 5 is devoted to a study of time changes of symmetric Markov processes, their Dirichlet form characterization, and applications. In the second half, Feller measures are introduced. They are used to characterize trace Dirichlet forms and to identify jump and killing measures of the timechanged process. The reflected Dirichlet space of a Dirichlet form is introduced and investigated in Chapter 6. It is first introduced under the regular Dirichlet form setting. The transient and recurrent cases are treated separately. It is then extended to the quasi-regular Dirichlet form setting by using quasi-homeomorphism. Concrete examples of reflected Dirichlet spaces are exhibited for a number

xiv

PREFACE

of Dirichlet forms including most of those appearing in Section 2.2. The important role that reflected Dirichlet spaces would play in the boundary theory for symmetric Markov processes is indicated by the fact that the active reflected Dirichlet form is the maximal Silverstein extension of the Dirichlet form. In Chapter 7, we present some recent developments of boundary theory of symmetric Markov processes, emphasizing the role of reflected Dirichlet spaces and the function spaces of finite Douglas integrals. In the second half of Chapter 7, we develop the theory when the boundary is countable and give many concrete illustrative examples. In Appendix A, we present basic materials on (not necessarily symmetric) right processes that are utilized in the text. For readers’ convenience, an index of some useful results is provided in the Catalogue of Some Useful Theorems at the end of the book. The material starting from Section 5.5 to the end of Chapter 7 appears here for the first time in a book. Except for a few sections, most results in other parts of the book are not new. However, their proofs and presentations can be found to be novel in many places. As compared to the book by M. Fukushima, Y. Oshima, and M. Takeda published in 1994 [73], the approach presented in this book is more probabilistic and the framework of a quasi-regular Dirichlet form is employed for the first time in parallel with a regular one. On the other hand, [73] contains more detailed expositions of analytic properties of a regular Dirichlet form as well as some other relevant topics including a construction of an associated Hunt process and a Girsanovtype transformation. We refer readers to the book by Z. M. Ma and M. Röckner [119] published in 1992 for further readings on quasi-regular (non-symmetric) Dirichlet forms, in particular for basic concrete examples of quasi-regular Dirichlet forms in infinite dimensions that are not touched upon in the present volume. The materials in Sections 3.5 and 5.4 and Appendix A owe a lot to the work by P. J. Fitzsimmons, R. K. Getoor, P. A. Meyer, and M. J. Sharpe on right processes, additive functionals, and energy functionals. The notes at the end of this book provide information on other closely related books, sources of materials, and related literature. In the bibliography, we list only literature that is directly linked to the topics of this book. But we admit with apology that it is still far from being complete partly due to a great diversity and vast literature of related areas. In August 2003, the second-named author gave an invited lecture series of London Mathematical Society at the University of Wales Swansea, kindly arranged by N. Jacob. The lecture notes contained a time-change theory as well as a preliminary account of the Douglas integrals for diffusions. They eventually grew into the present book, in collaboration with the first author. We are indebted to N. Jacob for creating an opportunity to write the preliminary version of the book.

PREFACE

xv

Thanks are due to M. Takeda for allowing us to use several ingredients from the recently published Japanese book by Fukushima and Takeda [74]. We are grateful to K. Burdzy, W. T. Fan, M. Hino, N. Kajino, P. Kim, K. Kuwae, S. Lou, and Y. Oshima for reading parts of the manuscript and providing us with helpful comments and lists of typos. We thank the reviewers of this book for their helpful comments. We also thank Vickie Kearn and Stefani Wexler at Princeton University Press for their truly kind assistance during the preparation of this manuscript. The research of the authors was supported in part by NSF grants in the United States and a Grant-in-Aid for Scientific Research in Japan. Zhen-Qing Chen Seattle Masatoshi Fukushima Osaka


Chapter One SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS

1.1. DIRICHLET FORMS AND EXTENDED DIRICHLET SPACES The concepts of Dirichlet form and Dirichlet space were introduced in 1959 by A. Beurling and J. Deny [8] and the concept of the extended Dirichlet space was given in 1974 by M. L. Silverstein [138]. They all assumed that the underlying state space E is a locally compact separable metric space. Concrete examples of Dirichlet forms (bilinear form, weak solution formulations) have appeared frequently in the theory of partial differential equations and Riemannian geometry. However, the theory of Dirichlet forms goes far beyond these. In this section, we work with a σ -finite measure space (E, B(E), m) without any topological assumption on E and establish the correspondence of the above-mentioned notions to the semigroups of symmetric Markovian linear operators. The present arguments are a little longer than the usual ones under the topological assumption found in [39] and [73, §1.4] but they are quite elementary in nature. Only at the end of this section, we shall assume that E is a Hausdorff topological space and consider the semigroups and Dirichlet forms generated by symmetric Markovian transition kernels on E. Let (E, B(E)) be a measurable space and m a σ -finite measure on it. Let Bm (E) be the completion of B(E) with respect to m. Numerical functions f , g on E are said to be m-equivalent (f = g [m] in notation) if m({x ∈ E : f (x) = g(x)}) = 0. For p ≥ 1 and a numerical function f ∈ B m (E), we put f p =

1/p |f (x)|p m(dx)

.

E

The family of all m-equivalence classes of f ∈ B m (E) with f p < ∞ is denoted by Lp (E; m), which is a Banach space with norm · p , namely, a complete normed linear space. We denote by L∞ (E; m) the family of all mequivalence classes of f ∈ Bm (E) which are bounded m-a.e. on E. L∞ (E; m) is

2

CHAPTER ONE

a Banach space with norm f ∞ :=

sup |f (x)|.

inf

N: m(N)=0 x∈E\N

Note that L2 (E; m) is a real Hilbert space with inner product (f , g) =

f (x)g(x)m(dx),

f , g ∈ L2 (E; m).

E

For a moment, √ let us consider an abstract real Hilbert space H with inner product (·, ·). (f , f ) for f ∈ H is denoted by f H . As is summarized in Section A.4, there are mutual one-to-one correspondences among four objects on the Hilbert space H: the family of all closed symmetric forms (E, D(E)), the family of all strongly continuous contraction semigroups {Tt ; t ≥ 0}, the family of all strongly continuous contraction resolvents {Rα ; α > 0}, and the family of all non-positive definite self-adjoint operators A. Here we mention the correspondences among the first three objects only. E or (E, D(E)) is said to be a symmetric form on H if D(E) is a dense linear subspace of H and E is a non-negative definite symmetric bilinear form defined on D(E) × D(E) in the sense that for every f , g, h ∈ D(E) and a, b ∈ R E(f , g) = E(g, f ),

E(f , f ) ≥ 0,

and

E(af + bg, h) = aE(f , h) + bE(g, h). For α > 0, we define Eα (f , g) = E(f , g) + α(f , g),

f , g ∈ D(E).

We call √ a symmetric form (E, D(E)) on H closed if D(E) is complete with norm E1 (f , f ). D(E) is then a real Hilbert space with inner product Eα for each α > 0. A family of symmetric linear operators {Tt ; t > 0} on H is called a strongly continuous contraction semigroup if, for any f ∈ H, Ts Tt f = Ts+t f ,

Tt f H ≤ f H ,

lim Tt f − f H = 0. t↓0

We call a family of symmetric linear operators {Gα ; α > 0} on H a strongly continuous contraction resolvent if for every α, β > 0 and f ∈ H, Gα f − Gβ f + (α − β)Gα Gβ f = 0, lim αGα f − f H = 0.

α→∞

αGα f H ≤ f H ,

3

SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS

The semigroup {Tt ; t ≥ 0} and the resolvent {Gα ; α > 0} as above correspond to each other by the next two equations:

∞

Gα f =

e−αt Tt fdt,

f ∈ H,

(1.1.1)

0

the integral on the right hand side being defined in Bochner’s sense, and Tt f = lim e−tβ β→∞

∞ (tβ)n n=0

n!

(βGβ )n f ,

f ∈ H.

(1.1.2)

{Gα ; α > 0} determined by (1.1.1) from {Tt ; t > 0} is called the resolvent of {Tt ; t ≥ 0}. Given a strongly continuous contraction symmetric semigroup {Tt ; t > 0} on H, for each t > 0, E (t) (f , g) :=

1 (f − Tt f , g), t

f,g ∈ H

(1.1.3)

defines a symmetric form E (t) on H with domain H. For each f ∈ H, E (t) (f , f ) is non-negative and increasing as t > 0 decreases (this can be shown, for example, by using spectral representation of {Tt ; t > 0}). We may then set D(E) = {f ∈ H : lim E (t) (f , f ) < ∞}, t↓0

E(f , g) = lim E (t) (f , g), t↓0

f , g ∈ D(E),

(1.1.4) (1.1.5)

which becomes a closed symmetric form on H called the closed symmetric form of the semigroup {Tt ; t > 0}. We call E (t) of (1.1.3) the approximating form of E. Conversely, suppose that we are given a closed symmetric form (E, D(E)) on H. For each α > 0, f ∈ H and v ∈ D(E), we have |(f , v)| ≤ f 2 v2 ≤ (1/α)1/2 f 2 Eα (v, v), which means that (v) = (f , v) is a bounded linear functional on the Hilbert space (D(E), Eα ). By the Riesz representation theorem, there exists a unique element of D(E) denoted by Gα f such that for every f ∈ H and v ∈ D(E), Gα f ∈ D(E)

and

Eα (Gα f , v) = (f , v).

(1.1.6)

{Gα ; α > 0} so defined is a strongly continuous contraction resolvent on H, which in turn determines a strongly continuous contraction semigroup

4

CHAPTER ONE

{Tt ; t > 0} on H by (1.1.2). They are called the resolvent and semigroup generated by the closed symmetric form (E, D(E)), respectively. The above-mentioned correspondences from {Tt ; t > 0} to (E, D(E)) and from (E, D(E)) to {Tt ; t > 0} are mutually reciprocal. From now on, we shall take as H the space L2 (E; m) on a σ -finite measure space (E, B(E), m). In this book, we need to consider extensions of the domain D(E) of a closed symmetric form E on L2 (E; m). For this purpose, we shall designate D(E) by F so that a closed symmetric form on L2 (E; m) will be denoted by (E, F). We now proceed to introduce the notions of Dirichlet form and extended Dirichlet space. D EFINITION 1.1.1. For 1 ≤ p ≤ ∞, a linear operator L on Lp (E; m) with domain of definition D(L) is called Markovian if f ∈ D(L) with 0 ≤ f ≤ 1 [m]

=⇒

0 ≤ Lf ≤ 1 [m].

A real function ϕ, namely, a mapping from R to R, is said to be a normal contraction if ϕ(0) = 0

and

|ϕ(s) − ϕ(t)| ≤ |s − t| for every s, t ∈ R.

A function defined by ϕ(t) = (0 ∨ t) ∧ 1, t ∈ R, is a normal contraction which is called the unit contraction. For any ε > 0, a real function ϕε satisfying the next condition is a normal contraction: ϕε (t) = t for t ∈ [0, 1]; 0 ≤ ϕε (t) − ϕε (s) ≤ t − s

−ε ≤ ϕε (t) ≤ 1 + ε for t ∈ R, for s < t.

(1.1.7)

D EFINITION 1.1.2. A symmetric form (E, D(E)) on L2 (E; m) is called Markovian if, for any ε > 0, there exists a real function ϕε satisfying (1.1.7) and f ∈ D(E)

=⇒

g := ϕε ◦ f ∈ D(E) with E(g, g) ≤ E(f , f ).

(1.1.8)

A closed symmetric form (E, F) on L2 (E; m) is called a Dirichlet form if it is Markovian. In this case, the domain F is said to be a Dirichlet space. T HEOREM 1.1.3. Let (E, F) be a closed symmetric form on L2 (E; m) and {Tt }t>0 , {Gα }α>0 be the strongly continuous contraction semigroup and resolvent on L2 (E; m) generated by (E, F), respectively. Then the following conditions are mutually equivalent: (a) Tt is Markovian for each t > 0. (b) αGα is Markovian for each α > 0. (c) (E, F) is a Dirichlet form on L2 (E; m).

5


(d) The unit contraction operates on (E, F): f ∈ F =⇒ g := (0 ∨ f ) ∧ 1 ∈ F and E(g, g) ≤ E(f , f ). (e) Every normal contraction operates on (E, F ): for any normal contraction ϕ f ∈ F =⇒ g = ϕ ◦ f ∈ F and E(g, g) ≤ E(f , f ). Proof. The implications (a) ⇒ (b) and (b) ⇒ (a) follow from (1.1.1) and (1.1.2), respectively. The implication (e) ⇒ (d) ⇒ (c) is obvious. (c) ⇒ (b): We fix an α > 0 and a function f ∈ L2 (E; m) with 0 ≤ f ≤ 1 [m], and introduce a quadratic form on F by (v) = E(v, v) + α v − αf , v − αf , v ∈ F. It follows from (1.1.6) that (Gα f ) + Eα (Gα f − v, Gα f − v) = (v),

v ∈ F,

namely, Gα f is a unique element of F minimizing (v). Suppose (E, F) is a Dirichlet form on L2 (E; m). There exists then for any ε > 0 a real function ϕε (t) = (1/α)ϕαε (αt), u = ϕε ◦ Gα f to ϕε satisfying (1.1.7) and (1.1.8). We let obtain u ∈ F and E(u, u) ≤ E(Gα f , Gα f ). Since | ϕε (t) − s| ≤ |t − s| for every s ∈ [0, 1/α] and t ∈ R, we have |u(x) − f (x)/α| ≤ |Gα f (x) − f (x)/α| [m] and (u − f /α, u − f /α) ≤ (Gα f − f /α, Gα f − f /α). Therefore, (u) ≤ (Gα f ) and consequently u = Gα f [m], which means that −ε ≤ Gα f ≤ 1/α + ε [m]. Letting ε → 0, we get (b). It remains to prove the implication (a) ⇒ (e), which will follow from a more general theorem formulated below. In what follows, we occasionally use for a symmetric form (E, F ) on L2 (E; m) the notations f E := E(f , f ), f Eα := Eα (f , f ), f ∈ F, α > 0 D EFINITION 1.1.4. Let (E, F ) be a closed symmetric form on L2 (E; m). Denote by Fe the totality of m-equivalence classes of all m-measurable functions f on E such that |f | < ∞ [m] and there exists an E-Cauchy sequence {fn , n ≥ 1} ⊂ F such that limn→∞ fn = f m-a.e on E. {fn } ⊂ F in the above is called an approximating sequence of f ∈ Fe . We call the space Fe the extended space attached to (E, F). When the latter is a Dirichlet form on L2 (E; m), the space Fe will be called its extended Dirichlet space.

6

CHAPTER ONE

T HEOREM 1.1.5. Let (E, F) be a closed symmetric form on L2 (E; m) and Fe be the extended space attached to it. If the semigroup {Tt ; t > 0} generated by (E, F) is Markovian, then the following are true: (i) For any f ∈ Fe and for any approximating sequence {fn } ⊂ F of f , the limit E(f , f ) = limn→∞ E(fn , fn ) exists independently of the choice of an approximating sequence {fn } of f . (ii) Every normal contraction operates on (Fe , E): for any normal contraction ϕ f ∈ Fe =⇒ g := ϕ ◦ f ∈ Fe ,

E(g, g) ≤ E(f , f ).

(iii) F = Fe ∩ L2 (E; m). In particular, (E, F) is a Dirichlet form on L2 (E; m). Assertion (ii) of this theorem implies the implication (a) ⇒ (e) in Theorem 1.1.3, completing the proof of Theorem 1.1.3. For f , g ∈ Fe , clearly both f + g and f − g are in Fe . Define E(f , g) = 1 (E(f + g, f + g) − E(f − g, f − g)), which is a symmetric bilinear form over 4 Fe . (E, Fe ) is called the extended Dirichlet form of (E, F). If a given closed symmetric form (E, F) on L2 (E; m) is a Dirichlet form, then the corresponding semigroup {Tt ; t > 0} is Markovian by virtue of the already proven implication (c) ⇒ (a) of Theorem 1.1.3. So the extended Dirichlet space Fe satisfies all properties mentioned in Theorem 1.1.5. Before giving the proof of Theorem 1.1.5, we shall fix a Markovian contractive symmetric linear operator T on L2 (E; m) and make some preliminary observations on T. By the linearity and the Markovian property of T on L2 (E; m) ∩ L∞ (E; m), f1 , f2 ∈ L2 ∩ L∞ , 0 ≤ f1 ≤ f2 [m] =⇒ 0 ≤ Tf1 ≤ Tf2 ≤ f2 ∞ [m]. Due to the σ -finiteness of m, we can construct a Borel function η ∈ L1 (E; m) which is strictly positive on E. If we put ηn (x) = (nη(x)) ∧ 1, then 0 < ηn ≤ 1, ηn ↑ 1, n → ∞. Hence we can define an extension of T from L2 (E; m) ∩ L∞ (E; m) to L∞ (E; m) as follows: ∞ Tf (x) := limn→∞ T(f · ηn )(x) [m], f ∈ L+ (E; m), (1.1.9) + − ∞ Tf := Tf − Tf , f ∈ L (E; m). By the symmetry of T, (g, T(f · ηn )) = (Tg, f · ηn ) for g ∈ bL1 (E; m). Letting n → ∞, we see that the function Tf , f ∈ L∞ (E; m), defined by (1.1.9), satisfies the identity (1.1.10) g, Tf = Tg, f for every g ∈ bL1 (E; m),

where g, f denotes the integral E gfdm for g ∈ L1 (E; m), f ∈ L∞ (E; m). Consequently, Tf is uniquely determined up to the m-equivalence for

7


f ∈ L∞ (E; m). T becomes a Markovian linear operator on L∞ (E; m) and satisfies ∞ fn , f ∈ L+ (E; m), fn ↑ f [m] =⇒ lim Tfn = Tf [m]. n→∞

(1.1.11)

Further, if a sequence {fn } ⊂ L∞ (E; m) is uniformly bounded and converges to f m-a.e. as n → ∞, lim g, Tfn = g, Tf

n→∞

for every g ∈ bL1 (E; m).

(1.1.12)

L EMMA 1.1.6. (i) For any g ∈ L∞ (E; m), T(g2 ) − 2gTg + g2 T1 ≥ 0 [m]. (ii) For any g ∈ L∞ (E; m), define 1 2 AT (g) = T(g ) − 2gTg + g2 T1 dm + g2 (1 − T1)dm. 2 E E

(1.1.13)

It holds for g ∈ L2 (E; m) ∩ L∞ (E; m) that AT (g) = (g − Tg, g).

(1.1.14)

(iii) For any g ∈ L∞ (E; m) and for any normal contraction ϕ, AT (ϕ ◦ g) ≤ AT (g).

(1.1.15)

(iv) For any f , g ∈ L∞ (E; m), AT (f + g)1/2 ≤ AT (f )1/2 + AT (g)1/2 .

(1.1.16)

Proof. (i) For a simple function on E expressed by s=

n

ai 1Bi ,

(1.1.17)

i=1

where ai ∈ R, Bi ∈ B(E) with Bi ∩ Bj = ∅ for i = j and ∪ni=1 Bi = E, we have T(g2 ) − 2sTg + s2 T1 =

n

1Bi T (g − ai )2 ≥ 0

[m].

(1.1.18)

i=1

Hence it suffices to choose an increasing sequence of simple functions {s , ≥ 1} of this type such that lim s = g

→∞

[m].

(ii) Recall that {ηn , n ≥ 1} is an increasing sequence of positive functions that is defined preceding (1.1.9). For g ∈ L2 ∩ L∞ , we have (Tg2 , ηn ) = (g2 , Tηn )

8

CHAPTER ONE

by the symmetry of T. By letting n → ∞, we get E Tg2 dm = E g2 T1dm <

∞ and (g − Tg, g) = 12 E 2g2 T1 − 2gTg dm + E g2 (1 − T1)dm = AT (g). (iii) For g ∈ L∞ (E; m) and k = 1, 2, . . . , we put 1 T(g2 ) − 2gTg + g2 T1, ηk + g2 (1 − T1), ηk . 2 When g is a simple function of the type (1.1.17), 1 (ai − aj )2 Jijk + a2i κik . AkT (s) = 2 1≤i,j≤n 1≤i≤n AkT (g) =

Here Jijk = E (T1Bi )1Bj ηk dm, κik = E 1Bi (1 − T1)ηk dm. For any normal contraction ϕ, it holds that (ϕ(ai ) − ϕ(aj ))2 ≤ (ai − aj )2

ϕ(ai )2 ≤ a2i .

and

Thus for a simple function s, AkT (ϕ ◦ s) ≤ AkT (s). For any g ∈ L∞ (E; m), we can take uniformly bounded simple functions s with lim→∞ s = g [m] to obtain AkT (ϕ ◦ s ) ≤ AkT (s ). Letting → ∞ and then k → ∞, we have by (1.1.12) that (1.1.15) holds. (iv) It suffices to show the triangular inequality (1.1.16) for AkT for each fixed k instead of AT . Since 0 ≤ AkT (g) < ∞, g ∈ L∞ (E; m), the bilinear form defined by 1 k AT (f + g) − AkT (f − g) , 4 satisfies the Schwarz inequality AkT (f , g) =

f , g ∈ L∞ (E; m),

|AkT (f , g)| ≤ AkT (f )1/2 · AkT (g)1/2 ,

from which follows the desired triangular inequality. Let {ϕ , > 0} be a specific family of normal contractions defined by ϕ (t) = ((−) ∨ t) ∧ ,

t ∈ R.

(1.1.19)

For any m-measurable function g on E with |g| < ∞ [m], AT (ϕ ◦ g) is increasing as increases. This is clear from ϕ ◦ (ϕ +1 ◦ g) = ϕ ◦ g and Lemma 1.1.6(iii). We can then extend AT (g) to g by letting AT (g) = lim AT (ϕ ◦ g) (≤ ∞). →∞

(1.1.20)

L EMMA 1.1.7. (i) For g ∈ L2 (E; m), AT (g) = (g − Tg, g). (ii) (Fatou’s property) For any m-measurable functions gn , g on E with |gn | < ∞, |g| < ∞ [m], limn→∞ gn = g [m], AT (g) ≤ lim inf AT (gn ). n→∞

(1.1.21)


9

(iii) For any m-measurable function g on E with |g| < ∞ [m] and for any normal contraction ϕ, AT (ϕ ◦ g) ≤ AT (g). (iv) The triangular inequality (1.1.16) holds for every m-measurable functions f and g that are finite m-a.e. on E. Proof. (i) follows from Lemma 1.1.6(ii) and the contraction property of T on L2 (E; m). (ii) We first give a proof when |gn | ≤ M, |g| ≤ M for some M and limn→∞ gn = g [m]. From the linearity, the Markovian property of T on L∞ (E; m), and (1.1.11), we have for b ∈ R

T((g − b) ) = lim T inf(gn − b) 2

2

j→∞

n≥j

≤ lim inf T((gn − b)2 ). n

Since the identity (1.1.18) holds when s is a simple function like (1.1.17), we get from the above inequality 0 ≤ T(g2 ) − 2sTg + s2 T1 ≤ lim inf(T(g2n ) − 2sTgn + s2 T1). n

On the other hand, | T(g2n ) − 2gn Tgn + g2n T1 − T(g2n ) − 2sTgn + s2 T1 | ≤ 2|Tgn | |gn − s| + |g2n − s2 | T1 ≤ 2M|gn − s| + |g2n − s2 | hence 0 ≤ T(g2 ) − 2sTg + s2 T1 ≤ lim inf T(g2n ) − 2gn Tgn + g2n T1 + 2M|g − s| + |g2 − s2 |. n

Taking a sequence of simple functions s such that s → g [m], 0 ≤ T(g2 ) − 2gTg + g2 T1 ≤ lim inf T(g2n ) − 2gn Tgn + g2n T1 . n

Integrating both sides with respect to m and taking the defining formula (1.1.13) into account, we arrive at the desired (1.1.21) using the Fatou’s lemma in the Lebesgue integration theory. When gn and g are not necessarily uniformly bounded, we can use the results obtained above to get AT (ϕ ◦ g) ≤ lim inf AT (ϕ ◦ gn ) ≤ lim inf AT (gn ). n

n

By letting → ∞, we still have the inequality (1.1.21).

10

CHAPTER ONE

(iii) It holds for f = ϕ ◦ g, f = ϕ ◦ ϕ ◦ g that lim→∞ f = f . Equations (1.1.15) and (1.1.21) then lead us to AT (f ) ≤ lim inf AT (f ) ≤ lim inf AT (ϕ ◦ g) = AT (g). →∞

→∞

(iv) If we let fn := ϕ ◦ f and gn := ϕ ◦ g, then limn→∞ (fn + gn ) = f + g[m] so that (1.1.16) and (1.1.21) yield n

n

AT (f + g)1/2 ≤ lim inf AT (fn + gn )1/2

n→∞

≤ lim AT (fn )1/2 + AT (gn )1/2 = AT (f )1/2 + AT (g)1/2 . n→∞

Proof of Theorem 1.1.5. (i) For any f ∈ Fe , take its approximating sequence {fn } ⊂ F. fn being E-Cauchy, the triangular inequality guarantees the existence of the limit E(f , f ) = limn→∞ E(fn , fn ). Let us prove that 1 (1.1.22) AT (f ) ↑ E(f , f ) as t ↓ 0, t t which in particular implies that E(f , f ) does not depend the choice of the approximating sequence. Since f − f ∈ Fe for each and {fn − f } ⊂ F is its approximating sequence, we have from Lemma 1.1.7 and (1.1.4) 1 1 ATt (f − f ) ≤ lim inf ATt (fn − f ) ≤ lim fn − f 2E . n→∞ n→∞ t t Therefore, lim→∞ ATt (f − f ) = 0, and by the triangular inequality ATt (f ) = lim→∞ ATt (f ), which particularly implies that 1t ATt (f ) increases as t decreases to 0. Since limt↓0 1t ATt (f ) = f 2E , we can get from the triangular inequality and the inequality obtained above that

1 1

AT (f ) − f E ≤ lim AT (f − f ) ≤ lim fn − f E .

lim n→∞

t↓0 t t

t↓0 t t The last term tends to 0 as → ∞. The proof of (1.1.22) is complete. (ii) For any f ∈ Fe and any normal contraction ϕ, we are led from Lemma 1.1.7(iii) and (1.1.22) to 1 1 ATt (ϕ ◦ f ) ≤ ATt (f ) ≤ E(f , f ) for every t > 0. t t Hence it suffices to show ϕ ◦ f ∈ Fe . For an approximating sequence {fn } ⊂ F of f , we obtain by Lemma 1.1.7 and (1.1.4) 1 1 ATt (ϕ ◦ fn ) ≤ ATt (fn ) ≤ E(fn , fn ). t t


11

Thus ϕ ◦ fn ∈ F with E(ϕ ◦ fn , ϕ ◦ fn ) ≤ E(fn , fn ). This means that ϕ ◦ fn are elements of F with uniformly bounded E-norm. Therefore, the Cesàro mean gk = (1/k) kj=1 ϕ ◦ fnj of its suitable subsequence {fnj } is an E-Cauchy sequence by Theorem A.4.1. Since limk→∞ gk = ϕ ◦ f [m], we arrive at ϕ ◦ f ∈ Fe . (iii) The first identity follows from (1.1.4), Lemma 1.1.7, and (1.1.22). Since every normal contraction operates on (E, F ) by (ii), (E, F) is Markovian, namely, a Dirichlet form. Remark 1.1.8. Property (1.1.22) in particular implies that if {fk , k ≥ 1} ⊂ F is an E-Cauchy sequence and fk → 0 [m], then E(fk , fk ) → 0.

C OROLLARY 1.1.9. (Fatou’s lemma) Suppose {fk , k ≥ 1} ⊂ Fe and f ∈ Fe . If fk → f [m], then E(f , f ) ≤ lim inf E(fk , fk ). k→∞

Proof. It follows from (1.1.21) and (1.1.22) that 1 E(f , f ) ≤ lim inf lim inf ATt (fk , fk ) ≤ lim inf E(fk , fk ). t→0 k→∞ t k→∞ In the remainder of this section, (E, F) is a Dirichlet form on L2 (E; m). Exercise 1.1.10. Show that for f , g ∈ Fe ∩ L∞ (E; m), f · g ∈ Fe f · gE ≤ g∞ · f E + f ∞ · gE .

and

We state two lemmas for later use. L EMMA 1.1.11. (i) Let {ψ }≥1 be a sequence of normal contractions satisfying lim→∞ ψ (t) = t for every t ∈ R. Then lim→∞ ψ (f ) − f E1 = 0 for any f ∈ F. (ii) Suppose {fn } ⊂ F is E1 -convergent to f ∈ F. Then, for any normal contraction ϕ, {ϕ(fn )} is E1 -weakly convergent to ϕ(f ). If further ϕ(f ) = f , then the convergence is E1 -strong. Proof. (i) If we let ψ (f ) = f , then f ∈ F and f E1 is uniformly dominated by f E1 . Since G1 (L2 ) is E1 -dense in F by (1.1.6) and E1 (f , G1 g) = (f , g) → (f , g) = E1 (f , G1 g) for every g ∈ L2 , we can conclude that f converges as → ∞ to f weakly in (F, E1 ). But f − f 2E1 ≤ 2f 2E1 − 2E1 (f , f ) means that the convergence is strong as well.

12

CHAPTER ONE

(ii) E1 -norm of ϕ(fn ) is uniformly bounded and, for any g ∈ L2 (E; m), E1 (G1 g, ϕ(fn ) − ϕ(f )) = (g, ϕ(fn ) − ϕ(f )) → 0, n → ∞. Hence the first assertion follows. If ϕ(f ) = f , then as n → ∞, E1 (ϕ(fn ) − f , ϕ(fn ) − f ) ≤ E1 (fn , fn ) + E1 (f , f ) − 2E1 (f , ϕ(fn )) → 0. L EMMA 1.1.12. Let f be an m-measurable function on E with |f | < ∞ [m]. If, for the contractions ϕ of (1.1.19), f := ϕ ◦ f ∈ Fe for every ≥ 1, and sup f E < ∞, then f ∈ Fe . Proof. Without loss of generality, we assume that f is non-negative. For each , choose an approximating sequence f,k ∈ F for f such that supk f,k 2E ≤ + ∧ f . Then v,k ∈ Fe ∩ L2 (E; m) = F and it conf 2E + 1. We put v,k = f,k verges to f m-a.e. as k → ∞ for each . Furthermore, v,k 2E ≤ f,k 2E + f 2E ≤ 2 sup f 2E + 1 < ∞.

Take a strictly positive m-measurable function g with E gdm ≤ 1 and put g(x) = g(x)/(f (x) ∨ 1). Since 0 ≤ v,k (x) ≤ f (x) for x ∈ E, v,k is convergent gdx) as k → ∞ and the latter converges to f in L1 (E; gdx) to f in L1 (E; gdx) as well as m-a.e. as → ∞. Hence w = v,k converges to f in L1 (E; on E if we choose a suitable subsequence {k } of {k}. According to the boundedness of v,k E obtained above, we can conclude that f ∈ Fe admits the Cesàro mean of a subsequence of w ∈ F as its approximating sequence by Theorem A.4.1. A numerical function K(x, B) of two variables x ∈ E, B ∈ B(E), is said to be a kernel on the measurable space (E, B(E)) if, for each fixed x ∈ E, it is a (positive) measure in B and, for each fixed B ∈ B(E), it is a B(E)-measurable function in x. We then put (1.1.23) Kf (x) := f (y)K(x, dy), x ∈ E. E

Kf ∈ B+ (E) for f ∈ B+ (E) because the latter is an increasing limit of simple functions. A kernel K is called Markovian if K(x, E) ≤ 1 for every x ∈ E. A Markovian kernel K defines a linear operator on the space of bounded B(E)measurable functions by (1.1.23). A Markovian kernel K on E is said to be conservative or a probability kernel if K(x, ·) is a probability measure for each x ∈ E.


13

We call a kernel K(x, ·) (or an operator K) on (E, B(E)) m-symmetric if (Kf )(x)g(x)m(dx) = f (x)(Kg)(x)m(dx) for f , g ∈ B+ (E). (1.1.24) E

E

Let K be an m-symmetric Markovian kernel on (E, B(E)) and f ∈ bB(E) ∩ L2 (E; m). We then have from (1.1.23) (Kf )2 (x) ≤ (Kf 2 )(x), which yields by integrating with respect to m and using (1.1.24) the contraction property Kf 22 ≤ K1(x)f (x)2 m(dx) ≤ f 22 . E

This means that K can be regarded as a bounded linear operator on the space of m-essentially bounded m-measurable functions in L2 (E; m), which is dense in L2 (E; m). Hence K is uniquely extended to a linear contraction symmetric operator on L2 (E; m). So far we have assumed that (E, B(E)) is only a measurable space. In the rest of this section, we assume that E is a Hausdorff topological space. In this case, we shall use the notation B(E) exclusively for the Borel field, namely, the σ -field of subsets of E generated by open sets. The space of B(E)measurable real-valued functions will be denoted by B(E). We sometimes ∗ need to consider a larger σ -field Bµ (E); the family of universally measurable ∗ subsets of E: B (E) = µ∈P(E) B (E), where P(E) denotes the family of all probability measures on E and Bµ (E) is the completion of B(E) with respect to µ ∈ P(E). D EFINITION 1.1.13. (i) A family {Pt ; t ≥ 0} is called a transition function on (E, B(E)) (resp. (E, B∗ (E))) if Pt is a Markovian kernel on (E, B(E))(resp. (E, B∗ (E))) for each t ≥ 0 and the following four conditions are satisfied: (t.1) Ps Pt f = P s+t f for s, t ≥ 0 and f ∈ bB(E) (resp. f ∈ bB ∗ (E)). Here Pt f (x) := E f (y)Pt (x, dy). (t.2) For each B ∈ B(E), Pt (x, B) is B([0, ∞)) × B(E)-measurable (resp. B([0, ∞)) × B∗ (E)-measurable) in two variables (t, x) ∈ [0, ∞) × E. (t.3) For each x ∈ E, P0 (x, ·) = δx (·), where δx denotes the unit mass concentrated at the one-point set {x}. (t.4) limt↓0 Pt f (x) = f (x) for any f ∈ bC(E) and x ∈ E. A transition function {Pt ; t ≥ 0} is called a transition probability if Pt is conservative for every t > 0. (ii) A family {Rα ; α > 0} is called a resolvent kernel on (E, B(E)) (resp. (E, B∗ (E))) if, for each α > 0, αRα is a Markovian kernel on (E, B(E))

14

CHAPTER ONE

(resp. (E, B∗ (E))) and Rα f − Rβ f + (α − β)Rα Rβ f = 0, lim αRα f (x) = f (x),

α→∞

α, β > 0, f ∈ bB(E). x ∈ E,

f ∈ bC(E).

(1.1.25) (1.1.26)

Property (t.1) is called the semigroup property or Chapman-Kolmogorov equation. Identity (1.1.25) is called the resolvent equation. For a transition function {Pt ; t ≥ 0} on (E, B(E)) (resp. (E, B∗ (E))), it is easy to verify that ∞ e−αt Pt f (x)dt, α > 0, f ∈ B(E), (1.1.27) Rα f (x) = 0

determines uniquely a resolvent on (E, B(E)), (resp. (E, B∗ (E))), which is called the resolvent kernel of the transition function {Pt ; t ≥ 0}. A topological space E is called a Lusin space (resp. Radon space) if it is homeomorphic to a Borel (resp. universally measurable) subset of a compact metric space F. For a topological space E, a measure m on (E, B(E)) is said to be regular if, for any B ∈ B(E), m(B) = inf{m(U) : B ⊂ U, U open} = sup{m(K) : K ⊂ B, K compact}. Any Radon measure on a locally compact separable metric space is regular. Any finite measure on a Lusin space or on a Radon space is regular. L EMMA 1.1.14. Let {Pt ; t ≥ 0} be a family of Markovian kernels on a Lusin space E equipped with the Borel field B(E) or on a Radon space equipped with the σ -field B∗ (E) of its universally measurable subsets. (i) Suppose {Pt ; t ≥ 0} satisfies (t.1), (t.3) and (t.4) For every f ∈ bC(E), Pt f (x) is right continuous in t ∈ [0, ∞) for each x ∈ E. Then {Pt ; t ≥ 0} is a transition function. (ii) Suppose {Pt ; t ≥ 0} satisfies (t.1), (t.4) and, for a σ -finite measure m on E, {Pt ; t ≥ 0} is m-symmetric in the sense that Pt is m-symmetric for each t > 0. Let Tt be the symmetric linear operator on L2 (E; m) uniquely determined by Pt . Then {Tt ; t ≥ 0} is a strongly continuous contraction semigroup on L2 (E; m). Proof. We give a proof for a family {Pt ; t ≥ 0} of Markovian kernels on a Lusin space (E, B(E)). The proof for a Radon space (E, B∗ (E)) is the same. (i) It suffices to establish (t.2). Let H be the collection of functions in bB(E) such that Pt f (x) is measurable in two variables (t, x). H is then a linear space closed under the operation of taking uniformly bounded increasing limits. By (t.4) , it holds that bC(E) ⊂ H. Hence (t.2) follows from Proposition A.1.3.


15

(ii) We may assume that E is a Borel subset of a compact metric space (F, d) and identify L2 (E; m) with L2 (F; m) by setting m(F \ E) = 0. We first show that bC(F) ∩ L2 (F; m) is dense in L2 (F; m). Since m is σ -finite, it suffices to assume that m is a finite measure and that the indicator function of a set B ∈ B(F) can be L2 -approximated. For any ε, there exist a compact set K and an open set U such that K ⊂ B ⊂ U, m(U \ K) < ε. If we let g(x) = d(x,√ U c )/(d(x, U c ) + 2 d(x, K)), x ∈ F, then g ∈ bC(F) ∩ L (F; m) and g − 1B 2 < ε. For any f ∈ L2 (E; m) and ε > 0, take a function g ∈ bC(F) ∩ L2 (E; m) such that f − g2 < ε. Because of the contraction property of {Tt ; t > 0}, we then have Tt f − f 2 ≤ Pt g − g2 + 2ε. Further, Pt g − g22 ≤ 2g22 − 2(g, Pt g), which tends to 0 as t ↓ 0 by (t.4) and the Lebesgue-dominated convergence theorem. By virtue of Lemma 1.1.14, any m-symmetric transition function {Pt ; t ≥ 0} on a Lusin space (E, B(E)) or a Radon space (E, B∗ (E)) determines a unique strongly continuous contraction semigroup {Tt ; t ≥ 0} on L2 (E; m), which in turn decides a Dirichlet form (E, F ) on L2 (E; m) according to Theorem 1.1.3. (E, F) is called the Dirichlet form of the transition function {Pt ; t ≥ 0}. In this case, the resolvent {Gα ; α > 0} of {Tt ; t ≥ 0} is the unique extension of the resolvent kernel {Rα ; α > 0} of {Pt ; t ≥ 0} from bB(E) ∩ L2 (E; m) to L2 (E; m). Moreover, we have from (1.1.6) that for f ∈ bB(E) ∩ L2 (E; m), Rα f ∈ F

with Eα (Rα f , v) = (f , v) for every v ∈ F.

(1.1.28)

Conversely, if the resolvent kernel {Rα ; α > 0} of a transition function {Pt ; t ≥ 0} satisfies (1.1.28) for a Dirichlet form (E, F) on L2 (E; m), then {Pt ; t ≥ 0} is m-symmetric and its Dirichlet form coincides with (E, F ). In the rest of this chapter, we give a quick introduction to the basic theory of quasi-regular Dirichlet forms. The importance of a quasi-regular Dirichlet form is that they are in one-to-one correspondence with symmetric Markov processes having some nice properties. We will show that any quasi-regular Dirichlet form is quasi-homeomorphic to a regular Dirichlet form on a locally compact separable metric space. Thus the study of quasi-regular Dirichlet forms can be reduced to that of regular Dirichlet forms.

1.2. EXCESSIVE FUNCTIONS AND CAPACITIES In this section, let E be a Hausdorff topological space with the Borel σ -field B(E) being assumed to be generated by the continuous functions on

16

CHAPTER ONE

E and m be a σ -finite measure with supp[m] = E. Here for a measure ν on E, its support supp[ν] is by definition the smallest closed set outside which ν vanishes. Let (E, F) be a symmetric Dirichlet form on L2 (E; m), and {Tt ; t ≥ 0} and {Gα ; α > 0} be its associated semigroup and resolvents on L2 (E; m). D EFINITION 1.2.1. For α > 0, u ∈ L2 (E; m) is called α-excessive if e−αt Tt u ≤ u m-a.e. for every t > 0. Remark 1.2.2. (i) If u is α-excessive, then u ≥ 0. This is because e−αt Tt u2 = e−αt Tt u2 ≤ e−αt u2 and so u ≥ limt→∞ e−αt Tt u = 0. 2 (E; m), (ii) The constant function 1 is α-excessive if m(E) < ∞. For f ∈ L+ Gα f is α-excessive. (iii) If u1 ≥ 0, u2 ≥ 0 are α-excessive functions, then so are u1 ∧ u2 and u1 ∧ 1. 2 (E; m) be α-excessive for α > 0. Assume there is L EMMA 1.2.3. Let u ∈ L+ v ∈ F such that u ≤ v. Then u ∈ F and Eα (u, u) ≤ Eα (v, v).

Proof. By the symmetry and contraction property of Tt in L2 (E; m), for each t > 0, (f , g − e−αt Tt g) is a non-negative symmetric bilinear form on L2 (E; m). So it satisfies the following Cauchy-Schwarz inequality: |(f , g − e−αt Tt g)| ≤ (f , f − e−αt Tt f )1/2 · (g, g − e−αt Tt g)1/2 . Thus we have by the α-excessiveness of u, (u − e−αt Tt u, u) ≤ (u − e−αt Tt u, v) ≤ (u, u − e−αt Tt u)1/2 · (v, v − e−αt Tt v)1/2 , and so (u − e−αt Tt u, u) ≤ (v, v − e−αt Tt v). It follows then that 1 1 1 lim (u − Tt u, u) = lim (u − e−αt Tt u, u) + lim (e−αt − 1)(Tt u, u) t→0 t t→0 t t→0 t 1 ≤ lim (v − e−αt Tt v, v) − α(u, u) t→0 t = E(v, v) + α(v, v) − α(u, u) < ∞. We conclude from (1.1.4)–(1.1.5) that u ∈ F with Eα (u, u) ≤ Eα (v, v).

17


L EMMA 1.2.4. The following statements are equivalent for u ∈ F and α > 0: (i) u is α-excessive. (ii) Eα (u, v) ≥ 0 for every non-negative v ∈ F. Proof. (i) ⇒ (ii): It follow from (1.1.5) that 0≤

1 1−e−αt 1 (u − e−αt Tt u, v) = (u−Tt u, v) + (Tt u, v) → Eα (u, v) (1.2.1) t t t

as t ↓ 0. 2 (E; m) and t > 0, since (ii) ⇒ (i): For v ∈ L+ −αt

Gα v − e

Tt Gα v =

t

e−αs Ts vds ≥ 0,

0

we have (u − e−αt Tt u, v) = (u, v − e−αt Tt v) = Eα (u, Gα (v − e−αt Tt v)) ≥ 0. This implies that u − e−αt Tt u ≥ 0 and so (i) holds.

For a closed subset F of E, define FF := {f ∈ F : f = 0 m−a.e. on E \ F}.

(1.2.2)

T HEOREM 1.2.5. Let α > 0 and f be a non-negative function defined on E. For an open set D, denote LD,f = {u ∈ F : u ≥ f m-a.e. on D}. Suppose LD,f = ∅. Then (i) there is a unique fD ∈ LD,f such that Eα (u, u) ≥ Eα (fD , fD )

for every u ∈ LD,f .

(ii) fD is the unique function in LD,f such that Eα (u, fD ) ≥ Eα (fD , fD )

for every u ∈ LD,f .

(iii) Eα (fD , v) ≥ 0 for every v ∈ F with v ≥ 0 m-a.e. on D. In particular, fD is α-excessive and Eα (fD , v) = 0 for every v ∈ FDc . (iv) fD ≤ f if and only if fD ∧ f is an α-excessive function. In this case, fD = f m-a.e. on D. fD is the minimum element among α-excessive functions in LD,f in the sense that, if u ∈ LD,f is α-excessive, then fD ≤ u. (v) If open sets D1 ⊂ D2 and LD2 ,f = ∅, then fD1 ≤ fD2 and Eα (fD1 , fD1 ) ≤ Eα (fD2 , fD2 ).

18

CHAPTER ONE

(vi) For open sets D1 ⊂ D2 , if f ∧ fD2 is an α-excessive function, then (fD2 )D1 = fD1 . If further f ∧ fD1 is α-excessive, then Eα (fD1 , fD2 ) = Eα (fD1 , fD1 ). (vii) For open sets D1 ⊂ D2 , (fD1 )D2 = fD1 . Proof. (i) Because LD,f is a closed convex set in the Hilbert space (F, Eα ), it has a unique minimizer fD . (ii) For every u ∈ LD,f and 0 < ε < 1, fD + ε(u − fD ) = (1 − ε)fD + εu ∈ LD,f and so Eα (fD + ε(u − fD ), fD + ε(u − fD )) ≥ Eα (fD , fD ). This implies that Eα (fD , u − fD ) ≥ 0. Now suppose v ∈ LD,f is another function such that for every u ∈ LD,f , Eα (v, u − v) ≥ 0. As fD ∈ LD,f , Eα (v, fD − v) ≥ 0. But with Eα (fD , v − fD ) ≥ 0, we have Eα (fD − v, fD − v) ≤ 0. Therefore, v = fD . (iii) For any v ∈ F with v ≥ 0 m-a.e. on D, fD + εv ∈ LD,f for every ε > 0. One immediately deduces from Eα (fD + εv, fD + εv) ≥ Eα (fD , fD ) that Eα (fD , v) ≥ 0. (iv) This follows immediately from (iii) and Lemma 1.2.3. (v) The first part follows from (iv). The second part follows from (i). (vi) By (iv), f = fD2 on D2 and hence by definition, (fD2 )D1 = fD1 . The second assertion follows from (iii) and (iv). (vii) For every u ∈ LD2 ,fD1 , Eα (fD1 , u − fD1 ) ≥ 0 by (iii). We therefore have by (ii) that fD1 = (fD1 )D2 . The function fD is called the α-reduced function of f on D. Remark 1.2.6. (i) If f is α-excessive in F, then fD is the Eα -orthogonal projection of f into the Eα -orthogonal complement of FDc . This is because f = (f − fD ) + fD , where f − fD ∈ FDc by Theorem 1.2.5(iv) and fD is Eα orthogonal to FDc by Theorem 1.2.5(iii). (ii) By (iii) and (iv) of Theorem 1.2.5, if g ∈ F is α-excessive, then Eα (fD , g) = Eα (fD , gD ). D EFINITION 1.2.7. ((h, α)-capacity) Fix α > 0. Let h ≥ 0 be a function on E satisfying one of the following two conditions: (i) h ∈ F and h is α-excessive; (ii) h ∧ hD is a α-excessive function for every open set D ⊂ E with LD,h = ∅. (This is equivalent to, by Theorem 1.2.5(iv), that h ≥ hD for every open set D ⊂ E with LD,h = ∅). Define for open subset D ⊂ E, Eα (hD , hD ) if LD,h = ∅, (1.2.3) Caph,α (D) := +∞ otherwise,

19


and for an arbitrary subset A ⊂ E, Caph,α (A) := inf Caph,α (D) : open set D ⊃ A .

(1.2.4)

Remark 1.2.8. (i) Important cases are h = 1 and h = Gα ϕ for some strictly positive ϕ ∈ L2 (E; m). (ii) Under either of conditions (i) and (ii), h = hD [m] on D whenever LD,h = ∅. (iii) When h > 0 [m] on E, then Caph,α (A) = 0 implies that m(A) = 0. (iv) If 0 ≤ h(1) ≤ h(2) are two functions satisfying either condition (i) or (2) (ii) in Definition 1.2.7, we have by Theorem 1.2.5(iv) that h(1) D ≤ hD for any open set D with LD,h(2) = ∅. Therefore Caph(1) ,α (D) ≤ Caph(2) ,α (D) by Lemma 1.2.3. (v) We shall use the following comparison in α > 0 for the capacity: if h1 is 1-excessive, h2 is 2-excessive, and h2 ≤ h1 , then Caph2 ,2 (A) ≤ 2Caph1 ,1 (A),

A ⊂ E.

In fact, we have for an open set D, Caph2 ,2 (D) =

inf

u∈F, u≥h2

on D

E2 (u, u) ≤

inf

u∈F, u≥h1

on D

2E1 (u, u)

≤ 2Caph1 ,1 (D).

In the remainder of this section h ≥ 0 is a non-trivial function on E satisfying one of the conditions (i) or (ii) in Definition 1.2.7. T HEOREM 1.2.9. (i) For open sets D1 ⊂ D2 , Caph,α (D1 ) ≤ Caph,α (D2 ). (ii) For open sets D1 and D2 , Caph,α (D1 ∪ D2 ) + Caph,α (D1 ∩ D2 ) ≤ Caph,α (D1 ) + Caph,α (D2 ). (iii) For any increasing sequence of open sets {Dk , k ≥ 1}, Caph,α (∪k≥1 Dk ) = sup Caph,α (Dk ). k≥1

(iv) For any decreasing sequence of open sets {Dk , k ≥ 1} with LD1 ,h = ∅, {hDk ; k ≥ 1} is decreasing to as well as Eα -convergent to a function h∞ ∈ F, and infk≥1 Caph,α (Dk ) = Eα (h∞ , h∞ ). Proof. (i) follows from Theorem 1.2.5(v).

20

CHAPTER ONE

(ii) Without loss of generality, we may assume Caph,α (Di ) < ∞ for i = 1, 2. By the property of hD , Caph,α (D1 ∪ D2 ) + Caph,α (D1 ∩ D2 ) ≤ Eα (hD1 ∨ hD2 , hD1 ∨ hD2 ) + Eα (hD1 ∧ hD2 , hD1 ∧ hD2 ) = 12 Eα (hD1 + hD2 , hD1 + hD2 ) + 12 Eα (|hD1 − hD2 |, |hD1 − hD2 |) ≤ 12 Eα (hD1 + hD2 , hD1 + hD2 ) + 12 Eα (hD1 − hD2 , hD1 − hD2 ) = Eα (hD1 , hD1 ) + Eα (hD2 , hD2 ) = Caph,α (D1 ) + Caph,α (D2 ). (iii) Without loss of generality, assume that supk≥1 Caph,α (Dk ) < ∞. For j > k, we have from Theorem 1.2.5(vi) Eα (hDj − hDk , hDj − hDk ) = Eα (hDj , hDj ) − 2Eα (hDj , hDk ) + Eα (hDk , hDk ) = Eα (hDj , hDj ) − Eα (hDk , hDk ) = Caph,α (Dj ) − Caph,α (Dk ) → 0

as j, k → ∞.

So hDk is Eα -convergent to some h∞ ∈ F. As h∞ = hk = h [m] on Dk , we have h∞ = h [m] on ∪k≥1 Dk . For v ∈ L∪k≥1 Dk ,h , by Theorem 1.2.5(ii), Eα (h∞ , v) = lim Eα (hDk , v) ≥ lim Eα (hDk , hDk ) = Eα (h∞ , h∞ ). k→∞

k→∞

By Theorem 1.2.5(ii) again, h∞ = h∪k≥1 Dk and therefore sup Caph,α (Dk ) = lim Caph,α (Dk ) = lim Eα (hDk , hDk ) k→∞

k≥1

k→∞

= Eα (h∞ , h∞ ) = Caph,α (∪k≥1 Dk ). (iv) {hDk } is decreasing by Theorem 1.2.5(v). For j > k, we have from Theorem 1.2.5(vi) Eα (hDj − hDk , hDj − hDk ) = Eα (hDj , hDj ) − 2Eα (hDj , hDk ) + Eα (hDk , hDk ) = Eα (hDk , hDk ) − Eα (hDj , hDj ), which leads us to (iv).

Observe that the proof of Theorem 1.2.9(iii) shows that hDk converges to h∪k≥1 Dk both monotonously and in (F, Eα ).


21

T HEOREM 1.2.10. Caph,α is a Choquet K-capacity, where K denotes all the compact subsets of E; that is, (i) For any subsets A ⊂ B, Caph,α (A) ≤ Caph,α (B); (ii) For any increasing sequence of subsets {Aj , j ≥ 1}, Caph,α (∪j≥1 Aj ) = sup Caph,α (Aj ); j≥1

(iii) For any decreasing sequence of compact subsets {Kj , j ≥ 1}, Caph,α (∩j≥1 Kj ) = inf Caph,α (Kj ). j≥1

Proof. (i) follows immediately from Theorem 1.2.9(i) and the definition of Caph,α . (ii) Without loss of generality, we may assume that Caph,α (Aj ) < ∞ for every j ≥ 1. In view of (i), it suffices to show Caph,α (∪j≥1 Aj ) ≤ sup Caph,α (Aj ). j≥1

For any ε > 0, let an open set Oj ⊃ Aj be such that Caph,α (Oj ) < Caph,α (Aj ) + j 2−j ε. Define Dj := ∪k=1 Ok . Then {Dj , j ≥ 1} is an increasing sequence of open sets. We claim that Caph,α (Dj ) ≤ Caph,α (Aj ) + (1 − 2−j )ε

for every j ≥ 1.

(1.2.5)

We prove this by induction. Clearly this is true for j = 1. Suppose it is true for j ≥ 1. Since Dj+1 = Dj ∪ Oj+1 , we have by Theorem 1.2.9(ii), Caph,α (Dj+1 ) + Caph,α (Dj ∩ Oj+1 ) ≤ Caph,α (Dj ) + Caph,α (Oj+1 ). But as Aj ⊂ Dj ∩ Oj+1 , we have Caph,α (Dj+1 ) ≤ Caph,α (Dj ) + Caph,α (Oj+1 ) − Caph,α (Aj ) ≤ Caph,α (Oj+1 ) + (1 − 2−j )ε ≤ Caph,α (Aj+1 ) + 2−j−1 ε + (1 − 2−j )ε = Caph,α (Aj+1 ) + (1 − 2−j−1 )ε. This proves the claim (1.2.5). Therefore, we have Caph,α (∪j≥1 Aj ) ≤ Caph,α (∪j≥1 Dj ) = sup Caph,α (Dj ) j≥1

≤ sup Caph,α (Aj ) + ε. j≥1

Passing ε ↓ 0, we get Caph,α (∪j≥1 Aj ) ≤ supj≥1 Caph,α (Aj ), and so (ii) is established.

22

CHAPTER ONE

(iii) It suffices to show that Caph,α (∩j≥1 Kj ) ≥ infj≥1 Caph,α (Kj ). We may assume that Caph,α (∩j≥1 Kj ) < ∞. For any ε > 0, let D be an open set such that D ⊃ ∩j≥1 Kj and Caph,α (D) < Caph,α (∩j≥1 Kj ) + ε. Since Kj is compact for every j ≥ 1, there is n ≥ 1 such that D ⊃ Kn . Therefore, Caph,α (D) ≥ Caph,α (Kn ) ≥ infj≥1 Caph,α (Kj ). This yields that, after letting ε ↓ 0, Caph,α (∩j≥1 Kj ) ≥ infj≥1 Caph,α (Kj ). D EFINITION 1.2.11. A subset A ⊂ E is said to be C-capacitable for a set function C on E if C(A) = sup C(K). K⊂A

K:

compact

The celebrated Choquet’s theorem says that every K-analytic set is C-capacitable for any Choquet K-capacity C (cf. [37, III: 28]). It is known that any Borel subset of a compact metric space is K-analytic (cf. [37, III: 7,13]). In particular, for a Lusin space E, it holds from Theorem 1.2.10 that Caph,α (B) =

Caph,α (K) sup compact

for B ∈ B(E).

(1.2.6)

K⊂B, K

D EFINITION 1.2.12. (i) An increasing sequence {Fk , k ≥ 1} of closed sets of E is an E-nest if ∪k≥1 FFk is E1 -dense in F, where E1 = E + ( , )L2 (E;m) . (ii) A subset N of E is E-polar if there is an E-nest {Fk , k ≥ 1} such that N ⊂ ∩k≥1 (E \ Fk ). (iii) A statement depending on x ∈ A is said to hold E-quasi-everywhere (E-q.e. in abbreviation) on A if there is an E-polar set N ⊂ A such that the statement is true for every x ∈ A \ N. (iv) A function f on E is said to be E-quasi-continuous if there is an E-nest {Fk , k ≥ 1} such that f |Fk is finite and continuous on Fk for each k ≥ 1, which will be denoted in abbreviation as f ∈ C({Fk }). (v) An increasing sequence {Fk } of closed sets of E is Caph,α -nest if lim Caph,α (E \ Fk ) = 0.

k→∞

(vi) A subset N of E is Caph,α -polar if Caph,α (N) = 0. Obviously, if {Fn , n ≥ 1} of E is an E-nest, then so is {Kn , n ≥ 1} where Kn = supp[1Fn · m]. Since F is a dense linear subspace in L2 (E; m), every E-polar set is m-null. T HEOREM 1.2.13. Fix an arbitrary α > 0 and let h = Gα ϕ for some strictly positive ϕ ∈ L2 (E; m). Let {Fk , k ≥ 1} be an increasing sequence of closed subsets. Then


23

(i) {Fk , k ≥ 1} is an E-nest if and only if it is a Caph,α -nest. (ii) A set N ⊂ E is E-polar if and only if it is Caph,α -polar. (iii) If {Fk1 , k ≥ 1} and {Fk2 , k ≥ 1} are two E-nests, then {Fk1 ∩ Fk2 , k ≥ 1} is also an E-nest. Proof. Let hk := hFkc . By Theorem 1.2.9(iv), hk is decreasing to as well as Eα convergent to some non-negative h∞ ∈ F and lim Caph,α (Fkc ) = Eα (h∞ , h∞ ).

k→∞

(1.2.7)

In particular, for every v ∈ ∪k≥1 FFk , by Theorem 1.2.5(iii), Eα (h∞ , v) = lim Eα (hk , v) = 0. k→∞

(1.2.8)

Now suppose {Fk , k ≥ 1} is an E-nest. Then by (1.2.8), h∞ = 0 and so limk→∞ Caph,α (Fkc ) = 0 by (1.2.7). Conversely, suppose that limk→∞ Caph,α (Fkc ) = 0. Then h∞ = 0 by (1.2.7). For any α-excessive function v ∈ F, denote vFkc by vk so v − vk ∈ FFk . By the same reasoning as above for h, we see that vk is decreasing to as well as Eα -convergent to some v∞ ∈ F. By Remark 1.2.6(i), ϕ(x)v∞ (x)m(dx) = Eα (h, v∞ ) = lim Eα (h, vk ) k→∞

E

= lim Eα (hk , v) = Eα (h∞ , v) = 0. k→∞

This implies that v∞ = 0 [m] on E as ϕ > 0 [m] on E. Therefore, v = limk→∞ (v − vk ) is in the Eα -completion of ∪k≥1 FFk . Since Gα L2 (E; m) is Eα dense in F, we have that ∪k≥1 FFk is Eα -dense (and hence E1 -dense) in F; that is, {Fk , k ≥ 1} is an E-nest. The second assertion of the theorem is immediate from the first. The third follows from the first and Theorem 1.2.9. T HEOREM 1.2.14. Suppose that h > 0 is a function that satisfies one of the conditions in Definition 1.2.7 for α = 1. Suppose there is an increasing sequence of open sets {Dk , k ≥ 1} of finite (h, 1)-capacity such that Dk ⊂ Dk+1 , k ≥ 1, and {Dk , k ≥ 1} constitutes an E-nest. (i) An increasing sequence {Fk , k ≥ 1} of closed subsets of E is an E-nest if and only if limk→∞ Caph,1 (Dn \ Fk ) = 0 for every n ≥ 1. (ii) A set N ⊂ E is E-polar if and only if it is Caph,1 -polar. Proof. (i) Proof of the “only if” part: Suppose {Fk , k ≥ 1} is an E-nest. For a fixed n, let gk = hDn \Fk . By Theorem 1.2.9(iv), {gk , k ≥ 1} is then decreasing to and E1 -convergent to a function g∞ ∈ F. By Theorem 1.2.5(iii), gk is E1 -orthogonal to the space F{Dn \F }c ⊃ FF for any k ≥ . Hence g∞ is

24

CHAPTER ONE

E1 -orthogonal to ∪ FF , which is E1 -dense in F. Thus g∞ = 0 and so for each fixed n ≥ 1, Caph,1 (Dn \ Fk ) = E1 (gk , gk ) → 0 as k → ∞. Proof of the “if” part: By the assumption and Theorem 1.2.5(iv), h1 :=

∞

2−n hDn −1 2 hDn

n=1

is a 1-excessive function in L2 (E; m) with 0 < h1 ≤ hD1 −1 2 h [m] on E. h. As h is 1-excessive, h2 is 2Let h2 := G2 h1 . Clearly h2 ≤ h1 ≤ hD1 −1 1 2 excessive, and h2 ≤ h1 , we see from Remark 1.2.8(iv)–(v) that, for any open set D, Caph2 ,2 (D) ≤ 2Caph1 ,1 (D) ≤ 2hD1 −2 2 Caph,1 (D).

(1.2.9)

Now suppose that limk→∞ Caph,1 (Dn \ Fk ) = 0 for every n ≥ 1. Since Fkc ⊂ ∪ (Dn+1 \ Fk ), we see by Theorem 1.2.9(ii) that

c Dn

c

Caph2 ,2 (Fkc ) ≤ Caph2 ,2 (Dn ) + Cap(Dn+1 \ Fk ). By noting (1.2.9) and Theorem 1.2.13, we let k → ∞ and then n → ∞ to get limk→∞ Caph2 ,2 (Fkc ) = 0, which means that {Fk , k ≥ 1} is an E-nest by Theorem 1.2.13. (ii) If N is E-polar, then it is a subset of ∩k (E \ Fk ) for some E-nest {Fk }. By (i), Caph,1 (Dn ∩ N) = 0 for each n, and by letting n → ∞, we get Caph,1 (N) = 0 on account of Theorem 1.2.10(ii). Conversely, suppose Caph,1 (N) = 0. For the 2-excessive function h2 := G2 h1 defined in the “if” part of the proof of (i), the inequality (1.2.9) holds for any set D and so Caph2 ,2 (N) = 0, which implies that N is E-polar in view of Theorem 1.2.13. Remark 1.2.15. Let h > 0 be a function that satisfies one of the conditions in Definition 1.2.7. (i) Any Caph,1 -nest is an E-nest. (ii) If h ∈ F, then one can take Dn = E, n ≥ 1, in Theorem 1.2.14 and hence a Caph,1 -nest becomes a synonym of an E-nest. (iii) h = 1 is an important case for Theorem 1.2.14. 1.3. QUASI-REGULAR DIRICHLET FORMS We maintain the same assumptions on (E, B(E), m) as in the preceding section. Let (E, F) be a Dirichlet form on L2 (E; m). L EMMA 1.3.1. Let S be a countable family of E-quasi-continuous functions on E. Then there is an E-nest {Fk , k ≥ 1} such that S ⊂ C({Fk }).

25


Proof. Fix some ϕ ∈ L2 (E; m) with 0 < ϕ ≤ 1 and set h = G1 ϕ. Spell out S = {fk , k ≥ 1}. For each n ≥ 1, there is an E-nest {Fn,k , k ≥ 1} such that c ) ≤ 2−nk . Define Fk := ∩n≥1 Fn,k , which is closed. fn ∈ C({Fn,k }) and Caph,1 (Fn,k By Theorems 1.2.9 and 1.2.10, c )≤ Caph,1 (Fkc ) = Caph,1 (∪n≥1 Fn,k

c Caph,1 (Fn,k ) ≤ 2−k

for k ≥ 1.

n≥1

So {Fk , k ≥ 1} is an E-nest by Theorem 1.2.13 and clearly S ⊂ C({Fk }).

T HEOREM 1.3.2. Let h = G1 ϕ for some ϕ ∈ L2 (E; m) with 0 < ϕ ≤ 1. Suppose that u ∈ F has an E-quasi-continuous m-version u. Then u| > λ) ≤ E1 (u, u)/λ2 Caph,1 (|

for every λ > 0.

u ∈ C({Fk }). For λ > 0, Proof. Let {Fk , k ≥ 1} be an E-nest such that u(x)| > λ} ∪ Fkc , which is an open subset of E. Let let Dk := {x ∈ Fk : | u| + hFkc ∈ F. Since 0 < h ≤ 1 [m] on E, uk ≥ h on Dk . Thus uk := λ−1 | u| > λ) ≤ Caph,1 (Dk ) ≤ E1 (uk , uk ) Caph,1 (| ≤ λ−2 E1 (|u|, |u|) + 2λ−1 E1 (|u|, hFkc ) + Caph,1 (Fkc ). u| > λ) ≤ lim supk→∞ Caph,1 (Dk ) ≤ E1 (u, u)/λ2 . It follows then Caph,1 (|

T HEOREM 1.3.3. Suppose each uk ∈ F has an E-quasi-continuous m-version uk and that uk converges to u in (F , E1 ) as k → ∞. Then there unk converges to an E-quasiexists a subsequence {unk , k ≥ 1} such that continuous m-version u of u quasi uniformly; that is, there is an E-nest u, unj , j ≥ 1} ⊂ C({Fk }) and unk converges to u {Fk , k ≥ 1} such that { uniformly on each Fk . Proof. Taking a subsequence if necessary, we may assume that E1 (uk+1 − uk , uk+1 − uk ) < 2−3k for every k ≥ 1. Define Ak := x ∈ E : | uk+1 (x) − uk (x)| > 2−k . By Theorem 1.3.2, Caph,1 (Ak ) ≤ 22k E1 (uk+1 − uk , uk+1 − uk ) < 2−k . Let uk , k ≥ 1} ⊂ C({Ek }). Define Fk := Ek ∩ {E , ≥c 1} be an E-nest such that { ∩l≥k Al , which is closed. Since Caph,1 (Fkc ) ≤ Caph,1 (Ekc ) +

l≥k

Caph,1 (Al ) < Caph,1 (Ekc ) + 2−k+1 ,

26

CHAPTER ONE

which tends to 0 as k → ∞, {Fk , k ≥ 1} is an E-nest and clearly un converges u ∈ C({Fk }) and therefore it is an to some u uniformly on each Fk . Thus E-quasi-continuous m-version of u. D EFINITION 1.3.4. An E-nest {Fk , k ≥ 1} is called m-regular if for each k ≥ 1, supp[1Fk m] = Fk ; that is, for every x ∈ Fk and every neighborhood U(x) of x, m(U(x) ∩ Fk ) > 0. D EFINITION 1.3.5. A Hausdorff topological space is called a Lindelöf space if every open covering of the space has a countable subcover. Lindelöf theorem asserts that a topological space is Lindelöf if its topology has a countable base (see, e.g., [103, p. 49]). On a Lindelöf space, the topological support of any σ -finite measure µ is well defined to the smallest closed set F that µ does not charge on its complement F c . L EMMA 1.3.6. Let {Fk , k ≥ 1} be an E-nest. Suppose that the relative topolFk = supp[1Fk m]. Then { Fk , k ≥ 1} is an ogy on each Fk is Lindelöf. Let m-regular E-nest. Proof. Let h := G1 ϕ for some ϕ ∈ L2 (E; m) such that 0 < ϕ ≤ 1. Note that Fk \ Fk = {x ∈ Fk : there is an open neighborhood U(x) of x such that m(U(x) ∩ Fk ) = 0} By the Lindelöf property, m(Fk \ Fk ) = 0. Thus LFkc ,h = LFkc ,h and therec c fore Caph,1 (Fk ) = Caph,1 (Fk ). This proves that { Fk , k ≥ 1} is an m-regular E-nest. T HEOREM 1.3.7. Suppose {Fk , k ≥ 1} is an m-regular E-nest and f ∈ C({F k }). If f ≥ 0 [m] on an open set D, then f (x) ≥ 0 for every x ∈ D ∩ ∪k≥1 Fk ; i.e., f ≥ 0 E-q.e. on D. Proof. Since f ≥ 0 [m] on D and f is continuous on each Fk , f ≥ 0 on Fk ∩ D due to the assumption Fk = supp[1Fk m]. Thus f ≥ 0 on ∪k≥1 (Fk ∩ D). D EFINITION 1.3.8. A Dirichlet form (F, E) on L2 (E; m) is called quasiregular if: (i) there exists an E-nest {Fk , k ≥ 1} consisting of compact sets; (ii) there exists an E1 -dense subset of F whose elements have E-quasicontinuous m-versions; (iii) there exists {fk , k ≥ 1} ⊂ F having E-quasi-continuous m-versions fk : k ≥ 1} separates the points { fk , k ≥ 1} and an E-polar set N ⊂ E such that { of E \ N.


27

Remark 1.3.9. (i) Part (i) of Definition 1.3.8 implies that, for any α-excessive function h in F with α > 0, Caph,1 is tight; that is, there is an increasing sequence of compact sets {Kj , j ≥ 1} such that limj→∞ Caph,1 (E \ Kj ) = 0. (ii) Part (ii) of Definition 1.3.8 implies by Theorem 1.3.3 that every function in F has an E-quasi-continuous m-version, which will be denoted by f. (iii) We may assume, by parts (i) and (iii) of Definition 1.3.8 together with Theorem 1.2.13 and Lemma 1.3.1, that there is an E-nest {Fk , k ≥ 1} consisting fk , k ≥ 1} separates points of of compact sets so that { fk , k ≥ 1} ⊂ C({Fk }) and { ∪k Fk . Define ρ(x, y) :=

∞

2−j | fj (y)| ∧ 1 fj (x) −

for x, y ∈ ∪k≥1 Fk .

j=1

Then ρ(x, y) is a (separating) metric on each Fk , which by the compactness of Fk is compatible with the original topology on Fk inherited from E. Since the topology induced by ρ on each Fk has countable base, Fk is a separable metric space. Hence L2 (E; m) = L2 (∪j≥1 Fj ; m) is separable and therefore so is (F, E1 ). Thus Definition 1.3.8(ii) can be replaced by (ii). There exists an E1 -dense countable subset {uk , k ≥ 1} of F whose elements have E-quasi-continuous m-version. (iv) By the Lindelöf theorem, each compact set Fk in (iii) is Lindelöf. Hence for a quasi-regular Dirichlet form (E, F ), if f ≥ 0 [m] on an open set D and if f is E-quasi-continuous, then f ≥ 0 E-q.e. on D by Lemma 1.3.6 and Theorem 1.3.7. (v) By Corollary 2 on p.12 of [136] Y := ∪k≥1 Fk is a Lusin space (i.e., it is homeomorphic to a Borel subset of a compact metric space). Since L2 (E; m) can be identified with L2 (Y; m), when dealing with quasi-regular Dirichlet forms, we can assume that E is a topological Lusin space. For an m-measurable function f defined and finite m-a.e. on E, the support of f is defined to be the support of the measure f · m. When f is continuous, the support of f is just the closure of the set {f = 0}. When E is a locally compact separable metric space, we shall denote by Cc (E) the family of all continuous functions on E with compact support, and by C∞ (E) the family of all continuous functions f on E which vanishes at ∞, namely, there exists for any ε > 0 a compact set with |f (x)| < ε for every x ∈ E \ K. C∞ (E) is a Banach space with respect to the uniform norm f ∞ = supx∈E |f (x)|. D EFINITION 1.3.10. A Dirichlet form (E, F ) on L2 (E; m) is said to be regular if (i) E is a locally compact separable metric space and m is a Radon measure on E with full support;

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(ii) F ∩ Cc (E) is E1 -dense in F; (iii) F ∩ Cc (E) is uniformly dense in Cc (E). Remark 1.3.11. (i) By Stone-Weierstrass theorem (cf. [58, Theorem 4.45]), (iii) in Definition 1.3.10 is equivalent to (iii ) F ∩ Cc (E) separates the points of E. (ii) Clearly, a regular Dirichlet form is quasi-regular.

L EMMA 1.3.12. Let E be a locally compact separable metric space and m a Radon measure on E with full support. Suppose that (E, F) is a Dirichlet form on L2 (E; m). If the space F ∩ C∞ (E) is dense both in (F , · E1 ) and in (C∞ (E), · ∞ ), then (E, F) is regular. Proof. For a fixed f ∈ F ∩ C∞ (E), we consider its composition f = ϕ ◦ f with specific normal contractions defined by ϕ (t) := t − ((−1/) ∨ t) ∧ (1/),

t ∈ R, ≥ 1,

(1.3.1)

then f ∈ F ∩ Cc (E) and f − f ∞ ≤

1 for ≥ 1,

and we see that F ∩ Cc (E) is dense in the space (Cc (E), · ∞ ). On the other hand, Lemma 1.1.11 implies that f is E1 -convergent to f and hence F ∩ Cc (E) is dense in (F, E1 ). Exercise 1.3.13. Let (E, F) be a regular Dirichlet form on L2 (E; m). Show that, for any f ∈ Cc (E), there exist fn ∈ F ∩ Cc (E) such that supp[fn ] ⊂ supp[f ] for every n ≥ 1, and fn converges to f uniformly on E as n → ∞. For a regular Dirichlet form (E, F) on L2 (E; m), it is customary to use 1-capacity denoted by Cap1 , that is, (h, α)-capacity with h = 1 and α = 1. This is because in this case Cap1 (D) < ∞ for every relatively compact open subset D ⊂ E. Note that since E is a locally compact separable metric space, there is a sequence of relatively compact open subsets {Dk , k ≥ 1} with Dk ⊂ Dk+1 , k ≥ 1, and ∪k≥1 Dk = E. Thus Theorem 1.2.14 is applicable with h = 1. In particular, we have the following, which gives the equivalence of E-polar set, E-nest, and E-quasi-continuity with the notions of set of capacity zero, generalized nest, and quasi continuity, respectively, defined in the book [73]. T HEOREM 1.3.14. Suppose that (E, F) is a regular Dirichlet form on L2 (E; m). Then (i) A subset set of E is E-polar if and only if it is Cap1 -polar.


29

(ii) An increasing sequence of closed subsets {Fj , j ≥ 1} is an E-nest if and only if limj→∞ Cap1 (K \ Fj ) = 0 for every compact set K ⊂ E. (iii) A function f is E-quasi-continuous if and only

if for every ε > 0, there is an open set D ⊂ E with Cap1 (D) < ε such that f E\D is finite and continuous or, equivalently, there exists a Cap1 -nest {Fk } such that f ∈ C({Fk }). Proof. (i) and (ii) follow immediately from Theorem 1.2.14. For (iii), if f is E-quasi-continuous, then, in view of Theorem 1.2.13, there is an E-nest {Fk , k ≥ 1} consisting of closed sets so that f ∈ C({Fk }). Let {Dk , k ≥ 1} be an increasing sequence of relatively compact open subsets with ∪k≥1 Dk = E and Cap(Dk ) < ∞, k ≥ 1. By Theorem 1.2.14, for every ε > 0 and n ≥ 1, there is an integer kn ≥ 1 so that Cap1 (Dn \ Fkn ) 0 and n0 ≥ 1 so that B(x0 , r0 ) ⊂ Dn0 and that E \ D = ∩n≥1 (Dcn ∪ Fkn ), we have B(x0 , r0 ) ∩ (E \ D) ⊂ B(x0 , r0 ) ∩ Fkn0 . It follows that f |E\D is continuous at x0 . The sufficiency in (iii) is obvious because any Cap1 -nest is an E-nest by Remark 1.2.15. The last assertion of the above theorem can be strengthened as follows. Let E∂ = E ∪ {∂} be the one-point compactification of the locally compact metric space E. For a closed set F ⊂ E, we regard F ∪ {∂} as a topological subspace of E∂ . For an increasing sequence {Fk } of closed sets, we denote by C∞ ({Fk }) the collection of functions f on E such that, if f is extended to E∂ by setting f (∂) = 0, then f Fk ∪{∂} is finite and continuous for each k. Obviously the space C∞ (E) is contained in C∞ ({Fk }). Suppose, for a function f on E, there exists an E-nest {Fk } such that f ∈ C∞ ({Fk }). Then f is said to be quasi continuous in the restricted sense relative to the E-nest {Fk }. L EMMA 1.3.15. If (E, F ) is a regular Dirichlet form on L2 (E; m), then each element f ∈ F admits an m-version f which is quasi continuous in the restricted sense relative to a Cap1 -nest. Proof. For f ∈ F ∩ Cc (E) and λ > 0, the set Dλ = {x ∈ E : |f (x)| > λ} is an open set with f /λ ∈ LDλ ,1 so that Cap1 (Dλ ) ≤ E1 (f , f )/λ2 ,

(1.3.2)

which yields the above assertion as in Theorem 1.3.3 because F ∩ Cc (E) is E1 -dense in F.

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Exercise 1.3.16. Let (E, F) be a regular Dirichlet form. Show that the statement in Theorem 1.3.3 holds with an E-nest {Fk } and the space C({Fk }) being replaced by a Cap1 -nest {Fk } and C∞ ({Fk }), respectively. We give the following definition for future use. D EFINITION 1.3.17. Let E be a locally compact separable metric space, m be a Radon measure on E with supp[m] = E, and (E, F) be a Dirichlet form on L2 (E; m). (i) C ⊂ F ∩ Cc (E) is said to be a core of (E, F) if C is dense both in (F , · E1 ) and in (Cc (E), · ∞ ). Clearly the Dirichlet form (E, F) is regular if it has a core. (ii) A core C is said to be standard if it is a dense linear subspace of Cc (E), and for any ε > 0, there exists a normal contraction ϕε of (1.1.7) such that ϕε (C) ⊂ C. (iii) A standard core C is said to be special if C is a dense subalgebra of Cc (E), and for any compact set K and relatively compact open set G with K ⊂ G, there exists f ∈ C+ such that f = 1 on K and f = 0 on E \ G. (iv) (E, F) is called local if E(f , g) = 0 whenever f , g ∈ F have disjoint compact supports. (v) (E, F) is called strongly local if E(f , g) = 0 whenever f ∈ F has a compact support and g ∈ F is constant on a neighborhood of the support of f .

1.4. QUASI-HOMEOMORPHISM OF DIRICHLET SPACES E, B( E)) be a second Suppose (E, F) is a Dirichlet form on L2 (E; m). Let ( measurable space and j : (E, B(E)) → ( E, B( E)) be a measurable map. Define m := m ◦ j−1 , the push forward measure of m under map j; that is, for A ∈ E; m) → L2 (E; m) is an isometry, where B( E), m(A) = m(j−1 (A)). Then j∗ : L2 ( ∗ 2 ∗ 2 f ◦ j for f ∈ L ( E; m). j L ( E; m) is, in general, a closed subspace of j f := := { f ∈ F} and f ∈ L2 ( E; m): j∗ L2 (E; m). Define F E( f , g) := E(j∗ f , j∗ g)

for f , g ∈ F.

F) is a closed form on L2 ( Clearly (E, E; m). If j∗ maps L2 ( E, m) onto L2 (E; m), 2 m), which is called the image Dirichlet then (E, F) is a Dirichlet form on L (E; F) as j(E, F). form of (E, F) under j. We denote in the sequel (E, E) on L2 (E; m) D EFINITION 1.4.1. Given two Dirichlet forms (E, F) and (F, 2 m), respectively, where E and E are two Hausdorff topological spaces and L (E; and m and m are σ -finite measures on E and E respectively with supp[m] = E and supp[ m] = E. The Dirichlet form (E, F) is said to be quasi-homeomorphic


31

F) if there is an E-nest {Fn }n≥1 and an E-nest { Fn }n≥1 and a map to (E, j : ∪k≥1 Fk → ∪k≥1 Fk such that Fk for each k ≥ 1. (a) j is a topological homeomorphism from Fk onto (b) m = m ◦ j−1 . F) = j(E, F); that is, (E, F) is the image Dirichlet form of (E, F) under (c) (E, map j. f is uniquely defined on E modulo an m-null set For every function f on E, j∗ ∗ E; m) onto L2 (E; m). and j is an isometry from L2 ( F) on Exercise 1.4.2. Suppose two Dirichlet form (E, F ) on L2 (E; m) and (E, 2 m) are quasi-homeomorphic by a map j as in the above definition. Prove L (E ; that j is quasi notion preserving in the following sense: (i) An increasing sequence {Ek } of closed subsets of E is an E-nest if and only if {j(Ek ∩ Fk )} is an E-nest. (ii) N ⊂ E is E-polar if and only if j (∪k≥1 Fk ) ∩ N is E-polar. (iii) A function f defined E-q.e. on E is E-quasi-continuous if and only if f ◦ j−1 is E-quasi-continuous. The following theorem gives an important connection between quasiregular Dirichlet forms and regular Dirichlet forms, which enables us to transfer known results for regular Dirichlet forms to quasi-regular Dirichlet forms. T HEOREM 1.4.3. A Dirichlet space (E, F) on L2 (E; m) is quasi-regular if and only if (E, F) is quasi-homeomorphic to a regular Dirichlet space on a locally compact separable metric space. Proof. The “if” part is trivial. We only need to show the “only if” part. Take a strictly positive bounded ϕ ∈ L1 (E; m) and let h = G1 ϕ, which is strictly positive m-a.e. on E and is in F . Since (E, F) is quasi-regular on L2 (E; m), by Theorem 1.3.3, h has an E-quasi-continuous m-version h. We claim that h ∈ C({Kj }) there is an E-nest {Kj , j ≥ 1} consisting of compact sets so that and h ≥ 1/j on each Kj . By Theorems 1.2.13 and 1.3.3, there is an E-nest h ∈ C({ Kj }). For each j ≥ 1, { Kj , j ≥ 1} consisting of compact sets so that Kj ∩ { h ≥ 1/j}, which is compact. We show that {Kj , j ≥ 1} is an define Kj = Kjc ∪ {x ∈ Kj : h(x) < 1/j} and vj := h E-nest. Note that Kjc = Kjc + (1/j) ∧ h ∈ c LKj ,h . Thus lim Caph,1 (Kjc ) ≤ lim E1 h Kjc + (1/j) ∧ h, h Kjc + (1/j) ∧ h j→∞

j→∞

≤ 2 lim E1 h Kjc , h Kjc + 2 lim E1 (1/j) ∧ h, (1/j) ∧ h j→∞

= 0,

j→∞

32

CHAPTER ONE

where in the last equality we used Lemma 1.1.11(i) applied to normal contractions ψj (t) := t − ((−1/j) ∨ t) ∧ (1/j). This establishes that {Kj , j ≥} is an E-nest. Observe that for f ∈ L2 (E; m), ψj (f ) ∈ L1 (E; m). So by Lemma 1.1.11 and Remark 1.3.9, there exists a countable E1 -dense set B0 = {fn , n ≥ 1} of bounded E-quasi-continuous functions in F ∩ L1 (E; m) such that (i) h ∈ B0 and B0 is an algebra over the rational numbers, (ii) There is an E-nest {Fk , k ≥ 1} consisting of compact sets such that h ≥ 1/k on Fk . B0 ⊂ C({Fk }) and B0 separates points of ∪k≥1 Fk and We make functions in B0 to take value zero on E \ ∪k≥1 Fk . Define B := ·∞ B0 , which is a commutative Banach algebra. We now construct a regular F) on a locally compact separable metric space Dirichlet form (E, E via the Gelfand transform which will be quasi-homeomorphic to (E, F). Step 1. Construct a locally compact separable metric space E. Let E be a collection of non-trivial real-valued functionals γ on B which satisfy for f , g ∈ B and for rational numbers a and b, (a) |γ (f )| ≤ f ∞ , (b) γ (fg) = γ (f )γ (g), (c) γ (af + bg) = aγ (f ) + bγ (g). We equip E with the weakest topology so that the function f : γ → γ (f ) is continuous for every f ∈ B. It is well-known that E is a separable locally compact Hausdorff space which is compact if and only if 1 ∈ B, and {f , f ∈ E). The topological space E is metrizable with metric δ defined by B} ⊂ C∞ ( 2−n (|γ (fn ) − η(fn )| ∧ 1) , γ,η ∈ E. δ(γ , η) := n≥1

E such that Let j be the unique map from ∪k≥1 Fk into (jx)(f ) := f (x)

for f ∈ B and x ∈ ∪k≥1 Fk .

Fk := j(Fk ) is By (ii) above, j is a continuous one-to one map on each Fk . Hence Fk for every compact in E and j is a topological homeomorphism from Fk onto E is Borel measurable and m(E \ ∪k≥1 Fk ) = k ≥ 1. Note that j : ∪k≥1 Fk → Fk ) = 0. It follows from the mm( E \ ∪k≥1 0. Define m := m ◦ j−1 . Clearly m is a Radon measure, and it is easy to integrability of functions in B0 that check that supp[ m] = E (see [138, p. 23]). Since B0 is dense in L2 (E; m), j∗ is 2 m) onto L2 (E; m). a unitary map from L (E; E). Step 2. maps B onto C∞ ( E), where f (γ ) = γ (f ). Clearly, f ∞ = f ∞ . So For f ∈ B, f ∈ C∞ ( (B) is closed under uniform norm. Since h ∈ B and (B) is an algebra of realvalued functions that vanish at infinity and separates points in E with ( h) > 0


33

E). on E, by Stone-Weierstrass theorem (cf. [58, Theorem 4.52]), (B) = C∞ ( E; m). Step 3. The image Dirichlet form j(E, F ) is regular on L2 ( F) := j(E, F). Then (E, F ) is a Dirichlet form on L2 ( E; m) as j∗ Let (E, 2 2 ∩ C∞ ( is an isometry from L (E; m) onto L (E; m). Since F E) ⊃ (B0 ) and , (E, F) is a regular E) and E1 -dense in F the latter is uniformly dense in C∞ ( F = FFk for every k ≥ 1, { E; m). Since j∗ F Fk , k ≥ 1} is Dirichlet form on L2 ( k F). an E-nest and therefore j is a quasi-homeomorphism from (E, F) to (E, This completes the proof of the theorem.

1.5. SYMMETRIC RIGHT PROCESSES AND QUASI-REGULAR DIRICHLET FORMS T HEOREM 1.5.1. Let (E, F) be a regular Dirichlet form on L2 (E; m), where E is a locally compact separable metric space and m is a Radon measure on E with full support. There exists then a Hunt process X on E with an m-symmetric transition function so that (E, F) is the Dirichlet form of the transition function of X. This theorem was proved by the second-named author [62] in 1971. A rather different proof from that of [62] is presented in [73]. Theorem A.1.37 of Appendix A on the Feller semigroup and this theorem constitute basic existence theorems of Hunt processes on locally compact spaces. We can now combine Theorem 1.4.3 with Theorem 1.5.1 to show that there is a nice Markov process called an m-tight special Borel standard process associated with every quasi-regular Dirichlet form. See Section A.1.3 for the definition of a right process and a special Borel standard process. Let (E, B∗ (E)) be a Radon space, m be a σ -finite measure on it, and X a right process on it. If the transition function {Pt ; t ≥ 0} is m-symmetric, we say that X is m-symmetric. In this case, the Dirichlet form (E, F ) on L2 (E; m) of {Pt ; t ≥ 0} is called the Dirichlet form of the m-symmetric right process X. We say further that X is properly associated with (E, F) if Pt f is an E-quasicontinuous m-version of Tt f for any f ∈ B(E) ∩ L2 (E; m) and t > 0, where {Tt ; t > 0} is the L2 (E; m)-semigroup generated by (E, F ). A right process X is called m-tight if there is an increasing sequence of compact sets {Kj , j ≥ 1} so that Pm (limj→∞ τKj < ζ ) = 0. Here τKj := inf{t ≥ 0 : / Kj } is the first exit time from Kj by X and ζ is the lifetime of X. Xt ∈ T HEOREM 1.5.2. Suppose that (E, F ) is a quasi-regular Dirichlet form on L2 (E; m), where E is a Hausdorff topological space such that the Borel σ -field B(E) is generated by the continuous functions on E. Then there is an E-polar Borel set N ⊂ E and an m-symmetric, m-tight, special Borel standard process X on E \ N that is properly associated with (E, F).

34

CHAPTER ONE

Proof. By Theorem 1.4.3, (E, F) is quasi-homeomorphic to an m-symmetric F) on a locally compact separable metric space regular Dirichlet form (E, F) = E through quasi-homeomorphism j. More precisely, m = m ◦ j−1 , (E, j(E, F), and there are E-nest {Fk , k ≥ 1} and E-nest {Fk , k ≥ 1} so that j is a Fk for every k ≥ 1. j is a one-to-one topological homeomorphism from Fk onto ∞ map from E1 = ∪∞ k=1 Fk onto E1 = ∪k=1 Fk and it can be extended to a oneE1 ∪ { ∂}, where ∂ is an extra point adjoined to-one map from E1 ∪ {∂} onto to E and ∂ is the point at infinity of E. On account of Theorem 1.2.13 and Theorem 1.3.14, we may and do assume that each Fk is compact (conse if quently, each Fk is compact) by taking an intersection with another E-nest necessary. By virtue of Theorem 1.5.1, there is an m-symmetric Hunt process ζ, Xt , Px ) X = (, { Ft }, F) is the Dirichlet form of on E such that (E, X on L2 ( E, m). We shall make use of some theorems in Section 3.1 concerning the relations be (This is the only proof in the book that uses forward tween X and E. references.) In view of Proposition 3.1.9, X is automatically properly asso F). ciated with (E, Fk of X . By Theorem 3.1.4 and Denote by τFk the first exit time from m( N) = 0 Theorem 3.1.5, there exists a Borel set N containing E \ E1 such that ) = 1 for every x∈ E\ N , where and Px ( = ω ∈ : lim τFk = ζ, Xt , Xt− ∈ E∂ \ N for every t ≥ 0 . k→∞

The above set N is called a Borel properly exceptional set for the Hunt process X. E1 \ N ). We let We define an E-polar Borel set N ⊂ E by E \ N = j−1 ( = , Ft = Ft ∩ , t ∈ [0, ∞], and denote an element of (resp. F∞ ) by ω (resp. ). Finally we define X = (, {Ft }, Xt , ζ , Px ) by Xt (ω)) and ζ (ω) := ζ (ω) for ω ∈ and t ≥ 0, Xt (ω) := j−1 ( and Px () := Pj(x) () for x ∈ E \ N and ∈ F∞ . Observe that τFk = τFk for every k ≥ 1, where τFk is the first exit time from Fk by X. It is straightforward to check that X is an m-symmetric, m-tight, special Borel standard process on E \ N properly associated with (E, F). As will be shown in Theorems 3.1.12 and 3.1.13, the Hunt process (respectively, m-symmetric right process) associated with a regular Dirichlet form (respectively, quasi-regular Dirichlet form) is unique in distribution. Moreover, it will be shown in Theorem 3.1.13 that for a quasi-regular Dirichlet


35

form (E, F), {Fk , k ≥ 1} is an E-nest if and only if lim τFk = ζ

k→∞

Px -a.s. for E-q.e. x ∈ E,

where ζ is the lifetime of for the right process X associated with (E, F). Thus quasi-homeomorphism is not only an isometry at the Dirichlet form level but Fk also a topological isometry at the process level up to its lifetime, as j : Fk → is a topological homeomorphism. In view of Theorem 1.5.2, we can assume without loss of generality, in most of the rest of the book, that the Dirichlet form (E, F) is regular, as corresponding results for quasi-regular Dirichlet forms can be easily deduced via quasi-homeomorphism. In fact, the quasi-regularity of a Dirichlet form is not only sufficient but also necessary for the association of an m-tight special Borel standard process. More generally the following theorem holds: T HEOREM 1.5.3. Let E be a Radon space and m be a σ -finite measure on E with full support. Suppose that X is an m-symmetric and m-tight right process on E. Then the Dirichlet form (E, F) on L2 (E; m) of X is quasi-regular and X is properly associated with (E, F ). If in particular E is a Lusin space, then the Dirichlet form of an m-symmetric right process is quasi-regular and X is properly associated with (E, F). The second statement follows from the first because an m-symmetric right process on a Lusin space E is necessarily m-tight (see [119, Theorem IV.1.15]). When X is an m-tight, m-special Borel standard process on E, this result was proved by S. Albeverio and Z. M. Ma [2] in 1991 and its proof can be found in the book by Z. M. Ma and M. Röckner [119, Theorem IV.5.1] under a more general assumption on the state space E. The result under the current condition follows from the aforementioned result in [119] together with a result of P. J. Fitzsimmons [53, Theorem 3.22], who showed that the restriction of X on the complement of some m-inessential set is an m-special standard process and the Borel measurability assumption on the transition function can be weakened to the universal measurability. As a matter of fact, the stated results in [119] and [53] are formulated for a more general (not necessarily symmetric) sectorial Dirichlet form (E, F). Moreover, Theorem 1.4.3 also holds for more general sectorial Dirichlet forms; see [31]. It is important to consider a general right process in applications as it is invariant under variety of transformations (for example, time change, killing, h-transformations) while Borel measurability of the transition function is not. However, we shall prove in Theorem 3.1.13 of Section 3.1 that any m-symmetric right process properly associated with a quasi-regular Dirichlet form is, when restricted to the complement of an m-inessential set, a Borel special standard process properly associated with the form. This combined

36

CHAPTER ONE

with Theorem 1.5.3 means that any m-tight m-symmetric right process on a Radon space or any m-symmetric right process on a Lusin space can always be modified to be a Borel special standard process (see Corollary 3.1.14 below). We end this chapter by noting that any quasi-regular Dirichlet form (E, F) on L2 (E; m) admits the following expression in terms of an associated m-symmetric right process X = (Xt , Px , ζ ) on E: for any f ∈ Fe , 1 E(f , f ) = lim Em (f (Xt ) − f (X0 ))2 ; t < ζ t→0 2t 2 (1.5.1) +2 f (x) Px (ζ ≥ t)m(dx) , E

where (Fe , E) is the extended Dirichlet space of (E, F). To see this, let {Tt ; t > 0} be the semigroup on L∞ (E; m) determined by the transition function of X. Then by (1.1.13), for f ∈ L∞ (E; m), 1 2 ATt (f , f ) = Em (f (Xt ) − f (X0 )) ; t < ζ + f (x)2 Px (ζ ≥ t)m(dx). 2 E In view of (1.1.20), (1.1.22), 1 1 E(f , f ) = lim ATt (f , f ) = lim lim ATt (ϕ ◦ f , ϕ ◦ f ), t→0 t t→0 →∞ t

f ∈ Fe ,

for the normal contraction ϕ defined by (1.1.19), and consequently, we get (1.5.1) by the monotone convergence theorem.

Chapter Two BASIC PROPERTIES AND EXAMPLES OF DIRICHLET FORMS

2.1. TRANSIENCE, RECURRENCE, AND IRREDUCIBILITY In this section, we introduce the concepts of the transience, recurrence, and irreducibility of the semigroup for general Markovian symmetric operators and present their characterizations by means of the associated Dirichlet form as well as the associated extended Dirichlet space. These notions are invariant under the time changes of the associated Markov process. We shall also examine them in the concrete examples of the next section. As in Section 1.1, we let (E, B(E), m) be a σ -finite measure space and consider a strongly continuous contraction semigroup {Tt ; t > 0} of symmetric Markovian operators and a Dirichlet form (E, F ) on L2 (E; m), which are mutually related according to Theorem 1.1.3. Denote by {Gα ; α > 0} the resolvent of {Tt ; t > 0}. For f ∈ L2 (E; m), we define t Ts fds, t > 0, (2.1.1) St f = 0

where the integral is in the sense of Bochner. St is a linear operator on L2 (E; m) and satisfies St f 2 ≤ t f 2 for f ∈ L2 (E; m). Take f ∈ L2 (E; m) ∩ L1 (E; m). Choose Bn ∈ B(E) with m(Bn ) < ∞ and Bn ↑ E. By the symmetry and the Markov property of Tt , |Tt f (x)|m(dx) ≤ (Tt | f |, 1Bn ) = (| f |, Tt 1Bn ) ≤ | f (x)|m(dx). Bn

E

Letting n → ∞, we get Tt f 1 ≤ f 1 . Similarly, we get St f 1 ≤ t f 1 . Hence both {Tt , t ≥ 0} and {St , t ≥ 0} can be extended to linear operators on L1 (E; m) satisfying for f ∈ L1 (E; m), Ts Tt f = Ts+t f ,

Tt f 1 ≤ f 1

and

St f 1 ≤ t f 1 .

Moreover, Tt and are Markovian. The same can be said about {Gα }. A concrete way to extend Tt is to define Tt f = lim→∞ Tt (ϕ ◦ f ) for f ∈ L1 (E; m), by the function ϕ (t) of (1.1.19). 1 S t t

38

CHAPTER TWO

Since St and Gα so extended enjoy the positivity and the monotonicity for 1 (E; m): for t > s > 0 and β > α > 0, f ∈ L+ 0 ≤ Ss f ≤ St f [m]

and

0 ≤ Gβ f ≤ Gα f [m],

1 we can define for f ∈ L+ (E; m) a function Gf (x) by

Gf = lim SN f = lim G1/N f N→∞

N→∞

[m],

(2.1.2)

which is uniquely up to the m-equivalence. 1 (E; m) to Gf is called the potential operator The operation that maps f ∈ L+ of the semigroup {Tt ; t > 0}. When E is a Hausdorff topological space and {Tt ; t > 0} is determined by an m-symmetric transition function {Pt ; t ≥ 0} 1 (E; m) ∩ B(E), Gf has an m-equivalent as in Lemma 1.1.14, then for f ∈ L+ version Rf defined by ∞ Pt f (x)dt, x ∈ E. (2.1.3) Rf (x) = 0

This operator R is said to be the potential operator or 0-order resolvent kernel of the transition function {Pt ; t ≥ 0}. D EFINITION 2.1.1. (i) {Tt ; t > 0} is called transient if Gg < ∞ [m] for some 1 (E; m) with g > 0 [m]. g ∈ L+ 1 (E; m), Gf is either ∞ or (ii) {Tt ; t > 0} is called recurrent if, for any f ∈ L+ 0 [m], namely, m{x ∈ E : 0 < Gf (x) < ∞} = 0. (iii) A ∈ Bm (E) is called {Tt }-invariant if Tt (1Ac f ) = 0 m-a.e. on A for every t > 0 and f ∈ L2 (E; m). (iv) {Tt ; t > 0} is called irreducible if any {Tt }-invariant set A is trivial in the sense that either m(A) = 0 or m(Ac ) = 0. Inequality (2.1.4) in the first part of next lemma is called Hopf’s maximal inequality. L EMMA 2.1.2. (i) For f ∈ L1 (E; m) and t > 0, consider the set Et = {x ∈ E : supn Snt f (x) > 0}. Then St f (x)m(dx) ≥ 0. (2.1.4) Et 1 (ii) For f ∈ L+ (E; m) and B ∈ B(E), 1 St fdm ≥ fdm. lim inf t↓0 t B B

(2.1.5)

39

BASIC PROPERTIES AND EXAMPLES OF DIRICHLET FORMS

Proof. (i) If we let

Etn

:= x ∈ E : max Skt f (x) > 0 1≤k≤n

+

= x ∈ E : max (Skt f ) (x) > 0 , 1≤k≤n

then, for x ∈

Etn ,

St f (x) + max (S(k+1)t f − St f )+ (x) ≥ max (Skt f )+ (x). 1≤k≤n

1≤k≤n

On the other hand, since S(k+1)t f − St f = Tt (Skt f ) and Tt is positivity preserving (namely, sending a non-negative function to a non-negative function) linear operator, we have max (S(k+1)t f − St f )+ (x) ≤ Tt max (Skt f )+ (x). 1≤k≤n

1≤k≤n

Combining the above two inequalities, we obtain St f (x)m(dx) Etn

≥

Etn

+

+

max (Skt f ) (x) − Tt max (Skt f )

1≤k≤n

1≤k≤n

(x) m(dx)

+ + ≥ max (Skt f ) − Tt max (Skt f ) ≥ 0. 1≤k≤n 1≤k≤n 1

1

(ii) Since limt→0+ 1t St g − g2 = 0 for BN ∈ B(E) with m(BN ) < ∞ and BN ↑

g ∈ L2 (E; m), we can get, by choosing E, 1 1 St fdm ≥ 1B∩BN , St ( f ∧ N) → (1B∩BN , f ∧ N), t ↓ 0. t B t

It suffices then to let N → ∞.

P ROPOSITION 2.1.3. (i) {Tt ; t > 0} is transient if and only if Gf < ∞ [m] 1 (E; m). for every f ∈ L+ (ii) The recurrence of {Tt ; t > 0} is equivalent to either of the following two conditions: Gf = ∞ [m]

1 (E; m) with f > 0 [m]. for every f ∈ L+

1 There is some g ∈ L+ (E; m) so that Gg = ∞ [m].

(iii) If {Tt ; t > 0} is irreducible, then it is either transient or recurrent.

(2.1.6) (2.1.7)

40

CHAPTER TWO

1 (iv) {Tt ; t > 0} is irreducible and recurrent if and only if, for any f ∈ L+ (E; m), either Gf = ∞ [m] or Gf = 0 [m].

Proof. (i) Assuming the transience, we take a function g satisfying condition 1 (E; m), put B = {x ∈ E : Gf (x) = ∞}. It (i) in Definition 2.1.1. For any f ∈ L+ suffices to show that m(B) = 0. For any a > 0, h > 0, we have for the set A = x ∈ E : sup Snh ( f − ag)(x) > 0 , n

Sh ( f − ag)dm ≥ 0 by virtue of (2.1.4). Since B ⊂ A, m-a.e., h fdm ≥ Sh fdm ≥ a Sh gdm ≥ a Sh gdm,

the inequality

A

E

A

and hence

A

B

1 Sh gdm, (2.1.8) h B E which, combined with (2.1.5), implies B gdm ≤ 1a f 1 . By letting a → ∞, we get B gdm = 0 and accordingly m(B) = 0. (ii) If we put B = {x ∈ E : Gf (x) = 0} in (2.1.5), then we get f = 0 m-a.e. on B and hence the recurrence implies (2.1.6), which in turn implies 1 (E; m), we (2.1.7) trivially. For the function g satisfying (2.1.7) and f ∈ L+ put A := {x ∈ E : supn Sn (g − af )(x) > 0} for a > 0, h > 0, and B := {x ∈ E : Gf (x) < ∞}. We then obtain, by (2.1.4) and the same argument as in the proof of (i), the inequality (2.1.8) with f and g being interchanged. By letting a → ∞, we get Sh f = 0 and hence Gf = 0 on the set B. This shows {Tt ; t > 0} is recurrent. 1 (E; m), the set B = {x ∈ E : Gg(x) = ∞} is {Tt ; t > 0}(iii) For g ∈ L+ invariant. To see this, put Cn = {x ∈ En : Gg(x) ≤ n}, g = g ∧ for En ∈ B(E) such that m(En ) < ∞, En ↑ E. Then Cn ↑ Bc and, by the symmetry of 2 (E; m) Tt , we have for any f ∈ L+ 1 a

fdm ≥

(Tt (1Cn f ), G1/ g ) = (1Cn f , Tt G1/ g ) ≤ (1Cn f , Gg) ≤ n( f , 1Cn ) < ∞. By letting → ∞, we get (Tt (1Cn f ), Gg) < ∞ and consequently 1B · Tt (1Cn f ) = 0 [m]. We finally let n → ∞ to conclude 1B · Tt (1Bc f ) = 0 [m]. Therefore, under the irreducible assumption, the above set B for a strictly positive g ∈ L1 (E; m) satisfies either m(B) = 0 or m(Bc ) = 0. {Tt ; t > 0} is transient in the former case and recurrent in the latter case by (ii). (iv) Under the irreducibility, the above set B in the proof of (iii) for g ∈ 1 (E; m) satisfies either m(B) = 0 or m(Bc ) = 0. If we assume the recurrence L+ additionally, then Gg = 0 [m] in the former case.

41


Conversely, assume the condition in (iv). Then {Tt ; t > 0} is obviously recurrent. If A ∈ B(E) is {Tt }-invariant, then G(1Ac f ) = 0 m-a.e. on A for a strictly positive f ∈ L1 (E; m) ∩ L2 (E; m). If both m(A) and m(Ac ) are positive, then m(g > 0) > 0 and m(Gg = 0) > 0 for g = 1Ac f . By the assumption, Gg = 0 [m], which forces m(g > 0) = 0 by (2.1.5), a contradiction. We now give a transience characterization in terms of the Dirichlet form. L EMMA 2.1.4. (i) For any g ∈ L2 (E; m) and t > 0, St g ∈ F and E(St g, u) = (g − Tt g, u)

for every u ∈ F .

(ii) For any non-negative g ∈ L1 (E; m) ∩ L2 (E; m),

(|u|, g) sup √ g · Ggdm (≤ +∞). = E(u, u) u∈F E

(2.1.9)

(2.1.10)

t+s s Proof. (i) Since St g − Ts St g = − t Tv gdv + 0 Tv gdv and 1s (St g − Ts St g, St g) converges as s ↓ 0 to a finite limit (g, St g) − (Tt g, St g), we get the first conclusion. The same computation gives (2.1.9). (ii) Denote√by c the left √ hand side of (2.1.10). Suppose c < ∞, then √ (St g, g) ≤ c E(St g,St g) ≤ c (St g, g) by (i) and so (St g, g) ≤ c. Letting t ↑ ∞, we obtain E gGgdm ≤ c. Conversely, suppose that the right hand ∞ side of (2.1.10) is finite. Since E gGgdm = 0 (Ts g, g)ds and (Ts g, g) = (Ts/2 g, Ts/2 g) is non-increasing as s increases, we must have lims↑∞ (Ts g, g) = 0. By (i), we then have for any u ∈ F, (|u|, g) = E(|u|, St g) + (|u|, Tt g) ≤ St gE uE + Tt g2 u2

≤ (St g, g) uE + (T2t g, g) u2 → gGgdm uE E

as t ↑ ∞.

T HEOREM 2.1.5. (i) {Tt ; t > 0} is transient if and only if there exists a bounded m-integrable function g strictly positive on E such that |u(x)|g(x)m(dx) ≤ uE for every u ∈ F . (2.1.11) E

(ii) Suppose {Tt ; t > 0} is transient. Then the inequality (2.1.11) holds for every u ∈ Fe . Furthermore, the extended Dirichlet space Fe is a real Hilbert space with inner product E.

42

CHAPTER TWO

Proof. (i) If (2.1.11) holds, then the right hand side of (2.1.10) is no larger than 1. Consequently, Gg < ∞ [m]; namely, {Tt ; t > 0} is transient. Conversely, if {Tt ; t > 0} is transient, then the condition of Proposition 2.1.3(i) is fulfilled. By taking a strictly positive bounded function h on E with E hdm = 1 and putting g = h/(Gh ∨ 1), we have 0 < g ≤ h and g · Ggdm ≤ h · Ggdm ≤ Gh · (h/Gh)dm = hdm = 1, E

E

E

E

which, combined with (2.1.10), leads us to (2.1.11). (ii) Suppose {Tt ; t > 0} is transient. The first assertion is obvious due to the definition of the extended Dirichlet space Fe and indeed Fe is continuously embedded into the space L1 (E; g · m). As for the second assertion, it suffices to prove the completeness of the space (Fe , E). Observe that, for u ∈ Fe and its approximating functions {un } ⊂ F, u − un E = lim→∞ u − un E and hence {un } is E-convergent to u. Now take any E-Cauchy sequence {un } in Fe . Choose {vn } ⊂ F with limn→∞ un − vn E → 0. Then {vn } is E-Cauchy and hence L1 (E; g · m)-Cauchy by virtue of (2.1.11). There is subsequence nk so that vnk converges m-a.e. to a function u ∈ L1 (E; g · m) as k → ∞. Then u ∈ Fe and un − uE ≤ un − unk E + unk − vnk E + vnk − uE . Letting k → ∞ and then n → ∞, we see that un is E-convergent to u as n → ∞. A Dirichlet form possessing the property of Theorem 2.1.5(i) is called transient and a function g appearing there will be called a reference function for the transient Dirichlet form. A strictly positive bounded m-integrable function g on E will be called a reference function of a transient semigroup {Tt ; t > 0} if E g · Ggdm ≤ 1. A function g is a reference function of a transient semigroup if it is so for the associated Dirichlet form. We next formulate the restriction of the Dirichlet form to a {Tt }-invariant set. P ROPOSITION 2.1.6. A ∈ Bm (E) is {Tt }-invariant if and only if so is Ac . Furthermore, the following conditions for A ∈ Bm (E) are mutually equivalent. (i) A is {Tt }-invariant. (ii) Tt (1A f ) = 1A Tt f for every t > 0 and f ∈ L2 (E; m). (iii) Gα (1A f ) = 1A Gα f for every α > 0 and f ∈ L2 (E; m). (iv) For any f ∈ F, 1A · f ∈ F and E( f , g) = E(1A f , 1A g) + E(1Ac f , 1Ac g),

f , g ∈ F.

(2.1.12)

f , g ∈ Fe .

(2.1.13)

(v) For any f ∈ Fe , 1A · f ∈ Fe and E( f , g) = E(1A f , 1A g) + E(1Ac f , 1Ac g),

43


Proof. Due to the symmetry of Tt , (1A f , Tt (1Ac g)) = (Tt (1A f ), 1Ac g), for f , g ∈ L2 (E; m) and consequently the {Tt }-invariance of A is equivalent to that of Ac . Therefore, if (i) holds, then Tt (1A f ) = 1A Tt (1A f ) = 1A (Tt f − Tt (1Ac f )) = 1A Tt f , namely, (ii) is valid. The implications (ii) ⇒ (i) and (ii) ⇔ (iii) are obvious. If we assume (i), then ( f , (I − Tt )f ) = (1A f , (I − Tt )(1A f )) + (1Ac f , (I − Tt )(1Ac f )) and hence (iv) follows from (1.1.4), (1.1.5). We next assume (iv). Comparing the equations obtained by substituting 1A f , 1A g for f , g in (2.1.12), respectively, we get E(1A f , g) = E(1A f , 1A g) = E( f , 1A g) for f , g ∈ F , and accordingly we have for any f ∈ L2 (E; m), g ∈ F , Eα (Gα (1A f ), g) = (1A f , g) = ( f , 1A g) = Eα (Gα f , 1A g) = Eα (1A Gα f , g), yielding (iii). The implication (iv) ⇒ (v) follows from Theorem 1.1.5(i). The converse is trivial. We call a Dirichlet form or an extended Dirichlet space irreducible if, for any A ∈ Bm (E) satisfying condition (iv) or (v) of Proposition 2.1.6, either m(A) = 0 or m(Ac ) = 0. Let A ⊂ E be an m-measurable {Tt }-invariant set. The restrictions of a function f and a measure m on E to A will be denoted by f |A and m|A , respectively. If we let F A = { f |A : f ∈ F},

E A ( f |A , g|A ) = E(1A f , 1A g),

f , g ∈ F,

(2.1.14)

then (E A , F A ) is a closed symmetric form on L2 (A; mA ). The semigroup {TtA ; t ≥ 0} and the resolvent {GAα ; α > 0} generated by it can be verified to satisfy

TtA ( f |A ) = Tt (1A f ) A , GAα ( f |A ) = Gα (1A f ) A , f ∈ L2 (E; m), (2.1.15) and accordingly, (E A , F A ) becomes a Dirichlet form on L2 (A; mA ). We call this the restriction of the Dirichlet form (E, F) to the {Tt }-invariant set A. Let (FeA , E A ) be the space defined by (2.1.14) with Fe in place of F . (FeA , E A ) is called the restriction of the extended Dirichlet space (Fe , E) to A. (FeA , E A ) is easily seen to be the extended Dirichlet space of (E A , F A ). We now turn to a recurrence characterization. We employ a simple perturbation method for Dirichlet forms. We choose a function η on E satisfying η ∈ L1 (E; m) ∩ L∞ (E; m),

η > 0 [m],

(2.1.16)

44

CHAPTER TWO

and put

where ( f , g)η·m

E η ( f , g) = E( f , g) + ( f , g)η·m , = E f (x)g(x)η(x)m(dx). Since

f , g ∈ F,

E1 ( f , f ) ≤ E η ( f , f ) + ( f , f ) ≤ E( f , f ) + (1 + η∞ )( f , f ),

(2.1.17)

f ∈ F,

(E η , F) is clearly a Dirichlet form on L2 (E; m). The quantities related to this Dirichlet form will be designated by a superscript η. η

L EMMA 2.1.7. Suppose {Tt ; t > 0} is recurrent. If we let fn = G1/n η, then fn ∈ F, 0 ≤ fn ↑ 1 [m] as n → ∞, and limn→∞ E( fn , fn ) = 0. Proof. For any f ∈ L2 (E; m), g ∈ F and α > 0, it holds that Eα (Gηα f , g) = Eαη (Gηα f , g) − (Gηα f , g)η·m = ( f − ηGηα f , g).

(2.1.18)

Hence Gηα f = Gα ( f − ηGηα f ).

(2.1.19)

On the other hand, for any ε > 0, (E, F) can be viewed as a Dirichlet form on L2 (E; (η + ε)m)(= L2 (E; m)) and we have the identity E(Gηε (εf + f η), g) + (Gηε (εf + f η), g)(η+ε)m = Eεη (Gηε (εf + f η), g) = (εf + f η, g) = ( f , g)(η+ε)m , which means that Gηε (εf + f η) is nothing but the 1-order resolvent of f generated by the Dirichlet form (E, F) on L2 (E; (η + ε)m). Due to the Markov property of the resolvent (Theorem 1.1.3,) we have 0 ≤ Gηε (εf + f η) ≤ 1 for any f ∈ F with 0 ≤ f ≤ 1. Since Gηε is positivity preserving, it holds then that 0 ≤ Gηε ( f η) ≤ 1. By letting ε ↓ 0 and then f ↑ 1, we have 0 ≤ Gη η ≤ 1 [m].

(2.1.20)

Taking f = η in (2.1.20), we have by (2.1.20) that 0 ≤ G(η(1 − Gη η)) ≤ lim G(η − ηGηα η) = Gη η ≤ 1 [m]. α↓0

(2.1.21)

Since {Tt ; t > 0} is recurrent and η > 0 on E, (2.1.21) implies Gη η = 1 [m] on η account of (2.1.5). We now let fn = G1/n η. Then 0 ≤ fn ↑ 1, n → ∞, and we have from (2.1.18) E( fn , fn ) ≤ E1/n ( fn , fn ) = (η(1 − fn ), fn ) ≤ η(1 − fn )dm → 0 as n → ∞. E


45

T HEOREM 2.1.8. The following are mutually equivalent: (i) {Tt ; t > 0} is recurrent. (ii) There exists a sequence { fn } ⊂ F such that limn→∞ fn = 1 [m] and limn→∞ E( fn , fn ) = 0. (iii) 1 ∈ Fe and E(1, 1) = 0. Proof. By the definition of the extended Dirichlet space (Fe , E), (ii) and (iii) are equivalent. The implication (i) ⇒ (ii) has been proved by the preceding lemma. Assume that (ii) holds. If (i) is false, then (2.1.6) does not hold and there 1 (E; m) with g > 0 [m] such that the set A = {x ∈ E : exists a function g ∈ L+ Gg(x) < ∞} satisfies m(A) > 0. In view of the first statement in Proposition 2.1.6 and the proof of Proposition 2.1.3(iii), A is {Tt }-invariant so that we may consider the restriction (E A , FA ) of the Dirichlet form (E, F) to the set A defined by (2.1.14). (2.1.15) then implies GA g|A < ∞ [mA ] and Relation (2.1.15) then implies A G g|A < ∞ [mA ] and, accordingly, (E A , F A ) is transient as a Dirichlet form on L2 (A; mA ). By virtue of Theorem 2.1.5, there exists an mA -integrable bounded function h on A with h > 0 [mA ] such that | f |hdm ≤ 1A f E ≤ f E for every f ∈ F. A

Applying the above to fn , we have from the assumption (ii) and Fatou’s lemma that A hdm = 0. This contradiction establishes the implication that (ii) ⇒ (i). Based on this theorem, a Dirichlet form or an extended Dirichlet space satisfying condition (ii) or (iii) of Theorem 2.1.8, respectively, can be called recurrent. We now give characterizations of the transience in terms of the extended Dirichlet space. T HEOREM 2.1.9. The following conditions are mutually equivalent: (i) {Tt ; t > 0} is transient. (ii) The extended Dirichlet space (Fe , E) is a real Hilbert space. (iii) f = 0 [m] for every f ∈ Fe with E( f , f ) = 0. Proof. The implication (i) ⇒ (ii) has already been established in Theorem 2.1.5. The implication (ii) ⇒ (iii) is trivial. Assume (iii) holds. If (i) were not true, (2.1.6) of Proposition 2.1.3(ii) 1 (E; m) so that the set A = fails and thus there exists a function g ∈ L+ {x ∈ E : Gg(x) = ∞} has m(A) > 0. By the proof of Proposition 2.1.3(iii), A is {Tt }-invariant and we may consider the restricted Dirichlet form

46

CHAPTER TWO

(E A , FA ) to A defined by (2.1.14). By (2.1.15) and Proposition 2.1.6, GA (g|A ) = G(1A g)|A = (Gg)|A = ∞ [mA ]. So by (2.1.7), {TtA ; t > 0} is recurrent. Hence Theorem 2.1.8 yields that there is a sequence { fn } ⊂ F such that lim 1A fn = 1A

and

n→∞

lim E(1A fn , 1A fn ) = 0.

n→∞

Since 1A fn ∈ F for each n ≥ 1, the above means 1A ∈ Fe with E(1A , 1A ) = 0, a contradiction to the condition (iii). As was mentioned in Section 1.1, we can make use of an increasing 1 (E; m) that increases to 1 to deduce sequence of functions {ηn ; n ≥ 1} ⊂ L+ 2 from the operator Tt on L (E; m) a unique Markovian linear operator Tt on L∞ (E; m) by (1.1.9). If Tt 1 = 1 [m] for some (and hence for all) t > 0, then we call the semigroup {Tt ; t > 0} or the Dirichlet form (E, F) conservative. P ROPOSITION 2.1.10. If {Tt ; t > 0} is recurrent, then it is conservative. If ∪ {T 1 < 1} = E [m], then {Tt ; t > 0} is transient. Proof. If f ∈ L1 (E; m) ∩ L∞ (E; m), f > 0 [m], then for every t > 0, N (SN f , ηn − Tt ηn ) = f , Ts (ηn − Tt ηn )ds =

0

t

f, 0

Ts ηn ds −

N+t

t Ts ηn ds ≤ f , Ts ηn ds ≤ t fdm.

N

0

E

By letting n → ∞ and then N → ∞, we obtain Gf (x)(1 − Tt 1(x))m(dx) ≤ t fdm < ∞, E

E

which implies the first assertion (with the aid of Proposition 2.1.3(ii)) as well as the second one. The next theorem gives a criterion for a recurrent Dirichlet form to be irreducible. According to Theorem 2.1.8, the recurrence of the Dirichlet form (E, F) is equivalent to the following property when m(E) < ∞: 1∈F

and

E(1, 1) = 0.

(2.1.22)

T HEOREM 2.1.11. Suppose that m(E) < ∞ and that a Dirichlet form (E, F) is recurrent. Then the following conditions are mutually equivalent. (i) (E, F) is irreducible. (ii) f is constant m-a.e. for every f ∈ F with E( f , f ) = 0. (iii) f is constant m-a.e. for every f ∈ Fe with E( f , f ) = 0. (iv) f is constant m-a.e. for every f ∈ L2 (E; m) with Tt f = f for every t > 0.


47

Proof. (i) ⇒ (ii). We first note that f is {Tt }-invariant whenever f ∈ F with E( f , f ) = 0. Indeed, by the Cauchy-Schwarz inequality holding for the nonnegative definite symmetric form E, we have E( f , g) = 0 and hence Eα ( f , g) = α( f , g) for every g ∈ F and α > 0. Therefore by (1.1.6), αGα f = f for every α > 0, and so by (1.1.2) Tt f = f for every t > 0. Assume that f ∈ F with E( f , f ) = 0. By (2.1.22), it holds that for any λ ∈ R that f − λ ∈ F and E( f − λ, f − λ) = 0. Since ϕ + (t) = t ∨ 0, t ∈ R, is a normal contraction, fλ = ϕ + ◦ ( f − λ) ∈ F with E( fλ , fλ ) = 0. Consequently, fλ is {Tt }-invariant. Put Bλ = {x ∈ E : fλ (x) = 0}. Since (1Bλ , Tt (1Bcλ fλ )) = (1Bλ , Tt ( fλ )) = (1Bλ , fλ ) = 0, we have by the Markovian property of Tt , (1Bλ , Tt (1Bcλ 1{ fλ ≥1/n} )) = 0

for every n ≥ 1.

By letting n → ∞, we get 1Bλ Tt (1Bcλ ) = 0 [m], which gives the {Tt }-invariance of the set Bλ . By the irreducibility assumption (i), either m(Bλ ) = 0 or m(Bcλ ) = 0. If we let λ0 = sup{λ : m(Bλ ) = 0}, then, for any λ > λ0 , m(Bλ ) = 0 and hence m(Bcλ ) = 0, namely, m({ f > λ0 }) = 0. On the other hand, we have, for any λ < λ0 , m(Bλ ) = 0 and m({ f < λ0 }) = 0. We have derived f = λ0 [m]. (ii) ⇒ (iv). If f ∈ L2 (E; m) is {Tt }-invariant, then f ∈ F with E( f , f ) = 0 in view of (1.1.4) and (1.1.5). (iv) ⇒ (i). If A ∈ Bm (E) is Tt -invariant, then, by Proposition 2.1.6 and Proposition 2.1.10, we have Tt 1A = 1A Tt 1 = 1A . Therefore either m(A) = 0 or m(Ac ) = 0 by (iv). This shows that (E, F ) is irreducible. (ii) ⇒ (iii). If f ∈ Fe with E( f , f ) = 0, then by Theorem 1.1.5(ii), f = ϕ ◦ f ∈ F = Fe ∩ L2 (E; m) and E( f , f ) = 0. Here ϕ is the normal contraction given by (1.1.19). By the assumption (ii), f is constant m-a.e. Since ∈ N is arbitrary, f is constant m-a.e. The converse implication is trivial. The finiteness assumption of m(E) in the above theorem will be removed in Theorem 5.2.16 under the (quasi) regularity assumption on the Dirichlet form (E, F). In the remainder of this section, we consider a relation of the extended Dirichlet space of a transient semigroup to the potential operator G and give a characterization of the former by means of the latter. 1 (E; m) by (2.1.2). We extend The function Gf has been defined for f ∈ L+ the potential operator G for any non-negative m-measurable function f on E by Gf = lim G( f ∧ (nη)) [m], n→∞

(2.1.23)

where η is a fixed strictly positive bounded m-integrable function on E. We 1 (E; m), the right hand side of (2.1.23) coincides with Gf (x) note that for f ∈ L+ of (2.1.2) m-a.e. due to the exchangeability of monotone limits. We also note

48

CHAPTER TWO

that when E is a Hausdorff topological space and {Tt ; t > 0} is determined by an m-symmetric transition function, Gf defined by (2.1.23) has R f of (2.1.3) as its version for any Borel version f of f . T HEOREM 2.1.12. Assume that {Tt } is transient. (i) If a non-negative m-measurable function f has E f · Gfdm < ∞, then Gf ∈ Fe

and

f · v ∈ L1 (E; m)

for every v ∈ Fe ,

(2.1.24)

E(Gf , v) =

f · vdm

for every v ∈ Fe .

(2.1.25)

E

Furthermore, the function Gf does not depend on the choice of η in its definition (2.1.23) up to m-equivalence. (ii) For a reference function g of the transient Dirichlet form (E, F), let L = { f = h · g : h ∈ bB(E)}. Then E f · Gfdm < ∞ for any f ∈ L+ and G(L) is a dense linear subspace of the extended Dirichlet space (Fe , E). m-integrable function f with Proof. (i) First consider a non-negative bounded f · Gfdm < ∞. By (2.1.9), we have for t > t E St f − St f 2E ≤ (St f , f ) − (St f , f ) + (St f , Tt f − Tt f ), which converges to zero as t, t → ∞ because (St f , f ) → E f · Gfdm and t+t (St f , Tt f ) = t (Tu f , f )du → 0. Hence {Sn f } is an E-Cauchy sequence which converges to Gf m-a.e., and so Gf ∈ Fe . We saw in the proof of Lemma 2.1.4(ii) that Tt f 22 = (T2t f , f ) → 0 as t → ∞ for a function f as above. Hence by letting t → ∞ in (2.1.9) with g = f and u = v ∈ F, we arrive at the identity (2.1.25) for v ∈ F. Next take v ∈ Fe with its approximating sequence {vn } ⊂ F. Since |vn − vm | fdm = E(Gf , |vn − vm |) ≤ Gf E vn − vm E , we see that {vn } is L1 (E; f · m)-Cauchy and converges to v m-a.e. Therefore we have the second property of (2.1.24). By letting n → ∞ in (2.1.25) for v = vn , we get the same equation for v ∈ Fe . Now consider a non-negative m-measurable function f with E f · Gfdm < 1 (E; m). Then, by what has just been proved, we ∞. Put fn = f ∧ (nη) ∈ bL+ have Gfn − Gfk 2E ≤ fn · Gfn dm − fk · Gfk dm, n > k, E

E

which converges to zero as n, k → ∞. Therefore we are led to (2.1.24) and (2.1.25) for f from those for fn . Equation (2.1.25) in particular implies that the function Gf ∈ Fe depends only on f and does not depend on a particular choice of η in (2.1.23).

49


(ii) The first assertion is immediate from (2.1.10). The second is also clear because any f ∈ L satisfies (2.1.25). Exercise 2.1.13. Assume {Tt } is transient. Prove the following. (i) If un ∈ Fe is E-convergent to u ∈ Fe as n → ∞ and a real-valued function ϕ is a normal contraction, then ϕ(un ) converges to ϕ(u) E-weakly as n → ∞. If, in addition, ϕ(u) = u, then the convergence is E-strong. (ii) Let {ϕ }≥1 be a sequence of normal contractions satisfying lim→∞ ϕ (t) = t. Then for any u ∈ Fe , lim→∞ ϕ (u) − uE = 0. Now we present a useful characterization for a given function space to be the extended Dirichlet space of a transient Markovian semigroup. T HEOREM 2.1.14. Suppose that {Tt ; t > 0} is transient and (G, a) is a real Hilbert space satisfying the following conditions. (i) G is a collection of m-equivalence class of Bm (E)-measurable functions on E. (ii) There exists a linear subspace L of L1 (E; m) such that (a) f ∈ L =⇒ | f | ∈ L. 1 (b) If an m-measurable function v satisfies f · v ∈ L (E; m) and E f vdm = 0 for any f ∈ L+ , then v = 0 [m]. (iii) For any f ∈ L+ and any v ∈ G, Gf ∈ G

and

a(Gf , v) =

f vdm.

(2.1.26)

E

Then (G, a) = (Fe , E) the extended Dirichlet space of {Tt ; t > 0}. Proof. Put G(L) = {Gf1 − Gf2 : f1 , f2 ∈ L+ }. Then by (ii)(b) and (iii), G(L) is a dense linear subspace of (G, a) and f · Gfdm = a(Gf , Gf ) < ∞ for every f ∈ L+ . E

Hence, by virtue of Theorem 2.1.12, for every f ∈ L+ and v ∈ Fe , 1 Gf ∈ Fe , f · v ∈ L (E; m) and E(Gf , v) = f vdm. E

By the assumption (ii), we then see that G(L) is dense in (Fe , E). Since a = E on G(L), we get the desired conclusion.

50

CHAPTER TWO

A typical example of the space L satisfying conditions of the above theorem is L = { f · g : f ∈ bB(E)}, where g is a reference function for the transient semigroup {Tt ; t > 0}. Finally we present a useful lemma which will reduce some arguments for a general (not necessarily transient) Dirichlet form E to those for transient ones. Let us consider a collection of functions K0 = g : g ∈ bB(E) strictly positive m-a.e. with gdm ≤ 1 . (2.1.27) E

For g ∈ K0 , define

where (u, v)g·m Since

E g (u, v) = E(u, v) + (u, v)g·m , = E u(x)v(x)g(x)m(dx).

u, v ∈ F,

(2.1.28)

E1 (u, u) ≤ E g (u, u) + (u, u) ≤ E(u, u) + (gL∞ + 1) (u, u), g (E g , F) is a Dirichlet form on L2 (E; m) and the norm E1 is equivalent to √ E1 on F. In particular, (E g , F) shares the same quasi notions with (E, F). Furthermore, (E g , F) is transient and possesses g as a reference function because |u|gdm ≤ (u, u)g·m ≤ E g (u, u), u ∈ F. E

Denote by

g (Fe , E g )

the extended Dirichlet space of (F, E g ).

L EMMA 2.1.15. For any u ∈ Fe and for any E-approximating sequence g {un , n ≥ 1} ⊂ F for u, there exists g ∈ K0 such that u ∈ Fe and {un } is an g E -approximating sequence for u. Proof. Take a bounded strictly positive Borel function f with E fdm ≤ 1 and put g = f /(supn un (x)2 ∨ 1). Then g ∈ K0 and un is L2 (E; g · m)-convergent to u.

2.2. BASIC EXAMPLES In this section, we present some basic examples of Dirichlet forms. Special attention will be paid to their transience, recurrence, and irreducibility as well as explicit expressions of the corresponding extended Dirichlet spaces.

51


2.2.1. Pure Jump Step Processes Let E be a locally compact separable metric space and Q(x, dy) be a probability kernel on (E, B(E)). We assume that Q(x, {x}) = 0 for every x ∈ E. Let λ(x) be a Borel measurable function on E such that 0 < λ(x) < ∞. From these two objects one can construct a Markov process X on E in the following way. If X starts from x0 ∈ E, it remains there for an exponentially distributed holding time T1 with parameter λ(x0 ), then it jumps to some x1 according to the probability distribution Q(x0 , dy); it remains at x1 for an exponentially distributed holding time T2 with parameter λ(x1 ), which is independent of T1 , before jumping to x2 according to Q(x1 , dy), and so on. From the probabilistic role they play, sometimes we call Q(x, dy) the road map and λ(x) the speed function of the process X. Suppose now λ(x) ≡ 1 on E. Denote by P(k) t (x, dy) the distribution of Xt that starts from x and there are exactly k jumps that occurred during the time period [0, t]. Then P(k) t (x, dy) =

tk −t (k) e Q (x, dy), k!

t > 0, k ≥ 1,

where Q(k) is the kth iterated kernel of the probability kernel Q. So the transition function of X is given by Pt (x, dy) =

∞ k=0

P(k) t (x, dy)

∞ k t −t (k) = e Q (x, dy) k! k=0

with the convention that Q0 (x, dy) = δ{x} (dy), the Dirac measure at x. When λ(x) is not a constant function, it is difficult to write out the transition function explicitly. The process X, which was rigorously constructed in Section 12 of Chapter 1 in [13], is called the regular step process there and it is shown to be a standard process. It was proved there that the resolvent kernel {Rα ; α > 0} of Xt is given by ∞

f (y) Q(k) α (x, dy) α + λ(y) E k=0 f (x) λ(x) = Rα f (y)Q(x, dy) + α + λ(x) α + λ(x) E

Rα f (x) =

(2.2.1)

λ(x) for f ∈ B+ (E), where Qα (x, dy) := α+λ(x) Q(x, dy). We now assume that there exists a σ -finite measure m0 on E with supp[m0 ] = E such that

Q(x, dy)m0 (dx) = Q(y, dx)m0 (dy).

(2.2.2)

52

CHAPTER TWO

We call m0 a symmetrizing measure of the road map Q. Define m(dx) =

1 m0 (dx). λ(x)

(2.2.3)

1 of m is the mean holding time of the process X The density function λ(x) at x so we may call m the speed measure for X. We are going to show that under the condition (2.2.2) the process X is m-symmetric. For f ∈ B+ (E), let (k) f (x) := f (y)Q Q(k) α α (x, dy). E

L EMMA 2.2.1. For f , g ∈ B+ (E), α > 0 and k ≥ 0, g f (k) (k) f (x)Qα (x)m(dx) = g(x)Qα (x)m(dx). α+λ α+λ E E

(2.2.4)

Proof. We prove it by mathematical induction on k. When k = 0, (2.2.4) is obviously true. For k = 1, by (2.2.2), g f (x)Qα (x)m(dx) α+λ E f (x) g(y) λ(x)Q(x, dy)m(dx) = E×E α + λ(x) α + λ(y) f (x) g(y) = λ(y)Q(y, dx)m(dy) E×E α + λ(x) α + λ(y) f = g(x)Qα m(dx). α+λ E So (2.2.4) holds for k = 1. Now suppose that (2.2.4) holds for k = j, then g (j+1) f (x)Qα (x)m(dx) α+λ E α + λ (j) g Q (x)m(dx) = f (x)Qα α+λ α α+λ E f g (j) (x) · (α + λ(x))Qα (x)m(dx) = Qα α+λ α+λ E f (j) (x)m(dx) = g(x)Qα Qα α+λ E f (j+1) (x)m(dx). = g(x)Qα α+λ E So (2.2.4) holds for k = j + 1 and the lemma is proved.

53


It follows from (2.2.1) and (2.2.4) that for f , g ∈ B+ (E) and α > 0,

f (x)Rα g(x)m(dx) =

g(x)Rα f (x)m(dx).

E

E

Consequently, the strong Markov process X is m-symmetric. Next we are going to figure out its associated Dirichlet form (E, F) on L2 (E; m). We know from Section 1.1 that each αRα extends uniquely to a Markovian contraction operator αGα in L2 (E; m) and {Gα ; α > 0} forms a strongly continuous resolvent in L2 (E; m). T HEOREM 2.2.2. Assume that the speed function λ is bounded and that (2.2.2) holds. Then the Dirichlet form (E, F) of the regular step process X on L2 (E; m) is given by F = L2 (E; m), 1 (u(x) − u(y))2 Q(x, dy)m0 (dx) E(u, u) = 2 E×E

for u ∈ L2 (E; m).

Proof. By (2.2.2) and the Cauchy-Schwarz inequality, for f ∈ L2 (E; m),

| f (x)f (y)|λ(x)Q(x, dy)m(dx) ≤

E×E

f (x)2 λ(x)Q(x, dy)m(dx)

E×E

f (x)2 λ(x)m(dx) < ∞

= E

and, since αGα f converges to f in L2 (E; m) as α → ∞,

α f (x) (αGα f (y) − f (y)) λ(x)Q(x, dy)m(dx)

α + λ(x) E×E 1/2 ≤ lim sup f (x)2 λ(x)Q(x, dy)m(dx)

lim α→∞

α→∞

E×E

·

1/2 (αGα f (y) − f (y)) λ(x)Q(x, dy)m(dx) 2

E×E

1/2

≤ c lim sup α→∞

(αGα f (x) − f (x))2 λ(x)Q(x, dy)m(dx) E×E

= c lim sup α→∞

1/2

(αGα f (x) − f (x)) λ(x)m(dx) 2

E

= 0.

54

CHAPTER TWO

These together with (2.2.1) and the dominated convergence theorem yield lim α( f − αRα f , f )L2 (E;m)

α→∞

= lim α α→∞

=

λ α+λ

f −α

E

f (x) λ(x)m(dx) − lim 2

E

α→∞ E×E

f (x)2 m0 (dx) −

= E

=

Gα f (y)Q(·, dy) , f

1 2

α f (x)f (y)λ(x)Q(x, dy)m(dx) α + λ(x)

f (x)f (y)Q(x, dy)m0 (dx) E×E

( f (x) − f (y))2 Q(x, dy)m0 (dx). E×E

This proves the theorem.

When the road map Q(x, dy) is a Markovian kernel, that is, when Q(x, dy) is a measure on E with Q(x, E) ≤ 1 for every x ∈ E, we can extend it to be a probability kernel on E∂ := E ∪ {∂}, the one-point compactification of E, by setting Q(x, {∂}) := 1 − Q(x, E) and Q(∂, {∂}) := 1. Extend the definition of λ to E∂ by setting λ(∂) = 0. Then by the same argument as in Section 12 of Chapter 1 in [13], there exists a regular step process X on E with ∂ as the trap. Under the hypothesis of (2.2.2), X is still an m-symmetric standard process on E. In this case, we have for f ∈ L2 (E; m) 1 lim α( f − αRα f , f )L2 (E;m) = ( f (x) − f (y))2 Q(x, dy)m0 (dx) α→∞ 2 E×E + f (x)2 (1 − Q(x, E))m0 (dx), E

provided that λ is bounded. Thus we have the following. T HEOREM 2.2.3. Assume that Q(x, dy) is a Markovian kernel, λ is bounded, and (2.2.2) holds. Then the Dirichlet form (E, F) of the regular step process X on L2 (E; m) is given by F = L2 (E; m), 1 (u(x) − u(y))2 Q(x, dy)m0 (dx) E(u, u) = 2 E×E + u(x)2 (1 − Q(x, E))m0 (dx) for u ∈ F.

(2.2.5)

E

Remark 2.2.4. Notice that only the road map Q and its symmetrizing measure m0 are involved in the description (2.2.5) of the Dirichlet form E. The role of λ is reflected in the reference measure m.


55

Note further that in Theorems 2.2.2 and 2.2.3 the speed function λ is assumed to be bounded. For an unbounded speed function, one can still obtain an explicit characterization of the Dirichlet form of X by time change. Assume that the symmetrizing measure m0 of the road map Q(x, dy) is a positive Radon measure and that λ is locally bounded away from zero in the sense that infx∈K λ(x) > 0 for any compact set K ⊂ E. The speed measure m defined by (2.2.3) is then a positive Radon measure again. Let X be the m-symmetric regular step process corresponding to (Q, λ) and (E, F ) be the Dirichlet form of X on L2 (E; m). Let Z be the m0 -symmetric regular step process on E corresponding to (Q, 1). Denote by (E Z , F Z ) and FeZ the Dirichlet form of Z on L2 (E; m0 ) and its extended Dirichlet space, respectively. Since Theorem 2.2.3 is applicable to (E Z , F Z ), we see that F Z = L2 (E; m0 ) and E Z is given by (2.2.5). In particular, (E Z , F Z ) is a regular Dirichlet form on L2 (E; m0 ). Its extended Dirichlet space FeZ is a linear subspace of G containing L2 (E; m0 ), where G = u : |u| < ∞ m0 -a.e. with (u(x) − u(y))2 Q(x, dy)m0 (dx) E×E

u(x) (1 − Q(x, E))m0 (dx) < ∞ .

+

2

(2.2.6)

E

It is clear that X is a time s change of Z by using the new time clock τt := inf{s : As > t} with As = 0 λ(Z1 r ) dr; that is, Xt = Zτt . By invoking (5.2.17) of Section 5.2, we can conclude that  F = FeZ ∩ L2 (E; m),      1 (u(x) − u(y))2 Q(x, dy)m0 (dx) E(u, u) = (2.2.7) 2 E×E      + u(x)2 (1 − Q(x, E))m (dx) for u ∈ F . 0

E

As we shall see in Section 6.5, the function space G is the reflected Dirichlet space of (E Z , F Z ) that will be introduced in Chapter 6. We shall also see there that, when Q is a probability kernel, the identity FeZ = G holds if and only if Z is recurrent. Exercise 2.2.5 concerns a simple example of Z, a (continuous time) symmetric simple random walk on Zn , which is recurrent when n ≤ 2 and transient when n ≥ 3.

2.2.2. Translation-Invariant Dirichlet Forms Let Rn be the n-dimensional Euclidean space and Lp (Rn ) the Lp space with respect to the Lebesgue measure dx on Rn . The inner product in L2 (Rn ) is denoted by ·, ·. For x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) in Rn , we adopt the

56

CHAPTER TWO

notations x, y :=

n

|x| :=

xi yi ,

x, x.

n=1

Define the convolution of two finite measures µ and ν on Rn by 1B (x + y)µ(dx)ν(dy), B ∈ B(Rn ). µ ∗ ν(B) = Rn ×Rn

We call a system of probability measures {νt , t ≥ 0} on Rn a continuous symmetric convolution semigroup on Rn if ν0 = δ0 , νt ∗ νs = νt+s for t, s > 0, νt (A) = νt (−A) for t > 0 and A ∈ B(Rn ), and limt↓0 νt = δ0 weakly. Here δ0 is the Dirac measure concentrated at the origin {0}. The celebrated Lévy-Khinchin formula1 for such {νt ; t ≥ 0} reads as follows: eix,y νt (dy) = e−tψ(x) (2.2.8) νt (x) := Rn

ψ(x) =

1 Sx, x + 2

Rn \{0}

(1 − cosx, y)J(dy),

(2.2.9)

where S is a (constant) non-negative definite n × n symmetric matrix and J is |x|2 a symmetric measure on Rn \ {0} with Rn \{0} 1+|x| 2 J(dx) < ∞. ψ(x) and J are called the Lévy exponent and the Lévy measure, respectively, for {νt , t ≥ 0}. We extend J to Rn by setting J({0}) = 0. Let {νt ; t ≥ 0} be a continuous symmetric convolution semigroup on Rn with the Lévy exponent ψ(x) given by (2.2.9). We put (2.2.10) Pt (x, B) = νt (B − x), t ≥ 0, x ∈ Rn , B ∈ B(Rn ). Then Pt f (x) = Rn f (x + y)νt (dy) for t ≥ 0 and x ∈ Rn , and we readily see that {Pt ; t ≥ 0} is a transition probability on Rn in the sense of Definition 1.1.13 which is also symmetric with respect to dx. Let (E, F) be the Dirichlet form on L2 (Rn ) of {Pt ; t ≥ 0}. It is defined by (1.1.4) and (1.1.5) by the strongly continuous contraction semigroup {Tt ; t > 0} of Markovian symmetric operators on L2 (Rn ) determined by {Pt ; t ≥ 0} according to Lemma 1.1.14. Since the resolvent {Rα ; α > 0} of {Pt ; t ≥ 0} admits the expression ∞ f (x + y)wα (dy), wα (B) = e−αt νt (B)dt Rα f (x) = Rn

0

for B ∈ B(R ), we see that Rα (Cc (R )) is not only a · ∞ -dense subset of C∞ (Rn ) but also an E1 -dense subset of F on account of equation (1.1.28). n

1 Cf.

[132].

n


57

Hence the Dirichlet form (E, F) is regular in the sense of Definition 1.3.10 by virtue of Lemma 1.3.12. The associated Hunt process X on Rn is called a symmetric Lévy process. The process X has independent stationary increments and hence can start from every point in Rn . We shall show that  2 n  F = {u ∈ L (R ) : | u(x)|2 ψ(x)dx < ∞}, Rn (2.2.11)  E(u, v) = u(x) v(x)ψ(x)dx for u, v ∈ F. Rn

Here the Fourier transform f of a function f is defined first by n ei(x,y) f (y)dy, f ∈ S, f (x) = (2π)− 2 Rn

where S is the space of rapidly decreasing functions, namely, infinitely differentiable functions on Rn whose derivatives of any order are bounded when multiplied by polynomials. The definition of f is then extended to any f ∈ L2 (Rn ) so that the Parseval formula f (y) g(y)dy, f , g ∈ L2 (Rn ), ( f , g) = Rn

remains valid. νt (x) u(x) = e−tψ(x) u(x), x ∈ Rn , u ∈ S, we have by the Since P t u(x) = Parseval formula 1 − e−tψ(x) 1 | u(x)|2 (u − Pt u, u) = dx, t > 0, u ∈ S, (2.2.12) t t Rn which readily extends to any u ∈ L2 (Rn ) by choosing uk ∈ S converging to u in L 2 (Rn ) as k → ∞. The right hand side of (2.2.12) for u ∈ L2 (Rn ) increases u(x)|2 ψ(x)dx(≤ ∞) as t ↓ 0 and so we get (2.2.11). Obviously the to Rn | Dirichlet form (2.2.11) is translation-invariant in the sense that if u ∈ F, then for any y ∈ Rn , the translated function x → uy (x) := u(x − y) is in F and uy E = uE . When S is the identity matrix and J vanishes in (2.2.9), namely, when ψ(x) = 12 |x|2 , the associated convolution semigroup is νt (dx) = gt (x)dx,

gt (x) =

|x|2 1 e− 2t , n/2 (2π t)

(2.2.13)

which corresponds to the transition probability of the n-dimensional standard Brownian motion by (2.2.10). The associated Dirichlet form (2.2.11) on L2 (Rn ) can be written as (2.2.14) (E, F ) = 12 D, H 1 (Rn ) ,

58

CHAPTER TWO

where  n ∂f (x) ∂g(x)    dx, D( f , g) =  n ∂xi ∂xi i=1 R  ∂f   H 1 (Rn ) = f ∈ L2 (Rn ) : ∈ L2 (Rn ) for 1 ≤ i ≤ d , ∂xi

(2.2.15)

where ∂x∂ i is the derivative in the sense of the Schwartz distribution. D( f , g) is the Dirichlet integral of functions f , g, and H 1 (Rd ) is called the Sobolev space on Rd of order 1. A. Beurling had this example in mind when he gave the name “Dirichlet space” to a function space on which every normal contraction operates. When S in the formula (2.2.9) vanishes, we can also derive from (2.2.11) the following expression of the Dirichlet form (E, F) in terms of the Lévy measure J:  1   ( f (x + y) − f (x))(g(x + y) − g(x))J(dy)dx, E( f , g) =  2 Rn ×Rn  2 2 n  F = f ∈ L (R ) : ( f (x + y) − f (x)) J(dy)dx < ∞ .

(2.2.16)

Rn ×Rn

In fact, we have from (2.2.9) and (2.2.11) | f (x)|2 (1 − cosx, y)J(dy)dx. E( f , f ) = Rn ×Rn

On the other hand, for each y ∈ Rn , the Fourier transform of the function gy (x) = f (x)(e−ix,y − 1) and the Parseval formula gy (x) = f (x + y) − f (x) is gives gy (x)2 J(dy)dx = 2 | f (x)|2 (1 − cosx, y)dxJ(dy). Rn ×Rn

Rn ×Rn

The transience of a symmetric Markovian semigroup {Tt ; t > 0} on an L2 space was defined in Definition 2.1.1. In the present case where Tt = Pt determined by (2.2.10), the following five conditions can be verified to be mutually equivalent (cf. [73, Example 1.5.2]): (1) {Tt ; t > 0} is transient. (2) w(K) < ∞ for any compact K ⊂ Rn , where ∞ νt (B)dt, B ∈ B(Rn ). w(B) = (3)

∞ 0

0

(Pt f , f )dt < ∞ for any non-negative f ∈ Cc (Rn ).


(4) For any compact K ⊂ Rn , there exists a constant CK such that |u(x)|dx ≤ CK uE for every u ∈ F .

59

(2.2.17)

K

(5) ψ(x)−1 is locally integrable on Rn . Exercise 2.2.5. Consider the special case where S = 0 and J is a symmetric probability measure on Rn \ {0} in (2.2.9). The associated transition semigroup {Tt ; t > 0} defined by (2.2.10) corresponds to a compound Poisson process which is a special case of the regular step process X of Section 2.2.1 with E = Rn , λ ≡ 1, Q(x, B) = J(B − x) and m0 being the Lebesgue measure 1 (cf. [132]). Let J be a discrete distribution on Zn with J({ek }) = J({−ek }) = 2n where ek denotes a vector with 1 in the kth coordinate and 0 elsewhere. Show that {Tt ; t > 0} is transient if and only if n ≥ 3. The restriction of X to its invariant set Zn is the (continuous-time) symmetric simple random walk on Zn . Assume that {Tt ; t > 0} is transient. Then all the conditions (2)–(5) are fulfilled. Let us show that the extended Dirichlet space (Fe , E) of the Dirichlet form (2.2.11) admits an explicit expression2  1  (Rn ) ∩ S : u ∈ L2 (Rn ; ψ · dx) , Fe = u ∈ Lloc (2.2.18)  u(x) v(x)ψ(x)dx, u, v ∈ Fe , E(u, v) = Rn

where S denotes the space of tempered distributions, namely, continuous u ∈ S linear functionals on the space S, and, for u ∈ S , its Fourier transform is defined by the formula u, φ = u, φ for every φ ∈ S. Notice that a positive Radon measure µ on Rn can be considered as an element of S if and only if Rn (1 + |x|2 )−m µ(dx) < ∞ for some m ∈ N. For φ ∈ S, define φˇ by ˇ ˇ It is φ(x) := φ(−x) for x ∈ Rn , and denote by φ the Fourier transform of φ. easy to see that φ is the Fourier transform of φ. By integrating the identity (2.2.8) in t and using the Fubini theorem, we get ˇ ψ(x)−1 φ ∗ φ(x)dx = | φ |2 (y)w(dy) ≥ 0 for φ ∈ Cc∞ (Rn ), Rn

Rn

where w is the measure defined in (2). This means that the Radon measure µ0 (dx) = ψ(x)−1 dx is of positive type and hence is the Fourier transform of a tempered distribution.3 Accordingly, µ0 ∈ S , which in turn implies that, for 2 Due 3 Cf.

to J. Deny [39]. L. Schwartz [135].

60

CHAPTER TWO

any v ∈ L2 (Rn ; ψ · dx), v · dx is tempered because 1/2 |v(x)| 2 −m dx ≤ vL2 (Rn ;ψ·dx) (1 + |x| ) µ0 (dx) . 2 m/2 Rn (1 + |x| ) Rn Thus L2 (Rn ; ψ · dx) ⊂ S . Further we see from the above estimate that if v converges to v in L2 (Rn ; ψ · dx) as → ∞, then lim→∞ v , φ = v, φ for every φ ∈ S. Take any u ∈ Fe . There exists then a sequence {u }≥1 ⊂ F which is E-convergent as well as a.e. on Rn to u. Notice that, on account of (4), u 1 then converges to u in the space Lloc (Rn ) as → ∞. By (2.2.11), { u }≥1 2 n is convergent in L (R ; ψ · dx) to a function v ∈ L2 (Rn ; ψ · dx) and uE = vL2 (Rn ;ψ·dx) . Since v ∈ S by the above observation, v is the Fourier transform of v ∈ S and moreover we have φ = lim u , φ = v, φ = v, φ lim u ,

→∞

→∞

for φ ∈ S.

Hence v = u and we get v = u and uE = uL2 (Rn ;ψ·dx) . Thus the inclusion ⊂ holds in (2.2.18). The converse inclusion is readily verifiable. An explicit expression of the extended Dirichlet space (Fe , E) in the recurrence case will be given in Section 6.5. The continuous symmetric convolution semigroup corresponding to ψ(x) = |x|α with 0 < α ≤ 2 is called the rotation-invariant stable semigroup of index α, which corresponds in the formula (2.2.9) to4 S=0

and

J(dy) = A(n, −α)|y|−n−α dy,

(2.2.19)

α2α−1 ((α+n)/2) . π n/2 ((2−α)/2)

The corresponding Lévy process X on Rn where A(n, −α) := is called a rotationally symmetric α-stable process. According to (5), the associated L2 -semigroup is transient if and only if α < n. In this case, its extended Dirichlet space is described as (2.2.18) with ψ(x) = |x|α . We shall now present two other characterizations of its extended Dirichlet space in terms of the Riesz convolution kernels. In what follows, we shall fix α with 0 < α ≤ 2 and α < n, the rotationinvariant stable semigroup {νt , t ≥ 0} of index α and the corresponding extended Dirichlet space (Fe , E). The first characterization is as follows:  Fe = u = Iα/2 ∗ f : f ∈ L2 (Rn ) , (2.2.20) E(u, v) = f (x)g(x)dx for u = Iα/2 ∗ f , v = Iα/2 ∗ g, Rn

where Iα denotes the Riesz convolution kernel of index α given by ((n − α)/2) . Iα (x) = γn,α |x|−(n−α) with γn,α = α n/2 2 π (α/2)

(2.2.21)

So Fe is the space of all Riesz potentials of index α/2 of L2 -functions. 4 Cf.

[132, E18.9].


Characterization (2.2.20) is readily obtained from  1 Iβ ∗ f ∈ Lloc (Rn ) ∩ S , −β I β ∗ f = |x| f ,

61

(2.2.22)

for β < n/2 and f ∈ L2 (Rn ). Indeed, using (2.2.22), it follows from (2.2.18) with ψ(x) = |x|α that Iα/2 ∗ f ∈ Fe for f ∈ L2 (Rn ) and f (x) |x|−α g(x)|x|α dx = ( f , g), u = Iα/2 ∗ f , v = Iα/2 ∗ g, E(u, v) = Rn

which mean that the space defined by the right hand sides of (2.2.20) is a closed subspace of (Fe , E). Suppose u ∈ Fe is E-orthogonal to this subspace, then u(x) g(x)|x|α/2 dx = 0 for g ∈ L2 (Rn ). Rn

g(x) = u(x)|x|α/2 ∈ L2 (Rn ), Choosing a complex function g ∈ L2 (Rn ) such that α/2 we have u(x)|x| = 0 a.e. and consequently u = 0. To prove (2.2.22), let B = {x ∈ Rn : |x| < 1} be the unit open ball in Rn and define K1 := Iβ · 1B and K2 := Iβ − K1 . For β < n/2, K2 ∈ L2 (Rn ). Thus for f ∈ L2 (Rn ), K1 ∗ f ∈ L2 (R2 ) and K2 ∗ f ∈ C∞ (Rn ). This proves the first half of (2.2.22). On the other hand, for δ >2 0, since the Fourier transform of the function ), we have exp(− 2δ |x|2 ) is δ −n/2 exp(− |x| 2δ |x|2 δ 2 e− 2 |x| e− 2δ φ(x)dx, φ ∈ S. φ (x)dx = δ −n/2 Rn

Rn

n−β 2

Multiply the both hand sides by δ −1 and then integrate in δ over [0, ∞) to get n Iβ (x) φ (x)dx = (2π )− 2 |x|−β φ(x)dx, φ ∈ S, β < n. (2.2.23) Rn

Rn

(y − x). Multiply the resulting equaφ (x) = ψ Take φ(x) = e−iy,x ψ(−x), then tion by f (y) ( f ∈ L2 (Rn )) and then integrate in y on Rn to get the second half of (2.2.22), namely, (x)dx = f (x)ψ(x)dx, ψ ∈ S, β < n/2. (Iβ ∗ f )(x)ψ |x|−β Rn

Rn

As a direct application of (2.2.20), we can rewrite the Sobolev inequality5 Iα/2 ∗ f Lp0 (Rn ) ≤ C f L2 (Rn ) , 5 Cf.

E. M. Stein [142, Chap. V].

62

CHAPTER TWO

where 1/p0 = 1/2 − α/2d, to obtain uLp0 (Rn ) ≤ uE ,

u ∈ Fe ,

(2.2.24)

where C is a positive constant. Inequality (2.2.24) is stronger than (2.2.17). In fact, it implies that the space (Fe , E) is continuously embedded into the space 2 (Rn ). Lloc We next observe that the Radon measure w defined in (2) admits the Riesz kernel of index α as a density function, namely, w(dx) = Iα (x)dx, because we have for any φ ∈ S ∞ α φ(x) e−t|x| dtdx φ (x)w(dx) = (2π)−n/2 Rn

= (2π)−n/2

Rn

Rn

0

|x|−α φ(x)dx,

which coincides with the left hand side of (2.2.23). Therefore, for non-negative f ∈ Cc (Rn ), its Riesz potential Iα ∗ f is a version of Rf defined by (2.1.3) for the transition function (2.2.10) for {νt ; t ≥ 0}. By virtue of Theorem 2.1.12, it holds that for every f ∈ Cc (Rn ) and v ∈ Fe , f (x)v(x)dx. (2.2.25) Iα ∗ f ∈ Fe and E(Iα ∗ f , v) = Rn

Furthermore, by taking Cc (Rn ) as the space L in Theorem 2.1.14, we can conclude that the extended Dirichlet space (Fe , E) is characterized as a Hilbert space consisting of equivalence classes (in the sense of a.e.) of measurable functions on Rn for which (2.2.25) holds. We also note here that the irreducibility of the present semigroup {Tt ; t > 0} can be deduced as follows. Suppose that a measurable set A ⊂ Rn is {Tt }invariant. By Proposition 2.1.6, 1Ac Rβ (1A f ) = 0 a.e. for any non-negative f ∈ Cc (Rn ) and β > 0. Letting β → 0, we get 1Ac R(1A f ) = 0 a.e., which forces A to be either Rn or ∅ a.e. because R has a strictly positive convolution kernel Iα . When ψ(x) = 12 |x|2 and n ≥ 3, we have w(dx) = 2I2 (x)dx so that the associated potential operator R is given by the Newtonian convolution kernel 2I2 (x) =

( 2n − 1) 2−n |x| . 2π n/2

(2.2.26)

The extended Dirichlet space of the Dirichlet form (2.2.14) can be characterized by (2.2.25) using the Newtonian potentials of Cc functions. We now present a brief discussion of symmetric convolution semigroups on the unit circle and their associated Dirichlet forms. The definition of a

63


continuous symmetric convolution semigroup {µt , t > 0} on the unit circle S is analogous to that on Rn above. We can identify S with [0, 2π ). By the LévyKhinchin formula [78], {µt , t > 0} is characterized by 2π µt (k) := eikx µt (dx) = e−tak , k ∈ Z, 0

with

ak = βk +

2π

2

(1 − cos(kx))J(dx),

(2.2.27)

0

where β ≥ 0 is a constant and J is a non-negative Radon measure on the 2π circle S = [0, 2π) satisfying 0 sin2 1(x/2) J(dx) < ∞. Let {Tt ; t > 0} be the

L2 -strongly continuous semigroup generated by {µt , t > 0}, that is, Tt f := 2π 2π −ikx 1 2 f (x)dx, k ∈ 0 f (x + y)µt (dy). For f ∈ L ([0, 2π); dx), let f (k) := 2π 0 e Z, denote its Fourier coefficients. Then by Parseval’s formula, 2π 2 2π 2 1 | f (k)| (1 − µt (k)) = | f (k)| (1 − e−tak ), ( f − Tt f , f ) = t t t k∈Z

k∈Z

f (k)|2 . Hence the corresponding Dirichlet which increases to 2π k∈Z ak | space is    2 2  ak | f (k)| < ∞ F = f ∈ L ([0, 2π); dx) : k∈Z   f (k) ak g(k) for f , g ∈ F .  E( f , g) = 2π k∈Z

It is easy to see that if we take β = 0 and |x + 2kπ|−α−1 dx J(dx) = c k∈Z

with α ∈ (0, 2) in (2.2.27), then ak = c0 |k|α for k ∈ Z. The measures {µt , t > 0} are called the symmetric stable semigroup of index α on the unit circle. As can be easily checked, the corresponding Dirichlet form is regular. It uniquely determines a Hunt process which is called the symmetric α-stable process on the unit circle. When α = 1, it is called the symmetric Cauchy process on the unit circle. 2.2.3. One-Dimensional Strongly Local Dirichlet Forms The second-order ordinary differential operator Au(x) = a(x)u (x) + b(x)u (x) with real-valued functions a > 0 and b can be converted into Feller’s canonical

64 form

CHAPTER TWO d du dm ds

with −B(x)

ds = e

dx,

eB(x) dm = dx, a(x)

B(x) =

x

x0

b(y) dy. a(y)

Further, formal computation gives du du dv − Au · v dm = − v · d = ds. ds ds ds This example leads us to the following general consideration. Let I ⊂ R be a one-dimensional open interval and s = s(x), x ∈ I, be a strictly increasing continuous function on I. Following [95], we call s a canonical scale. Consider the space F (s) = {u : u is absolutely continuous on I with respect to ds and E (s) (u, u) < ∞}, where

E (u, v) = (s)

I

(2.2.28)

du dv (x) (x)ds(x). ds ds

(2.2.29)

Let m be a positive Radon measure on I with full support. Such a measure on an interval will be called a canonical measure. The aim of this subsection is to study the function space (E, F) defined by E = E (s) ,

F = F (s) ∩ L2 (I; m).

(2.2.30)

First of all, we make a simple observation. From an elementary equality y du u(z) − u(y) = (x)ds(x), y, z ∈ I, z ds we get for u ∈ F (s) , (u(z) − u(y))2 ≤ |s(z) − s(y)|E (s) (u, u),

y, z ∈ I,

(2.2.31)

which also implies that for any compact set K ⊂ I, there exists a positive constant CK with sup u(x)2 ≤ CK E1(s) (u, u).

(2.2.32)

x∈K

In fact, if [α, β] ⊂ I, we get from (2.2.31) sup u(y)2 ≤ 2(s(β) − s(α))E (s) (u, u) + 2u(x)2 ,

α≤y≤β

α ≤ x ≤ β.

Integrating both sides by dm(x) on [α, β], we arrive at (2.2.32). For any ∈ N, we consider the normal contraction ϕ defined by (1.3.1).


65

L EMMA 2.2.6. (i) Assume that {un , n ≥ 1} ⊂ F (s) is E (s) -Cauchy and that un converges to a function u m-a.e. Then u ∈ F (s) and un is E (s) -convergent to u. (ii) For any u ∈ F (s) , there exists a subsequence {n ; n ≥ 1} such that, if we denote by {un ; n ≥ 1} the Cesàro mean sequence of {ϕk (u); k ≥ 1}, then un ∈ F (s) and un is E (s) -convergent to u. n Proof. (i) If {un } is E (s) -Cauchy, { du } is L2 (I; ds)-convergent to some v ∈ ds 2 L (I; ds). Since un is convergent to u m-a.e., we see from (2.2.31) that {un } is convergent uniformly on each compact subinterval of I to a continuous version of u and hence v(x)ψ(s(x))ds(x) = − lim un (x)ψ (s(x))ds(x)

n→∞ I

I

u(x)ψ (s(x))ds(x),

=− I

for any ψ ∈ Cc1 (J), where J = s(I) and Cck (J) denotes the space of k times continuously differentiable functions on J with compact support. This means = v and un is E (s) -convergent to u. that u ∈ F (s) with du ds (s) (ii) For u ∈ F and ∈ N, ϕ (u) ∈ F (s) ,

E (s) (ϕ (u), ϕ (u)) ≤ E (s) (u, u).

Hence by the Banach-Saks Theorem A.4.1, we can find a subsequence {n } n is L2 (I; ds)-convergent to such that, for the Cesàro mean un of {ϕn (u)}, du ds 2 some v ∈ L (I; ds). Since un converges to u pointwisely, the same argument as in the proof of (i) works to get the desired conclusion. L EMMA 2.2.7. The bilinear form (E, F ) defined by (2.2.30) is a Dirichlet form on L2 (I; m). Proof. The closedness of the bilinear form (E, F ) follows from Lemma 2.2.6(i) immediately. F (s) is dense in L2 (I; m) as it contains the space {u(s) : u ∈ Cc1 (s(I))}. For any ε > 0, choose a smooth function ϕε satisfying (1.1.7) to obtain 2 (s) 2 du ϕε (u(x)) (x) ds(x) ≤ E (s) (u, u), E (ϕε (u), ϕε (u)) = ds R the Markovian property of (E, F).

Let I = (r1 , r2 ). We say that the boundary r1 (respectively, r2 ) is approachable if (respectively, s(r2 ) < ∞). −∞ < s(r1 ) By (2.2.31), we see that if r1 is approachable, then for any u ∈ F (s) , the limit u(r1 ) = limx→r1 ,x∈I u(x) exists, u ∈ C([r1 , r2 )), and (2.2.31) holds for any

66

CHAPTER TWO

a, b ∈ [r1 , r2 ). The same statement holds for r2 . In particular, if both r1 and r2 are approachable, then F (s) ⊂ C([r1 , r2 ]),

1, s ∈ F (s) ,

(2.2.33)

and consequently F (s) is a uniformly dense subalgebra of C([r1 , r2 ]). The point r1 is called regular if r1 approachable

and m((r1 , c)) < ∞

for every c ∈ (r1 , r2 ).

The regularity of r2 is defined similarly. We now study the properties of the Dirichlet form (E, F) defined by (2.2.30) in three cases separately. P ROPOSITION 2.2.8. Suppose both r1 and r2 are regular. Then F = Fe = F (s) ,

(2.2.34)

and (E, F) is a regular, strongly local, recurrent, and irreducible Dirichlet form on L2 ([r1 , r2 ]; 1I · m). Proof. Property (2.2.33) and the finiteness of m(I) imply that F coincides with F (s) , which also equals Fe by virtue of Lemma 2.2.6. In view of Lemma 2.2.7 and (2.2.33), (E, F) is a regular Dirichlet form on L2 ([r1 , r2 ]; 1I · m). Since 1 ∈ F and E(1, 1) = 0, it is recurrent by Theorem 2.1.8. It is obvious that (E, F) is strongly local. Suppose there is a Borel set A ⊂ [r1 , r2 ] such that 1A equals some function u ∈ F m-a.e. Since u is continuous and takes values 0 or 1 only, u−1 ({1}) is a closed and open subset of [r1 , r2 ] and hence either A or Ac is m-negligible, yielding the irreducibility by virtue of Proposition 2.1.6. P ROPOSITION 2.2.9. Assume that both r1 and r2 are approachable but nonregular. If we let F0(s) = u ∈ F (s) : u(r1 ) = 0 = u(r2 ) , (2.2.35) then F ⊂ Fe = F0(s) ,

(2.2.36)

and (E, F) is a regular, strongly local, transient, and irreducible Dirichlet form on L2 (I; m). Proof. Since m diverges in neighborhoods of r1 and r2 , we see from (2.2.33) that F ⊂ F0(s) . Take any u ∈ F0(s) . For any ∈ N, ϕ (u) has compact support in I and so ϕ (u) ∈ F. Hence the functions un in Lemma 2.2.6(ii) are in F and converge to u both pointwise and in E (s) -metric. This implies that u ∈ Fe . So we have

67


established F0(s) ⊂ Fe . Conversely, assume un ∈ F(⊂ F0(s) ) is E (s) -Cauchy and un → u m-a.e. Then u ∈ F (s) and un is E (s) -convergent to u by Lemma 2.2.6. Since inequality (2.2.31) holds on [r1 , r2 ], u(ri ) = lim un (ri ) = 0, n→∞

i = 1, 2.

Consequently, u ∈ F0(s) . We have thus established that F0(s) = Fe . For u ∈ F , the functions un of Lemma 2.2.6(ii) are in F ∩ Cc (I) and E1 -convergent to u. Since F (s) is dense in C([r1 , r2 ]), F ⊂ F0(s) is dense in C∞ (I) in view of Lemma 1.3.12, proving the regularity of the space (E, F). Its strongly local property is obvious. Since, for u ∈ Fe , E(u, u) = 0 implies u = 0, the transience follows from Theorem 2.1.9. The irreducibility can be proved in the same way as the proof of the preceding theorem because, for any a < b with [a, b] ⊂ I, F admits a function taking value 1 on [a, b]. P ROPOSITION 2.2.10. Suppose both r1 and r2 are non-approachable. Then Fe = F (s)

(2.2.37)

and (E, F) is a regular, strongly local, recurrent, and irreducible Dirichlet form on L2 (I; m). Proof. Lemma 2.2.6(i) implies the inclusion Fe ⊂ F (s) . To prove the converse, take any u ∈ F (s) . Since the truncation ϕ ◦ u is E (s) -convergent to u as → ∞, in view of Lemma 1.1.12, we may assume without loss of generality that u is bounded, that is, |u| ≤ M for some constant M. We consider a sequence of functions ψn ∈ Cc1 (R+ ) such that for 0 ≤ x < n; ψn (x) = 0 for x > 2n + 1; ψn (x) = 1 (2.2.38) 1 |ψn (x)| ≤ n , n ≤ x ≤ 2n + 1; 0 ≤ ψn (x) ≤ 1, x ∈ R+ . Put un (x) = u(x) · ψn (|s(x)|) for x ∈ I. Then, owing to the non-approachability of the both boundaries, un ∈ F (s) ∩ Cc (I)(⊂ F) and E (s) (u − un , u − un ) 2 du 2 ≤2 (1 − ψn (|s(x)|) ds(x) + 2 u2 (x)(ψn (|s(x)|)2 ds(x) ds I I 2 du ds(x) + 2M 2 (ψn (|s(x)|)2 ds(x) ≤2 ds |s(x)|≥n n≤|s(x)| 0

and

inf Cap1 ({a}) > 0.

a∈K

(2.2.40)

Consequently, a function is quasi continuous if and only if it is continuous. In the transient case, the inequality (2.1.11) for (Fe , E) can be also derived directly from (2.2.31). Suppose for instance that s(r2 ) < ∞ and m(c, r2 ) = ∞ for some c ∈ I. Then |u(x)| ≤ |s(x) − s(r2 )| E (s) (u, u), x ∈ I, u ∈ Fe . Consequently, inequality (2.1.11) holds for every u ∈ Fe and for any mintegrable strictly positive bounded function g with |s(x) − s(r2 )|g(x)dm(x) ≤ 1. I


69

When s(x) = x and m is the Lebesgue measure on I, then the space (2.2.28) (resp. (2.2.30)) is denoted by BL(I) (resp. H 1 (I)) and called the space of BLfunctions (resp. the Sobolev space of order 1). The space Fe is then denoted by He1 (I). 2.2.4. Space of BL Functions Let D be a domain in the n-dimensional Euclidean space Rn . For p ≥ 1, the Lp -space of real-valued functions on D with respect to the Lebesgue measure dx will be denoted as Lp (D). We focus our attention on the space ∂T BL(D) = T : ∈ L2 (D), 1 ≤ i ≤ n (2.2.41) ∂xi of Schwartz distributions T. Any distribution T ∈ BL(D) can be identified with 2 (D)6 so that a function in Lloc ∂u 2 (D) : ∈ L2 (D), 1 ≤ i ≤ n , BL(D) = u ∈ Lloc ∂xi

(2.2.42)

where the derivatives are taken in the Schwartz distribution sense. Members in BL(D) are called BL(Beppo-Levi)-functions on D. For u, v ∈ BL(D), we put n ∂u ∂v dx, (2.2.43) D(u, v) = ∂xi ∂xi i=1 D The space BL(D) is known to enjoy the following properties (cf. [40], [121]): ˙ (BL.1) The quotient space BL(D) of BL(D) by the subspace of constant functions is a Hilbert space with inner product D. Any D-Cauchy sequence un ∈ BL(D) admits u ∈ BL(D) and constants cn such that un 2 is D-convergent to u and un + cn is Lloc -convergent to u. (BL.2) A function u on D is in BL(D) if and only if, for each i (1 ≤ i ≤ n), there is a version u(i) of u such that it is absolutely continuous on almost all straight lines parallel to xi -axis and the derivative ∂u(i) /∂xi in the ordinary sense (which exists a.e. on D) is in L2 (D). In this case, the ordinary derivatives coincide with the distribution derivatives of u. The Sobolev space of order 1 on the domain D ⊂ Rn is defined by H 1 (D) = BL(D) ∩ L2 (D). 6 Cf.

L. Schwartz [135], J. Deny and J. L. Lions [40].

(2.2.44)

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Then (E, F) =

1 2

D, H 1 (D)

(2.2.45)

is a closed symmetric form on L2 (D). To see this, suppose {un } ⊂ F is E1 n is L2 (D)-convergent to some vi ∈ L2 (D) for each 1 ≤ i ≤ n Cauchy. Then ∂u ∂xi and un is L2 (D)-convergent to some u ∈ L2 (D). Then for any f ∈ Cc∞ (D), ∂f ∂un ∂f , f = − lim un , (vi , f ) = lim = − u, . n→∞ n→∞ ∂xi ∂xi ∂xi ∂u Hence vi = ∂x for 1 ≤ i ≤ n. It is a Dirichlet form on L2 (D) because its i Markov property can be verified in the same way as the proof of Lemma 2.2.7 by using the property (BL.2), which also implies the strongly local property of (2.2.45). Denote by He1 (D) the extended Dirichlet space of (2.2.45). We shall be concerned with its relationship to the space BL(D). Let us first consider the case that D = Rn . The Dirichlet form (2.2.46) (E, F) = 12 D, H 1 (Rn )

on L2 (Rn ) has already appeared in (2.2.14) of Subsection 2.2.2, where we saw that it is a regular Dirichlet form, namely, the space H 1 (Rn ) ∩ Cc (Rn ) is a core of it. We can further see that it has Cc∞ (Rn ) as a special standard core by making convolutions with mollifiers ε−n ϕ(x/ε), ε > 0, where ϕ is a non-negative infinitely differentiable function supported by B = {|x| < 1} with B ϕ(x)dx = 1. The Dirichlet form (2.2.46) on L2 (Rn ) is associated with the transition ∞ density gt (x) of (2.2.13). Since, for x = 0, 0 gt (x)dt is divergent when n = 1, 2, but convergent and equal to the Newtonian kernel (2.2.26) when n ≥ 3, the corresponding L2 -semigroup is recurrent in the former case and transient in the latter case. When n ≥ 3, the extended Dirichlet space (He1 (Rn ), E) of (2.2.46) is a real Hilbert space on account of Theorem 2.1.5 and, in particular, u ∈ He1 (Rn ) with E(u, u) = 0 =⇒ u = 0.

(2.2.47)

T HEOREM 2.2.12. Assume that n ≥ 3. Then He1 (Rn ) ⊂ BL(Rn ), E(u, u) = D(u, u) for u ∈ He1 (Rn ), and BL(Rn ) is the linear space spanned by He1 (Rn ) and constant functions. The space (He1 (Rn ), E) is isometric with the space n 1 n ˙ ˙ ), 2 D) by the canonical map BL(Rn ) → BL(R ). Furthermore, (BL(R 1 He1 (Rn ) = {u ∈ BL(Rn ) ∩ S : u ∈ Lloc (Rn )}.

(2.2.48)

Proof. For u ∈ ⊂ H (R ) which is D-Cauchy and convergent to u a.e. D(un , un ) then converges to E(u, u). By (BL.1), there exist v ∈ BL(Rn ) and constants cn such that {un } is D-convergent to He1 (Rn ), there is a sequence {un }

1

n


71

2 v and the sequence {un + cn } is convergent to v in Lloc (Rn ). By choosing a subsequence if necessary, we may assume that the latter sequence converges to v a.e. Then limn→∞ cn = c exists, u = v − c and, consequently, u ∈ BL(Rn ) and E(u, u) = D(u, u). Further, we see from (2.2.47) that the Hilbert space n 1 ˙ ), 2 D) (He1 (Rn ), E) is isometrically embedded into a closed subspace of (BL(R n n ˙ by the canonical map BL(R ) → BL(R ). We denote by u˙ the equivalence class represented by u ∈ BL(Rn ). n ˙ ) is D-orthogonal to this closed subspace, then, since If u˙ ∈ BL(R ∞ n 1 Cc (R ) ⊂ H (Rn ), we have

(u, f ) = −D(u, f ) = 0

for every f ∈ Cc∞ (Rn ),

which implies that u = 0, namely, (a version of) u is harmonic on Rn . Since ∂u , 1 ≤ i ≤ n, are also harmonic on Rn , we get from the ordinary derivatives ∂x i the mean-value theorem the estimate

1/2

∂u

∂u 1

dx ≤

(x) ≤ 1 , x ∈ Rn , D(u, u)

∂x

∂x |B (x)| |B (x)| i r i r Br (x) where Br (x) is the ball of radius r centered at x and |Br (x)| denotes its volume. By letting r → ∞, we see that all derivatives of u vanish and hence u is n ˙ ). constant, and consequently u˙ is the 0 element of BL(R Identity (2.2.48) can be derived from the identity (2.2.18) with ψ(x) = 12 |x|2 , namely,  1  (Rn ) ∩ S : u ∈ L2 (Rn ; |x|2 dx) He1 (Rn ) = u ∈ Lloc (2.2.49) 1  u(x) v(x)|x|2 dx, u, v ∈ He1 (Rn ). E(u, v) =  2 Rn Indeed, from the Plancherel theorem7 , we see for u ∈ S that u ∈ BL(Rn ) u, 1 ≤ i ≤ n, are functions in L2 (Rn ). On if and only if the distributions xi the other hand, the last condition combined with the condition that u is a 1 (Rn ) can be seen to be equivalent to that u is a function in function in Lloc L2 (Rn ; |x|2 dx). T HEOREM 2.2.13. Assume that n ≤ 2. Then, for any domain D ⊂ Rn , the Dirichlet form ( 12 D, H 1 (D)) on L2 (D) is recurrent. Furthermore, denoting its extended Dirichlet space by (E, He1 (D)), it holds that (2.2.50) (E, He1 (D)) = 12 D, BL(D) . Proof. When n = 1, 2, the first statement is true for D = Rn and, by restricting to D those functions in H 1 (Rn ) appearing in the recurrence criterion in 7 K.

Yosida [154].

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Theorem 2.1.8(ii), we see that the same criterion holds for the Dirichlet form (2.2.45) on L2 (D). When n = 1, the identity (2.2.50) is established in Theorem 2.2.11 under a more general context. A similar method to the proof of Proposition 2.2.10 works for n = 2. Consider any domain D ⊂ R2 and take any function u ∈ BL(D) such that |u| ≤ for some constant. Let ψn ∈ Cc1 (R+ ) be functions satisfying properties (2.2.38) and put un (x) = u(x)ψn (|x|) for x ∈ D. Then un ∈ BL(D) ∩ L2 (D) = H 1 (D) and D(un , un ) ≤ 2 |∇u|2 (x)ψn (|x|)2 dx + 2 u2 (x)ψn (|x|)2 dx D

D

≤ 2D(u, u) + 22 ≤ 2D(u, u) +

{x∈R2 :|x|≤2n+1}

ψn (r)2 rdrdθ

22 π(2n + 1)2 ≤ 2D(u, u) + 182 π. n2

Hence by Theorem A.4.1, a Cesàro mean of a subsequence of {un } is D-convergent. Since un converges to u pointwise, we conclude that u ∈ He1 (D) and E(u, u) = 12 D(u, u). Next take any u ∈ BL(D) and put u = ϕ ◦ u, ∈ N, for the normal contraction ϕ of (1.1.19). By (BL.2), u ∈ BL(D) and 2 |∇u|2 dx → 0, → ∞. u − u D = {|u|>}

So in particular u D is bounded. We have just shown that u ∈ He1 (D) with E(u , u ) = 12 D(u , u ). Thus we conclude by Lemma 1.1.12 that u ∈ He1 (D). We next consider, on a general domain D ⊂ Rn , a finite measure m(dx) = m(x)dx with a density function m(x) satisfying m(x) > 0 for every x ∈ D,

m ∈ bC(D) ∩ L1 (D)

(2.2.51)

and an associated form (E, F) =

1 D, BL(D) ∩ L2 (D; m) , 2

(2.2.52)

which is obtained just by replacing L2 (D) with L2 (D; m) in (2.2.44) and 2 (D), (2.2.45). Since the convergence in L2 (D; m) implies the convergence in Lloc 2 (2.2.52) can be readily seen to be a Dirichlet form on L (D; m).


73

T HEOREM 2.2.14. The Dirichlet form (2.2.52) on L2 (D; m) is recurrent and irreducible. Its extended Dirichlet space (E, Fe ) coincides with the space ( 21 D, BL(D)). Proof. Since m is assumed to be a finite measure on D, the present Dirichlet form enjoys the recurrence condition (2.1.22) as well as the condition (ii) of Theorem 2.1.11. According to the same theorem, it is irreducible and we further have u ∈ Fe with E(u, u) = 0 =⇒ u is constant a.e.

(2.2.53)

Denote by F˙e the quotient space of Fe by the subspace of constant functions. Just as in the proof of the preceding theorem but using (2.2.53) in place of (2.2.47), we conclude that the space (F˙ e , E) is isometrically embedded into the 1 ˙ D). space (BL(D), 2 For any u ∈ BL(D), the functions u ∈ BL(D), ≥ 1, defined as in the last part of the proof of the preceding theorem is D-convergent to u. Since u ∈ F = BL(D) ∩ L2 (D; m) and u converges to u pointwise, u must be an element of Fe . Hence the above isometric embedding is an onto map and Fe = BL(D). When the Lebesgue measure of the domain D is finite, then we can take m(dx) to be the Lebesgue measure in (2.2.52) in reducing F to H 1 (D). Hence C OROLLARY 2.2.15. If the domain D is of finite Lebesgue measure, then ( 21 D, H 1 (D)) is irreducible recurrent and He1 (D) = BL(D).

(2.2.54)

Recall that the Dirichlet form (2.2.46) on L2 (Rn ) is regular and possesses as its core. Therefore each function u ∈ H 1 (Rn ) admits a quasi continuous version u based on the capacity Cap1 on Rn evaluated by (1.2.3)– (1.2.4) with h = 1 and α = 1 there using the form (2.2.46). In what follows, “q.e.” means “except for an E-polar set N ⊂ Rn ” (or, equivalently, in view of Theorem 1.3.14, except for a set N ⊂ Rn with Cap1 (N) = 0). The Dirichlet form ( 12 D, H 1 (D)) on L2 (D) is not necessarily regular. But if the boundary ∂D is so regular that Cc∞ (D), the space of functions in Cc∞ (Rn ) restricted to D, is E1 -dense in H 1 (D), then (2.2.45) can be regarded as a regular Dirichlet form on L2 (D; 1D · dx)(= L2 (D)). Such a property holds, for example, when the domain D has continuous boundary in the following sense:8 any x ∈ ∂D has a neighborhood U such that Cc∞ (Rn )

D ∩ U = {(x1 , . . . , xn ) : xn > F(x1 , . . . , xn−1 )} ∩ U 8 Cf.

Theorem 2 on page 14 of V. G. Maz’ja [121].

(2.2.55)

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in some coordinate (x1 , . . . , xn ) and with a continuous function F. In this case, ( 21 D, H 1 (D)) can be regarded as a regular, strongly local Dirichlet form on L2 (D)(= L2 (D; 1D · dx)) rather than on L2 (D). A counterexample concerning the above property is provided by a planar domain D = {(r, θ) : 1 < r < 2, 0 < θ < 2π }, which is obtained from the annulus D1 = {(r, θ ) : 1 < r < 2, 0 ≤ θ < 2π} by removing a segment on the positive x-axis. The function u(r, θ ) = θ is in BL(D) but not in BL(D1 ) because it violates the property (BL.2) on D1 . Hence it cannot be D-approximated by functions in Cc∞ (D) = Cc∞ (D1 ). Finally we consider the very special case where D is the unit disk D = {x ∈ R2 : |x| < 1} in R2 . Then the Poincaré inequality u − uD 2 ≤ CuD

for u ∈ C1 (D)

(2.2.56)

is known to be true for some constant C > 0, where uD := π1 D u(x)dx. In fact, (2.2.56) holds for every u ∈ He1 (D). This is because for u ∈ He1 (D), we can find an approximating sequence {un } for u from C1 (D). Then un − (un )D is L2 -convergent and un converges to u a.e. In particular, H 1 (D) = He1 (D) = BL(D).

(2.2.57)

The second identity follows from Corollary 2.2.15. The boundary of D, the unit circle, is denoted by T and parameterized by T = {θ : 0 ≤ θ < 2π}. Let us introduce a Dirichlet form (C, G) on L2 (T) by   C(ϕ, ψ) = 1 (ϕ(θ ) − ϕ(θ ))(ψ(θ ) − ψ(θ ))U(θ − θ )dθ dθ 2 T×T (2.2.58)  G = {ϕ ∈ L2 (T) : C(ϕ, ϕ) < ∞}, where U(θ ) = (4π(1 − cos θ ))−1 .

(2.2.59)

The above integral is called the Douglas integral.9 We shall show that   1 D(Hϕ, Hψ) = C(ϕ, ψ) 2 (2.2.60) ϕ ∈ L2 (T) : Hϕ ∈ BL(D) = G, where Hϕ(x) = T K(x, θ )ϕ(θ )dθ , x ∈ D, is the integral with the Poisson kernel K(x, θ ) =

9 Cf.

1 1 − ρ2 , 2π 1 − 2ρ cos(θ − θ ) + ρ 2

J. Douglas [42] and J. L. Doob [41].

x = ρeiθ .

(2.2.61)


75

To see (2.2.60), we express ϕ ∈ L2 (T) by the Fourier series ϕ(ξ ) = 2π −iνθ ∞ 1 iνξ ϕ(θ )dθ . A change of variables η = θ − θ , ν=−∞ cν e , cν = 2π 0 e ξ = θ and an evaluation of residue yield 2π 2π 1 C(ϕ, ϕ) = (ϕ(ξ + η) − ϕ(ξ ))2 (1 − cos η)−1 dξ dη 8π 0 0 2π 1 2 |cν | (1 − cos νη)(1 − cos η)−1 dη = 2 ν 0 |cν |2 |ν|(≤ ∞). =π ν

Now take ϕ ∈ L2 (T). Using Green’s formula on each disk of radius ρ, 1 1 2π ∂(Hϕ) iθ Hϕ(ρeiθ ) D(Hϕ, Hϕ) = lim (ρe )ρdθ. ρ↑1 2 0 2 ∂ρ 1 |ν| iν(θ−θ ) leads us to On the other hand, the expression K(ρeiθ , θ ) = 2π νρ e Hϕ(ρeiθ ) = ν cν ρ |ν| eiνθ , which converges uniformly in θ and in ρ ≤ ρ0 for each fixed ρ0 < 1. Hence ∞ 1 |ν|ρ 2|ν| |cν |2 = C(ϕ, ϕ)(≤ ∞). D(Hϕ, Hϕ) = lim π ρ↑1 2 ν=−∞

We note the relation −

1 d K(ρeiθ , θ ) ρ=1− = U(θ − θ ). 2 dρ

(2.2.62)

As we shall see in Example (3◦ ) of Section 5.3, (C, G) is the Dirichlet form of the reflecting Brownian motion X on D time-changed by its local time on ∂D corresponding to the Lebesgue surface measure of ∂D. In Section 7.2 we shall encounter a general situation where the trace Dirichlet form of a symmetric Markov process has the Douglas integral representation analogous to (2.2.58). 2.2.5. Brownian Motions on Manifolds Let (M, g) be an n-dimensional Riemannian manifold. A Brownian motion B = {Bt , t ≥ 0} is the minimal M-valued strong Markov process with continuous sample paths with infinitesimal generator 12 g , where g is the LaplaceBeltrami operator on M. There are several ways to construct a Brownian motion on M. One can solve stochastic differential equations in local coordinates and then piece together the resulting diffusions in local coordinates. This was done by R. Gangolli in [79]. It can also be constructed by solving an M-valued stochastic differential equation using Itô’s map (see [91]). In this subsection,

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we illustrate that Brownian motion on M can be constructed through a regular Dirichlet form via Theorem 1.5.1. For p ∈ M, gp is an inner product on the tangent space Tp (M) of M at p, which is of dimension n. Hence gp establishes a one-to-one correspondence between Tp (M) and its dual Tp∗ (M), the co-tangent space at p. For every f ∈ C(M), df , the C∞ co-vector field, is in T ∗ (M). It uniquely determines an Xf ∈ T(M) by g(Xf , Y) = df , Y = Yf for every Y ∈ T(M). In local coordinates,   n n n ∂f ∂f   ∂ , dxi and Xf = gij df = ∂xi ∂xj ∂xi i=1 i=1 j=1 where (gij ) is the matrix of the inner product g in these local coordinates and (gij ) is its inverse matrix. Let V(dp) be the volume element of (M, g) and define, for u, v ∈ Cc∞ (M), 1 gp (Xu , Xv )V(dp). E(u, v) = 2 M √ In local coordinates, V(dx) = g(x)dx, where g(x) is the determinant of (gij (x)) and, with ∇u(x) := ( ∂u(x) , . . . , ∂u(x) ) denoting the (Euclidean) gradient of u, ∂x1 ∂xn gp (Xu , Xv )V(dp) = ((gij )∇u)(x) · (gij (x))((gij )∇v)(x) g(x)dx = ∇u(x) · (gij (x))∇v(x) g(x)dx. In the same way as the proof of Lemma 2.2.7, the Markovian property of the symmetric form (E, Cc∞ (M)) in the sense of Definition 1.1.2 is readily verifiable. Note that for u, v ∈ Cc∞ (M), E(u, v) = (− 12 g u, v)L2 (M; dV) . This can be easily seen since in local coordinates, the Laplace-Beltrami differential operator g on (M, g) has the expression n 1 ∂ ∂ ij g = √ g(x)g (x) . ∂xj g(x) i,j=1 ∂xi As, for {u, vk , k ≥ 1} ⊂ Cc∞ (M) with vk → 0 in L2 (M; V(dp)), 1 = 0, lim E(u, vk ) = lim − g u, v k→∞ k→∞ 2 L2 (M; dV) the quadratic form (E, Cc∞ (M)) is closable. Define H01 (M) to be the closure of Cc∞ (M) under the inner product E1 := E + (·, ·)L2 (M; dV) . Then (E, H01 (M)) is a regular Dirichlet form on L2 (M; dV) as the closure of a closable Markovian symmetric form (cf. [73, Theorem 3.1.1]). The above regularity can also be checked directly by using Theorem 1.1.3(d).


77

The Hunt process B associated with this Dirichlet form is a Brownian motion on M. That B has continuous sample paths follows from the fact that the Dirichlet form (E, H01 (M)) is strongly local and Theorem 4.3.4 below. In Example 5.1.11 below, we will present a quasi-regular but not regular Dirichlet form on Rn that has a Hunt process properly associated with it.

2.3. ANALYTIC POTENTIAL THEORY FOR REGULAR DIRICHLET FORMS Let E be a locally compact separable metric space and m be a positive Radon measure on E satisfying supp[m] = E. Let (E, F) be a regular Dirichlet form on L2 (E; m) in the sense of Definition 1.3.10. In this section, we present an analytic potential theory for (E, F ) primarily due to A. Beurling and J. Deny [8] and J. Deny [39], which will be used in the sequel. The E-quasi-notions (E-nest, E-polarity, E-quasi-continuity) are introduced in Definition 1.2.12 for a Dirichlet form (E, F) on a general topological space (E, m). These notions for the regular Dirichlet form (E, F) on L2 (E; m) have been interpreted in terms of the capacity Cap1 which is defined to be Caph,α with h ≡ 1 and α = 1 in Section 1.2. For reader’s convenience, we recall the definition of Cap1 below. Denote by O the family of all open subsets of E and, for A ∈ O, we put LA,1 = { f ∈ F : f ≥ 1 m-a.e. on A}. The function eA ∈ LA,1 minimizing {E1 (u, u), u ∈ LA,1 } is called the (1-order) equilibrium potential of the open set A. (Deviating from the notation used in Section 1.2, we write it as eA instead of 1A in order to distinguish it from the indicator of A). For open set A, define inf{E1 ( f , f ) : f ∈ LA,1 }, A ∈ O0 Cap1 (A) = (2.3.1) ∞ A∈ / O0 where O0 = {A ∈ O : LA,1 = ∅}. The capacity Cap1 (B) for an arbitrary set B ⊂ E is defined by Cap1 (B) = inf{Cap1 (A) : A ∈ O, A ⊃ B}.

(2.3.2)

Theorem 1.3.14 states that a set N ⊂ E is E-polar if and only if Cap1 (N) = 0, an increasing sequence of closed sets {Fk } is an E-nest if and only if limk→∞ Cap1 (K \ Fk ) = 0 for any compact set K ⊂ E, and a function f is E-quasi-continuous if and only if for any ε > 0, there is an open set O with Cap1 (O) < ε such that f |E\O is finite and continuous. Since we are dealing with a fixed regular Dirichlet form (E, F), for convenience we drop “E-” from the terminology an “E-nest” and “E-quasi-continuous” and will simply call them a nest and quasi continuous, respectively. We call an increasing sequence {Fk } of closed sets a Cap1 -nest if limk→∞ Cap1 (E \ Fk ) = 0. Any

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Cap1 -nest is a nest but not vice versa. However, a function f is quasi continuous if and only if there is a Cap1 -nest {Fk } such that f ∈ C({Fk }). Moreover, in view of Lemma 1.3.15, any element f ∈ F admits a quasi continuous version f in the restricted sense relative to a Cap1 -nest, namely, there is a Cap1 -nest {Fk } such that f ∈ C∞ ({Fk }). We also use the term quasi everywhere or q.e. to mean “except for an E-polar set,” or, equivalently, “except for a Cap1 -polar set,” in view of Theorem 1.3.14. Theorem 1.3.3 enables us to introduce the equilibrium potentials for sets which are not necessarily open. For an arbitrary set B ⊂ E. we let LB,1 = { f ∈ F : f ≥ 1 q.e. on B}. If LB,1 = ∅, then LB,1 is a closed convex subset of (F , E1 ) and there exists a unique element eB ∈ LB,1 minimizing E1 (u, u). We call eB the (1-order) equilibrium potential of B. By Theorem 1.3.7, if B is an open set, then eB coincides with the equilibrium potential of B mentioned at the beginning of this section. eB enjoys the following properties. T HEOREM 2.3.1. Fix an arbitrary set B ⊂ E with LB,1 = ∅. (i) It holds that Cap1 (B) = E1 (eB , eB ).

(2.3.3)

(ii) 0 ≤ eB ≤ 1 [m] and eB = 1 q.e. on B. (iii) eB is the unique element of F satisfying the following properties: eB = 1 q.e. on B and E1 (eB , f ) ≥ 0 for any f ∈ F with f ≥ 0 q.e. on B. Proof. In view of Theorem 1.3.3, (ii) and (iii) can be shown similarly to the proof of Theorem 1.2.5. Let us show (i). For any ε > 0, there exists A ∈ O0 such that B ⊂ A and Cap1 (B) > Cap1 (A) − ε. Since the equilibrium potential eA of A belongs to LB,1 by Theorem 1.3.7, Cap1 (A) = E1 (eA , eA ) ≥ E1 (eB , eB ), and so we get Cap1 (B) ≥ E1 (eB , eB ). To prove the converse inequality, we fix a quasi continuous version eB of eB . eB |Acε is For any ε > 0, we can find an open set Aε with Cap(Aε ) < ε such that continuous and eB (x) ≥ 1 for every x ∈ B ∩ Acε . Then Gε = {x ∈ Acε : eB > 1 − ε} ∪ Aε is an open set containing B. If we denote the equilibrium potential of Aε by eε , it holds that eB + eε ≥ 1 − ε m-a.e. on Gε . Therefore Cap1 (B) ≤ Cap1 (Gε ) ≤ (1 − ε)−2 eB + eε 2E1 ≤ (1 − ε)−2 (eB E1 + eε E1 )2 ≤ (1 − ε)−2 (eB E1 + We get (2.3.3) by letting ε ↓ 0.

√ 2 ε) .


79

Next suppose that the semigroup generated by Dirichlet form (E, F) is transient in the sense of Definition 2.1.1. Then the extended Dirichlet space (Fe , E) is a Hilbert space and by Theorem 2.1.5 uL1 (g·m) ≤ uE ,

u ∈ Fe

for some reference function g. (0) For A ∈ O, let L(0) A,1 = { f ∈ Fe : f ≥ 1 m-a.e. on A}. If LA,1 = ∅, there exists (0) (0) (0) a unique element e(0) A ∈ LA,1 minimizing {E(u, u), u ∈ LA,1 }. eA is called the (0) 0-order equilibrium potential of A. The 0-order capacity Cap (A) of an open set A is defined by (2.3.1) with E1 and LA,1 being replaced by E and L(0) A,1 , (0) respectively. Cap (B) for any set B ⊂ E is defined analogously to (2.3.2). An increasing sequence {Fk } of closed sets is said to be a Cap(0) nest if limk→∞ Cap(0) (E \ Fk ) = 0. A set N ⊂ E is called Cap(0) -polar if Cap(0) (N) = 0. T HEOREM 2.3.2. Suppose (E, F ) is transient. (i) Any Cap(0) -nest is a nest. (ii) A subset of E is Cap(0) -polar if and only if it is E-polar. (iii) Any element of Fe admits a quasi continuous m-version in the restricted sense relative to a Cap(0) -nest. If a sequence of quasi continuous functions { fn , n ≥ 1} ⊂ Fe is E-convergent to f ∈ Fe , then there exist a subsequence {n } and a Cap(0) -nest {Fk } with fn ∈ C∞ ({Fk }) such that fn converges as → ∞ uniformly on each set Fk to a quasi continuous version of f . Proof. (i) For any relatively compact open set D ⊂ E, choose a non-negative function h ∈ F ∩ Cc (E) such that h ≥ 1 on D \ F1 . Define vk = e(0) E\Fk ∧ h. 2 Then vk ∈ F(= Fe ∩ L (E; m)) and sup E1 (vk , vk ) ≤ sup Cap(0) (E \ Fk ) + E1 (h, h) < ∞. k≥1

k≥1

Hence by Theorem A.4.1, a Cesàro mean sequence {wk } of a subsequence of {vk } is E1 -convergent. On the other hand, by Theorem 2.1.5(i), vk converges to 0 m-a.e. as k → ∞, so is wk . Since wk ∈ LD\Fk ,1 , this together with lim Cap1 (D \ Fk ) ≤ lim E1 (wk , wk ) = 0

k→∞

k→∞

yields via Theorem 1.3.14(ii) that {Fk } is a nest. (ii) The “Only if” part is a consequence of (i). If a set is E-polar, then it is Cap1 -polar and consequently Cap(0) -polar. (iii) The second assertion can be proved in the same way as that of Theorem 1.3.3 by using a 0-order counterpart of Exercise 1.3.16. The first assertion follows from the second.

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Exercise 2.3.3. Let 0 < α ≤ 2, α < n. Then the rotation-invariant stable semigroup of index α on Rn is regular and transient. Moreover, its extended Dirichlet space is characterized as (2.2.20) in Section 2.2.2: (2.3.4) Fe = u = Iα/2 ∗ f : f ∈ L2 (Rn ) with E(u, v) = Rn f (x)g(x)dx for u = Iα/2 ∗ f , v = Iα/2 ∗ g. Prove that every function appearing on the right hand side of (2.3.4), namely, the α/2-order Riesz potential of an L2 -function, is quasi continuous. The next theorem says that Theorem 2.3.2(iii) in fact holds for any (not necessarily transient) Dirichlet form (E, F). T HEOREM 2.3.4. Any f ∈ Fe admits its quasi continuous version. If { fn } ⊂ F is an approximating sequence of f ∈ Fe and fn , n ≥ 1, are quasi continuous, then there exists a subsequence n such that fn converges to a quasi continuous version of f q.e. as → ∞. If f ∈ Fe is bounded on E, its approximating sequence { fn } ⊂ F can be taken to be uniformly bounded on E. Proof. Owing to Lemma 2.1.15, there exists a function g belonging to the g family K0 defined by (2.1.27) such that f ∈ Fe and { fn } is E g -convergent to f , where (E g , F) is the perturbed Dirichlet form on L2 (E; m) defined by (2.1.28) g and Fe is its extended Dirichlet space. Since (E g , F) is transient and the norm √ g E1 (u, u) is equivalent to E1 (u, u) for u ∈ F, we get the desired conclusions from Theorem 2.3.2 and Exercise 2.1.13. A positive Radon measure µ on E is called a measure of finite energy integral if there exists a constant Cµ > 0 such that |g(x)|µ(dx) ≤ Cµ gE1 for every g ∈ F ∩ Cc (E). (2.3.5) E

Let S0 denote the family of all measures of finite energy integrals. For µ ∈ S0 , the linear functional µ on F ∩ Cc (E) defined by µ (g) = E g(x)µ(dx) has the property |µ (g)| ≤ Cµ gE1

for every g ∈ F ∩ Cc (E).

Since (E, F) is regular, for each α > 0, µ determines a unique bounded linear functional on the Hilbert space (F , Eα ). By Riesz representation theorem (cf. [154]), there exists a unique element Uα µ ∈ F so that Eα (Uα µ, g) = g(x)µ(dx) for every g ∈ F ∩ Cc (E). (2.3.6) E

Uα µ is called the α-potential of µ ∈ S0 .

81


Let {Tt ; t > 0} be the L2 -semigroup generated by the Dirichlet form (E, F ). For α > 0, the α-excessiveness of a function f ∈ L2 (E; m) relative to {Tt ; t > 0} is defined by Definition 1.2.1. L EMMA 2.3.5. Let f ∈ F and α > 0. The following are equivalent. (i) f = Uα µ for some µ ∈ S0 . (ii) Eα ( f , g) ≥ 0 for every g ∈ F ∩ Cc (E)+ . (iii) Eα ( f , g) ≥ 0 for every g ∈ F with g ≥ 0 [m]. (iv) f is α-excessive relative to {Tt ; t ≥ 0}. Proof. The equivalence of (iii) and (iv) has already been established by Lemma 1.2.4. (i) ⇒ (ii) is trivial. Assuming (ii), take, for g ∈ F with g ≥ 0 [m], a sequence {gn , n ≥ 1} ⊂ F ∩ Cc (E) that is E1 -convergent to g. Since the sequence {g+ n , n ≥ 1} ⊂ F ∩ Cc (E) is uniformly bounded in E1 -norm and a subsequence of {g+ n , n ≥ 1} converges m-a.e. to g, by Theorem A.4.1, there aro mean sequence {hn , n ≥ 1} is is a subsequence of {g+ n , n ≥ 1} whose Ces` E1 -convergent to g. We thus get (iii) from Eα ( f , hn ) ≥ 0 by letting n → ∞. (iii) ⇒ (ii) is trivial. We now show (ii) ⇒ (i). For any compact set K ⊂ E, we can find a non-negative function gK ∈ F ∩ Cc (E) with gK ≥ 1 on K. If we define f (g) = Eα ( f , g), g ∈ F ∩ Cc (E), then, by the assumption, we have |f (g)| ≤ g∞ f (gK ) for any g ∈ F ∩ Cc (E) with supp[g] ⊂ K. According to Exercise 1.3.13, any h ∈ Cc (E) is a uniform limit of a sequence gn ∈ F ∩ Cc (E) with supp[gn ] ⊂ supp[h]. Therefore f can be extended uniquely to a positive linear functional on Cc (E). Therefore there is a positive Radon measure µ on E such that f (g) = E gdµ for every g ∈ F ∩ Cc (E). This establishes (i). By Remark 1.2.2, we get C OROLLARY 2.3.6. If f1 , f2 are α-potentials, then so are the functions f1 ∧ f2 and f1 ∧ 1. the totality of quasi continuous functions belonging to F. Denote by F T HEOREM 2.3.7. Let µ ∈ S0 . Then (i) µ charges no E-polar set. ⊂ L1 (E; µ) and, for every α > 0 and g ∈ F, (ii) F Eα (Uα µ, g) = g(x)µ(dx).

(2.3.7)

E

Proof. (i) Define for every n ≥ 1, gn := n(U1 µ − e−1/n T1/n (U1 µ)). Note that by Lemma 2.3.5, gn ≥ 0 [m]. So it follows from (1.2.1) and (2.3.6) that, for

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CHAPTER TWO

any h ∈ F ∩ Cc (E), h(x)gn (x)m(dx) = E1 (U1 µ, h) = h(x)µ(dx). lim n→∞ E

E

Therefore, for any compact set K and any relatively compact open set G containing K, we have by Theorem 1.2.5 µ(K) ≤ lim inf gn (x)m(dx) ≤ lim inf(gn , eG ) n→∞

n→∞

G

= E1 (U1 µ, eG ) ≤ U1 µE1 Cap1 (G), which proves (i). (ii) For any g ∈ F, choose gn ∈ F ∩ Cc (E) so that it is E1 -convergent to g as g q.e. n → ∞. By virtue of Theorem 1.3.3, a subsequence gnk converges to Since (2.3.7) is valid for each gn , it follows from (i) and Fatou’s lemma that | g(x) − gn (x)|µ(dx) = lim inf |gnk (x) − gn (x)|µ(dx) E k→∞

E

≤ Cµ · lim inf gnk − gn E1 . k→∞

g in L1 (E; µ). It now This shows that g ∈ L1 (E; µ) and that gn converges to suffices to let n → ∞ in the equation (2.3.7) for gn . For µ ∈ S0 , Eα (µ) = Eα (U α µ, Uα µ) is called the α-energy integral of µ, which is equal to the integral E U! α µ(x)µ(dx) by virtue of the above theorem. Exercise 2.3.8. Prove the following equation: Uα µ − Uβ µ + (α − β)Gα Uβ µ = 0

for µ ∈ S0 and α, β > 0.

(2.3.8)

L EMMA 2.3.9. (i) Suppose, for µ ∈ S0 and a constant C ≥ 0, U! α µ ≤ C µa.e. Then Uα µ ≤ C [m]. (ii) For any µ ∈ S0 , there exists a Cap1 -nest {Fk } such that U1 (1Fk µ)∞ < ∞

for every k ≥ 1.

Proof. (i) The function f = Uα µ ∧ C is an α-potential by Corollary 2.3.6. By virtue of the preceding theorem f , µ = U! Eα ( f , Uα µ) = α µ, µ = Eα (Uα µ, Uα µ). Since f ≤ Uα µ [m], we get by the above identity and Lemma 2.3.5 that f − Uα µ2Eα = Eα ( f , f − Uα µ) ≤ 0. Hence Uα µ = f ≤ C [m].


83

! (ii) For µ ∈ S0 , choose a quasi continuous version of U 1 µ and an associated 0 0 ! Cap1 -nest {Fk }. We let Fk = {x ∈ Fk : U1 µ(x) ≤ k}, k = 1, 2, . . . . For each ! (1Fk · µ) ≤ U k, U1 1 µ ≤ k q.e. on Fk , and consequently U1 (1Fk · µ) ≤ k [m] ! by (i). Furthermore, Cap1 (E \ Fk ) ≤ Cap1 (E \ Fk0 ) + Cap1 ({U 1 µ > k}), which tends to zero as k → ∞ in view of Exercise 1.3.16. L EMMA 2.3.10. The following conditions are mutually equivalent for f ∈ F and a closed set F ⊂ E: (i) f = Uα µ for some µ ∈ S0 with supp[µ] ⊂ F. g ≥ 0 q.e. on F. (ii) Eα ( f , g) ≥ 0 for any g ∈ F with (iii) Eα ( f , g) ≥ 0 for any g ∈ F ∩ Cc (E) with g ≥ 0 on F. Proof. (i) ⇒ (ii) follows from Theorem 2.3.7(ii). (ii) ⇒ (iii) is trivial. (iii) ⇒ (i) follows from the proof of Lemma 2.3.5 and Exercise 1.3.13. Recall the 1-order equilibrium potential eB defined for a set LB,1 = ∅. By Theorem 2.3.1(iii) and the above lemma, there measure µB ∈ S0 such that eB = U1 µB with supp[µB ] ⊂ B. µB equilibrium measure of the set B. For a compact set K ⊂ E, particular Cap1 (K) = E1 (eK , eK ) = µK (K)

B ⊂ E with is a unique is called 1we have in (2.3.9)

on account of Theorem 2.3.1 and Theorem 2.3.7. Let us define a subfamily S00 of S0 by S00 = {µ ∈ S0 : µ(E) < ∞, U1 µ∞ < ∞}.

(2.3.10)

C OROLLARY 2.3.11. The following conditions for B ∈ B(E) are mutually equivalent: (i) Cap1 (B) = 0. (ii) µ(B) = 0 for every µ ∈ S0 . (iii) µ(B) = 0 for every µ ∈ S00 . Proof. The implication (i) ⇒ (ii) is given by Theorem 2.3.7(i) and the implication (ii) ⇒ (iii) is trivial. If Cap1 (B) > 0, then there is a compact set K ⊂ B with Cap1 (K) > 0 by (1.2.6) with h = α = 1 there. Since µK ∈ S00 has µK (K) > 0 by (2.3.9), this establishes the implication (iii) ⇒ (i). Fix an α > 0 and an arbitrary set B ⊂ E. Let f ∈ F be α-excessive relative to {Tt } and put g ≥ f q.e. on B}. Lf ,B = {g ∈ F :

(2.3.11)

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CHAPTER TWO

Since Lf ,B is a closed convex subset of (F, Eα ) by Theorem 1.3.3, there exists a unique function fB in Lf ,B minimizing gEα . Since Eα ( fB , g) ≥ 0 for g ∈ F with g ≥ 0 q.e. on B,

(2.3.12)

fB is again α-excessive relative to {Tt } by Lemma 2.3.5. fB is said to be the α-reduced function of the α-excessive function f ∈ F with respect to B. By Lemma 2.3.10, there exist unique measures µ, ν ∈ S0 so that f = Uα µ, fB = Uα ν, and supp[ν] ⊂ B. The correspondence µ ∈ S0 → ν ∈ S0 is said to be the α-sweeping out of µ on the set B. L EMMA 2.3.12. fB is an α-reduced function of α-excessive function f ∈ F relative to a set B if and only if fB is an element of F satisfying (2.3.12) and fB = f q.e. on B.

(2.3.13)

Proof. We need only show the necessity of (2.3.13). To this end, we observe that h = fB ∧ f is again α-excessive relative to {Tt ; t > 0} with h ≤ fB ∈ F, and hence Lemma 1.2.3 implies that h ∈ F with hEα ≤ fB Eα , yielding h ∈ Lf ,B and fB = h ≤ f . This lemma leads us to another important interpretation of a reduced function. If we let FE\B = {g ∈ F : g = 0 q.e. on B},

(2.3.14)

then FE\B is a closed linear subspace of the Hilbert space (F, Eα ). Denote by HBα its orthogonal complement, and so F = FE\B ⊕ HBα . Denoting by PHBα the orthogonal projection to the space HBα , we have PHBα f = fB ,

(2.3.15)

because f = ( f − fB ) + fB gives the Eα -orthogonal decomposition of f in view of Lemma 2.3.12. We notice that, in a particular case where B is an open set, fB is nothing but the α-reduced function of the α-excessive function f on the set B in the sense of Section 1.2 on account of Theorem 1.2.5. D EFINITION 2.3.13. A positive measure µ on (E, B(E)) is called smooth if it satisfies the following two conditions: (S.1) µ charges no E-polar set. (S.2) There exists a nest {Fk } such that µ(Fk ) < ∞ for every k ≥ 1. The nest {Fk } appearing in the " above is said to be attached to the smooth measure µ. We note that µ(E \ k Fk ) = 0 in this case. We denote by S the totality of smooth measures. Any positive Radon measure on E charging no E-polar set is an element of S. In this case, for an


85

increasing sequence {Gk } of relatively compact open sets, we can let Fk = Gk to get a nest {Fk } satisfying (S.2). In particular, S00 ⊂ S0 ⊂ S. L EMMA 2.3.14. (i) Let ν be a finite measure on (E, B(E)). If ν(A) ≤ C · Cap1 (A) for every A ∈ B(E) for some positive constant C, then ν ∈ S0 . (ii) Let ν be a finite measure on (E, B(E)) charging no E-polar set. There exists then a decreasing sequence {Gn } of open sets such that lim Cap1 (Gn ) = 0,

n→∞

lim ν(Gn ) = 0

n→∞

and ν(A) ≤ 2n Cap1 (A)

∀A ∈ B(E), A ⊂ E \ Gn , n ≥ 1.

Proof. (i) For any non-negative g with g ∈ F ∩ Cc (E) and E1 (g, g) = 1, we have by (1.3.2) and Exercise 1.3.16, ∞ g(x)ν(dx) ≤ ν(E) + 2k+1 ν({x : 2k < g(x) ≤ 2k+1 }) E

k=0

≤ ν(E) + C

∞

2k+1 Cap1 ({x : g(x) > 2k })

k=0

≤ ν(E) + C

∞

2k+1 2−2k = ν(E) + 4C,

k=0

which means ν ∈ S0 . (ii) Fix n ≥ 1 and let α := inf{2n Cap1 (A) − ν(A) : A ∈ B(E)}. Clearly −ν(E) ≤ α. If α < 0, then choose an open set B1 with 2n Cap1 (B1 ) − ν(B1 ) ≤ α/2 and let α1 = inf{2n Cap1 (A) − ν(A) : A ∈ B(E), A ⊂ E \ B1 }. Since for any A ∈ B(E) with A ⊂ E \ B1 we have α ≤ 2n Cap1 (A ∪ B1 ) − ν(A ∪ B1 ) ≤ 2n Cap1 (A) − ν(A) + 2n Cap1 (B1 ) − ν(B1 ), it holds that α/2 ≤ α1 . If α1 < 0, then choose a relatively open set B2 ⊂ E \ B1 with 2n Cap(B2 ) − ν(B2 ) ≤ α1 /2. By repeating the same procedures, we find a sequence {B1 ∪ B2 ∪ · · · ∪ Bk } of open sets such that 2n Cap(B1 ∪ · · · ∪ Bk ) − ν(B1 ∪ · · · ∪ Bk ) ≤ 0 2n Cap(A) − ν(A) ≥ 2−k α

for every A ⊂ E \ (B1 ∪ · · · ∪ Bk ).

If we let Gn = ∪nk=1 Bk , then for every A ⊂ E \ Gn , 2n Cap(A) ≥ ν(A)

and

2n Cap(Gn ) ≤ ν(Gn ) ≤ ν(E) < ∞.

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Gn = ∪∞ =n G gives a desired sequence of decreasing open sets. In fact, −n+1 ν(E) → 0, n → ∞. Since ν charges no E-polar set, 0 = Cap(G # n) ≤ 2 ν( n Gn ) = limn→∞ ν(Gn ).

T HEOREM 2.3.15. The following assertions are mutually equivalent for a positive measure µ on (E, B(E)): (i) µ ∈ S. (ii) There exists a nest {Fk } satisfying $ µ(Fk ) < ∞, 1Fk · µ ∈ S0 , ∀k ≥ 1,

µ E\

∞ %

& = 0.

Fk

(2.3.16)

k=1

(iii) There exists a nest {Fk } satisfying $ 1Fk · µ ∈ S00 , ∀k ≥ 1,

µ E\

∞ %

& Fk

= 0.

(2.3.17)

k=1

Proof. (i) ⇒ (ii): If µ is a finite measure charging no E-polar set, then, by letting Fk be the complement of the open set Gk in Lemma 2.3.14(ii), we see that {Fk } is a Cap1 -nest satisfying the second part of (2.3.16) as well as its first half on account of Lemma 2.3.14(i). For a general smooth measure µ, let {E , ≥ 1} be a nest associated with µ in Definition 2.3.13. For each , µ = 1E · µ is finite and charging no Epolar set, there exists a Cap1 -nest {Fk() } satisfying (2.3.16) for µ . Let Fk = "k () =1 {E ∩ Fk }, then {Fk } is an increasing sequence of closed sets satisfying the first half of (2.3.16). Furthermore, for any compact set K, Cap1 (K \ Fk ) ≤ Cap1 (K \ E ) + Cap1 (E \ Fk() ),

≤ k,

and we conclude from Theorem 1.2.14 that {Fk } is a nest by letting k → ∞ and then → ∞. Since µ charges no E-polar set, it satisfies the last part of (2.3.16) as well. (ii) ⇒ (iii): For µ, take a nest {E } satisfying (2.3.16) but with {E } in place of {Fk }. For each , we associate with µ = 1E · µ a Cap1 -nest {Fk() } satisfying condition of Lemma 2.3.9(ii) and then construct a nest {Fk } in the similar way as in the proof of (i). Then 1Fk · µ ∈ S00 for every k ≥ 1. (iii) ⇒ (i): If µ satisfies (2.3.17) for some nest {Fk }, then µ charges no E-polar set by Theorem 2.3.7 and it is smooth.

87


In the rest of this section, we assume that the Dirichlet form (E, F) is transient. We will be concerned with 0-order counterparts of Theorem 2.3.7 and Lemma 2.3.12. Note that functions in the extended Dirichlet space Fe are not necessarily in L2 (E; m) and so some modifications of the proofs are called for. We say that a positive Radon measure µ on E is of finite 0-order energy integral (µ ∈ S0(0) in notation) if the inequality (2.3.5) holds with E1 on the right hand side being replaced by E. Then the equation (2.3.6) with Eα being replaced by E determines a unique function Uµ ∈ Fe , which is called the 0order potential of µ. Evidently S0(0) ⊂ S0 and in particular any µ ∈ S0(0) charges no E-polar set by Theorem 2.3.7(i). For µ ∈ S0(0) , we have also the 0-order counterpart of Theorem 2.3.7(ii) 1 Fe ⊂ L (E; µ), E(Uµ, g) = g(x)µ(dx), g ∈ Fe , (2.3.18) E

e denotes the totality of quasi continuous functions belonging to Fe , as where F well as the 0-order counterpart of a part of the characterization Lemma 2.3.5: f = Uµ, ∃µ ∈ S0(0) ⇐⇒ E( f , g) ≥ 0 for g ∈ F+ ∩ Cc (E).

(2.3.19)

Furthermore, Lemma 2.3.10 is still valid with F, Uα µ, S0 , and Eα being replaced by Fe , Uµ, S0(0) , and E, respectively. L EMMA 2.3.16. If f1 , f2 ∈ Fe are (0-order) potentials, then so are f1 ∧ f2 and f1 ∧ 1. Proof. This is the 0-order counterpart of Corollary 2.3.6. But analogous reasoning does not work because elements of Fe need not be in L2 (E; m). Instead we first observe that equations (2.3.7) and (2.3.18) imply for µ ∈ S0(0) ! Uα µ − Uβ µ2Eα ≤ U! α µ, µ − U β µ, µ ! µ, µ − U! Uµ − Uα µ2E ≤ U α µ, µ

for β > α > 0, for α > 0,

which means that U! α µ, µ increases to a finite limit, and Uα µ is E-convergent to some f ∈ Fe as α ↓ 0. Moreover, since ! µ, µ, α(Uα µ, Uα µ) ≤ Eα (Uα µ, Uα µ) = U! α µ, µ ≤ U limα↓0 α(Uα µ, g) = 0 for every g ∈ F. We then let α ↓ 0 in (2.3.6) to conclude that f = Uµ. For µ, ν ∈ S0(0) , set f1 := Uµ and f2 := Uν. Then Uα µ ∧ Uα ν is E-weakly convergent to f1 ∧ f2 ∈ Fe as α ↓ 0 by the above observation and Exercise 2.1.13. Hence by letting α ↓ 0 in the inequality Eα (Uα µ ∧ Uα ν, g) ≥ 0

for g ∈ F+ ∩ Cc (E),

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we get E( f1 ∧ f2 , g) ≥ 0. This shows that f1 ∧ f2 is a potential in view of (2.3.19). For an arbitrary set B ⊂ E and a (0-order) potential f ∈ Fe , the (0-order) reduced function fB(0) of f relative to B can be specified as the unique element of L(0) g ≥ f q.e. on B} minimizing gE . Since fB(0) satisfies B = {g ∈ Fe : E( fB(0) , g) ≥ 0,

∀g ∈ Fe , g ≥ 0 q.e. on B,

(2.3.20)

fB(0) is a potential by virtue of the 0-order version of Lemma 2.3.10. L EMMA 2.3.17. fB(0) is a reduced function of a potential f ∈ Fe relative to a set B if and only fB(0) is an element of Fe satisfying (2.3.20) and (2.3.13). Proof. We need to argue differently from Lemma 2.3.12. To get the necessity of (2.3.13), we observe that, together with fB(0) , h = fB(0) ∧ f is a potential on account of Lemma 2.3.16. Thus there are µ, ν ∈ S0(0) so that fB(0) = Uµ and h = Uν. We get from (2.3.18) E(h, h) = h, ν ≤ fB(0) , ν = E( fB(0) , h) = h, µ ≤ fB(0) , µ = E( fB(0) , fB(0) ). Consequently, fB(0) = h ≤ f .

If we let Fe,E\B = {g ∈ Fe : g = 0 q.e. on B},

(2.3.21)

then Fe,E\B is a closed linear subspace of the Hilbert space (Fe , E). Denote by HB its orthogonal complement. Denoting by PHB the orthogonal projection to the space HB , we have PHB f = fB(0) ,

(2.3.22)

because f = ( f − fB(0) ) + fB(0) gives the orthogonal decomposition of f in view of Lemma 2.3.17.

2.4. LOCAL PROPERTIES Let E be a Hausdorff topological space with the Borel σ -field B(E) being generated by the continuous functions on E and m be a σ -finite measure with supp[m] = E. For an m-measurable function f defined and finite m-a.e. on E, we denote by supp[f ] its support, namely, the support of the measure f · m. Let (E, F) be a Dirichlet form on L2 (E; m). In this section, we present some equivalent conditions for (E, F) to be local.

89


For a closed subset F ⊂ E, recall the subspace FF of F defined by (1.2.2), which is a closed linear subspace of the Hilbert space (F, E1 ). Let HF1 c be its E1 -orthogonal complement. The E1 -orthogonal projection of f ∈ F to the space HF1 c is denoted by fFc , which is the unique element of F such that fFc = f m-a.e. on F c and E1 ( fFc , g) = 0 for any g ∈ FF . f − fFc is the E1 -orthogonal projection of f into FF . We start with the following lemma. L EMMA 2.4.1. For every f ∈ F and an E-nest {Fk , k ≥ 1}, f − fFkc is E1 convergent to f as k → ∞. Proof. For notational convenience, let fk := f − fFkc ∈ FFk . By the above characterizations, we see for j > k that ( fFjc )Fkc = fFjc , fk = fj − ( fj )Fkc and E1 ( fk , fj ) = E1 ( fk , fk ). Therefore E1 ( fj − fk , fj − fk ) = E1 ( fj , fj ) − E1 ( fk , fk ). It follows that { fk , k ≥ 1} is a Cauchy sequence in (F , E1 ) and thus it is E1 convergent to some g ∈ F . Since f − fk is E1 -orthogonal to FFk , we have % E( f − g, ϕ) = lim E( f − fk , ϕ) = 0 for every ϕ ∈ FFk . k→∞

k≥1

As ∪k≥1 FFk is E1 -dense in F, we conclude from above that f = g and this completes the proof of the lemma. T HEOREM 2.4.2. Let (E, F) be a quasi-regular Dirichlet form on L2 (E; m) and (C, F) be a symmetric form with 0 ≤ C(u, u) ≤ E(u, u)

for every u ∈ F.

Then the following two conditions are equivalent: (i) For every u, v ∈ F with disjoint compact support, C(u, v) = 0, (ii) For every u, v ∈ F with uv = 0 m-a.e. on E, C(u, v) = 0. Proof. Clearly (ii) implies (i). Since (E, F ) is quasi-regular, there exist a Borel E-polar set N ⊂ E and a Borel standard process X on E \ N properly associated with (E, F) in view of Theorem 1.5.2. Denote by {Rα ; α > 0} the resolvent kernel of X. We let h = R1 f for a strictly positive bounded m-integrable function f on E. Then h is an E-quasi-continuous element of F and strictly positive on E \ N. Suppose (i) holds. By considering u+ , u− , v + and v − separately, without loss of generality we may and do assume that u, v ∈ F are two non-negative functions with uv = 0 m-a.e. on E. Also we may take as u, v E-quasicontinuous m-versions. There is an m-regular E-nest {Fk , k ≥ 1} of compact sets so that {u, v, h} ⊂ C({Fk }) with N ⊂ E \ ∪k Fk . Then δk := infx∈Fk h(x) is positive for every k ≥ 1.

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By Lemma 2.4.1, u − uFkc and v − vFkc are E1 -convergent to u and v, respectively, as k → ∞. Let uk := (u − uFkc )+ ∧ u and vk := (v − vFkc )+ ∧ v. Then it follows from Lemma 1.1.11(ii) that uk and vk are E1 -convergent to u and v, respectively. For every ε > 0, it is clear that supp[(uk − εh)+ ] ⊂ Fk ∩ {u ≥ εδk } and supp[(vk − εh)+ ] ⊂ Fk ∩ {v ≥ εδk }. Since uv = 0 m-a.e., u(x)v(x) = 0 for every x ∈ Fk in view of Theorem 1.3.7. It follows that (uk − εh)+ and (vk − εh)+ have disjoint compact support. Thus by (i), C((uk − εh)+ , (vk − εh)+ ) = 0

for every k ≥ 1 and ε > 0.

Letting ε → 0 and then k → ∞, we obtain C(u, v) = 0.

T HEOREM 2.4.3. Let (E, F) be a regular Dirichlet form on L2 (E; m) where E is a locally compact separable metric space and m is a positive Radon measure on E with supp[m] = E. Let (C, F) be a symmetric form with 0 ≤ C(u, u) ≤ E(u, u)

for every u ∈ F.

Then the following are equivalent: (i) C(u, v) = 0 whenever u, v ∈ F have compact support with v being constant in an open neighborhood of supp[u]. (ii) For u, v ∈ F with v being constant in an open neighborhood of supp[u], C(u, v) = 0. (iii) For u, v ∈ F with u(v − c) = 0 m-a.e. on E for some constant c ∈ R, C(u, v) = 0. Proof. Clearly the implications (iii) ⇒ (ii) ⇒ (i) are valid so that it suffices to show the implication (i) ⇒ (iii). When c = 0, this is already shown in the preceding theorem. Let u, v ∈ F with u(v − c) = 0 m-a.e. on E for c = 0. Without loss of generality, assume c = 1, u is bounded and u ≥ 0 (otherwise consider u = u+ ∧ n and u− ∧ n, respectively). Since (E, F) is regular, there is a sequence {uk , k ≥ 1} ⊂ F ∩ Cc (E) such that uk is E1 -convergent to u. Fix k ≥ 1. Let h ∈ F+ ∩ Cc (E) so that h = 1 on an open neighborhood of supp[uk ]. By (i), C(uk ∧ u, h) = 0. C(u+ k ∧ u, h) = 0. On the other hand, + + |(u+ k ∧ u)(v − h)| ≤ |(uk ∧ u)(v − 1)| + |(uk ∧ u)(1 − h)| = 0 m-a.e. on E.


91

So by the proof of Theorem 2.4.2, we have C(u+ k ∧ u, v − h) = 0 and hence + C(u+ k ∧ u, v) = 0 for every k ≥ 1. As uk ∧ u is E1 -convergent to u, after taking k → ∞ we have C(u, v) = 0. Remark 2.4.4. We may define a symmetric form (E, D(E)) to be strongly local if E(u, v) = 0 whenever u, v ∈ D(E) with u(v − c) = 0 m-a.e. on E for some constant c. Note that this definition is invariant under quasi-homeomorphism. So under this definition, Beurling-Deny decomposition (see Theorem 4.3.3 in Chapter 4 for its statement) carries over to quasi-regular Dirichlet forms by using the fact that it is quasi-homeomorphic to a regular Dirichlet form on a locally compact separable metric space. See Remark 4.3.5(iii) below.

Chapter Three SYMMETRIC HUNT PROCESSES AND REGULAR DIRICHLET FORMS

As is clearly embodied by Theorem 1.4.3, three theorems of Section 1.5, and Theorem 3.1.13 below, the study of general symmetric Markov processes can be essentially reduced to the study of a symmetric Hunt process associated with a regular Dirichlet form. So without loss of generality, we assume throughout this chapter, except for the last parts of Sections 3.1 and 3.5, that E is a locally compact separable metric space, m is a positive Radon measure on E with supp[m] = E, and X = (Xt , Px ) is an m-symmetric Hunt process on (E, B(E)) whose Dirichlet form (E, F) is regular on L2 (E; m). (By redefining the process X at a properly exceptional set if needed, we may assume that X starts from every point in E.) We shall adopt without any specific notices those potential theoretic terminologies and notations that are formulated in Section 2.3 for the regular Dirichlet form (E, F). Throughout this chapter, we adopt the convention that any numerical function on E is extended to the one-point compactification E∂ = E ∪ {∂} by setting its value at ∂ to be zero.

3.1. RELATIONS BETWEEN PROBABILISTIC AND ANALYTIC CONCEPTS Denote by O0 the family of all open subsets of E with finite capacity Cap1 with respect to the Dirichlet form (E, F). For A ∈ O0 , let eA be its 1-equilibrium potential and p1A the 1-order hitting probability of A for the Hunt process X on E defined by p1A (x) := Ex e−σA , x ∈ E. L EMMA 3.1.1. eA = p1A [m] for A ∈ O0 . Proof. According to Lemma 1.1.14, the L2 -semigroup {Tt ; t > 0} generated by (E, F) is related to the transition function {Pt , t ≥ 0} of X by Tt f = Pt f [m] for every t > 0 and f ∈ B+ (E) ∩ L2 (E; m). eA is 1-excessive relative to {Tt } in

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view of Theorem 1.2.5, while Lemma A.2.4 says that p1A is 1-excessive relative to {Pt , t ≥ 0}, and so it is relative to {Tt }. Further, p1A (x) = 1 for every x ∈ A. Therefore, it suffices to show the inequality p1A ≤ eA

[m],

(3.1.1)

because Lemma 1.2.3 then implies that p1A ∈ F and E1 (p1A , p1A ) ≤ E1 (eA , eA ), and the variational characterization Theorem 1.2.5(i) of eA yields the desired identity. eA (x) = 1 for To prove (3.1.1), we take a Borel modification eA of eA with eA (Xt (ω)), t ≥ 0, ω ∈ . Then, for any nonevery x ∈ A and let Yt (ω) = e−t negative Borel measurable function h with E hdm = 1, the stochastic process (Yt , Ft0 , Ph·m )t≥0 is a supermartingale. In fact, by the Markov property of X, we have for any 0 ≤ s < t, Eh·m Yt F s0 = e−s e−(t−s) Pt−s eA (Xs ) ≤ e−s eA (Xs ), Ph·m -a.s. The last inequality follows from e−(t−s) Pt−s eA ≤ eA [m]. For a finite set ⊂ (0, ∞), denote the minimum and maximum of by a, b, respectively, and define σ (; A) = min{t ∈ : Xt ∈ A}

with convention min ∅ := b.

By the optional sampling theorem for the supermartingale, Eh·m e−σ (;A) ; σ (; A) < b ≤ Eh·m Yσ (;A) ≤ Eh·m [Ya ] ≤ (h, eA ). Letting ↑ Q+ ∩ (0, b) and then b ↑ ∞, we get (h, p1A ) ≤ (h, eA ), which yields (3.1.1). L EMMA 3.1.2. The following two conditions are equivalent for a decreasing sequence {An } of open sets of finite capacity: lim Cap1 (An ) = 0,

n→∞

lim p1 (x) n→∞ An

= 0,

m-a.e. x ∈ E.

(3.1.2) (3.1.3)

Proof. By the preceding lemma, Cap1 (An ) = p1An 2E1 ≥ p1An 22 . Moreover, Theorem 1.2.9(iv) says that {p1An } is E1 -Cauchy. Hence we readily see the equivalence of the two stated conditions. Notice that the m-symmetry of X implies that the measure m is excessive with respect to {Pt ; t ≥ 0} in the sense of Section A.2.2. We say a subset N ⊂ E is m-polar if there is a nearly Borel set N1 ⊃ N such that Pm (σN1 < ∞) = 0, where σ N := inf{t > 0 : Xt ∈ N1 }.

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T HEOREM 3.1.3. A subset of E is E-polar if and only if it is m-polar for X. Proof. If N ⊂ E is E-polar, then there exists a decreasing sequence {An } of open sets containing N satisfying (3.1.2). The set B = ∩∞ n=1 An then contains N and satisfies p1B (x) = 0 [m] by the preceding lemma. Hence N is m-polar. Here we make a general remark that, for a compact set K ⊂ E and a sequence {Dn } of relatively compact open sets such that Dn ⊃ Dn+1 , n ≥ 1,

∞

Dn = K,

(3.1.4)

n=1

we have lim σDn = σ˙ K Px -a.s. for any x ∈ E n→∞

and

σ˙ K = σK Pm -a.e.,

(3.1.5)

where σ˙ K is the entrance time of K defined by (A.1.20). Indeed, if σ = limn→∞ σDn = limn→∞ σ˙ Dn , then clearly σ ≤ σK . On the other hand, by the quasi-left-continuity (A.1.31) of the Hunt process X, we have Xσ = limn→∞ XσDn ∈ ∩n Dn = K, almost surely on {σ < ∞}, and consequently σ˙ K = σ . If x ∈ / K \ K r , then σ˙ K = σK , Px -a.s. But, by Lemma A.2.18, the set K \ K r is semipolar and thus of potential zero. Hence it is m-negligible by Theorem A.2.13 and we get the second conclusion in (3.1.5). Now, if K is a compact m-polar set, then, for a sequence {Dn , n ≥ 1} of open sets as above, we have limn→∞ p1Dn = p1K = 0 [m] by (3.1.5). Since Cap1 (Dn ) < ∞, we can conclude from Lemma 3.1.2 that Cap1 (K) ≤ lim Cap1 (Dn ) = 0. n→∞

If N is an arbitrary m-polar set, then there exists an m-polar Borel set containing N by Theorem A.2.13. Observe that every compact subset of is m-polar, and hence is E-polar. On account of (1.2.6), N as well as N E-polar.

N N is

Based on Theorem 3.1.3, we can and we will use the term quasi everywhere (q.e. in abbreviation) for the two equivalent meanings: “except for an m-polar set” or “except for an E-polar set”. We call an increasing sequence {Fk } of closed sets a strong nest if Px lim σE\Fk < ∞ = 0 for q.e. x ∈ E. (3.1.6) k→∞

By virtue of the second statement of the next theorem, any strong nest is indeed a nest. T HEOREM 3.1.4. Let {Fk } be an increasing sequence of closed subsets of E. (i) {Fk } is a Cap1 -nest if and only if Cap1 (E \ Fn ) < ∞ for some n and {Fk } is a strong nest.

95


(ii) {Fk } is a nest if and only if Px lim σE\Fk < ζ = 0 k→∞

for q.e. x ∈ E.

(3.1.7)

Proof. (i) Denote the complement of Fk by Ak and let p(x) := lim p1Ak (x) k→∞

for x ∈ E.

By Lemma 3.1.2, the condition that {Fk } is a Cap1 -nest is, under the finiteness of Cap1 (An ) for some n, equivalent to p = 0 [m], which is also equivalent to the validity of (3.1.6) for m-a.e. x ∈ E. Hence it suffices to derive p = 0 q.e. from p = 0 m-a.e. Since {Ak } are open sets, p1Ak (x) are Borel measurable functions by Theorem A.1.9 and so is p(x). For any ε > 0, consider a compact subset K of the Borel set {x ∈ E : p(x) ≥ ε} and let H1K be the 1-order hitting distribution of K defined by (A.2.2). From Lemma A.2.4, H1K p1Ak ≤ p1Ak , k ≥ 1, and we get H1K p ≤ p by letting k → ∞. Hence εp1K (x) ≤ p(x) for x ∈ E, which means that K is m-polar if p = 0 m-a.e. Therefore, Theorem 3.1.3 and (1.2.6) imply that the set {p ≥ ε} as well as the set {p > 0} is E-polar. (ii) On account of Theorem 1.3.14 and Lemma 3.1.2, {Fk } is a nest if and only if, for any relatively compact open set D, the decreasing sequence {Ak = D \ Fk , k ≥ 1} of open sets satisfies (3.1.3), which, in view of the proof of (i), is also equivalent to the validity of (3.1.8) Px lim σD\Fk < ∞ = 0 for q.e. x ∈ E. k→∞

It suffices to show the equivalence of condition (3.1.7) and property (3.1.8) holding for any relatively compact open set D. Assume (3.1.7) holds. We fix an arbitrary x ∈ E for which (3.1.7) is true. Suppose (3.1.8) fails to be true for this fixed x and for some relatively compact open set D, namely, by setting σ = limk→∞ σD\Fk , Px (σ < ∞) = δ, we assume δ > 0. On account of the quasi-left-continuity (A.1.31) of the Hunt process X, we then have Px (Xσ ∈ D, σ < ∞) = δ, and consequently Px (limk→∞ σE\Fk ≤ σ < ζ ) ≥ δ, arriving at a contradiction. Conversely, (3.1.7) can be derived from (3.1.8) by taking a sequence {D } of relatively compact open sets increasing to E and by using the inequality σE\Fk ≥ σD \Fk ∧ σE\D . A nearly Borel set A ⊂ E is called X-invariant for the Hunt process X if σE\A < ∞) = 0 Px (σE\A ∧

for every x ∈ A,

where σE\A := inf{t > 0 : Xt− ∈ E \ A} with the convention that inf ∅ = ∞. According to Lemma A.1.27 and Remark A.1.30, the restriction XA of X to its invariant set A is a Hunt process again.

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Parallel to Definition A.2.12 of an m-inessential set for a Borel right process, we say that a set N ⊂ E is properly exceptional for the Hunt process X if N is m-negligible, nearly Borel and E \ N is X-invariant. Clearly a properly exceptional set for X is m-polar. On account of Lemma A.1.32, the proper exceptionality of a set with respect to a Hunt process X is actually a synonym for its m-inessentiality with respect to X regarded as a Borel right process. Therefore, the next theorem is a special case of Theorem A.2.15 formulated for a Borel right process. Yet we give an alternative direct proof of it for the present symmetric Hunt process. T HEOREM 3.1.5. Any m-polar set N is contained in a Borel properly exceptional set for X. Proof. If N is m-polar, then Cap1 (N) = 0 by Theorem 3.1.3 and hence there exists a Cap1 -nest {Fk } such that N is contained in the set B0 = ∩∞ k=1 (E \ Fk ), which is a Gδ -set of zero capacity. Since σE\Fk ≤ σB0 , k ≥ 1, and also σE\Fk = σB0 for k ≥ 1, it follows from Theorem 3.1.4(i) and the property σE\Fk ≤ σB0 < ∞) = 0 q.e. Therefore, we can find a strong nest (3.1.6) that Px ( σB0 ∧ {Fk } such that B1 = ∩∞ k=1 (E \ Fk ) contains B0 and the above identity holds for any x ∈ E \ B1 . Repeating this procedure, we can construct an increasing sequence Bn of Gδ -sets of zero capacity satisfying σBn < ∞) = 0 Px (σBn ∧

for every x ∈ E \ Bn+1 .

B = ∪∞ n=1 Bn is then a Borel properly exceptional set containing N.

D EFINITION 3.1.6. A numerical function f defined q.e. on E is called finely continuous q.e. if there exists a properly exceptional set N such that f E\N is nearly Borel measurable and finely continuous with respect to the restricted Hunt process X E\N . For simplicity, we phrase this property as f is nearly Borel and finely continuous on E \ N for a properly exceptional set N. We note that if f has this property, then so does it for any properly exceptional set containing N on account of Exercise A.1.31. T HEOREM 3.1.7. (i) If a function f on E is quasi continuous, then f is finely continuous q.e. and finite q.e. Furthermore, there exists a Borel properly exceptional set N such that f is a finite Borel measurable function on E \ N and, for any x ∈ E \ N, f (Xt ) = f (Xt ) for every t ≥ 0 = 1, (3.1.9) Px lim t ↓t


Px lim f (X ) = f (X ) for every t ∈ (0, ζ ) = 1, t t− t ↑t

97 (3.1.10)

Px (lim f (Xt ) = f (Xζ − ), ζ < ∞, Xζ − ∈ E) t ↑ζ

= Px (ζ < ∞, Xζ − ∈ E).

(3.1.11)

If in addition f ∈ F , then the time interval (0, ζ ) in (3.1.10) can be strengthened to (0, ∞). (ii) If f is finely continuous q.e. and f ∈ F, then f is quasi continuous. Proof. (i). If f is quasi continuous, then by Theorem 1.3.14(iii) there is a Cap1 -nest {Fk } such that f ∈ C({Fk }). Denote by N0 the m-polar set in (3.1.6) for {Fk }. We can then find a Borel properly exceptional set N containing N0 ∪ (∩∞ k=1 (E \ Fk )) by Theorem 3.1.5. Clearly f is finite and Borel measurable on E \ N. For x ∈ E \ N, limk→∞ σE\Fk = ∞ Px -a.s., and thus (3.1.9), (3.1.10), and (3.1.11) are valid. Applying Theorem A.2.9 to the Hunt process X E\N , we conclude from (3.1.9) that f is finely continuous on E \ N. If f is quasi continuous and f ∈ F , then f equals q.e. to a quasi continuous function in the restricted sense relative to a Cap1 -nest in view of Lemma 1.3.15. This together with Theorem 3.1.4(i) yields the last assertion of (i). (ii). Suppose f is finely continuous q.e. and f ∈ F. By Lemma 1.3.15, f admits a quasi continuous m-version f . By (i), f is finely continuous q.e. and f = f [m]. On account of Theorem A.2.15, there exists a Borel m-inessential set N for X as a right process such that both f and f are finely continuous with respect f = f q.e. by Theorem A.2.13(iv). Since f to the right process XE\N . Thus equals the quasi continuous function f except for an E-polar set, f itself is quasi continuous. The condition “f ∈ F” in statement (ii) of the above theorem will be removed in Theorem 3.3.3. Remark 3.1.8. For a right process X having m as an excessive measure, it follows from Theorem A.2.9, Theorem A.2.13, Lemma A.2.14, and Theorem A.2.15 that a function f is finite q.e. and finely continuous q.e. if and only if there exists an m-inessential set N for the right process X such that f is a finite Borel measurable function on E \ N and f satisfies (3.1.9) for every x ∈ E \ N. For the present m-symmetric Hunt process X, f also satisfies (3.1.10), a reflection of the symmetry of X. This property is not necessarily true for non-symmetric right processes. For instance, the deterministic uniform motion X to the right on R is a right process possessing the Lebesgue measure m as an excessive measure. In this case, a real function f on R is finely continuous q.e. with respect to X if and only if f is right continuous as a real function and (3.1.10) is not satisfied unless f is left continuous.

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If two numerical functions f , g on E satisfy f = g [m], then f is said to be a version or an m-version of g. Let us consider the resolvent kernel {Rα ; α > 0} (resp. {Gα ; α > 0}) of {Pt } (resp. {Tt ; t ≥ 0}) defined by (1.1.27) (resp. (1.1.1)). According to (1.1.28), Rα f is a version of Gα f for f ∈ bB(E) ∩ L2 (E; m). Hence, for a non-negative universally measurable function f ∈ L2 (E; m), Rα f is a version of Gα f ∈ F. Since Rα f is α-excessive for X and finely continuous by Theorem A.2.9, Rα f is a quasi continuous version of Gα f by virtue of Theorem 3.1.7(ii), yielding (ii) of the following proposition. P ROPOSITION 3.1.9. Let f be a non-negative universally measurable function on E belonging to L2 (E; m). (i) For each t > 0, Pt f is a quasi continuous version of Tt f . (ii) For each α > 0, Rα f is a quasi continuous version of Gα f . Proof. For a non-negative universally measurable function f , there are f1 , f2 ∈ B+ (E) so that 0 ≤ f1 ≤ f ≤ f2 and m({f2 > f1 }) = 0. Thus Pt f1 ≤ Pt f ≤ Pt f2 with m({Pt f2 > Pt f1 }) = 0. If Pt fi , i = 1, 2, are quasi continuous, then so is Pt f by Lemma 1.3.6 and Theorem 1.3.7. Hence it suffices to prove (i) for every Borel measurable function f . Denote by G the family of all non-negative Borel measurable functions f ∈ L2 (E; m) for which Pt f is quasi continuous. G contains the family Cc+ (E) of all non-negative continuous functions with compact support. In fact, if f ∈ Cc+ (E), then for x ∈ E, αRα (Pt f )(x) = Pt (αRα f )(x) → Pt f (x)

as α → ∞.

Since this convergence also takes place strongly in F by virtue of Section A.4(v) and Rα (Pt f ) ∈ G by (ii), we get f ∈ G in view of Theorem 1.3.3. If {fn , n ≥ 1} is a sequence of functions in G increasing pointwise to f ∈ L2 (E; m), then Pt fn is E1 -convergent to Pt f in the space G by Section A.4(v) again, and thus f ∈ G. We can now conclude that G = B+ (E) ∩ L2 (E; m), because, for any relatively compact open set D ⊂ E, the family C = {A ⊂ D : A ∈ B(E), 1A ∈ G} becomes a Dynkin class containing all open subsets of D and so C = B(D) by virtue of Proposition A.1.2. We present three important implications of Proposition 3.1.9. T HEOREM 3.1.10. Any semipolar set of X is E-polar. Proof. For any open set D ∈ O0 , the 1-excessive function p1D is an m-version of the 1-equilibrium potential eD ∈ F in view of Lemma 3.1.1. By the preceding proposition, e−t Pt p1D is quasi continuous for each t > 0. As t ↓ 0, it increases to p1D pointwisely and it is also E1 -convergent to eD by Section A.4(v). Hence p1D is a quasi continuous version of eD by Theorem 1.3.3.


99

According to Definition A.2.6, a semipolar set is a set contained in a countable union of thin sets. Hence it is enough to show that any Borel measurable thin set is E-polar. In view of (1.2.6), we may show this for a compact thin set K so that K r = ∅. Choose for K relatively compact open sets {Dn } satisfying (3.1.4). By (3.1.5), p1Dn (x) decreases to p˙ 1K (x) := Ex [e−σ˙K ] as n → ∞ for each x ∈ E. Since E1 (p1Dn , p1Dn ) = Cap1 (Dn ), p1Dn is E1 -convergent as n → ∞ by Theorem 1.2.9. Therefore, in view of Theorem 1.3.3, p˙ 1K is a quasi continuous function in F. Since e−t Pt p˙ 1K (x) increases to p1K (x) as t ↓ 0 for each x ∈ E, we see as above that p1K is also a quasi continuous version of p˙ 1K . Consequently, E-polar. p˙ 1K = p1K < 1 E-q.e., yielding that K is E-polar. Theorem 3.1.10 is not necessarily valid for non-symmetric right processes. For instance, when X is the deterministic uniform motion to the right on R, each one-point set K = {x}, x ∈ R, has no regular point for X and hence K is semipolar, while K is neither polar nor m-polar (m is the Lebesgue measure) because it can be hit by the path starting from the left of x. Recall the absolute continuity conditions (AC) and (AC) for the transition function {Pt ; t ≥ 0} and the resolvent kernel {Rα ; α > 0} of X introduced in Definition A.2.16. P ROPOSITION 3.1.11. Conditions (AC) and (AC) for X are equivalent. Proof. Obviously (AC) implies (AC) . Conversely, assume condition (AC) holds. Then any m-polar set is polar by Theorem A.2.17. If A ∈ B(E) satisfies m(A) = 0, then Pt (·, A) = 0 m-a.e. by the symmetry of Pt . Since Pt (x, A) is quasi continuous in x by Proposition 3.1.9, we have Pt (·, A) = 0 q.e. by Lemma 1.3.6 and Theorem 1.3.7. Hence there exists a Borel polar set N such that Pt (x, A) = 0 for every x ∈ E \ N. Therefore, for any x ∈ E, we get / N] = 0, yielding (AC). P2t (x, A) = Ex [Pt (Xt , A); Xt ∈ Proposition 3.1.11 also reflects the symmetry of X. The deterministic uniform motion to the right on R does not satisfy (AC) but it does satisfy (AC) . As an application of Proposition 3.1.9 and Theorem 3.1.5, we formulate a uniqueness statement. T HEOREM 3.1.12. Let (E, F) be a regular Dirichlet form on L2 (E; m). Let X (1) and X (2) be m-symmetric Hunt processes on E associated with a common Dirichlet form (E, F). There exists then a Borel measurable properly exceptional set N common for X (1) and X (2) such that the transition functions of X (1) and X (2) are identical on E \ N.

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Proof. Denote the transition function of X (i) by {P(i) t ; t ≥ 0}. Let C1 (E) be a countable uniformly dense subfamily of Cc (E). On account of Proposition 3.1.9, Lemma 1.3.1, Lemma 1.3.6, and Theorem 1.3.7, there exists a Borel measurable E-polar set B0 such that (2) P(1) t f (x) = Pt f (x)

for x ∈ E \ B0 , t ∈ Q+ and f ∈ C1 (E).

By the right continuity of the sample paths of X all t ≥ 0. Accordingly, (2) P(1) t (x, B) = Pt (x, B)

(1)

(3.1.12)

(2)

and X , (3.1.12) holds for

for t ≥ 0, x ∈ E \ B0 and B ∈ B(E).

By Theorem 3.1.5, we can choose a properly exceptional Borel set N of X (2) that contains B0 . Then P(1) t (x, N) = 0 for every x ∈ E \ N, and the Markov processes X (1) E\N and X (2) E\N have the same distribution. In particular, N is also a properly exceptional set for X (1) . We call two m-symmetric Hunt processes on E equivalent if their transition functions are the same outside some common properly exceptional set. Theorem 3.1.12 says that an m-symmetric Hunt process on E associated with a regular Dirichlet form on L2 (E; m) is unique up to the equivalence. The relationship between an m-symmetric Hunt process and an associated regular Dirichlet form studied so far in this section can be generalized to that between an m-symmetric right process and an associated quasi-regular Dirichlet form in the following fashion. Let E be a Radon space, m be a σ -finite measure on E, and (E, F) be a quasi-regular Dirichlet form on L2 (E; m) in the sense of Definition 1.3.8. Let N0 ∈ B∗ (E) be an E-polar set and X be an m-symmetric right process on E \ N0 whose Dirichlet form on L2 (E; m) is equal to (E, F). Denote by {Pt ; t ≥ 0} and {Tt ; t > 0} the transition function of X and the corresponding semigroup on L2 (E; m). The process X is said to be properly associated with the Dirichlet form (E, F) if Pt f for any t > 0, f ∈ B∗ (E) ∩ L2 (E; m), is an E-quasi-continuous m-version of Tt f . In view of the proof of Proposition 3.1.9, this condition is equivalent to the condition that the resolvent kernel Rα f of X is an E-quasi-continuous m-version of Gα f for any α > 0 and f ∈ B∗ (E) ∩ L2 (E; m). Proposition 3.1.9 says that an m-symmetric Hunt process associated with a regular Dirichlet form is automatically properly associated with it. Hence X ) constructed for a quasi-regular the special Borel standard process j−1 ( Dirichlet form (E, F) in Theorem 1.5.2 is also properly associated with it. The next theorem gives the uniqueness of an m-symmetric right process properly associated with a quasi-regular Dirichlet form, which can be proved in a way similar to the proof of Theorem 3.1.12. We shall use the term a properly exceptional set for a Borel standard process X = (Xt , Px , ζ ) in a slightly weaker sense than for a Hunt process. A nearly

101


Borel measurable set A is said to be X-invariant if for every x ∈ A Px (Xt ∈ A and Xt− ∈ A for every t ∈ [0, ζ )) = 1. A nearly Borel set N is called properly exceptional for X if m(N) = 0 and E \ N is X-invariant. T HEOREM 3.1.13. Suppose that X is an m-symmetric right process properly X := associated with a quasi-regular Dirichlet form (E, F) on L2 (E; m). Let X ) be the special Borel standard process constructed in Theorem 1.5.2 j−1 ( F) that is quasifrom a Hunt process X associated with a regular Dirichlet (E, homeomorphic to (E, F) through a quasi-homeomorphism j. There exists then a Borel properly exceptional set N for X such that the transition functions of X and X are identical on [0, ∞) × (E \ N) × B(E). Consequently, X E\N and X E\N have the same distribution. This in particular implies that X E\N is a special Borel standard process on E \ N. The process X is properly E\N

associated with (E, F) and it enjoys those properties in Theorem 3.1.3, Theorem 3.1.4, and Theorem 3.1.5. This theorem combined with Theorem 1.5.3 yields the following: C OROLLARY 3.1.14. Let X be an m-tight m-symmetric right process X on a Radon space E or an m-symmetric right process X on a Lusinspace E. (i) There exists an m-inessential Borel set N ⊂ E such that X E\N is a special Borel standard process. (ii) Conditions (AC) and (AC) for X are equivalent. We close this subsection by presenting some useful results relating a general symmetric right process to the associated Dirichlet form without assuming the quasi-regularity of the latter. Let (E, B∗ (E)) be a Radon space and m be a σ -finite measure on it. Let X = (Xt , ζ , Px ) be a right process on (E, B∗ (E)) with an m-symmetric transition function {Pt ; t ≥ 0}. We denote the associated Dirichlet form on L2 (E; m) by (E, F). The resolvent kernel of X is denoted by {Rα ; α > 0}. L EMMA 3.1.15. Let F be a nearly Borel finely closed subset of E and α > 0. Put C = E \ F and

σC F −αt e f (Xt )dt , x ∈ E, f ∈ B(E). Rα f (x) = Ex 0

Then

RFα (bB(E) ∩ L2 (E; m)) ⊂ F , Eα (RFα f , RFα g) = (RFα f , g),

f , g ∈ bB(E) ∩ L2 (E; m).

(3.1.13)

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Proof. We put for α > 0, γ > 0

γC Rα f (x) = Ex

∞

−αt−γ

e

t 0

0

Since C is finely open, σC = inf{t > 0 : γC limγ →∞ Rα f (x). Therefore, we have

1C (Xs )ds

t 0

f (Xt )dt .

1C (Xs )ds > 0} and RFα f (x) =

E(RFα f , RFα f ) = lim lim β(Rγα C f − βRβ Rγα C f , Rγα C f ) β→∞ γ →∞

(3.1.14)

in the sense that, if the right hand side is finite, then RFα f ∈ F and the limit coincides with the left hand side. We compute the right hand side by using a generalized resolvent equation Rβ f − Rγα C f + (β − α)Rβ Rγα C f − γ Rβ (1C · Rγα C f ) = 0.

(3.1.15)

This equation for β = α implies Rγα C f ∈ F, Eα (Rγα C f , v) + γ (Rγα C f , v)1C ·m = (f , v). γC

In particular, Rα is m-symmetric. Keeping this in mind, we can rewrite the right hand side of (3.1.14) as lim lim β(Rβ f − αRβ Rγα C f − γ Rβ (1C · Rγα C f ), RFα f )

β→∞ γ →∞

= (f , RFα f ) − α(RFα f , RFα f ) − lim lim γ (f , Rγα C (1C · βRβ RFα f )). β→∞ γ →∞

s If we put τt = inf s > 0 : 0 1C (Xv )dv > t , then we have from Lemma A.3.7

∞ γC −ατt −γ t γ Rα (1C · g)(x) = γ Ex e g(Xτt )dt , 0

−ατ0

g(Xτ0 )] as γ → ∞ provided that g is bounded and which converges to Ex [e finely continuous. Since Xτ0 = XσC ∈ Cr by (A.2.4), and RFα f (x) = 0, x ∈ Cr , we arrive at the desired identity lim lim γ (f , Rγα C (1C · βRβ RFα f )) = Ef ·m e−ατ0 RFα f (Xτ0 ) = 0. β→∞ γ →∞

Exercise 3.1.16. Derive equation (3.1.15) from (4.1.7) of chapter 4. Prove further that (3.1.13) holds for any open set D in place of the finely closed set F when X is an m-symmetric Hunt process on a locally compact separable metric space E where m is a positive Radon measure on E with full support.


103

L EMMA 3.1.17. Let {Fn } be an increasing sequence of closed subsets of E. If Pm ( lim σE\Fn < ζ ) = 0, n→∞

(3.1.16)

then {Fn } is an E-nest. Proof. Take any increasing sequence {Fk } of closed sets satisfying (3.1.16). Then for f ∈ bB(E) ∩ L2 (E; m) the function RF1 k f (x) increases to R1 f (x) as k → ∞ for m-a.e. x ∈ E. Since (3.1.13) implies R1 f − RF1 k f 2E1 = (R1 f − RF1 k f , f ), RF1 k f ∈ FFk is E1 -convergent to R1 f . {R1 f } being E1 -dense in F, ∪∞ k=1 FFk is E1 -dense in F, namely, {Fk } is an E-nest.

3.2. HITTING DISTRIBUTIONS AND PROJECTIONS I From this section until the first half of Section 3.5, we shall again work with a fixed m-symmetric Hunt process X and an associated regular Dirichlet form (E, F) on L2 (E; m) as is formulated in the beginning of Chapter 3. For α > 0, the α-order hitting distribution HαB of X for a nearly Borel measurable set B is defined as HαB g(x) := Ex e−ασB g(XσB ) . According to Theorem A.1.22, we have, for any non-negative universally measurable function f , the identity

∞ α −αt HB (Rα f )(x) = Ex e f (Xt )dt , x ∈ E. (3.2.1) σB

g for Our aim in this section is to identify in Theorem 3.2.2 the function HαB g ∈ F with the projection of g on the orthogonal complement HBα of the closed subspace FE\B of (F, Eα ). For a nearly Borel measurable set B ⊂ E, recall from (2.3.14) the space FE\B = {f ∈ F : f =0

q.e. on B}.

(3.2.2)

FE\B is a closed subspace of the Hilbert space (F , Eα ) for every α > 0. Its orthogonal complement is denoted by HBα and the projection on HBα by PHBα . L EMMA 3.2.1. If f is α-excessive with respect to X and f ∈ F , then PHBα f = HαB f . Proof. If f ∈ F is α-excessive with respect to X, then so it is with respect to {Tt } and, by (2.3.15), PHBα f coincides with the α-reduced function fB of f

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relative to B. We need to prove HαB f = fB but it suffices to show the inequality HαB f ≤ fB

[m].

(3.2.3)

Indeed, since f ∈ F and f is finely continuous, f is quasi continuous by Theorem 3.1.7(ii). Hence the space defined by (2.3.11) can be rewritten g ≥ f q.e. on B} and fB is the unique element of Lf ,B as Lf ,B = {g ∈ F : minimizing the norm gEα . Since HαB f is α-excessive, (3.2.3) combined with Lemma 1.2.3 implies HαB f ∈ F, HαB f Eα ≤ fB Eα . Furthermore, HαB f = f q.e. on B because B \ Br is m-polar by virtue of Lemma A.2.18, Theorem 3.1.3, and Theorem 3.1.10. Hence HαB f ∈ Lf ,B and the inequality (3.2.3) turns out to be an equality. The proof of the inequality (3.2.3) is essentially analogous to that of Lemma 3.1.1. We fix a non-negative Borel quasi continuous version fB of fB . There exists then a Borel measurable properly exceptional set N such that fB (x) ≤ fB (x) e−αt Pt

for every t > 0 and x ∈ E \ N,

fB (x) = f (x) Px

for every x ∈ B \ N,

lim fB (Xs ) = fB (Xt ) for every t ≥ 0 = 1 s↓t

for x ∈ E \ N.

(3.2.4) (3.2.5) (3.2.6)

This can be verified in the following way. Since fB is α-excessive relative to fB is quasi continuous by Proposition 3.1.9, the inequality in (3.2.4) {Tt } and Pt holds q.e. for each t > 0 in view of Lemma 1.3.6 and Theorem 1.3.7. Further, taking (2.3.13) and Theorem 3.1.7 into account, we can use Theorem 3.1.5 to find a Borel measurable properly exceptional set N such that (3.2.5) and (3.2.6) are satisfied and (3.2.4) is valid for all t ∈ Q+ . By (3.2.6) and Fatou’s lemma, (3.2.4) is then valid for all t > 0. We fix x ∈ E \ N. Owing to (3.2.4) and (3.2.6), the stochastic process fB (Xt ), Ft , Px )t≥0 is a non-negative right continuous supermartingale and (e−αt consequently, by the optional sampling theorem, we have fB (x) = Ex e−ασB fB (XσB ) ≤ fB (x). HαB On the other hand, we can see from (3.2.5), (3.2.6), and the right continuity of f fB (x) = HαB f (x), arriving at the desired inequality along the sample path that HαB (3.2.3). T HEOREM 3.2.2. Let B ⊂ E be a nearly Borel measurable set. For any f is a quasi continuous version of PHBα f . f |(x) < ∞ q.e. and HαB f ∈ F, HαB | Proof. First we assume that f ∈ F is bounded and we fix a bounded Borel f is a difference measurable quasi continuous version f of f . For each β > 0, Rβ


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f ) is a of bounded α-excessive functions belonging to F and hence HαB (Rβ quasi continuous version of PHBα (Gβ f ) by virtue of the preceding lemma. Since βGβ f is E1 -convergent to f as β → ∞, so is βPHBα (Gβ f ) to PHBα f . On the other hand, we can find a properly exceptional set N such that f satisfies f )(x) → HαB f (x), β → ∞, and accordingly (3.2.6). If x ∈ E \ N, then βHαB (Rβ f is a quasi continuous version of PHBα f on account of we can conclude that HαB Theorem 1.3.3. For a general f ∈ F , functions f = ((− ) ∨ f ) ∧ ∈ F are E1 -convergent to f as → ∞ and so are the functions PHBα f to PHBα f . Hence we get the desired conclusion by repeating the same argument as above. We now state two important corollaries of Theorem 3.2.2. The first one extends Lemma 3.1.1 from open sets to Borel sets. The 1-equilibrium potential eB for a Borel set B with LB,1 = ∅ is determined in the paragraph preceding Theorem 2.3.1. p1B denotes the 1-order hitting probability of B for X defined by (A.2.3). C OROLLARY 3.2.3. For any Borel set B with LB,1 = ∅, p1B is a quasi continuous version of the 1-equilibrium potential eB . Proof. On account of Theorem 2.3.1(iii) and (2.3.15), we have eB = PHB1 eB . eB is a quasi continuous version of eB . Since By the above theorem, H1B eB (x) = 1 for q.e. x ∈ B, we conclude from Theorem 3.1.5 and Theorem 3.1.7 that, for q.e. x ∈ E, eB (x) = Ex e−σB eB (XσB ) = Ex e−σB = p1B (x). H1B For a nearly Borel measurable subset B of E, its complement E \ B will be denoted by D for simplicity. FD is the subspace of F defined by the right hand side of (3.2.2). (Sometimes we also denote it by F D .) For f ∈ B+ (E), we put

σB D −αt Rα f (x) = Ex e f (Xt )dt , x ∈ E. (3.2.7) 0

By (3.2.1), we then have α Rα f (x) = RD α f (x) + HB (Rα f )(x),

x ∈ E,

(3.2.8)

which still make sense for q.e. x ∈ E when f ∈ B(E) ∩ L2 (E; m). Further, we see from Lemma A.2.18 and Theorem 3.1.10 that RD α f ∈ FD in this case. Therefore, we obtain the following corollary immediately from Theorem 3.2.2. C OROLLARY 3.2.4. (i) For α > 0, f ∈ B(E) ∩ L2 (E; m), (3.2.8) represents the orthogonal decomposition of Rα f as a sum of elements of FD and HBα in

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the Hilbert space (F , Eα ). RD α f ∈ FD is quasi continuous and satisfies Eα (RD f (x)v(x)m(dx) for every v ∈ FD . (3.2.9) α f , v) = D

{RD α;α

(ii) > 0} is m-symmetric in the sense that, for any f , g ∈ B(E) ∩ L2 (E; m), f (x)RD g(x)m(dx) = RD (3.2.10) α α f (x)g(x)m(dx). D

D

3.3. QUASI PROPERTIES, FINE PROPERTIES, AND PART PROCESSES Corollary 3.2.4 enables us to study the quasi properties for the regular Dirichlet form (E, F) more thoroughly in relation to the fine topology for the Hunt process X. D EFINITION 3.3.1. (i) For Bi ⊂ E, i = 1, 2, we write as B1 ⊂ B2 q.e. if B1 \ B2 is E-polar. If B1 = B2 q.e., then we say that B1 , B2 are q.e. equivalent. (ii) A set D ⊂ E is called quasi open if there exists a nest {Fk } such that D ∩ Fk is open subset of Fk with respect to the relative topology for each k ≥ 1. The complement of a quasi open set is called quasi closed. (iii) A set D ⊂ E is called q.e. finely open if there exists a properly exceptional set N such that D \ N is a nearly Borel measurable finely open set of X E\N . For simplicity, we phrase the last property in the above definition as D \ N is nearly Borel finely open on E \ N for a properly exceptional set N. Note that if a set D has this property, then so does it for any properly exceptional set containing N on account of Exercise A.1.31. We further notice that, as in the proof of Theorem 1.3.14(iii), we can replace a nest {Fk } with a Cap1 -nest in the above definition of a quasi open set. Exercise 3.3.2. Prove the following: (i) A set q.e. equivalent to a quasi open set is again quasi open. (ii) A function f defined q.e. on E is quasi continuous if and only if f is finite q.e. and f −1 (I) is quasi open for any open set I ⊂ R. (iii) Let f be a quasi continuous function on E and D be a quasi open subset of E. Using Lemma 1.3.6, prove that if f ≥ 0 m-a.e. on D, then f ≥ 0 q.e. on D. The second assertion in the next theorem removes the condition “f ∈ F” from Theorem 3.1.7 (ii).


107

T HEOREM 3.3.3. (i) D ⊂ E is quasi open if and only if it is q.e. finely open. (ii) The following conditions are mutually equivalent for a numerical function f defined q.e. on E. (a) f is quasi continuous. (b) f is finite q.e. and finely continuous q.e. (c) There exists a Borel properly exceptional set N such that f is finite, Borel measurable and finely continuous on E \ N. (iii) B ⊂ E is quasi closed if and only if there exists a non-negative quasi continuous function g in F with B = g−1 ({0}) q.e. Proof. (i) Let D be a quasi open set and {Fk } be a nest such that D ∩ Fk is relatively open in Fk for each k ≥ 1. By Theorem 3.1.3, Theorem 3.1.4, and Theorem 3.1.5, we can find a Borel properly exceptional set N containing ∩k (E \ Fk ) such that (3.1.7) holds for any x ∈ E \ N. Then D \ N is Borel finely open on E \ N. Conversely, assume that D1 = D \ N is nearly Borel finely open on E \ N for some properly exceptional set N. We let B1 = E \ D1 and we consider 1 the function g = RD 1 f defined by (3.2.7) for α = 1, B = B1 and f ∈ bB(E) ∩ 2 L (E; m) strictly positive on E. Then {x ∈ E : g(x) > 0} = D1 ∪ (B1 \ Br1 ). Since g is quasi continuous by Corollary 3.2.4 and B1 \ Br1 is E-polar by Theorem 3.1.3 and Theorem 3.1.10, D1 and hence D are quasi open by Exercise 3.3.2. (ii) The implication (a) ⇒ (c) was shown in Theorem 3.1.7(i). (c) ⇒ (b) is trivial. If f is finite q.e. and finely continuous q.e., then, for any open set I ⊂ R, f −1 (I) is q.e. finely open and hence quasi open by (i). Hence f is quasi continuous by Exercise 3.3.2 (ii). (iii) The “if” part is obvious. If B is quasi closed, then, by (i), B1 = B \ N is nearly Borel finely closed on E \ N for some properly exceptional set N. Then the function g appearing in the proof of (i) has the desired property. The third statement of the above theorem will now be applied to obtain a useful characterization of the quasi support of a measure. D EFINITION 3.3.4. Let µ be a positive Borel measure on E charging no E-polar set. A set F ⊂ E is called the quasi support of µ if the following two conditions are satisfied: (a) F is quasi closed and µ(E \ F) = 0. (b) If F is another set with property (a), then F ⊂ F q.e. A quasi support of a positive Borel measure charging no E-polar set is unique up to q.e. equivalence. T HEOREM 3.3.5. For a positive Borel measure µ on E charging no E-polar set and a quasi closed set F ⊂ E, the following conditions are equivalent.

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(a) F is a quasi support of µ, (b) u = 0 µ-a.e. if and only if u = 0 q.e. on F for any quasi continuous function u ∈ F. (c) u = 0 µ-a.e. if and only if u = 0 q.e. on F for any quasi continuous function u on E. Proof. (b) ⇒ (a): We put

Nµ = u ∈ F :

| u|dµ = 0

E

and we assume that Nµ = FE\F where FE\F is the space defined by the right hand side of (3.2.2) for B = F. By Theorem 3.3.3(iii), there exists a non-negative quasi continuous function g ∈ F with F = g−1 ({0}) q.e. Then g ∈ FE\F and hence g ∈ Nµ , which means µ(E \ F) = 0. Consider another quasi closed set F1 with µ(E \ F1 ) = 0 and choose a corresponding function g1 for F1 by Theorem 3.3.3(iii). Then g1 ∈ Nµ and hence g1 ∈ FE\F , which implies that F ⊂ F1 q.e., yielding that F is a quasi support of µ. (a) ⇒ (c): Suppose F is a quasi support of µ. Then the “if” part of condition (b) is obviously satisfied for any Borel function u. If u is quasi continuous and u = 0 µ-a.e., then the set F1 = {u = 0} satisfies condition (a) in Definition 3.3.4 and consequently F ⊂ F1 q.e. and u = 0 q.e. on F. The implication (c) ⇒ (b) is trivial. C OROLLARY 3.3.6. If µ = f · m for a measurable function f strictly positive m-a.e., then µ has the full quasi support E. We shall prove in Theorem 5.2.1 that any smooth measure µ in the sense of Definition 2.3.13 admits its quasi support, which actually coincides q.e. with the support of a PCAF with Revuz measure µ. So much for the study of E-quasi notions with a specific use of the function RD α f in Corollary 3.2.4. We come back again to the general setting of Corollary 3.2.4 where B is nearly Borel measurable subset of E and D = E \ B and investigate its implications in some details. The space L2 (D; m) consisting of all m-square integrable real functions on D can be identified with the subspace {f ∈ L2 (E; m) : f = 0 m-a.e. on B} of L2 (E; m). FD can be viewed as a linear subspace of L2 (D; m) under this identification but it is not necessarily a dense subspace. However, D(E D ) = FD

and

E D (f , g) = E(f , g) for f , g ∈ FD

(3.3.1)

defines a bilinear form E on L (D; m) satisfying all other requirements as a Dirichlet form on L2 (D; m). E D is called the part of the Dirichlet form (E, F) on the set D. We will see that E D becomes a genuine Dirichlet form on L2 (D; m) if D is finely open. D

2


109

Let us now assume that D is a nearly Borel measurable and finely open with respect to the Hunt process X = (, M, Xt , ζ , Px ). We let for ω ∈

Xt (ω), 0 ≤ t < τD (ω) XtD (ω) := and ζ D (ω) := τD (ω). ∂, t ≥ τD / D} is the first time the process X exits D. Then Here τD := inf{t > 0 : Xt ∈ X D = (, M, XtD , ζ D , Px ) can be verified to be a standard process on the Radon space (D, B∗ (D)) in the sense of Definition A.1.39. Exercise 3.3.7. (i) Prove the last statement using Theorem A.1.37. (ii) Prove that X D is a Hunt process on (D, B(D)) if D is an open set. We call X D the part process of X on D. The transition function {PD t ; t ≥ 0} D ; α > 0} of X have the following expressions, and the resolvent kernel {RD α respectively: for f ∈ B+ (D), x ∈ D,

τD D −αt PD f (x) = E ); t < τ f (x) = E e f (X )dt . (3.3.2) (X , R [f ] x t D x t t α 0

≥ 0} is a transition function on (D, B∗ (D)) in the sense of In particular, Definition 1.1.13. If we denote the restriction of f ∈ B+ (E) to D by fD , then we have the D relation RD α f (x) = Rα fD (x) for x ∈ D. Thus (3.2.9) and (3.2.10) can be rewritten D as follows: for f ∈ B(D) ∩ L2 (D; m), RD α f ∈ D(E ) and EαD (RD f (x)v(x)m(dx) for v ∈ D(E D ). (3.3.3) α f , v) = {PD t ;t

D

Moreover,

D

f (x)RD α g(x)m(dx)

= D

RD α f (x)g(x)m(dx)

(3.3.4)

for every g ∈ B(D) ∩ L2 (D; m). The last identity means the m-symmetry of {RD α ; α > 0} on the set D. is also symmetric on D with respect to m. By Lemma 1.1.14, Consequently, PD t ; t ≥ 0} determines uniquely a strongly continuous the transition function {PD t contraction semigroup {TtD ; t > 0} of Markovian symmetric operators on L2 (D; m). Further, equation (3.3.3) confirms that E D is just the Dirichlet form on L2 (D; m) generated by {TtD ; t ≥ 0}. T HEOREM 3.3.8. Let D = E \ B be a nearly Borel measurable finely open set with respect to X, X D be the part process of X on D, and E D be the part of E on D. (i) If an increasing sequence {Fk } of closed subsets of E is a nest, then {Fk ∩ D} is an E D -nest.

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(ii) (E D , FD ) is a quasi-regular Dirichlet form on L2 (D; m) and X D is a standard process properly associated with it. (iii) N ⊂ D is E D -polar if and only if it is E-polar. (iv) For u ∈ FD , u is E D -quasi-continuous if and only if u is the restriction to D of a quasi continuous function on E. Proof. (i) Denote by σAD the hitting time of the part process X D for a set A ⊂ D. For any nest {Fk }, we then have D D Px lim σD\D∩F < ζ σ < ζ , x ∈ D, ≤ P lim x E\Fk k k→∞

k→∞

which equals zero by Theorem 3.1.4. Hence {Fk ∩ D} is an E D -nest by virtue of Lemma 3.1.17. (ii) By Exercise 3.3.7, X D is a standard process. (Moreover, by [13, 1 D IV:(4.33)] −σ X is special.) Consider the 1-order 1hitting probability pB (x) := Ex e B , x ∈ E, for the set B = E \ D. Then {pB > 0} = D ∪ N where N = B \ Br . Since N is E-polar by Theorem 3.1.3 and Theorem 3.1.10 and further p1B is quasi continuous by Corollary 3.2.3, there isa compact nest {Kk } such that ∪k Kk ⊂ E \ N and p1B ∈ C({Kk }). We let Dk := p1B ≥ 1k and KkD := Kk ∩ Dk for k ≥ 1. Obviously each KkD is a compact subset of D. Let σ = limk→∞ σD\Dk . By the quasi-left-continuity of X and the left continuity (3.1.10) of the quasi continuous function p1B , we have for q.e. x ∈ D Px (σ < σB ∧ ζ ) = Px p1B (Xσ ) = 0, σ < σB ∧ ζ = 0. D D D Since σD\K D = σD\Kk ∧ σD\Dk , it holds that, for q.e. x ∈ D, k

Px

lim σ D D k→∞ D\Kk

0 : Xt ∈

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Observe that in view of (3.2.2) and Theorem 1.3.7, we have (FD )Fk = {g ∈ FD : g = 0 m-a.e. on D \ Fk } = {g ∈ F : g = 0 q.e. on E \ Fk } = FFk . Therefore, ∪k≥1 FFk is E1 -dense in FD . Let k ≥ 1 and u ∈ bFFk . Since (E, F) is a regular Dirichlet form on L2 (E; m), there is a sequence {uj , j ≥ 1} ⊂ Cc (E) ∩ F that is E1 -convergent to u. On account of Lemma 1.1.11(ii), replacing uj by ((−u∞ ) ∨ uj ) ∧ u∞ if necessary, we may and do assume that supj≥1 uj ∞ < ∞. Since Cc (E) ∩ F is uniformly dense in Cc (E), there is some φ ∈ Cc (D) ∩ F so that φ = 1 on Fk . Clearly uj φ ∈ Cc (D) ∩ F and by Exercise 1.1.10, supj≥1 E1 (uj φ, uj φ) < ∞. As uj φ is L2 -convergent to uφ = u, we have by Banach-Saks Theorem (Theorem A.4.1), the Cesàro mean sequence of a subsequence of {uj φ} is E1 -convergent to u. This proves that u ∈ G and so bFFk ⊂ G for every k ≥ 1. Consequently, ∪k≥1 FFk ⊂ G. We then have FD ⊂ G as ∪k≥1 FFk is E1 -dense in FD . (ii) For any v ∈ F ∩ Cc (E) with supp[v] ⊂ D, there exists {vn } ⊂ CD which is E1 -convergent to v. To see this, it suffices to assume 0 ≤ v ≤ 1. Take w ∈ CD with w = 1 on supp[v] and choose wn ∈ C to be E1 -convergent to v and let, for a fixed ε ∈ (0, 1), vn = ϕε (wn ) · w for ϕε appearing in Definition 1.3.17. Then vn ∈ CD , vn converges to v as n → ∞, and E1 (vn , vn ) can be easily verified to be uniformly bounded in n. Hence, by Theorem A.4.1, the Cesàro mean sequence of a subsequence of {vn } is E1 -convergent to v. The concept of quasi continuity can be localized as follow. Let D be an open set. We call a function u defined q.e. on D quasi continuous on D if there exists a decreasing sequence {Gn , n ≥ 1} of open subsets of D such that limn→∞ Cap1 (Gn ) = 0 and uD\Gn is continuous for each n. Of course, the restriction to D of a quasi continuous function on E is quasi continuous on D. We notice that, if u is quasi continuous on an open set D, then u is E D -quasicontinuous, because, for {Gn } as above, {E \ Gn } is an E-nest so {D \ Gn } is an E D -nest by Theorem 3.3.8(i). The part process X D of X on an open set D is a Hunt process on D (Exercise 3.3.7). All the results obtained so far are directly applicable to X D and its associated regular Dirichlet form (E D , FD ). In particular, we have the following. P ROPOSITION 3.3.10. Let µ be a measure on E charging no E-polar set and F be its quasi support. Consider an open set D ⊂ E and a quasi continuous function u on D. If u = 0 µ-a.e. on D, then u = 0 E-q.e. on D ∩ F. Proof. Let ν = µD . Then ν(D \ (D ∩ F)) ≤ µ(E \ F) = 0. Suppose v ∈ FD is E D -quasi-continuous and v = 0 ν-a.e. on D. Then v is the restriction to D of an E-quasi-continuous function w on E that vanishes E-q.e. on E \ D. It


113

follows then that w = 0 µ-a.e. on E and so we have by Theorem 3.3.5 w = 0 E-q.e. on F. Accordingly, v = 0 E D -q.e. on D ∩ F. This means that D ∩ F is a quasi support of ν relative to the Dirichlet form (E D , FD ) by virtue of Theorem 3.3.5 applied to the regular Dirichlet form (E D , FD ). If u satisfies the stated condition, then u is E D -quasi-continuous on D. By Theorem 3.3.5(c) applied to (E D , FD ), we have u = 0 E D -q.e. on D ∩ F, and consequently E-q.e. on D ∩ F. 3.4. HITTING DISTRIBUTIONS AND PROJECTIONS II We keep the setting in the preceding section. The purpose of this section is to establish 0-order counterparts of Theorem 3.2.2. Let B ⊂ E be a nearly Borel measurable set. By Theorem 2.3.4, any function u. We u in the extended Dirichlet space Fe admits its quasi continuous version consider a linear subspace of Fe defined by Fe, E\B = {u ∈ Fe : u = 0 q.e. on B}.

(3.4.1)

The hitting distribution HB of X for the set B is defined by (A.2.2): HB g(x) = Ex g(XσB ); σB < ∞ for x ∈ E.

P ROPOSITION 3.4.1. If u ∈ bFe , then HB u is a quasi continuous element of Fe and E(HB u, v) = 0,

∀v ∈ Fe, E\B .

(3.4.2)

Proof. Suppose u ∈ Fe and |u| ≤ M for some M > 0. Take an approximating sequence {un } ⊂ F of u. By Theorem 2.3.4, we may assume that |un | ≤ M, u q.e. as n → ∞. n ≥ 1, and un → Choose αn ↓ 0 such that αn (un , un ) ≤ 1. By virtue of Theorem 3.2.2, HαBn un 2Eαn ≤ un 2Eαn ≤ un 2E + 1. Thus both HαBn un 2E and αn HαBn un 22 are uniformly bounded in n. In particular, un } such that its Cesàro by Theorem A.4.1, we can find a subsequence of {HαBn mean sequence, denoted by {fn }, is E-Cauchy. un = HB u q.e. on E and HαBn un is quasi continuous by Since limn→∞ HαBn u, which is therefore a Theorem 3.2.2, {fn } is an approximating sequence of HB quasi continuous element of Fe in view of Theorem 2.3.4. To prove the identity (3.4.2) for v ∈ FE\B , we observe from Theorem 3.2.2 Eαn HαBn un , v = 0, v ∈ FE\B .

114 Since

CHAPTER THREE

lim αn HαBn un , v ≤ lim αn HαBn un 2 v2 = 0,

n→∞

we have

n→∞

un , v) limn→∞ E(HαBn

= 0 and so

u, v) = lim E(fn , v) = 0 E(HB n→∞

for v ∈ FE\B .

We next show (3.4.2) for bounded v ∈ Fe,E\B . By applying the same argument as above to v instead of u, we can find vn , gn ∈ F, n ≥ 1, such v = v. Hence we that vn − gn ∈ FE\B and vn − gn is E-convergent to v − HB get (3.4.2) for v from that for vn − gn by letting n → ∞. Finally, for any v ∈ Fe,E\B , we put v = ϕ (v) by the contraction ϕ of (1.1.19). When (E, F) is transient, v is E-convergent to v as → ∞ by Exercise 2.1.13 and we obtain (3.4.2) for v from that for v . In general, we g can use Lemma 2.1.15 to find g ∈ K0 with v ∈ Fe so that v converges to v in g E and hence in E. If the Dirichlet form (E, F) is transient, then by the E (0-order) version of Theorem 1.3.3, the space Fe,E\B defined by (3.4.1) is a closed subspace of the Hilbert space (Fe , E). Denote its orthogonal complement by HB and the orthogonal projection on HB by PHB . u|(x) < ∞ T HEOREM 3.4.2. If (E, F) is transient, then, for any u ∈ Fe , HB | u is a quasi continuous version of PHB u. q.e. and HB Proof. If u ∈ Fe is bounded, this is contained in Proposition 3.4.1. It suffices to prove this for a non-negative u ∈ Fe . We then put u = u ∧ . By Proposition u is a quasi continuous element of HB and HB u E ≤ u E ≤ 3.4.1, HB uE . In particular, by Theorem A.4.1, the Cesàro mean sequence of a u } is E-convergent as → ∞. Since HB u increases subsequence of {HB u as → ∞, HB u is a quasi continuous element of HB and pointwise to HB u) + HB u give the E-orthogonal decomposition of u. u = (u − HB When (E, F) is transient, we have the 0-order version of Theorem 2.3.1: for any set B ⊂ E with the space L(0) B = {f ∈ Fe : f ≥ 1 q.e. on B} being (0) (0) non-empty, there exists a unique function eB ∈ LB minimizing E(f , f ) and 2 Cap(0) (B) = e(0) B E ,

(3.4.3)

where Cap(0) denotes the 0-order capacity defined in (2.3.1) with E1 and LA,1 being replaced by E, L(0) A,1 respectively. The hitting probability pB of X for a nearly Borel set B is defined by (A.2.3): pB (x) = Px (σB < ∞), x ∈ E.

115


Exactly in the same way as Corollary 3.2.3 is derived from Theorem 3.2.2 combined with Theorem 2.3.1 and (2.3.15), we can get the following corollary from Theorem 3.4.2 combined with the 0-order version of Theorem 2.3.1 and (2.3.22). C OROLLARY 3.4.3. Assume that (E, F) is transient. For any Borel set B with L(0) B = ∅, pB is a quasi continuous version of the equilibrium potential e(0) . B C OROLLARY 3.4.4. Assume that (E, F) is transient. An increasing sequence {Fk } of closed subsets of E is a nest if and only if ∪∞ k=1 (Fe )Fk is E-dense in Fe , where (Fe )Fk = {u ∈ Fe : u = 0 m-a.e. on E \ Fk }.

(3.4.4)

Proof. Recall that a nest is an abbreviation of an E-nest in the sense of Definition 1.2.12. Hence, the “only if” part is trivial. In order to prove the “if” part, we assume that an increasing sequence {Fk } of closed sets satisfies the stated denseness property. We shall derive the stochastic property (3.1.7) of {Fk } to deduce that it is a nest. Let limk→∞ σFk = σ and, for a strictly positive f ∈ bB(E) with E f · Rf dm < ∞,

u(x) = lim HE\Fk (Rf )(x) = Ex k→∞

ζ σ ∧ζ

f (Xt )dt , x ∈ E.

Since Rf ∈ Fe by Theorem 2.1.12 and Rf is excessive and hence quasi continuous by Theorem 3.3.3, we see from Theorem 3.4.2 that for k ≤ , HE\Fk (Rf ) − HE\F (Rf )2E = (HE\Fk (Rf ) − HE\F (Rf ), f ), and hence {HE\Fk (Rf )} is E-Cauchy and E-convergent to u ∈ Fe . Consequently, u is E-orthogonal to the space ∪∞ k=1 (Fe )Fk , and quasi continuous. Accordingly, u vanishes q.e. and, by the above expression of u, we get Px (σ < ζ ) = 0 q.e., namely, (3.1.7). In a general (not necessarily transient) case, we can still formulate a theorem which is obtained from Proposition 3.4.1 just by removing the boundedness u for requirement for u. To this end, we need to ensure the finiteness of HB u ∈ Fe in advance. Let us fix a nearly Borel measurable set B ⊂ E. We put D = {x ∈ E : pB (x) > 0}. Since pB is excessive, D is nearly Borel measurable and finely open.

(3.4.5)

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P ROPOSITION 3.4.5. (i) Px (σD < ∞) = 0 for x ∈ Dc . (ii) There exists a Borel properly exceptional set N ⊂ D such that both D \ N and Dc are X-invariant. Proof. (i) For n ≥ 1, define Dn := {x ∈ E : pB (x) ≥ 1/n}, which is finely closed. Then, for x ∈ Dc , pDn (x) = Px (σB = ∞, σDn < ∞) = Ex PXσDn (σB = ∞); σDn < ∞ 1 ≤ 1− pDn (x). n It follows that pDn (x) = 0. We then get (i) by letting n → ∞. (ii) Note that (i) and Lemma A.1.32 imply that the set Dc is X-invariant. Further, (i) implies that D is {Tt }-invariant in view of Proposition 2.1.6 and that Dc is finely open. Thus D is both finely open and finely closed. Consequently, the function 1D is finely continuous and therefore quasi continuous by Theorem 3.3.3. By the {Tt }-invariance of D and Proposition 3.1.9, we have Pt (1D u) = 1D · Pt u q.e. for any t > 0 and u ∈ Cc (E). Taking a sequence of non-negative functions {un , n ≥ 1} ⊂ Cc (E) that increases to 1, we obtain Pt 1D = 1D · Pt 1 q.e.; namely, for every t > 0, there is an mpolar set Nt so that Px (Xt ∈ D) = Px (t < ζ ) for x ∈ D \ Nt . Choose a properly exceptional set N1 containing ∪t∈Q+ Nt and let N := D ∩ N1 . As 1D is finely continuous, Px (Xt ∈ D \ N) = Px (t < ζ ) for any t ≥ 0 and any x ∈ D \ N, namely, Px (σE\(D\N) < ∞) = 0 for any x ∈ D \ N. This combined with Lemma A.1.32 yields the X-invariance of the set D \ N. We denote by D the X-invariant set D \ N in the above proposition and by D \ B: D1 the fine interior of D \ B : Px (σB > 0) = 1}. D1 := {x ∈

(3.4.6)

D1 is then nearly Borel measurable and finely open. By virtue of Theorem 3.3.8, the part (E D1 , FD1 ) of (E, F) on D1 is a Dirichlet form on L2 (D1 ; m) which is associated with the part process X D1 of X on D1 . L EMMA 3.4.6. The part E D1 of E on the set D1 is a transient Dirichlet form on L2 (D1 ; m). Proof. If we denote the resolvent kernel of the part process X D1 by 1 {RD α ; α > 0}, then

1 RD 1 1(x)

σB ∧ζ

= Ex

e dt ≤ 1 − Ex e−σB < 1, x ∈ D1 , −t

0

and we get the transience of E D1 on account of Proposition 2.1.10.


117

L EMMA 3.4.7. For u ∈ Fe , HB | u|(x) is finite m-a.e. Proof. We may assume that u ∈ Fe is non-negative. Since HB u(x) = u(x) < u(x) = 0 for every x ∈ Dc , it ∞ q.e. x ∈ B ∪ Br by Theorem 3.1.10 and HB suffices to show that u(x) < ∞ HB

m-a.e. on D1 .

(3.4.7)

Since E D is transient, it admits a reference function g on D1 : g positive bounded m-integrable function on D1 such that is a strictly D1 g(x)R g(x)m(dx) ≤ 1. By replacing g with g/(u ∨ 1) if necessary, we may D1 assume that D1 gudm < ∞. Recall that the domain of E D1 is defined by FD1 = {v ∈ F : v = 0 q.e. on E \ D1 }. We put un = u ∧ n for a fixed n. By Theorem 2.3.4, we can find a sequence vk = un q.e. In {vk } ⊂ F such that 0 ≤ vk ≤ n, {vk } is E-Cauchy and limk→∞ α (v − H v view of Theorem 3.2.2 and Proposition 2.1.6, we see that 1 D k B k ) ∈ FD1 and 1 α g|vk − HαB vk |dm = Eα RD vk | D |vk − HB α g, 1 D1

≤ D1

1/2 1 gRD gdm vk − HαB vk Eα ≤ vk Eα . α

By letting α ↓ 0 and k → ∞, we have g|un − HB un |dm ≤ un E ≤ uE . D1

Hence D1 g · HB un dm ≤ D1 g · un dm + uE , and, by letting n → ∞, we obtain D1 g · HB udm ≤ G g · udm + uE < ∞, proving (3.4.7). T HEOREM 3.4.8. For any u ∈ Fe and for any nearly Borel measurable u| is finite q.e. and HB u is a quasi continuous element of Fe set B ⊂ E, HB | satisfying (3.4.2). Proof. We may assume that u ∈ Fe is non-negative. We put un = u ∧ n. By Proposition 3.4.1, un then satisfies the desired properties and in particular un E ≤ un E ≤ uE . Since HB un increases pointwise to HB u as n → ∞ HB u is finite m-a.e. by Lemma 3.4.7, in view of Theorem A.4.1, we and HB u is an element of Fe admitting a Cesàro mean of a can conclude that HB un } as its approximating sequence. Equation (3.4.2) for subsequence of {HB u then follows from that for HB un . The quasi continuity of HB u follows HB un and Theorem 2.3.4. from that of HB

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T HEOREM 3.4.9. Suppose that D is a nearly Borel measurable finely open subset of E. Then the extended Dirichlet space FD,e of (E D , FD ) can be characterized as u = 0 q.e. on Dc }. FD,e = {u ∈ Fe :

(3.4.8)

Proof. We denote the right hand side of (3.4.8) by Fe,D . Clearly the extended Dirichlet space FD,e of (E D , FD ) is a subspace of Fe and it follows from Theorem 2.3.4 that FD,e ⊂ Fe,D . Let us prove the inverse inclusion. Assume first that (E, F) is transient. By Theorem 2.3.2, Fe,D is a closed linear subspace of the Hilbert space Fe with inner product E. Let R and RD be the 0-order resolvent kernel for the Hunt process X associated with (E, F) and its subprocess X D , respectively. Any f ∈ L2 (D; m) is extended to E by setting f (x) = 0, x ∈ Dc . For f ∈ L2 (D; m) with D f (x)R(1D f )(x)m(dx) < ∞, we know by Theorem 2.1.12 that Rf ∈ Fe and RD f ∈ FD,e . As HDc Rf = Rf − RD f , we have by Theorem 3.4.8 E(RD f , v) = E(Rf , v) = (f , v) for every v ∈ Fe,D . This implies that the subspace of FD,e defined by {RD f : f ∈ L2 (D; m) with f (x)R(1D f )(x)m(dx) < ∞} D

is E-dense in Fe,D . This proves that FD,e = Fe,D when (E, F) is transient. In the general case, for u ∈ Fe,D , let g ∈ K0 be the function in Lemma g 2.1.15 so that u ∈ Fe . Note that (E g , F) is a transient Dirichlet form with E g (u, u) ≥ E(u, u) for u ∈ F and shares the same quasi notions with (E, F). g g We conclude from the above transient case that u ∈ Fe,D = FD,e ⊂ FD,e . This completes the proof that FD,e = Fe,D . In view of Theorem 3.4.9, there is no ambiguity to use notation FeD to denote either of FD,e and Fe,D .

3.5. TRANSIENCE, RECURRENCE, AND PATH BEHAVIOR We say that the m-symmetric Hunt process X on E is transient, recurrent, and irreducible if so is its Dirichlet form (E, F) on L2 (E; m) in the sense of Section 2.1, respectively. In this section, we present basic stochastic features of transience, recurrence, and irreducibility of X. In dealing with the transience and its applications, we continue to work under the present setting that X is an m-symmetric Hunt process on a locally compact separable metric space E whose Dirichlet form (E, F) on L2 (E; m) is regular. However, the recurrence and irreducibility will be discussed for a

119


more general symmetric right process without assuming the regularity of the associated Dirichlet form in the second part of this section. We first state an application of Corollary 3.4.3. L EMMA 3.5.1. Assume that X is transient. If, for a decreasing sequence {An } of open sets, {E \ An } is a Cap(0) -nest in the sense that limn→∞ Cap(0) (An ) = 0, then Px

∞

{σAn = ∞} = 1,

q.e. x ∈ E.

(3.5.1)

n=1

In particular, {E \ An } is a strong nest in the sense of (3.1.6). Proof. By Corollary 3.4.3 and (3.4.3), pAn is quasi continuous and Cap(0) (An ) = pAn 2E for an open set An with Cap(0) (An ) < ∞. If this quantity tends to zero as n → ∞, then limn→∞ pAn (x) = 0 q.e. x ∈ E, which in turn implies (3.5.1). Obviously any Cap1 -nest {E \ An } is a Cap(0) -nest and hence {An } enjoys the property (3.5.1) in transient case. Property (3.5.1) is stronger than (3.1.6) and it reflects a specific property of a transient process X that escapes to infinity ∂ of E when time goes to infinity, as will be formulated below. T HEOREM 3.5.2. Assume that X is transient. Then the path wanders out to infinity whenever its lifetime is infinite: (3.5.2) Px ζ = ∞ and lim Xt = ∂ = Px (ζ = ∞) for q.e. x ∈ E. t→∞

Proof. Let R be the 0-order resolvent kernel of X defined by (2.1.3) and f be a strictly positive Borel measurable function on L1 (E; m). Then Rf is excessive and the set N = {x ∈ E : Rf (x) = ∞} is m-negligible in view of Proposition 2.1.3. Consequently, N is m-polar by Theorem A.2.13(v). On account of Theorem 3.1.5, we can find a Borel properly exceptional set N0 such that Rf (x) > 0

for x ∈ E

and

Rf (x) < ∞

for x ∈ E \ N0 .

By virtue of Theorem 3.3.3, Rf is quasi continuous and hence by Theorem 1.3.14(iii) there is a Cap1 -nest {E \ An } such that the restriction of Rf to each set E \ An is continuous. We may assume N0 ⊂ An for every n ≥ 1. Furthermore, on account of Lemma 3.5.1, the identity (3.5.1) holds for x ∈ E \ N1 , N1 being some Borel properly exceptional set containing N0 . Take an increasing sequence of compact sets {Kn , n ≥ 1} that increases to E, and let Fn = Kn \ An = Kn ∩ (E \ An ). Since Fn is compact and Rf is

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continuous on Fn , cn = infx∈Fn Rf (x) is positive and cn pFn (x) ≤ Ex (Rf )(XσFn ) ≤ Rf (x) < ∞ for x ∈ E \ N1 . Therefore, for x ∈ E \ N1 , Pt pFn (x) ≤ c−1 n Pt Rf (x) ↓ 0 as t ↑ ∞, and consequently,   ∞ Px   = lim Px j = 0 for j := {σFn ◦ θj < ∞} ∩ {j < ζ }. j=1

j→∞

Since ∞ j=1

j =

∞

{σFn ◦ θj < ∞}

{ζ = ∞},

j=1

we get   ∞ Px  {σFn ◦ θj = ∞} {ζ = ∞} = Px (ζ = ∞) for x ∈ E \ N1 . j=1

This holds for every n ≥ 1 and so for x ∈ E \ N1 ,    ∞ ∞  {X(j, ∞) ⊂ E \ Fn } = Px (ζ = ∞). Px {ζ = ∞} n=1 j=1

On account of (3.5.1), we can replace Fn with Kn in the above to get the desired identity (3.5.2). C OROLLARY 3.5.3. Suppose X is transient. Then any quasi continuous function f ∈ Fe satisfies (3.5.3) Px lim f (Xt ) = 0; c = Px (c ) for q.e. x ∈ E, t→ζ

where = {ζ < ∞, Xζ − ∈ E}. Proof. By Theorem 3.5.2, Px (Xζ − = ∂; c ) = Px (c ) for q.e. x ∈ E. By the 0-order counterpart of Lemma 1.3.15, any f ∈ Fe admits a quasi continuous version in the restricted sense with respect to Cap(0) : there exists a decreasing open sets {Ak } with limk→∞ Cap(0) (Ak ) = 0 such that f ∈ C∞ ({E \ Ak }). If f ∈ Fe is quasi continuous, then f differs from a function of this property by a set of zero capacity and f itself enjoys this property. Accordingly, (3.5.3) follows from Lemma 3.5.1. We can now combine Lemma 3.5.1 with Theorem 2.3.2 and Lemma 2.1.15 to deduce an important statement stronger than a part of Theorem 2.3.4 for


121

a general (not necessarily transient) m-symmetric Hunt process X = (Xt , Px ) whose Dirichlet form (E, F) on L2 (E; m) is regular. By Theorem 2.3.4, any element of the extended Dirichlet space Fe admits a quasi continuous m-version. T HEOREM 3.5.4. Assume that {fn , n ≥ 1} ⊂ Fe are both m-a.e. convergent fn , n ≥ 1, and f be quasi and E-convergent to f ∈ Fe as n → ∞. Let continuous versions of fn , n ≥ 1, and f , respectively. There exist then a f, fn , ≥ 1} ⊂ C∞ ({Fk }) subsequence {n } and a strong nest {Fk } such that { f uniformly on each set Fk as → ∞. In particular, it and fn converges to holds for every T > 0 that fn (Xt ) − f (Xt ) > 0 = 0 for q.e. x ∈ E. (3.5.4) Px lim sup →∞ 0≤t≤T

Proof. Property (3.5.4) is a consequence of the first assertion in view of Definition 3.1.6 of a strong nest. Furthermore, the first assertion follows from Theorem 2.3.2 and Lemma 3.5.1 when X is transient. In a general (not necessarily transient) case, we shall prove the first assertion by a reduction to a transient case. By Lemma 2.1.15, we can find g ∈ K0 such g that {f , fn , n ≥ 1} ⊂ Fe and fn is E g -convergent to f as n → ∞, where (E g , F) is the perturbed Dirichlet form on L2 (E; m) defined by (2.1.28). It is a transient f are E g -quasi-continuous. regular Dirichlet form and fn , n ≥ 1, Px ) be the canonical subprocess of the Hunt process X = Let X = ( Xt , (Xt , Px ) with respect to its multiplicative functional e−At for t At = g(Xs )ds, t ≥ 0, 0

as is defined by (A.3.28) and (A.3.29). Since e−At ∈ Ft0 , the transition function Pt (x, B) = Ex [e−At 1B (Xt )] of X is B(E)-measurable in x ∈ E for B ∈ B(E). Hence X is a Hunt process on (E, B(E)) in view of Theorem A.3.13. Moreover, X is m-symmetric and its Dirichlet form on L2 (E; m) coincides with (E g , F) by virtue of Theorem 5.1.3. Therefore, the first assertion of the theorem holds true for some increasing X , namely, sequence {Fk } of closed sets that is a strong nest with respect to Px (σ < ∞) = 0,

σ = lim σE\Fk k→∞

for q.e. x ∈ E.

But, for each T > 0, we have from (A.3.29) Px (σ < T) = Ex e−At ; σ < T , and consequently Px (σ < T) = 0 Px (σ < T) ≤ eTg∞

for q.e. x ∈ E,

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which means that {Fk } is a strong nest with respect to X, as was to be proved. We now turn to the probabilistic descriptions of irreducibility and recurrence under a more general setting. Let (E, B(E)) be a Lusin space and m be a σ -finite measure on it. Let X = (Xt , Px ) be a right process on (E, B(E)) whose transition function {Pt ; t ≥ 0} is m-symmetric. We do not consider the associated Dirichlet form. The term “q.e.” will mean “except for an m-polar set”. L EMMA 3.5.5. (i) If X is recurrent, then any bounded excessive function f is {Pt }-invariant: for q.e. x ∈ E, Pt f (x) = f (x)

for t > 0.

(3.5.5)

(ii) If X is irreducible recurrent, then any excessive function is constant q.e. Proof. (i) Assume that X is recurrent and take a strictly positive function g ∈ L1 (E; m). Then for any bounded excessive function f , the resolvent equation gives (Rβ g, f − αRα f ) = (g, Rβ f − αRβ Rα f ) = (g, Rα f − βRα Rβ f ) ≤ (g, Rα f ) < ∞. By letting β → 0, we get from Proposition 2.1.3 that αRα f = f m-a.e. and hence q.e. Since Pt f (x) is right continuous in t > 0 owing to the fine continuity of f , we obtain (3.5.5). (ii) If an excessive function f is not constant q.e., then it is not constant m-a.e. and there are constant 0 < a < b such that two finely open sets A = {x ∈ E : f (x) < a} and B = {x ∈ E : f (x) > b} are both of positive m-measures. Since f ∧ b is bounded and excessive by Exercise A.2.3, we have from (i) b = f (x) ∧ b = Ex [f (Xt ) ∧ b],

t > 0, x ∈ B \ N,

for some m-inessential set N. This implies for x ∈ B \ N that Px (Xt ∈ A) = 0 and consequently Px (Xt ∈ A, ∃t > 0) = 0 on account of the right continuity of f (Xt ) in t > 0. Hence R1A (x) = 0, x ∈ B \ N, while R1A (x) > 0, x ∈ A, arriving at a contradiction to the irreducible recurrence criterion Proposition 2.1.3(iv). T HEOREM 3.5.6. (i) If X is irreducible, then for any non-m-polar nearly Borel measurable set B Px (σB < ∞) > 0,

q.e. x ∈ E.

(3.5.6)


123

(ii) If X is irreducible and recurrent, then for any non-m-polar nearly Borel measurable set B Px (σB ◦ θn < ∞ for every n ≥ 1) = 1

for q.e. x ∈ E.

(3.5.7)

Proof. (i) The set D defined by (3.4.5) is of positive m-measure because of the non-m-polarity of B. By virtue of Proposition 3.4.5, D is {Tt }-invariant and hence m(Dc ) = 0 by the irreducibility assumption. Since Dc is finely open by Proposition 3.4.5(i), it is m-polar on account of Theorem A.2.13, yielding (3.5.6). (ii) In view of Lemma 3.5.5, pB (x) = c, x ∈ E \ N, for some positive constant c and a Borel m-inessential set N. We have then, for any t > 0, x ∈ E \ N, c = Px (σB < ∞) = Px (σB ≤ t) + Px (t < σB , σB < ∞) = Px (σB ≤ t) + Ex PXt (σB < ∞); Xt ∈ E \ N, t < σB = Px (σB ≤ t) + cPx (t < σB ). By letting t → ∞, we get c(1 − c) = 0 and hence c = 1. We are then readily led to (3.5.7) by the Markov property of X. At the end of this section, we present two examples of symmetric diffusions and study their transience, recurrence, and other properties. As is stated in Theorem 1.5.1, there exists for any regular Dirichlet form (E, F ) on L2 (E; m) an m-symmetric Hunt process X on E, which is unique up to E-polar sets in view of Theorem 3.1.12. If furthermore E is strongly local, then the Hunt process X can be taken to be a diffusion with no killing inside E, as will be proved in Theorem 4.3.4 at the end of the next chapter. We shall utilize these facts in the considerations of the following examples. Example 3.5.7. (One-dimensional diffusion) We consider the onedimensional local Dirichlet form studied in Section 2.2.3. Let I = (r1 , r2 ) be a one-dimensional open interval and (s, m) be a pair of a canonical scale and a canonical measure on I. Defining the Dirichlet integral E (s) based on s and the associated space F (s) by (2.2.29) and (2.2.28), respectively, we are concerned with the Dirichlet form (E, F ) = (E (s) , F (s) ∩ L2 (I; m))

(3.5.8)

on L2 (I; m). The boundary r1 (resp. r2 ) of I is said to be approachable if s(r1 ) > −∞ (resp. s(r2 ) < ∞), and it is called regular if it is approachable and m(r1 , c) < ∞ (resp. m(c, r2 ) < ∞) for c ∈ I. We denote by I ∗ the interval obtained from I by

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adjoining its boundary ri whenever ri is regular (i = 1, 2). m is extended to I ∗ by m(·) = m(· ∩ I). According to Theorem 2.2.11, (3.5.8) is a regular, strongly local and irreducible Dirichlet form on L2 (I ∗ ; m)(= L2 (I; m)). The Dirichlet form in (3.5.8) is transient if and only if either r1 or r2 is approachable but non-regular. Otherwise (3.5.8) is recurrent. Furthermore, In view of (2.2.40), the Dirichlet form (3.5.8) has a quite simple feature that every point of I ∗ is not E-polar. Therefore, by the remark made in advance, there exists a unique m-symmetric Hunt process X = (Xt , ζ , Px ) on I ∗ with continuous sample path and no killing inside I ∗ associated with (3.5.8). On account of Theorem 3.5.6(i), Px (σy < ∞) > 0

for every x, y ∈ I ∗ .

If (3.5.8) is recurrent, then X is point recurrent in the sense that Px (σy < ∞) = 1

for every x, y ∈ I ∗ ,

by Theorem 3.5.6(ii). When I ∗ \ I = ∅, X is called the reflecting diffusion on I ∗ associated with the pair (s, m) reflected at I ∗ \ I. Let X 0 = (Xt0 , ζ 0 , P0x ) be the part process of X on I. X 0 is obtained from X by killing upon the hitting time of I ∗ \ I. In view of Theorem 3.3.8, X 0 is m-symmetric and its Dirichlet form on L2 (I; m) equals the part (E I , FI ) of the Dirichlet form (3.5.8) on L2 (I) defined by (3.2.2) and (3.3.1). Accordingly,

FI = u ∈ F (s) ∩ L2 (I; m) : u(ri ) = 0 if ri is regular , (3.5.9) E I = E (s) . FI ×FI

By Theorem 3.3.9, (E I , FI ) is a regular Dirichlet form on L2 (I; m). Furthermore, by Theorem 3.4.9 combined with (2.2.39), its extended Dirichlet space (FI,e , E I ) can be described as

FI,e = u ∈ F (s) : u(ri ) = 0 if ri is approachable , (3.5.10) E I = E (s) . FI,e ×FI,e

Suppose u ∈ FI,e and E (s) (u, u) = 0. Then u is a constant, which vanishes if and only if either r1 or r2 is approachable. Hence (E I , FI ) is transient if either r1 or r2 is approachable and it is otherwise recurrent on account of Theorem 2.1.8 and Theorem 2.1.9. X 0 enjoys the following properties: (d.1) X 0 is a Hunt process on I. (d.2) X 0 is a diffusion process: Xt0 is continuous in t ∈ (0, ζ 0 ) almost surely.


125

(d.3) X 0 is irreducible: P0x (σy < ∞) > 0 for any x, y ∈ I. (d.4) X 0 admits no killing inside I: P0x (ζ 0 < ∞, Xζ00 − ∈ I) = 0, x ∈ I. It follows from (d.1) and (d.4) that P0x (Xζ00 − ∈ {r1 , r2 }, ζ 0 < ∞) = P0x (ζ 0 < ∞)

for every x ∈ I.

(3.5.11)

Let us call a process X 0 satisfying properties (d.1), (d.2), and (d.3) a minimal diffusion or an absorbing diffusion on an open interval I = (r1 , r2 ). We notice that under (d.1) and (d.2), the condition (d.3) is equivalent to the requirement for each point a ∈ I to be regular in the sense that Ea [e−σa+ ] = Ea [e−σa− ] = 1 where Ea [e−σa± ] = limb→±a Ea [e−σb ]. We have started with a pair (s, m) of a canonical scale and a canonical measure on I and constructed an m-symmetric minimal diffusion X 0 on I with no killing inside whose Dirichlet form and extended Dirichlet space are given by (3.5.9) and (3.5.10), respectively. Conversely, suppose we are given a minimal diffusion X 0 = (Xt0 , ζ 0 , P0x ) on I with no killing inside. It is well known [95, 97] that there is an associated pair (s, m) of a canonical scale and a canonical measure such that, for each J = (a, b), r1 < a < b < r2 , they coincide with sJ (x) = P0x (σa > σb ),

mJ (x) = −

dE0x [σa ∧ σb ] , x ∈ J, dsJ (x)

respectively, up to a linear transformation. The pair associated with X 0 is unique up to a multiplicative constant c > 0 in the sense that, for another associated pair ( s, m), d s = cds, d m = c−1 dm. The minimal diffusion X 0 is known to be symmetric with respect to an associated canonical measure m [95, 97]. We refer the reader to [69] for a proof of the following theorem. T HEOREM 3.5.8. Let X 0 be a minimal diffusion on I with no killing inside. Then its Dirichlet form on L2 (I; m) and its extended Dirichlet space are given by (3.5.9) and (3.5.10), respectively, in terms of an associated pair (s, m). In Example (5◦ ) of Section 5.3, we shall identify the Dirichlet form and the extended Dirichlet space of a general minimal diffusion allowing killings inside. Thus any minimal diffusion on an open interval can be studied entirely in the framework of the Dirichlet form. See also a remark stated at the end of that example. Feller’s classification of the boundaries stated in K. Itô [95] and K. Itô-H. McKean [97] particularly is concerned with the approachability of X 0 to r1 , r2 at a finite lifetime ζ 0 . For i = 1, 2, we call ri approachable in finite time if P0x (Xζ00 − = ri , ζ 0 < ∞) > 0,

for any x ∈ I.

(3.5.12)

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In view of (3.5.11), we see that X 0 is conservative, namely, P0x (ζ 0 = ∞) = 1 for every x ∈ I, if and only if neither r1 nor r2 is approachable in finite time. In Theorem 5.15.2 of [95], it is proved that r1 (resp. r2 ) is approachable in finite time if and only if, for c ∈ I, r2 c m((x, c))s(dx) < ∞ resp. m((c, x))s(dx) < ∞ . (3.5.13) c

r1

When the condition (3.5.13) is fulfilled, ri is called exit or regular (resp. exit) in the terminology of [95] (resp. [97]). Suppose now r1 is non-approachable, while r2 is approachable but nonregular. Then I ∗ = I, X = X 0 , and the Dirichlet form (3.5.8) is transient by Theorem 2.2.11. We can then show that (3.5.14) Px lim Xt = r2 = 1 for x ∈ I. t→ζ

In fact, combining (3.5.11) with Theorem 3.5.2, we have Px lim Xt ∈ {r1 , r2 } = 1 for every x ∈ I. t→ζ

On the other hand, the extended Dirichlet space Fe is equal to Fe = {u ∈ F (s) : u(r2 ) = 0} in the present case by Theorem 2.2.11(ii). In particular, Fe contains a function which is non-zero constant near r1 . If (3.5.14) were not true, we would encounter a contradiction to Corollary 3.5.3 which states that f (Xt ) converges to zero as t → ζ Px -a.s. for any f ∈ Fe . Property (3.5.14) legitimates our usage of the term “approachable”. For instance, if I = (0, ∞), ds = x1−n dx, dm = 2xn−1 dx, n ≥ 2, then 0 is non-approachable, while ∞ is non-approachable when n = 2 and approachable, when n ≥ 3. Hence the associated Dirichlet form E(u, v) = non-regular n−1 2 u v x dx on L (I; 2xn−1 dx) is recurrent when n = 2 and transient when I n ≥ 3. Furthermore, both integrals in (3.5.13) for r1 = 0, r2 = ∞ diverge for any n ≥ 2 and hence the associated diffusion X = (Xt , Px ) on (0, ∞) is always conservative. When n ≥ 3, we have from (3.5.14) Px ζ = ∞, lim Xt = ∞ = 1, t→∞

for any x ∈ (0, ∞).

d du X is called the Bessel process. The corresponding generator dm equals ds n−1 + x u ), which is nothing but the radial part of half of the n-dimensional Laplace operator . Accordingly, X is equivalent in law to the radial part |Bt | of the n-dimensional Brownian motion and the above-mentioned properties of X are obvious from those of Bt . 1 (u 2

127


As is directly derived at the end of Section 2.2.3 and will be used in the next example, we have

2

|u(x)|g(x)x

n−1

∞

dx ≤

2 n−1

(u ) x

1/2 dx

,

u ∈ G,

(3.5.15)

0

for some strictly positive bounded xn−1 dx-integrable function g, which expresses a transience inequality (2.1.11) for the Bessel process with n ≥ 3. Here G denotes the space F defined by (3.5.8) for the present choice of I, s, m. Example 3.5.9. (Brownian motion and related diffusions) For n ≥ 1, the n-dimensional standard Brownian motion (BM in abbreviation) X is a diffusion process on Rn with transition function (2.2.13). It is symmetric with respect to the Lebesgue measure and the associated Dirichlet form on L2 (Rn ) is ( 21 D, H 1 (Rn )) given by (2.2.15). Consider a domain D of Rn . The part process X D of X on D is called the absorbing Brownian motion on D. X D is obtained from X by killing upon its hitting time of Rn \ D. By Theorem 3.3.8, X D is symmetric with respect to the Lebesgue measure and its Dirichlet form on L2 (D) equals the part 1 D, H01 (D) (3.5.16) 2 on D of the Dirichlet form (2.2.15): u = 0 q.e. on Rn \ D}, (H01 (Rn ) = H 1 (Rn )). H01 (D) = {u ∈ H 1 (Rn ) : (3.5.17) The pair in (3.5.16) is a regular Dirichlet form on L2 (D) and actually it has Cc∞ (D) as a core by virtue of Theorem 3.3.9(ii). Accordingly, it can be identified with the (E1 -)closure of Cc∞ (D) in the Sobolev space H 1 (D) of (2.2.45), which is a customary way to introduce the space H01 (D). 1 (D) the extended Dirichlet spaces of the Dirichlet form Let us denote by H0,e (3.5.16). On account of Theorem 3.4.9 and (3.5.17), we then have 1 1 (D) = {u ∈ He1 (Rn ) : u = 0 q.e. on Rn \ D}, (H0,e (Rn ) = He1 (Rn )). H0,e (3.5.18)

P ROPOSITION 3.5.10. (i) When n = 1 or 2, X D is transient if and only if Cap1 (Rn \ D) > 0. In this case, the lifetime of X D is finite a.s. (ii) When n ≥ 3, X D is always transient. Proof. (i) When n = 1, Cap1 (R1 \ D) > 0 if and only if R1 \ D = ∅, which is in turn equivalent to the transience of X D , as seen in the preceding example. Assume that n = 2. In view of (3.5.18) and Theorem 2.2.13, the extended

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Dirichlet space of (3.5.16) equals 1 (D) = {u ∈ BL(R2 ) : u = 0 q.e. on R2 \ D}, E = H0,e

1 D. 2

1 (D) satisfies E(u, u) = 0, then u is constant a.e. on R2 and Hence, if u ∈ H0,e hence u is constant q.e. on R2 . If R2 \ D is of positive capacity, u must be zero q.e. on R2 , yielding the transience of the form by virtue of Theorem 2.1.9. If 1 (D) = He1 (R2 ) = BL(R2 ), the recurrent extended Cap1 (R2 \ D) = 0, then H0,e Dirichlet space. The second assertion follows from the irreducible recurrence of X and Theorem 3.5.6. (ii) When n ≥ 3, the transience property (2.2.47) is fulfilled by He1 (Rn ). Hence X D is transient by (3.5.18) and Theorem 2.1.9.

In what follows, we only consider a domain D possessing a continuous boundary in the sense of Section 2.2.4. Then the Dirichlet form ( 12 D, H 1 (D)) becomes a regular strongly local Dirichlet form on L2 (D) (= L2 (D; 1D · dx)). Accordingly, there exists an associated diffusion process X r on D uniquely up to the equivalence in the sense of Theorem 3.1.12. X r is called a reflecting Brownian motion on D. X r is irreducible because its transition function dominates that of the absorbing Brownian motion on D and the latter is known to possesses a strictly positive density function. If either n ≤ 2 or D is of finite Lebesgue measure, X r = (Xt , Px ) is recurrent by virtue of Theorem 2.2.13 and Corollary 2.2.15. In particular, by Proposition 2.1.10, X r is conservative in the sense that its lifetime is infinity Px -a.s. for q.e. x ∈ D. Further, for any non-Epolar nearly Borel set B ⊂ D, it follows from Theorem 3.5.6 that Px (σB ◦ θn < ∞, ∀n) = 1,

q.e. x ∈ D.

(3.5.19)

Suppose that n ≥ 3 and D is unbounded. Owing to Takeda’s test (cf. [146], [73, Example 5.7.1]), we can verify that X r is still conservative. But the transience of X r depends on the geometric shape of the unbounded domain D. For instance, X r is transient if D = R3 , while X r is recurrent if D = {x ∈ R3 : x1 ∈ R, x22 + x32 ≤ 1} the infinite tube. In the latter case, X r is a direct product of the one-dimensional Brownian motion and the reflecting Brownian motion on the closed unit disk so that the sample path of X r cannot escape to infinity as time goes to infinity. We notice that if D1 ⊂ D2 and the reflecting Brownian motion on D1 is transient, then so is the reflecting Brownian motion on D2 because of the recurrence criteria in Theorem 2.1.8. We show that for a domain D ⊂ Rn with n ≥ 3, the reflecting Brownian motion X r on D is transient if D contains an infinite cone CA = {rξ ∈ Rn : r > 0, ξ ∈ A} where A is a non-void open subset of the unit sphere 1 = {x ∈ Rn : |x| = 1}. By the above observation, it suffices to derive the transience


129

inequality (2.1.11) of the Dirichlet form ( 12 D, H1 (D)) for D = CA by assuming that CA has a continuous boundary. Let G denote the space that appeared at the end of Example 3.5.7. Take any u ∈ Cc∞ (CA ). Then u(x) = u(rξ ) belongs to G as a function of r for each ξ ∈ A and the inequality ur (rξ )2 ≤ |∇u(x)|2 with x = rξ is readily verified using the polar coordinate. Hence, denoting the surface measure on 1 by σ (dξ ), it follows from (3.5.15) that ∞ |u(x)|g(|x|)dx = |u(rξ )|g(r)rn−1 drσ (dξ ) CA

A

0

√ 1/2 √ ∞ σ (1 ) σ (1 ) ≤ ur (rξ )2 rn−1 drdσ (ξ ) ≤ · D(u, u)1/2 , 2 2 A 0 which readily extends to u ∈ H 1 (CA ). When the reflecting Brownian motion X r = (Xt , Px ) on D is transient, Xt approaches to the point at infinity ∂ of D as t → ∞ Px -a.s. for q.e. x ∈ D by virtue of Theorem 3.5.2. Further, any quasi continuous function in the extended Sobolev space He1 (D) goes to zero along Xt as t → ∞ by Corollary 3.5.3. In the transient case, He1 (D) does not contain non-zero constant functions, while the larger space BL(D) always does. When D is a bounded domain and a Lipschitz domain in the sense that the function F in the condition (2.2.55) can be taken to be Lipschitz continuous, we can choose a representative X r of the equivalence class of reflecting Brownian motions such that its transition function {P∗t ; t ≥ 0} is strong Feller in the sense that P∗t (bB(D)) ⊂ C(D) for every t > 0.1 Since X r then satisfies the absolute continuity condition (AC), we see from Remark 4.3.5 that X r is a conservative diffusion starting at every point of D and satisfies (3.5.19) for every x ∈ D.

1 [15].

Chapter Four ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES

This chapter is devoted to the study of additive functionals of symmetric Markov processes under the same setting as in the preceding chapter, namely, we let E be a locally compact separable metric space, B(E) be the family of all Borel sets of E, and m be a positive Radon measure on E with supp[m] = E, and we consider an m-symmetric Hunt process X = (, M, Xt , ζ , Px ) on (E, B(E)) whose Dirichlet form (E, F) on L2 (E; m) is regular on L2 (E; m). The transition function and the resolvent of X are denoted by {Pt ; t ≥ 0}, {Rα , α > 0}, respectively. B∗ (E) will denote the family of all universally measurable subsets of E. Any numerical function f defined on E will be always extended to E∂ by setting f (∂) = 0.

4.1. POSITIVE CONTINUOUS ADDITIVE FUNCTIONALS AND SMOOTH MEASURES We shall adopt the notions and results in Section A.3.1 of Appendix A, where, under a more general setting that X is a Borel right process on a Lusin space E with an excessive measure m, the notion of an additive functional (admitting an exceptional set) of X, the concept of the m-equivalence of two additive functionals as well as the notion of a positive continuous additive functional (PCAF in abbreviation) are introduced. It is further proved in Theorem A.3.5 that for any PCAF A of X, a measure µA on (E, B(E)) called the Revuz measure of A is uniquely determined. Under the present more special setting, m is of course excessive with respect to the m-symmetric Hunt process X, and accordingly we can use those notions and results in Section A.3.1 without any change except for one necessary modification mentioned below. Recall that a set N ⊂ E is said to be an m-inessential set (resp. a properly exceptional set) for a Borel right process (resp. for a Hunt process) X if N is an m-negligible nearly Borel set such that E \ N is X-invariant. See Section A.1.3 for the X-invariance. By Remark A.1.30, the restriction of a Borel right process

ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES

131

(resp. a Hunt process) to its invariant set is again a Borel right process (resp. a Hunt process). According to Lemma A.1.32, N is properly exceptional for a Hunt process X if and only if it is m-inessential for X regarded as a Borel right process. Therefore, as the exceptional set N appearing in Definition A.3.1 of an additive functional A, an m-inessential set for a Borel right process X can now be replaced by a properly exceptional set for the Hunt process X. In what follows, we shall use the notion of an additive functional for the present Hunt process X with this replacement. Let us denote by A+ c the totality of positive continuous additive functionals of the m-symmetric Hunt process X. A, B ∈ A+ c are called m-equivalent if Pm (At = Bt ) = 0 for every t > 0. We write A ∼ B in this case. By virtue of Lemma A.3.2 and the above remark, we see that A ∼ B if and only if there are a common defining set and a common Borel exceptional set N such that At (ω) = Bt (ω) for every t ≥ 0 and ω ∈ . We are concerned with an analytic characterization of the family A+ c /∼ of all equivalence classes. The notion of a smooth measure with respect to a regular Dirichlet form (E, F ) is introduced in Definition 2.3.13. The family of all smooth measures is denoted by S. A purpose of this section is to prove the next theorem on the one-to-one correspondence of A+ c /∼ and S formulated in terms of the notion of the Revuz measure. For A ∈ A+ c , its Revuz measure will be denoted by µA . According to Theorem A.3.5, µA is characterized by the following formula: for any f ∈ B+ (E), t 1 f (Xs )dAs µA , f = lim Em t↓0 t 0 ∞ = lim αEm e−αt f (Xs )dAs . (4.1.1) α→∞

0

T HEOREM 4.1.1. (i) For any A ∈ A+ c , µA ∈ S. (ii) For any µ ∈ S, there exists A ∈ A+ c satisfying

µA = µ uniquely up to the m-equivalence. (iii) For A ∈ A+ c and µ ∈ S, the following three conditions are mutually equivalent. (a) µA = µ. (b) For any f , h ∈ B+ (E) and t > 0, t t f (Xs )dAs = Ps h, f · µds. (4.1.2) Eh·m 0

0

(c) For any f , h ∈ B+ (E) and α > 0, ∞ −αs e f (Xs )dAs = Rα h, f · µ. Eh·m 0

(4.1.3)

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CHAPTER FOUR

We shall use the following notations in the sequel: for A ∈ A+ c , f ∈ bB(E), ∞ e−αt f (Xt )dAt , α > 0, x ∈ E \ N, (4.1.4) UAα f (x) = Ex 0

RAα

∞

f (x) = Ex

−αt −At

e

e

f (Xt )dt ,

α > 0, x ∈ E \ N,

(4.1.5)

0

where N is an exceptional set of A. Exercise 4.1.2. using e−βt − e−αt = Derive the Afollowing t identities −βt t −(α−β)s As t ds and e − 1 = 0 e dAs , respectively. For α, β > 0, (α − β)e 0e f ∈ bB(E), β

β

UAα f (x) − UA f (x) + (α − β)Rα UA f (x) = 0,

x ∈ E \ N,

(4.1.6)

RAα f (x) − Rα f (x) + UAα RAα f (x) = 0,

x ∈ E \ N.

(4.1.7)

1 L EMMA 4.1.3. Let A ∈ A+ c , f ∈ B+ (E). If UA 1 ≤ R1 f , [m], then µA (E) ≤ m, f .

Proof. By Lemma A.2.11, βmRβ ≤ m. From (4.1.6) and the resolvent equation (1.1.25), we have for β > 1, β

βm, Rβ f − UA 1 = βm, R1 f − UA1 1 − (β − 1)βm, Rβ (R1 f − UA1 1) ≥ βm, R1 f − UA1 1 − (β − 1)m, R1 f − UA1 1 = m, R1 f − UA1 1 ≥ 0. β

By (4.1.1), µA (E) = limβ→∞ βm, UA 1 ≤ limβ→∞ βmRβ , f = m, f .

Proof of Theorem 4.1.1(i). For A ∈ A+ c , with a properly exceptional set N and a strictly positive bounded function f ∈ B(E) ∩ L1 (E; m), we let ϕ be the function RA1 f defined by (4.1.5). Then ϕ(x) > 0 for every x ∈ E \ N. By (4.1.7), ϕ is a difference of finite 1-excessive functions R1 f , UA1 (RA1 f ) relative to the Hunt process XE\N and hence finely continuous q.e. relative to X, and accordingly quasi continuous by virtue of Theorem 3.3.3. Since N is E-polar by Theorem 3.1.3, by Theorem 1.3.14(iii), there is a Cap1 -nest {En } so that N ⊂ ∩n (E \ En ) and ϕ|En is continuous for each n. If we put 1 , Fn = x ∈ En : ϕ(x) ≥ n then the increasing sequence {Fn } of closed sets becomes a nest. To see this, put Bn = {x ∈ E \ N : ϕ(x) ≤ 1n }, σn = σBn , σ = limn→∞ σn . Since ϕ is finely

133


continuous relative to XE\N , we have for x ∈ E \ N ζ 1 Ex e−t f (Xt )e−At dt = Ex e−σn e−Aσn ϕ(Xσn ) ≤ . n σn By letting n → ∞, we get Px (σ < ζ ) = 0, x ∈ E \ N. Hence, by the inclusion E \ Fn ⊂ (E \ En ) ∪ Bn and Theorem 3.1.4, we can conclude that {Fn } is a nest. On account of Theorem A.3.5, the Revuz measure µA of A charges no mpolar set and hence it charges no E-polar set. By the same theorem, we see that for each n, the Revuz measure of the positive continuous additive functional An = 1Fn · A equals 1Fn · µA . Since

we get µA (Fn ) ≤ n

UA1 n 1 = UA1 1Fn ≤ nUA1 ϕ ≤ nR1 f ,

E

fdm < ∞ by Lemma 4.1.3. Thus µA ∈ S.

We next prove Theorem 4.1.1(iii). We prepare a lemma to this end. L EMMA 4.1.4. Suppose A ∈ A+ c , µA (E) < ∞. Then Rα h, µA ≤ h · m, UAα 1

for any α > 0 and h ∈ B+ (E).

(4.1.8)

Proof. We may assume that h ∈ bB+ (E) ∩ L (E; m). By virtue of Theorem A.3.5(iv), ∞ e−βs Rα h(Xs )dAs . Rα h, µA = lim βEm (4.1.9) 1

β→∞

0

Since the discontinuous points of the sample path s → Xs are at most countable and A is continuous, we have ∞ ∞ Em e−βs Rα h(Xs )dAs = Em e−βs Rα h(Xs− )dAs . (4.1.10) 0

0

We notice that if we put Zn := −βk (Ak+1 − Ak ), then Zn ≤ k≥0 e Em [Y] =

−β nk Rα h(X k − )(A k+1 k≥0 e n n h∞ Y for n ≥ 1 and α

n

−βk e−βk Em EXk (A1 ) ≤ e Em [A1 ]

k≥0

≤

− A k ) and Y :=

k≥0

1 µA (E) < ∞. 1 − e−β

Since Rα h is an α-excessive function belonging to F , Rα h(Xs− ) is left continuous in s ∈ (0, ζ ) Pm -a.e. on account of Theorem 3.1.7. Moreover, the measure dAs concentrates on (0, ζ ), Rα h is uniformly bounded, and e−βs dAs

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CHAPTER FOUR

∞ is integrable Pm -a.e. Consequently, Zn converges to 0 e−βs Rα h(Xs− )dAs as n → ∞ Pm -a.e. Combining this with the above notice, we can see that limn→∞ Em [Zn ] coincides with the right hand side of (4.1.10), and hence limβ→∞ limn→∞ βEm [Zn ] is identical with the left hand side of (4.1.9). Because of the quasi-left-continuity of X, we have X(k/n)− = Xk/n Pm -a.e., and so we get by the Markov property Rα h, µA = lim lim βEm [Zn ] β→∞ n→∞

= lim lim β β→∞ n→∞

β→∞ n→∞

β→∞ n→∞

Em e−βk/n Rα h(Xk/n )EXk/n [A1/n ]

k≥0

≤ lim lim β = lim lim

−βk/n

e

Rα h(y)Ey [A1/n ]m(dy) E

k≥0

β (h, Rα (E· [A1/n ])). 1 − e−β/n

β 1 n 1−e−β/n

→ 1, n → ∞, for each β > 0, we are led to Rα h, µA ≤ lim n h(y)Rα (E· [A1/n ])(y)m(dy)

Since

n→∞

E

∞

= lim

n→∞ 0

= lim

∞

n→∞ 0

= lim

n→∞

=α 0

∞

ne−αt Eh·m EXt [A1/n ] dt ne−αt Eh·m [At+(1/n) − At ]dt α/n

n(e

− 1)

∞

−αs

e

1/n

Eh·m [As ]ds − n

1/n

−αs

e

Eh·m [As ]ds

0

e−αs Eh·m [As ]ds = h · m, UAα 1.

The last equality is obtained by a computation similar to (A.3.10).

Proof of Theorem 4.1.1(iii). Since the Laplace transform of both sides of (4.1.2) gives (4.1.3), the implication (c) ⇒ (b) follows from the uniqueness of the Laplace transform. Substituting h = 1 in the equality (4.1.2), we get the first identity in (4.1.1) with µA being replaced by µ. This means the implication (b) ⇒ (a). Hence it suffices to show the implication (a) ⇒ (c). To this end, we only need to prove the identity Rα h, µA = h · m, UAα 1 for any α > 0

and h ∈ B+ (E),

(4.1.11)


135

holding for any A ∈ A+ c , because, if (4.1.11) is established, then we know the validity of the same identity with µA , 1 being replaced by f · µA , f ( f ∈ B+ (E)), respectively, owing to Theorem A.3.5(iii). We have already shown in Lemma 4.1.4 that when A ∈ A+ c and µA (E) < ∞, (4.1.11) holds with the equality being replaced by the inequality ≤. Using this, let us first derive the equality (4.1.11) when A ∈ A+ c satisfies µA (E) < ∞,

µA ∈ S0 .

(4.1.12)

For µ ∈ S0 , its α-potential Uα µ ∈ F is defined by (2.3.6). By (1.1.28), (2.3.7), and Proposition 3.1.9, we have for any α > 0 and h ∈ bB+ (E) ∩ L1 (E; m) Rα h, µ = Eα (Rα h, Uα µ) = h · m, Uα µ.

(4.1.13)

Therefore, by the preceding lemma, we get the inequality Uα µA ≤ UAα 1 [m] holding for any α > 0. Let us put gα = UAα 1 − Uα µA , α > 0. From equations (2.3.8) and (4.1.6), 1 1 m, gα = βm, Rα gβ+α + m, gβ+α ≤ + βm, gβ+α . α β α+β

If we let β → ∞, then βm, UA 1 → µA (E) by Theorem A.3.5(iv) and βm, Uα+β µA = βRα+β 1, µA → µA (E) by (4.1.13). Hence we get gα = 0 [m], which combined with (4.1.13) again implies (4.1.11). For any A ∈ A+ c , µA ∈ S by the already shown statement (i) of Theorem 4.1.1 and hence, in accordance with Theorem 2.3.15, there exists a nest {Fk } such that the measure 1Fk · µA satisfies the condition (4.1.12) for each k. If we put Ak = 1Fk · A, then µAk = 1Fk · µA by Theorem A.3.5(iii) and we have the identity (4.1.11) for Ak . By letting k → ∞ and using the stochastic characterization Theorem 3.1.4, we arrive at (4.1.11) for A. Theorem 4.1.1(ii) follows from two propositions presented below. L EMMA 4.1.5. Consider a measure µ on E satisfying µ ∈ S0 ,

µ(E) < ∞.

(4.1.14)

A necessary and sufficient condition for A ∈ A+ c to satisfy µA = µ is UA1 1 = U1 µ

[m].

In this case, it also holds that UAα 1 = Uα µ

[m]

where Uα µ denotes the α-potential of µ.

for any α > 0,

(4.1.15)

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CHAPTER FOUR

Proof. Suppose µ satisfies (4.1.14). If A ∈ A+ c satisfies µA = µ, then, by substituting this into the left hand side of (4.1.11), we get by (4.1.13) that (h, U1 µ) = (h, UA1 1) for every h ∈ bB+ (E) ∩ L1 (E; m), from which follows (4.1.15). Conversely, if the equality (4.1.15) holds, then, by equations (2.3.8) and (4.1.6), we have UAα 1 = Uα µ [m] for any α > 0. By substituting this into the right hand side of (4.1.11), we obtain αRα h, µA = αRα h, µ for every h ∈ B+ (E). We put h = 1 and let α → ∞ to get µA (E) = µ(E) < ∞. Then the same procedure for h ∈ bC+ (E) gives h, µA = h, µ, yielding µA = µ.

P ROPOSITION 4.1.6. For any µ ∈ S00 , there exists A ∈ A+ c such that µA = µ. Proof. Since µ ∈ S00 satisfies (4.1.14), it suffices to construct A ∈ A+ c satisfying (4.1.15) by virtue of Lemma 4.1.5. The 1-potential U1 µ of µ is in F and 1-excessive relative to {Tt }, and hence we can find a finite Borel measurable quasi continuous version f of U1 µ and a Borel properly exceptional set N such that

nRn+1 f (x) ↑ f (x), n → ∞ for any x ∈ E \ N (4.1.16) f (x) = 0, for any x ∈ N. We put

gn (x) =

n( f (x) − nRn+1 f (x)), x ∈ E \ N x ∈ N.

0,

Then R1 gn (x) ↑ f (x), x ∈ E \ N, n → ∞ and further R1 gn is E1 -convergent to f because R1 gn − f 2E1 ≤ −(gn , f ) + f 2E1 → 0, n → ∞. For each n, we define the functional An by An (t, ω) =

t

e−s gn (Xs (ω))ds,

t ≥ 0, ω ∈ .

0

Let us show for any ν ∈ S00 that Eν ( A (∞))2 ≤ 2Mν E1 (µ) R1 gn − R1 g E1 An (∞) −

(4.1.17)

with Mν = U2 ν∞ . Without loss of generality, we assume that ν ∈ S00 is a probability measure on E.

137


If we put gn, = gn − g , n > , then the left hand side of (4.1.17) equals ∞ ∞ 2Eν e−s gn, (Xs )ds e−u gn, (Xu )du 0

= 2Eν

s

∞

e−2s (gn, · R1 gn, )(Xs )ds = 2ν, R2 (gn, R1 gn, )

0

= 2U2 ν, gn, R1 gn, ) ≤ 2U2 ν, gn R1 gn, ) ≤ 2Mν (gn , R1 gn − R1 g ) = 2Mν E1 (R1 gn , R1 gn − R1 g ). Then the Schwarz inequality and the bound R1 gn 2E1 = (gn , R1 gn ) ≤ (gn , f ) = µ, R1 gn ≤ µ, f = E1 (µ) lead us to (4.1.17). On the other hand, we have Eν An (∞)Ft = An (t) + e−t EXt [ An (∞)] = An (t) + e−t R1 gn (Xt ). Accordingly, Mn (t) = An (t) + e−t R1 gn (Xt ),

0≤t≤∞

(4.1.18)

is a martingale relative to ({Ft }, Pν ), ν ∈ S00 , and, by Doob’s inequality, it holds for any ε > 0 that 1 An (∞) − A (∞))2 . (4.1.19) Pν sup |Mn (t) − M (t)| > ε ≤ 2 Eν ( ε 0≤t≤∞ Choose a subsequence {nk } satisfying R1 gnk+1 − R1 gnk E1 ≤ 2−3k and − Mnk+1 (t)| > 2−k }. It follows from (4.1.17) and put k = {sup0≤t≤∞ |Mnk (t)√ (4.1.19) that Pν (k ) ≤ 2Mν E1 (µ)2−k . By the Borel-Cantelli lemma, we are led to Pν (lim supk→∞ k ) = 0 for every ν ∈ S00 , which further implies that Px (lim supk→∞ k ) = 0 for q.e. x ∈ E, on account of Corollary 2.3.11. Combining this with the expression (4.1.18) and Theorem 3.5.4, we arrive at the following conclusion: there exist a subsequence {nk } and a properly N , where exceptional set N ⊃ N such that Px () = 1 for every x ∈ E \ : = {ω ∈ Ank (∞, ω) < ∞, Ank (t, ω) is uniformly convergent on each finite subinterval of [0, ∞)},

(4.1.20)

= {ω ∈ : σ and σ N = ∞, N = ∞}. / we let Ank (t, ω), while, for ω ∈ For ω ∈ , we put A(t, ω) = limk→∞ t A(s, ω), t ∈ [0, ∞]. Then A A(t, ω) = 0. Finally, we define A(t, ω) = 0 es d is a positive continuous additive functional of X with defining set and exceptional set N.

138

CHAPTER FOUR

In order to complete the proof of Proposition 4.1.6, it suffices to show A(∞)] = ν, f for every ν ∈ S00 , on account of Corollary 2.3.11. Since Eν [ An (∞) is L2 (Pν )-convergent, so is Mn (t). Therefore, Mn (∞) = A(t)] + e−t ν, Pt f = lim Eν [Mn (t)] Eν [ n→∞

An (∞)] = lim ν, R1 gn = ν, f . = lim Eν [ n→∞

n→∞

Since µ ∈ S00 , f ∞ < ∞ and ν, Pt f ≤ ν(E) f ∞ < ∞. We get the desired equality by letting t → ∞. Exercise 4.1.7. Show that for A, B ∈ A+ c , the identity ∞ ∞ Ex e−t dAt · e−s dBs = UA2 (UB1 1)(x) + UB2 (UA1 1)(x) 0

(4.1.21)

0

holds for x ∈ E \ (NA ∪ NB ), NA , NB being exceptional sets of A, B, respectively. P ROPOSITION 4.1.8. For a given µ ∈ S00 , A ∈ A+ c satisfying µA = µ is unique up to the m-equivalence. Proof. Given µ ∈ S00 , suppose A(1) , A(2) ∈ A+ c satisfy µA(1) = µA(2) = µ. Denoting by f a Borel measurable quasi continuous version of U1 µ, we see At = from Lemma 4.1.5 that UA1 (1) 1 = UA1 (2) 1 = f q.e. We use the notation t −s + e dA , t ∈ [0, ∞] for A ∈ A . If we put s c 0 (i) (j) A∞ · A∞ , i, j = 1 or 2, gij (x) = Ex then, by Exercise 4.1.7, gij = UA2 (i) f + UA2 (j) f q.e. For any strictly positive bounded m-integrable function h on E, we get from (4.1.3) hm, gij = R2 h, f · µA(i) + R2 h, f · µA(j) = 2R2 h, f µ ≤ h∞ f ∞ µ(E) < ∞. Consequently, (1) 2 Eh·m ( A(2) A∞ − = hm, g11 − 2g12 + g22 = 0. ∞) −t(i) (i) Since A(i) ∞ = At + e A∞ ◦ θt and (1) (1) 2 (2) 2 = 0, Eh·m ( A(2) A∞ − ∞ ) ◦ θt = EPt h·m (A∞ − A∞ )

(2) 2 A(1) it follows that Eh·m [( t − At ) ] = 0 for every t ≥ 0, yielding the (1) (2) m-equivalence of A , A .


139

Proof of Theorem 4.1.1(ii). For any µ ∈ S, there exists a nest {Fk } such that µ(k) = 1Fk · µ ∈ S00 for each k by virtue of Theorem 2.3.15. (k) for By Proposition 4.1.6, there exists A(k) ∈ A+ c satisfying µA(k) = µ (k+1) equals each k. According to Theorem A.3.5, the Revuz measure of 1Fk · A 1Fk · µA(k+1) = 1Fk · µ(k+1) = µ(k) , and consequently, 1Fk · A(k+1) and A(k) are m-equivalent, k = 1, 2, . . . , on account of Proposition 4.1.8. By Lemma A.3.2, we can take a common defining set ⊂ and an exceptional set N ⊂ E for {A(k) } such that (1Fk · A(k+1) )t (ω) = A(k) t (ω) for any t ≥ 0, ω ∈ and k ≥ 1. Since {Fk } is a nest, we may assume that σ (ω) = lim σE\Fk (ω) ≥ ζ (ω) k→∞

by redefining , N if necessary. Putting F0 = ∅, we now let

(k) At (ω), At (ω) = Aσ (ω)− (ω),

for any ω ∈

σE\Fk−1 ≤ t < σE\Fk , k = 1, 2, . . . , t ≥ σ (ω).

A is then a positive continuous additive functional of X possessing , N as its defining set and exceptional set. When k < , A(k) = 1Fk · A() . By letting → ∞, we obtain A(k) = 1Fk · A, which means that 1Fk · µA = 1Fk · µ for every k ≥ 1, namely, µA = µ. satisfy µA = µB = µ. Finally we assume, for µ ∈ S, A, B ∈ A+ c Take a nest {Fk } satisfying µ(k) = 1Fk · µ ∈ S00 as above. Since µ1Fk ·A = µ1Fk ·B = µ(k) , Proposition 4.1.8 applies in concluding that 1Fk · A and 1Fk · B are m-equivalent. k being arbitrary, A, B are m-equivalent. The proof of Theorem 4.1.1 is now complete. Let D ⊂ E be a nearly Borel, finely open set and X D be the part process of X on D. X D is an m|D -symmetric standard process on D. The transition function D D {PD t ; t ≥ 0} and the resolvent {Rα ; α > 0} of X are given by (3.3.2). Exercise 4.1.9. (i) Define the shift operator θt0 on by

θt ω for t < τD (ω), θt0 ω = ω∂ for t ≥ τD (ω),

(4.1.22)

where ω∂ denotes a specific element of with Xt (ω∂ ) = ∂ for every t ≥ 0. Using the relation t + τD (θt ω) = τD (ω) whenever t < τD (ω), show that D (ω), s, t ≥ 0. XtD (θs0 ω) = Xs+t (ii) Denote the minimum admissible filtration σ {XsD : s ≤ t} of X D by G0t and, for µ ∈ P(D∂ ), let Gµt = σ {G0t , N } for each t ≥ 0, where N is the µ family of all Pµ -null sets. Further let Gt = ∩µ∈P(D∂ ) Gt . Define the σ -field of

140

CHAPTER FOUR µ

events strictly prior to time τD ∧ t by F(τD ∧t)− = σ { ∩ {s < τD ∧ t} : ∈ Fsµ , s ≥ 0}. Prove the inclusion µ

µ

µ

F(τD ∧t)− ⊂ Gt ⊂ FτD ∧t .

(4.1.23)

Show that, for any F ∈ bG∞ and µ ∈ P(D∂ ), F ◦ θt0 ∈ bG∞ , Eµ [F ◦ θt0 | Ft ] = EXtD [F] Pµ -a.s.

(4.1.24)

(iii) For an additive functional A of X, let Bt = At∧τD , t ≥ 0. Show that {Bt }t≥0 satisfies the additivity Bs+t = Bs + Bt ◦ θs0 ,

s, t ≥ 0,

with respect to the shift operator θs0 defined by (4.1.22). Show that if A is continuous, then B is a {Gt }-adapted continuous additive functional of X D . Consider A ∈ A+ c and µ ∈ S such that µ is the Revuz measure of A. P ROPOSITION 4.1.10. Bt := At∧τD , t ≥ 0, is a PCAF of X D with Revuz measure µD := µ|D . More specifically, it holds for any f , h ∈ B+ (D) that

t∧τD

Eh·m

f (Xs )dAs =

0

0

τD

Eh·m

−αs

e 0

t

PD s h, f · µD ds,

t > 0,

(4.1.25)

f (Xs )dAs = RD α h, f · µD ,

α > 0.

(4.1.26)

Proof. By Exercise 4.1.9, {Bt } is a PCAF of X D . It suffices to show (4.1.26) for a bounded f and µ ∈ S00 in view of Theorem 2.3.15. Let UA0,α f (x) = τ Ex 0 D e−αs f (Xs )dAs . Then UA0,α f = UAα f − HαE\D UAα f and UAα f is a quasi continuous version of the α-potential Uα ( f · µ) by Lemma 4.1.5. On account of Theorem 3.2.2, the left hand side of (4.1.26) is equal for a bounded h ∈ L2 (E; m) to (h, UA0,α f ) = Eα (Rα h, UAα f − HαE\D UAα f ) = Eα (Rα h − HαE\D Rα h, UAα f ) D = Eα (RD α h, Uα ( f · µ)) = Rα h, f · µD .

We close this section by showing that under the absolute continuity condition for X, (AC) the transition function Pt (x, ·) of X is absolutely continuous with respect to m for every t > 0, x ∈ E,

141


positive continuous additive functionals in the strict sense of X can also be characterized. If the exceptional set of A ∈ A+ c can be taken to be empty, namely, if its defining set can be chosen in such a way that Px () = 1 for every x ∈ E, A is said to be a positive continuous additive functional in the strict sense. Let us denote by A+ c1 the family of all positive continuous additive functionals of X in the strict sense. A, B ∈ A+ c1 are called equivalent in the strict sense if Px (At = Bt ) = 1,

for any t > 0

and x ∈ E.

(4.1.27)

In this case, we can find a common defining set such that Px () = 1 for every x ∈ E, and At (ω) = Bt (ω) for every t ≥ 0 and ω ∈ . A measure µ on (E, B(E)) is called smooth in the strict sense if there exists an ∞increasing sequence {En } of Borel measurable finely open sets such that n=1 En = E and 1En · µ ∈ S00 for every n ≥ 1. We denote by S1 the totality of smooth measures in the strict sense. T HEOREM 4.1.11. Suppose X satisfies the condition (AC). (i) For any A ∈ A+ c1 , µA ∈ S1 . (ii) For any µ ∈ S1 , there exists A ∈ A+ c1 satisfying µA = µ uniquely up to the equivalence in the strict sense. Proof of Theorem 4.1.11(i). Suppose A ∈ A+ c1 is given. For a strictly positive bounded Borel m-integrable function f on E, ϕ(x) = RAα f (x) is well defined for all x ∈ E by (4.1.5). Since equation (4.1.7) holds for all x ∈ E, ϕ is a difference of the 1-excessive functions Rα f , UAα RAα f , and is finely continuous. Further, ϕ is Borel measurable on account of Theorem A.2.17 and the assumption (AC). If we put En = {x ∈ E : ϕ(x) > 1/n} ,

n = 1, 2, . . . ,

(4.1.28)

then {En } is a sequence of Borel finely open sets increasing to E. For each n, the Revuz measure of An = 1En · A ∈ A+ c1 equals 1En · µA by virtue of Theorem A.3.5. Since UA1 n 1(x) ≤ nUA1 ϕ(x) ≤ nR1 f (x),

x ∈ E,

(4.1.29)

we get µAn (E) < ∞ in view of Lemma 4.1.3. Moreover, by the above inequality, Lemma 2.3.5, and Lemma 1.2.3, there exists µ ∈ S0 such that UA1 n 1 = U1 µ [m]. In the same way as in the proof of Lemma 4.1.5, the last identity implies that µ(E) = µAn (E) < ∞ and furthermore µ = µAn . Hence 1En · µA ∈ S00 The proof of Theorem 4.1.11(ii) can be reduced to the next proposition. Under the condition (AC), the resolvent kernel Rα (x, ·) is absolutely continuous

142

CHAPTER FOUR

with respect to m. It is known (cf. [73, Lemma 4.2.4]) that the density function rα (x, y) can be taken in such a way that it is symmetric in x and y, α-excessive in each variable, and satisfies the resolvent equation. In this case, if we let for µ ∈ S0 Rα µ(x) = rα (x, y)µ(dy), x ∈ E, E

then Rα µ can be verified to be a quasi continuous and α-excessive version of the potential Uα µ. In particular, a necessary and sufficient condition for µ to be in S00 is that µ(E) < ∞ and R1 µ is bounded. P ROPOSITION 4.1.12. Suppose X satisfies (AC). For any µ ∈ S00 , there exists A ∈ A+ c1 satisfying UA1 1(x) = R1 µ(x)

for every x ∈ E,

(4.1.30)

uniquely up to the equivalence in the strict sense. Proof. Such an A ∈ A+ c1 can be constructed by making use of Proposition 4.1.6. Indeed, according to that proposition, we can construct an A ∈ A+ c with a suitable defining set and exceptional set N satisfying (4.1.30) for m-a.e. x ∈ E. Since both sides of (4.1.30) are 1-excessive relative to XE\N , this identity holds for all x ∈ E \ N under the (AC). condition −1 and we let, for ω ∈ 0 , For this A, , we put 0 = n θ1/n

if the limit exists, limn→∞ At−(1/n) (θ1/n ω) At (ω) = 0 otherwise. We further define = ω ∈ 0 : At (ω) < ∞ for every t > 0 and A0+ (ω) = 0 . ) = 1 for every x ∈ E, A becomes a positive continuous additive Then Px ( and functional in the strict sense with a defining set A can be verified to satisfy (4.1.30). We refer to [73, Theorem 5.1.6] for the details. The uniqueness can be easily shown. If A, B ∈ A+ c1 satisfy the identity (4.1.30), then A, B are m-equivalent by Lemma 4.1.5 and Proposition 4.1.8, and consequently (4.1.27) holds for q.e. x ∈ E by Lemma A.3.2. Hence, by (AC), we have Px (At − A1/n = Bt − B1/n ) = Ex [PX1/n (At−(1/n) = Bt−(1/n) )] = 0 for any x ∈ E. It suffices to let n → ∞. Proof of Theorem 4.1.11(ii). For µ ∈ S1 , there is a sequence {En } of Borel finely open sets increasing to E such that 1En · µ ∈ S00 for every n ≥ 1. Then,


143

due to the quasi-left-continuity of the Hunt process X, it holds that Px lim σE\En ≥ ζ = 1, for any x ∈ E. n→∞

Noting this, we can make use of Proposition 4.1.12 to prove the existence and uniqueness of the desired positive continuous additive functional in the strict sense A just as in the proof of Theorem 4.1.1(ii).

4.2. DECOMPOSITIONS OF ADDITIVE FUNCTIONALS OF FINITE ENERGY The notion of an additive functional At (ω) accompanying a defining set and a (properly) exceptional set N of the m-symmetric Hunt process X is formulated in the beginning of the preceding section. In the preceding section, we dealt exclusively with a positive continuous additive functional. In the present and the next sections, we are concerned with an additive functional which is not necessarily positive but finite càdlàg. By this we mean that for each ω ∈ , At (ω) is finite, right continuous in t on [0, ∞), and has left limits in t on (0, ∞). Thus, in Sections 4.2 and 4.3, we always require an additive functional of X to be finite càdlàg without a specific notice. We define the energy e(A) of an additive functional At of X by 1 Em [A2t ], t→0 2t

e(A) = lim

(4.2.1)

whenever the limit on the right hand side exists. For two additive functionals At , Bt of X, their mutual energy e(A, B) is defined by e(A, B) = lim t→0

1 Em [At Bt ], 2t

(4.2.2)

whenever the limit exists. We shall consider three types of additive functionals mentioned and studied below. (1◦ ) Additive functionals generated by functions in Fe For u ∈ Fe , we let u(Xt ) − u(X0 ), A[u] t =

t ≥ 0.

(4.2.3)

Since the quasi continuous modification u is decided uniquely up to q.e., A[u] is uniquely determined up to the m-equivalence. But we always let u(∂) = 0 by convention. In view of Theorem 3.5.4, u is then an element of C∞ ({Fk }) u|Fk ∪{∂} is continuous and bounded for each k. for some strong nest {Fk } and

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Therefore, we can assume A[u] to be finite càdlàg on [0, ∞) by choosing its exceptional set appropriately. In order to compute the energy of the additive functional A[u] , we need first to draw from (E, F) a positive Radon measure κ which will be called its killing measure. For u ∈ F, since 0≤

1 1 1 2 = (u − Pt u, u) − (1 − Pt 1, u2 ), Em (A[u] t ) 2t t 2t

(4.2.4)

we have 1 2 ≤ E(u, u), Em (A[u] t ) 2t

1 (1 − Pt 1, u2 ) ≤ E(u, u). 2t

(4.2.5)

Hence it holds for any compact set K ⊂ E that 1 sup 0 0, Eh·m e−αζ f (Xζ − ) = Rα h, f · κ. Proof. It suffices to show this for f ∈ Cc (E) ∩ F and g ∈ bB(E) ∩ L1 (E; m). For a sequence {tn } decreasing to 0 and satisfying (4.2.6), we have ∞

−αζ −αktn e f (X(k−1)tn ) : (k − 1)tn < ζ ≤ ktn Eh·m e f (Xζ − ) = lim Eh·m n→∞

= lim

n→∞

= lim

n→∞

k=1 ∞

Eh·m f (X(k−1)tn )e−αktn (1 − Ptn 1)(X(k−1)tn )

k=1 ∞

e−αktn f · P(k−1)tn h, 1 − Ptn 1 .

k=1

Since we get from (4.2.7) 1 ( f · Rα h, 1 − Ptn 1) = n→∞ tn

lim

f (x)Rα h(x)κ(dx), E

the last expression in the above display equals lim In + f (x)Rα h(x)κ(dx), n→∞

E

where In =

∞

1 f · e−αktn tn P(k−1)tn h − e−αs Ps hds , (1 − Ptn 1) . tn (k−1)t n k=1

ktn

On the other hand, we obtain from |Rα 1(x) − Ps Rα 1(x)| ≤ 2s for s > 0, the estimates |P(k−1)tn h − Ps h| ≤ 2tn g∞

for (k − 1)tn ≤ s < ktn .

147


It follows −αkt e n P(k−1)t h · tn − n =

ktn

−αs

e

(k−1)tn

ktn

−αktn

e

−αs

−e

Ps hds

−αs

P(k−1)tn h + e

(k−1)tn

P(k−1)tn h − Ps h ds

≤ αe−α(k−1)tn h∞ tn2 + 2e−α(k−1)tn g∞ tn2 ≤ 3g∞ · e−α(k−1)tn tn2 . Therefore, |In | ≤ 3g∞ 1−etn−αtn (| f |, 1 − Ptn 1), which tends to 0 as n → ∞. The identity (4.2.8) particularly implies that the probability of X continuously escaping to ∂ is small compared to that of jumping directly to ∂. P ROPOSITION 4.2.3. t > 0. Then

Let rt (x) := Px (Xζ − = ∂ and ζ ≤ t) for x ∈ E and

1 for every u ∈ Fe . (4.2.11) lim (u2 , rt ) = 0 t→0 t Proof. Since 1 − Pt 1(x) = Px (Xζ − ∈ E and ζ ≤ t) + rt (x) for x ∈ E, we have 1 (1 − Pt 1, u2 ) = It + 1t (u2 , rt ), where t

1 1E (Xs− )1{∂} (Xs ) . It = Eu2 ·m t s≤t By (A.3.31) and (4.1.2), t 1 1 t N(Xs , {∂})dHs = Ps u2 , κds. It Eu2 ·m t t 0 0 Therefore, (4.2.8) and Fatou’s lemma yield u2 , κ ≤ lim inf It ≤ lim sup It ≤ u2 , κ, t↓0

t↓0

which then implies (4.2.11). (2◦ ) Martingale additive functionals

An additive functional Mt of X in the sense of Section 4.1 is called a martingale additive functional (MAF in abbreviation) if, for each t > 0, Ex [Mt2 ] < ∞,

Ex [Mt ] = 0,

q.e. x ∈ E.

(4.2.12)

Then from the additivity of M and the Markov property of X, Ex [Mt+s |Fs ] = Ex [Ms + Mt ◦ θs |Fs ] = Ms + EXs [Mt ] = Ms , Px -a.s.,

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CHAPTER FOUR

namely, Mt is a Px -martingale for q.e. x ∈ E. Denote by M the totality of martingale additive functionals of X. For any M ∈ M, there exists M ∈ A+ c satisfying Ex [Mt2 ] = Ex [Mt ],

∀t ≥ 0,

q.e. x ∈ E,

which is unique up to the m-equivalence. In fact, if we let N be an exceptional set of M, then M can be regarded as the martingale additive functional in the strict sense of the restricted Hunt process XE\N , and Section A.3.3(1) yields M as above. M is called the predictable quadratic variation of M. The above identity implies that the energy e(M) of M can be expressed using the Revuz measure µM of M as e(M) = lim t↓0

1 1 1 Em [Mt2 ] = lim Em [Mt ] = µM (E). t↓0 2t 2t 2

(4.2.13)

◦

In what follows, we denote M = {M ∈ M : e(M) < ∞}. L EMMA 4.2.4. For A ∈ A+ c with Revuz measure µ ∈ S and for ν ∈ S00 , Eν [At ] ≤ (1 + t) U1 ν∞ µ(E),

t ≥ 0.

(4.2.14)

Proof. On account of Theorem 2.3.15 and Theorem A.3.5, it suffices to derive (4.2.14) for µ ∈ S00 . In this case, ct (x) = Ex [At ] ∈ F. Indeed, we have from (4.1.2) t µ, Ps f ds, t ≥ 0, Efm [At ] = 0

and, for s < t, t 1 1 (ct − Ps ct , ct ) = µ, (Pr ct − Ps+r ct )dr s s 0 s t+s 1 1 = Pr ct dr − Pr ct dr . µ, µ, s s 0 t Since, by Lemma 4.1.5, µ, E.[At ] ≤ et µ, UA1 1 = et µ, U1 µ < ∞,

(4.2.15)

the limit µ, ct − Pt ct is finite and we obtain ct ∈ F together with E(ct , ct ) = µ, ct − Pt ct in view of (1.1.4) and (1.1.5). Similarly, we obtain u − µ, Pt u, E(ct , u) = µ,

u ∈ F.

In particular, it holds for ν ∈ S00 that Eν [At ] = ν, ct = E1 (ct , U1 ν) = µ, U1 ν − Pt U1 ν + (ct , U1 ν) ≤ U1 ν∞ (µ(E) + m, ct ), which leads us to (4.2.14) because m, ct ≤ tµ(E).

149

ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES ◦

T HEOREM 4.2.5. For any e-Cauchy sequence {Mt(n) } in M, there exists a ◦ unique M ∈ M such that limn→∞ e(M (n) − M) = 0. Further, there exists a subsequence {nk } such that Mt(nk ) converges to Mt uniformly in t on each compact subinterval of [0, ∞) Px -a.s. for q.e. x ∈ E. Proof. By (4.2.13), the total mass µM (E) of the Revuz measure of M ∈ A+ c equals 2e(M). Hence, by the martingale inequality and Lemma 4.2.4, we are led to the inequality 1 2 (4.2.16) Pν sup |Ms | > λ ≤ 2 Eν [MT2 ] ≤ 2 (1 + T)U1 ν∞ e(M) λ λ 0≤s≤T holding for any ν ∈ S00 and T > 0. Choosing a subsequence {nk } such (n ) that e(M (nk+1 ) − M (nk ) ) < 2−3k and taking λ = 2−k , Mt = Mt k+1 − Mt(nk ) in (4.2.16), we obtain (nk+1 ) (nk ) −k − Ms | > 2 ≤ 2(1 + T)U1 ν∞ 2−k . Pν sup |Ms 0≤s≤T

By the Borel-Cantelli lemma, we have Pν () = 0 for ∞ ! ∞ = sup |Ms(nk+1 ) − Ms(nk ) | > 2−k . n=1 k=n

0≤s≤T

Since the above holds for every ν ∈ S00 , we conclude by Corollary 2.3.11 that / , limk→∞ Ms(nk ) (ω) = Ms (ω) exists as a Px () = 0 for q.e. x ∈ E. For ω ∈ uniform convergent limit on [0, T]. By the inequality (4.2.16), this convergence also takes place in L2 (Pν ) so that Eν (Mt2 ) < ∞ and Eν [Mt ] = 0. Consequently, Ex (Mt2 ) < ∞, Ex [Mt ] = 0 for q.e. x ∈ E and M ∈ M. Furthermore, by Fatou’s lemma, " # " # 1 1 Em (Mt(n) − Mt )2 ≤ lim inf Em (Mt(n) − Mt(nk ) )2 k→∞ 2t 2t ≤ lim inf e(M (n) − M (nk ) ), k→∞

and accordingly, e(M ◦

(n)

− M) ≤ lim infk→∞ e(M (n) − M (nk ) ). In particular, M ∈

M and M (n) is e-convergent to M as n → ∞.

(3◦ ) Continuous additive functionals of zero energy A continuous additive functional N of X is called a continuous additive functional of zero energy if Ex (|Nt |) < ∞ q.e. x ∈ E for each t > 0, e(N) = 0.

(4.2.17)

Nc will denote the totality of continuous additive functionals of zero energy.

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CHAPTER FOUR

t A typical example of N ∈ Nc is given by Nt = 0 f (Xs )ds, t ≥ 0 for a nearly t Borel function f ∈ L2 (E; m). In fact, the right hand side of Ex [ 0 |f (Xs )|ds] ≤ et R1 |f |(x) is finite for q.e. x ∈ E by Proposition 3.1.9, so that N is a continuous additive functional. Moreover, t t 2 Em [Nt ] = 2Em f (Xs ) f (Xv )dvds 0

= 2Em =2

0

0

t−s

(Ps 1, f · Pv f )dvds

0 t

=2 s

s

t−s

f (Xs )Pv f (Xs )dvds 0

t 0

where Ss f (x) :=

t

(Ps 1, f St−s f )ds, 0

Pr f (x)dr, and so we have Em [Nt2 ] ≤ t2 · ( f , f ).

We are now in a position to prove the following decomposition theorem due to the second-named author by making use of Theorem 3.5.4, Theorem 4.2.5, and the above example. ◦

T HEOREM 4.2.6. For any u ∈ Fe , there exist M [u] ∈ M and N [u] ∈ Nc uniquely such that A[u] = M [u] + N [u] ,

Px -a.s. for q.e. x ∈ E.

(4.2.18)

◦

Proof. Uniqueness: If M ∈ M ∩ Nc , then (4.2.13) implies that µM (E) = 0 and M = 0, Px -a.s. for q.e. x ∈ E. Therefore, M = 0. Existence: For any nearly Borel function f ∈ L2 (E; m) and u = R1 f ∈ F, define Nt[u] by t Nt[u] = (u(Xs ) − f (Xs ))ds 0

[u] [u] ∈ Nc . It is easy to verify and let Mt[u] = A[u] t − Nt . By the above example, N ◦

that M [u] ∈ M. Moreover, M [u] ∈ M because e(A[u] ) < ∞. For a general u ∈ Fe , choose a sequence of functions un = R1 fn , fn ∈ L2 (E; m) such that E(un − u, un − u) → 0 and un → u m-a.e. as n → ∞. Let A[un ] = M [un ] + N [un ] be the decomposition given above. By virtue of [un ] Theorem 3.5.4, there exists a subsequence {nk } such that, as k → ∞, At k [u] converges to At uniformly on each compact time interval Px -a.s. for q.e.

151


x ∈ E. Since we have from (4.2.9) e(M [um ] − M [un ] ) = e(A[um ] − A[un ] ) ≤ E(um − un , um − un ), we find a suitable subsequence of {nk } (denoted by {nk } again) such that M [unk ] ◦

converges as k → ∞ to some M [u] ∈ M in e-metric as well as uniformly on each compact time interval Px -a.s. for q.e. x ∈ E by virtue of Theorem 4.2.5. [un ] As a result, Nt k converges to a continuous additive functional Nt[u] and it [u] holds that At = Mt[u] + Nt[u] . N [u] is of zero energy because " 1 3 [un ] 2 lim Em (Nt[u] )2 ≤ lim Em A[u] t − At t→0 2t t→0 2t 2 2 # + Mt[u] − Mt[un ] + Nt[un ] ≤ 6 E(u − un , u − un ) → 0

as n → ∞.

4.3. PROBABILISTIC DERIVATION OF BEURLING-DENY FORMULA Any regular Dirichlet form (E, F) admits the following representation. This was first announced in the seminal paper [8] of A. Beurling and J. Deny published in 1959 and its analytic proof was given more than a decade later. For u, v ∈ F, 1 ( u(x) − u(y))( v(x) − v(y))J(dxdy) E(u, v) = E (c) (u, v) + 2 E×E\d u(x) v(x)κ(dx), (4.3.1) + E

where u denotes a quasi continuous version of u ∈ F. Here J is a symmetric Radon measure on E × E \ d, where d denotes the diagonal set, and κ is a Radon measure on E. E (c) is a symmetric form possessing the strongly local property in the sense of Definition 1.3.17(v). In this section, we maintain the setting of the preceding two sections on E, m, X = (Xt , Px ), and the Dirichlet form (E, F ). For u ∈ Fe , we have drawn in Theorem 4.2.6 of the preceding section the martingale part M [u] from the u(Xt ) − u(X0 ). In this section, we shall derive the additive functional A[u] t = Beurling-Deny formula (4.3.1) holding for u, v ∈ Fe by means of a further decomposition of the MAF M [u] . As a result, we can give more specific probabilistic expressions of the measures J, k in terms of the Lévy system of the Hunt process X, which describes the jumping and killing behaviors of X. They will also readily imply the sample path characterizations of the local property and the strongly local property of the form.

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CHAPTER FOUR

To this end, we first make a general observation. As stated at the beginning of Section 4.1, an additive functional A of X is accompanied by a properly exceptional set N called an exceptional set of A so that A can be regarded as an additive functional in the strict sense (an AF with a null exceptional set) with respect to the restricted Hunt process X|E\N . The MAF M ∈ M of X already studied in the preceding section is, denoting its exceptional set by N, an element of the family M of the martingale additive functionals in the strict sense with respect to the Hunt process XE\N in the sense of Section A.3.3. Therefore, by invoking the statements (1), (2), (4) of Section A.3.3, we can produce uniquely its continuous part M c , purely discontinuous part M d , predictable quadratic variation M, and quadratic variation [M] as additive functionals in the strict sense of XE\N . By regarding them as additive functionals (admitting exceptional sets) of X again, they are determined by M in the following ways uniquely up to the m-equivalence: We introduce subfamilies of the space M of the MAFs of X by Mc = {M ∈ M; Px (Mt is continuous in t) = 1, q.e. x} Md = {M ∈ M; Px (M, L = 0) = 1 for q.e. x and for any L ∈ Mc }. Then M ∈ M admits a unique decomposition M = M c + M d , M c ∈ Mc , M d ∈ Md . The predictable quadratic variation M is a positive continuous additive functional (M ∈ A+ c ) characterized by the following property: Ex [Mt2 ] = Ex [Mt ], Further, [M]t = M c t +

(Ms )2 ,

q.e. x ∈ E, p

[M]t = Mt ,

for any t ≥ 0. for any t ≥ 0.

(4.3.2)

(4.3.3)

s≤t

Here Ms = Ms − Ms− and Ap denotes the additive functional obtained as the dual predictable projection of the integrable additive functional A of bounded variation. For u ∈ Fe , the decomposition A[u] = M [u] + N [u] ,

◦

M [u] ∈ M(⊂ M), N [u] ∈ Nc

holds by Theorem 4.2.6. Let us denote the continuous part and purely discontinuous part of M [u] by M [u],c and M [u],d , respectively. As Nt[u] is a finite continuous additive functional, we see from the above decomposition that the jumps of M [u] coincide with those of A[u] . In particular, we have from Theorem 3.1.7(i) u(Xζ ) − u(Xζ − ) = − u(Xζ − ). Mζ[u] =

(4.3.4)

If we let Kt = − u(Xζ − )1{ζ ≤t} , t ≥ 0, then Kt is an additive functional of X and it changes only by a jump when X jumps from E to ∂ at its lifetime ζ .

153


By (4.3.4) and Section A.3.3(3), K has its dual predictable projection K p with K − K p ∈ Md . Due to the quasi-left-continuity, Xt does not jump at any predictable stopping time. Hence K p is a continuous additive functional by virtue of Section A.3.3(3). We now let u(Xζ − )1{ζ ≤t} − (− u(Xζ − )1{ζ ≤t} )p , Mt[u],k = −

t ≥ 0.

(4.3.5)

We further let M [u],j = M [u],d − M [u],k . Then M [u],k , M [u],j ∈ Md , M [u],k jumps [u],j

only at t = ζ and Mζ be decomposed as

◦

= 0 in view of Section A.3.3(5), and M [u] ∈ M can

M [u] = M [u],c + M [u],j + M [u],k .

(4.3.6)

◦

We next define for M, N ∈ M, [ M, N ] =

1 ([ M + N ] − [ M ] − [ N ]) . 2

It then follows from the first identity of (4.3.3) that [ M [u],j , M [u],k ]t =

Ms[u],j · Ms[u],k = 0.

0<s≤t

The second identity of (4.3.3) then implies M [u],j , M [u],k = 0. Accordingly, we conclude that (4.3.6) is actually an orthogonal decomposition with respect to , . Consequently, if we denote the Revuz measure of the predictable quadratic variation M ∈ A+ c of M ∈ M by µM , we obtain from the decomposition (4.3.6) that µM[u] = µM[u],c + µM[u],j + µM[u],k .

(4.3.7)

In what follows, we write for u, v ∈ Fe µu,v = µM[u] ,M[v] ,

µiu,v = µM[u],i ,M[v],i with i = c, j, k,

(4.3.8)

When u = v, we simply write µu for µu,u and µiu for µiu,u with i = c, j, k. Observe that µu,v = 14 (µu+v − µu−v ) and similar relations hold for µiu,v with i = c, j, k. Now consider the Lévy system (N(x, dy), H) of the Hunt process X introduced in Section A.3.4. For u ∈ Fe , by the definitions of M [u],j , M [u],k

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CHAPTER FOUR

combined with (4.3.3) and (A.3.34), we have  p

2 [u],j [u],j p [u] Ms  M t = [ M ]t =  0<s≤t1/k} 1{t≥τ0 >0} and

k,p

t∧τ0

|v(y) − v(Xs )|1{|v(y)−v(Xs )|>1/k} N(Xs , dy)dHs .

Bt := 0

F∂

Then k,p

Ex [Bt ] = Ex [Bkt ] ≤ 2 v ∞ Px (t ≥ τ0 )

for x ∈ E0

and Bk,p is a PCAF of X 0 having Revuz measure µk with µk (E0 ) = |v(x) − v(y )|1{|v(x)−v(y)|>1/k} N(x, dy)µH (dx) E0 ×F∂

≤k

E0 ×F∂

(v(x) − v(y ))2 N(x, dy)µH (dx) < ∞.

(5.5.20)

214

CHAPTER FIVE

By the Markov property of X 0 , (5.5.20), and Theorem 4.1.1, t t " # " # k,p k,p k,p dBk,p Em0 (At )2 ≤ Em0 (Bt )2 = 2Em0 dB r s 0

" # k,p EXs0 Bt−s dBk,p s

t

= 2Em0 0

≤ 4 v ∞ Em0

t

0

t

≤ 4 v ∞ 0

s

(1 − P0t−s 1(Xs0 ))dBk,p s

P0s 1, µk − P0s 1, P0t 1 · µk ds.

It then follows from the dominated convergence theorem that " # 1 k,p lim sup Em0 (At )2 ≤ 4 v ∞ (µk (E0 ) − µk (E0 )) = 0. t t→0 This together with (5.5.19) establishes the claim (5.5.18). Next by Fukushima’s decomposition, (5.5.18), the stated martingale or[v] − M, the identity [M]t = A2t , and finally by thogonality between M and M·∧τ 0 (5.5.16) and the Lévy system formula (A.3.33), we have lim sup t→0

1 Em0 (v(Xt ) − v(X0 ))2 ; t < τ0 t

= lim sup t→0

= lim sup t→0

≤ lim sup t→0

[v] 2 1 ) ; t < τ0 Em0 (Mt∧τ 0 t [v] 1 p − Mt − At )2 ; t < τ0 Em0 (Mt∧τ 0 t [v] 1 − Mt )2 Em0 (Mt∧τ 0 t

1 1 [v] 2 ) ] − lim Em0 [A2t ] Em0 [(Mt∧τ 0 t→0 t t t→0 (v(x) − v(y))2 N(x, dy)µH (dx). = µcv (E0 ) +

= lim sup

E0 ×E0

This completes the proof of the lemma. L EMMA 5.5.5. For v = Hu with u ∈ Fe , we have v(Xt ) − v(X0 ) = Mt[v]

for t ∈ [0, τ0 ].

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TIME CHANGES OF SYMMETRIC MARKOV PROCESSES

Proof. Denote by F r the set of all regular points of F. Since F \ F r is E-polar by Theorem 3.1.10, we can choose a properly exceptional set N ⊃ F \ F r . It then holds Px -a.s. for x ∈ E \ N that Xτ0 ∈ F r ∪ {∂} and τ0 ◦ θτ0 (ω) = 0. This means that v(Xt∧τ0 ) − v(X0 ) = Ex u(Xτ0 ) Ft∧τ0 − v(X0 ) Px -a.s. x ∈ E \ N; namely, v(Xt∧τ0 ) − v(X0 ) is a martingale relative to {Ft∧τ0 } under Px for each x ∈ E \ N. [v] [v] , then Ct = Nt∧τ , t ≥ 0, and Thus if we let Ct = v(Xt∧τ0 ) − v(X0 ) − Mt∧τ 0 0 {Ct }{t≥0} is a continuous Px -martingale relative to the filtration {Ft∧τ0 } for q.e. x ∈ E. Since N [v] has zero energy, we have for each fixed t > 0, ! n 2 [v] [v] E1E0 ·m [Ct ; t < τ0 ] = E1E0 ·m lim Nkt/n − N(k−1)t/n ; t < τ0 n→∞

≤ lim Em

k=1

n

n→∞

[v] Nkt/n

−

[v] N(k−1)t/n

2

! = 0.

k=1

Hence, for every t > 0, Ct = 0 P1E0 ·m -a.e. on {t < τ0 }. By the continuity of C, we have Cτ0 = 0 P1E0 ·m -a.e. Thus P1E0 ·m -a.e., Ct = 0, namely, v(Xt∧τ0 ) − [v] v(X0 ) = Mt∧τ for every t ≥ 0. 0 T HEOREM 5.5.6. For v = Hu with u ∈ bFe , 1 lim Em0 (v(Xt ) − v(X0 ))2 ; t < τ0 t→0 t (v(x) − v(y))2 J(dx, dy). = µcv (E0 ) +

(5.5.21)

E0 ×E0

Proof. For f ∈ F 0 ⊂ F, let the Fukushima decomposition of f (Xt0 ) − f (X0 ) 0,[f ] 0,[f ] be denoted as Mt + Nt , while the Fukushima decomposition for f (Xt ) − [f ] [f ] f (X0 ) is denoted by Mt + Nt . Since f (Xt∧τ0 ) − f (X0 ) = f (Xt0 ) − f (X00 ), we have [f ]

0,[f ]

Mt∧τ0 − Mt

0,[f ]

0,[f ]

= Nt

[f ]

− Nt∧τ0 ,

t ≥ 0.

In view of Exercise 4.1.9, Mt is a square-integrable martingale with respect [f ] 0,[f ] to the filtration {Ft∧τ0 , t ≥ 0} and so is Mt∧τ0 − Mt . On the other hand, by the same argument as that in the proof of Lemma 5.5.5, we have " # [f ] E1E0 ·m M·∧τ0 − M 0,[f ] t ; t < τ0 " # [f ] = E1E0 ·m N·∧τ0 − N 0,[f ] t ; t < τ0 = 0.

216

CHAPTER FIVE [f ]

[f ]

By the continuity of M·∧τ0 − M 0,[f ] t , we conclude that M·∧τ0 − M 0,[f ] τ0 = 0 [f ] 0,[f ] [f ] 0,[f ] and therefore Mt∧τ0 = Mt . Consequently, Nt∧τ0 = Nt . Now let f = αR0α 1E0 ∩K ∈ F 0 for a fixed compact set K ⊂ E. Note that 0 ≤ f ≤ 1. By Fukushima’s decomposition and Proposition 4.1.10, 1 lim Em0 (v(Xt ) − v(X0 ))2 ; t < τ0 t→0 t [v] 2 1 ) ; t < τ0 = lim Em0 (Mt∧τ 0 t→0 t [v] 2 0 1 ) f (Xt ) ≥ lim Em0 (Mt∧τ 0 t→0 t [v] 2 [v] 2 1 1 ) + lim Em0 (Mt∧τ ) ( f (Xt0 ) − f (X00 )) = lim Ef ·m0 (Mt∧τ 0 0 t→0 t t→0 t [v] 2 [v] 2 1 1 ) + lim Em0 (Mt∧τ ) ( f (Xt∧τ0 ) − f (X0 )) = lim Ef ·m0 (Mt∧τ 0 0 t→0 t t→0 t " # 1 [f ] [v] 2 f (x)µv (dx) + lim Em0 (Mt∧τ ) Mt∧τ0 = 0 t→0 t E0 f (x)µv (dx) + I. (5.5.22) =: E0

In the second to the last equality, we used the fact that t∧τ0 [f ] 0,[f ] Nt∧τ0 = Nt = α( f − 1E0 ∩K )(Xs )ds, 0

whose absolute value is bounded by αt. By Itô’s formula (A.3.39), t∧τ0 1 v Ms− dM [v],c M [f ],c s I = lim Em0 t→0 t 0 [f ] [v] 2 [v] 2 + ((Ms ) − (Ms− ) )(Ms[f ] − Ms− ) s≤t∧τ0

t∧τ0 1 v = lim Em0 Ms− dM [v],c M [f ],c s t→0 t 0 [f ] [v] [v] + 2Ms− (Ms[v] − Ms− )(Ms[f ] − Ms− ) s≤t∧τ0

+

(Ms[v]

−

[v] 2 Ms− ) (Ms[f ]

−

[f ] Ms− )

.

s≤t∧τ0

Since Mt[v] = v(Xt ) − v(X0 ) for t ≤ τ0 by Lemma 5.5.5, we then have by the Revuz formula in Proposition 4.1.10, the Lévy system formula (A.3.33), and

217


the symmetry of J(dx, dy), t∧τ0 1 I = lim Em0 (v(Xs ) − v(X0 ))dM [v],c M [f ],c s t→0 t 0 + 2(v(Xs− ) − v(X0 ))(v(Xs ) − v(Xs− ))( f (Xs ) − f (Xs− )) s≤t∧τ0

+

(v(Xs ) − v(Xs− ))2 ( f (Xs ) − f (Xs− ))

s≤t∧τ0

t∧τ0 1 = 0 + lim Em0 2 v(Xs ) (v(Xs ) − v(y))( f (Xs ) − f (y ))N(Xs , dy)dHs t→0 t E∂ 0t∧τ0 1 (v(Xs ) − v(y))( f (Xs ) − f (y ))N(Xs , dy)dHs − lim Ev·m0 2 t→0 t t∧τ0 0 E∂ 1 (v(y) − v(Xs ))2 ( f (y) − f (Xs ))N(Xs , dy)dHs + lim Em0 t→0 t 0 E∂ (v(y) − v(x))2 ( f (y) − f (x))N(x, dy)dµH (dx) = E 0 ×E∂ =− f (x)(v(x) − v(y ))2 N(x, dy)dµH (dx). E0 ×F∂

Thus we have, by (5.5.22), 1 lim Em0 (v(Xt ) − v(X0 ))2 ; t < τ0 t→0 t f (x)µv (dx) − f (x)(v(x) − v(y ))2 N(x, dy)dµH (dx) ≥ E0 ×F∂

E0

=

E0

f (x)µcv (dx) +

E0 ×E0

f (x)(v(x) − v(y ))2 N(x, dy)dµH (dx).

Since this is true for all f = αR0α 1E0 ∩K where α > 0 and K is a compact subset of E, we conclude by first letting K ↑ E and then α ↑ ∞ that 1 lim Em0 (v(Xt ) − v(X0 ))2 ; t < τ0 t→0 t (v(x) − v(y))2 N(x, dy)dµH (dx). ≥ µcv (E0 ) + E0 ×E0

This together with Lemma 5.5.4 completes the proof of the theorem.

The proof of Theorem 5.5.6 can be refined to show the following. Exercise 5.5.7. Show that (5.5.21) holds for every v ∈ Fe . The next theorem relates Feller measures to the jumping measure J and the killing measure κ of (E, F ).

218

CHAPTER FIVE

T HEOREM 5.5.8. Assume that m(E0 ) < ∞. For any u ∈ Fe , 2 µHu (E0 ) + (Hu(x) − u(ξ )) J(dx, dξ ) + (Hu)2 (x)κ(dx) E0 ×F

=

E0

(u(ξ ) − u(η))2 U(dξ , dη) + 2 F×F

u(ξ )2 V(dξ ). F

Proof. Without loss of generality, we may assume that u ∈ bFe since otherwise we consider un = ((−n) ∨ u) ∧ n and then pass n → ∞. For α > 0, by Lemma 5.5.3, (u(ξ ) − u(η))2 Uα (dξ , dη) + α(Hα (u2 ), q)E0 F×F

= α(Hα 1, w)E0 + α

E0 ×F

(Hu(x) − u(ξ ))2 Hα (x, dξ )m(dx),

(5.5.23)

where w = H(u2 ) − (Hu)2 and q = 1 − H1. It follows from (5.5.14) that

lim

α→∞ F×F

(u(ξ ) − u(η))2 Uα (dξ , dη) =

(u(ξ ) − u(η))2 U(dξ , dη). F×F

(5.5.24)

By definition (5.5.7) and (5.4.6), we have α

lim α(H (u ), q)E0 = 2

α→∞

u(ξ )2 V(dξ ).

(5.5.25)

F

The limit in α of the first term of the right hand side of (5.5.23) has the expression as is exhibited in (5.5.8) under the assumption m(E0 ) < ∞. Moreover, the last term in (5.5.23) can be rewritten as Iα := αEm e−ατ0 (Hu(X0 ) − u(Xτ0 ))2 1{τ0 0} the Laplace transform of {µt , t > 0}. Then, for any µfs , vds = (P0t Hf , v) µ ¯ fα , v = (Hα f , v)

and

0

t

for every t > 0,

(5.7.3)

for α > 0,

µfs , vds = Ev·m0 f (XσF ); σF ≤ t

(5.7.4)

for every t > 0.

(5.7.5) f

(iii) For any f ∈ bB+ (E) and X 0 -excessive function v on E0 , µt , v is a right continuous decreasing function in t > 0 and f

L(0) (Hf , v) =↑ limµt , v. t↓0

In particular, f

U( f ⊗ g) =↑ limµt , Hg t↓0

and

f

V( f ) =↑ limµt , q. t↓0

(5.7.6)

Proof. (i) Since Hf ∈ S pur (E0 ), Hf · m0 is an excessive measure of X 0 which is pure in the sense that lim (Hf · m0 )P0t , v = lim (P0t Hf , v) = 0

t→∞

t→∞

for v ∈ L1 (E0 , m0 ).

226

CHAPTER FIVE

Since X 0 is transient, the assertion follows from a theorem due to P. J. Fitzsimmons which we refer the readers to [81, Theorem 5.21]. (ii) For v ∈ L1 (E0 ; m0 ), we have ∞ ∞ ∞ f f µs , vds = µt+s , vdt = µfs , P0t vds t

0

= Hf ·

0

m0 , P0t v

= (P0t Hf , v)

and f

µt , v = − Hence

∞

d 0 (P Hf , v) dt t

for a.e. t > 0.

d 0 (Pt Hf , v)dt dt 0

∞

= −e−αt (P0t Hf , v) − α

µ ¯ fα , v = −

e−αt

0

0

∞

e−αt (P0t Hf , v)dt

= (Hf − αG0α Hf , v) = (Hα f , v). Equation (5.7.5) follows from (5.7.2), (5.7.3), and (5.5.6). (iii) If v is X 0 -excessive, then µt+s , v = {µt , P0s v ↑ µt , v as s ↓ 0. For v ∈ L1 (E0 ; m0 ), we get from (i) and (ii) that t 0 µfs , vds, (Hf − Pt Hf , v) = 0

which extends to any X -excessive v as in the proof of Lemma 5.4.2(ii). Thus we get the desired identity. 0

For a set B ⊂ E, define σB = inf{t > 0 : Xt− ∈ B}. From [13, p. 59] or [73, Theorem A.2.3], we have then σF a.s., or, equivalently, L EMMA 5.7.2. σF ≤ inf{t > 0 : Xt ∈ F} = inf{t > 0 : Xt ∈ F or Xt− ∈ F} a.s. We will be concerned with a random subset M(ω) of [0, ζ (ω)) defined by M(ω) = {t ∈ [0, ζ (ω)) : Xt (ω) ∈ F or Xt− (ω) ∈ F},

ω ∈ ,

(5.7.7)

with the convention that X0− (ω) = X0 (ω). M(ω) is homogeneous on (0, ∞) in the sense that (M − s) ∩ (0, ∞) = (M ◦ θs ) ∩ (0, ∞),

s > 0.

(5.7.8)


227

L EMMA 5.7.3. M(ω) is a relatively closed subset of [0, ζ (ω)) Px -a.s. for x ∈ E \ N0 , where N0 is some Borel properly exceptional set. Proof. Let f (x) = Ex [e−σF ], x ∈ E. f is 1-excessive by Lemma A.2.4 and so it is quasi continuous by Theorem 3.3.3. By virtue of Theorem 3.1.7, f is right continuous and left continuous as well along the sample path Xt in the sense that (3.1.9), (3.1.10), and (3.1.11) hold true Px -a.s. for every x ∈ E \ N for some E-polar set N. Denote by F r the set of all regular points of F. F r ⊂ F and F \ F r is E-polar by Theorem 3.1.10 and Lemma A.2.18. Choose a Borel properly exceptional set N0 containing N ∪ (F \ F r ). Then F \ N0 = {x ∈ E \ N0 : f (x) = 1}.

(5.7.9)

∈ F∞ with Px ( ) = 1 for every X|E\N0 is a Hunt process and there is , t ∈ M(ω) if and only if either x ∈ E \ N0 , such that, for every ω ∈ f (Xt (ω)) = 1 or f (Xt− (ω)) = 1. Furthermore, if tn ∈ M(ω) decreases to t ≥ 0, then f (Xt (ω)) = 1, and if tn ∈ M(ω) increases to t < ζ (ω), then f (Xt− (ω)) = 1. This means that M(ω) is a relatively closed subset of [0, ζ (ω)). By the above lemma, the set [0, ζ (ω)) \ M(ω) is a disjoint union of relatively open subintervals of [0, ζ (ω)). Each subinterval is called an excursion interval. Each segment of the sample path X· (ω) corresponding to an excursion (time) interval is said to be an excursion away from the set F. Denote by I(ω) the collection of all left endpoints of excursion intervals. Since M(ω) is homogeneous on (0, ∞) in the sense of (5.7.8), so is I(ω). Define R = R(ω) = inf{s > 0 : s ∈ M(ω)}. We have then from Lemma 5.7.2 R = σF ,

(5.7.10)

and from (5.7.9) F \ N0 = {x ∈ E \ N0 : Px (R = 0) = 1}. D EFINITION 5.7.4. A system (P∗x , L + J) is called an exit system relative to the homogeneous random set M, or, for (X, F) if [(1)] P∗ is a kernel from E \ N0 to such that P∗x = Px for x ∈ E0 \ N0 and, for any x ∈ E \ N0 , P∗x (X0 = x) = 0, P∗x (R = 0) = 0, and E∗x 1 − e−R ≤ 1. [(2)] L is a PCAF in the strict sense of X|E\N0 carried on F \ N0 and dJt = s∈I:Xs ∈E\F εs (dt), where εs (dt) is the unit atomic measure on R concentrated at the point {s}.

228

CHAPTER FIVE

[(3)] For any positive optional process {Zs , s ≥ 0} and non-negative random variable , ! ∞ ∗ Zs · ( ◦ θs ) = Ex Zs · EXs ()d(Ls + Js ) (5.7.11) Ex 0

s∈I

for x ∈ E \ N0 , where E∗x stands for the integration with respect to the measure P∗x . We refer the readers to B. Maisonneuve [120] (see also [38]) for a proof of the existence of an exit system relative to the present homogeneous random set M. Since P∗x (R > t) ≤ (1 − e−t )−1 E∗x [1 − e−R ], t > 0, we see from (1) that P∗x is σ -finite for x ∈ F \ N0 . We also notice that the following Markov property can be deduced for an exit system (cf. [120, §5]): for any non-negative ∈ Fs0 0 and non-negative ∈ F∞ , E∗x [ · ◦ θs ] = Ex E∗Xs [] ,

s > 0, x ∈ E \ N0 .

(5.7.12)

Define the entrance law {Q∗t (x, ·), t > 0, x ∈ E \ N0 } for the exit system (P∗x , L + J) by Q∗t g(x) := E∗x [g(Xt ); t < R] ,

g ∈ B+ (E).

(5.7.13)

Note that Q∗t g(x) = P0t g(x) for x ∈ E0 \ N0 . By the Markov property, we see that for each x ∈ F \ N0 , νt := Q∗t (x, ·) enjoys the X 0 -entrance law property νt P0s = νt+s . Since X is a Hunt process, it has a Lévy system (N, H). According to Section A.3.4, N(x, dy) is a kernel on (E∂ , B(E∂ )) and H is a PCAF in the strict sense of X such that the identity (A.3.31) holds for any non-negative Borel function f on E × E∂ that vanishes on the diagonal and is extended to be zero elsewhere. The Revuz measure of H (with respect to the excessive measure m) will be denoted as µH . T HEOREM 5.7.5. For any Borel subset B ⊂ E0 and f ∈ Bb (F), f f (x)Q∗t (x, B)µL (dx) µt (B) = F

+ F×(E\F)

f (x)P0t 1B (y)N(x, dy)µH (dx),

(5.7.14)

where µL is the Revuz measure of the PCAF L with respect to the measure m.

229


∞ Proof. Put Q∗α g(x) = 0 e−αt Q∗t g(x)dt, g ∈ Bb (E). By virtue of the identity (5.7.11), we have for any v ∈ Bb (E) vanishing on F and for every x ∈ E \ N0 , R0α v(x)

Hα Rα v(x) = Rα v(x) −

= Ex

e R

s+R◦θs

−αt

e

∞

= Ex

= Ex

−αt

v(Xt )1Mc (t)dt

!

v(Xt )dt = Ex

s

s∈I

∞

−αs

e 0

E∗Xs

R

−αt

e

−αs

!

R

e

−αt

e

v(Xt )dt ◦ θs

0

s∈I

v(Xt )dt d(Ls + Js ) .

0

Therefore, we have for any f ∈ Bb+ (E),

∞

( f , Hα Rα v) = Ef ·m

−αs

e 0

Q∗α v(Xs )d(Ls

+ Js ) .

(5.7.15)

On the other hand, owing to the formula (4.1.3), we obtain

∞

−αs

e

Ef ·m 0

Q∗α v(Xs )dLs

= Rα f , Q∗α v · µL .

(5.7.16)

Furthermore, since Q∗α (x, ·) = R0α (x, ·) for x ∈ E0 , we have

∞

−αs

e

Ef ·m 0

Q∗α v(Xs )dJs





= Ef ·m 

e−αs R0α v(Xs )

s∈I,Xs ∈E\F

= Ef ·m

! −αs

e

1F (Xs− )1E\F (Xs )R0α v(Xs )

s

∞

= Ef ·m

e−αs 1F (Xs )

0

E\F

+ = Rα f , 1F (·) E\F

N(Xs , dz)R0α v(z)dHs

, N(·, dz)R0α v(z) · µH .

Here in the last equality we used (4.1.3) again.

(5.7.17)

230

CHAPTER FIVE

We get from (5.7.15), (5.7.16), and (5.7.17) + (Hα Rα f , v) = Rα f , Q∗α v · µL + 1F (·)

E\F

, N(·, dz)R0α v(z) · µH .

Since this identity holds for an arbitrary f ∈ Bb (E), we obtain for any β > 0, 1 (E0 ; m), f ∈ bC+ (E) ∩ F, and v ∈ L+ + , ∗ 0 N(·, dz)Rα v(z) · µH . (5.7.18) (Hα Rβ f , v) = Rβ f , Qα v · µL + 1F (·) E\F

Multiplying β on both side of (5.7.18) with f = 1 and then letting β → ∞, we have, by monotone convergence theorem, , + N(·, dz)R0α v(z) · µH 1, Q∗α v · µL + 1F (·) E\F

= Hα 1, v ≤

v(x)m(dx) < ∞. E0

Now multiplying both side of (5.7.18) by β and letting β → ∞ in the above equation, we have, by bounded convergence theorem, + , ∗ 0 N(·, dz)Rα v(z) · µH . (Hα f , v) = f , Qα v · µL + 1F (·) E\F

This combined with (5.7.4) proves the desired identity (5.7.14) since the above display is nothing but the Laplace transform of (5.7.14). The following theorem describes Feller measures as joint distributions of the starting and end points of excursions of X away from the quasi closed set F. T HEOREM 5.7.6. The Feller measures are expressed by the exit system as U(dx, dy) = µL (dx)P∗x (XσF ∈ dy)

+ µH (dx) F

N(x, dz)Pz (XσF ∈ dy)

(5.7.19)

E\F

and V(dx) = µL (dx)P∗x (ζ > 0, σF = ∞)

+ µH (dx) F

N(x, dz)Pz (σF = ∞). E\F

(5.7.20)

231


Proof. It follows from (5.7.6), (5.7.14), and the definition of Q∗t (x, dy) that for any f , g ∈ B+ (E), f

f (x)g(y )U(dx, dy) = limµt , Hg t↓0

F×F

= lim t↓0

F×E0

F

= lim t↓0

F

f (x)E∗x [g(XσF )

t↓0

= F

f (x)P0t Hg(y)N(x, dy)µH (dx)

F×E0

F

f (x)Hg(y)N(x, dy)µH (dx) F×E0

◦ θt ; t < R]µL (dx)+

f (x)E∗x [g(XσF ); t < R]µL (dx) +

= lim

f (x)E∗x [Hg(Xt ); t < R]µL (dx) +

= lim t↓0

f (x)Hg(y)Q∗t (x, dy)µL (dx)+

f (x)E∗x [g(XσF )]µL (dx) +



f (x)Ey [g(XσF )]N(x, dy)µH (dx). F×E0

In the third to the last equality, we used the Markov property (5.7.12). Identity (5.7.19) now follows. The proof for (5.7.20) is similar to that for (5.7.19). Note that for x ∈ E0 , q(x) = 1 − H1(x) = Px (σF = ∞). Thus by (5.7.6), f

f (x)V(dx) = limµt , q t↓0

F

= lim t↓0

F×E0

= lim t↓0

F

= lim t↓0

= F

F

f (x)q(y)Q∗t (x, dy)µL (dx) +

F×E0

f (x)E∗x [q(Xt ); t < ζ ∧ R]µL (dx) +

f (x)P0t q(y)N(x, dy)µH (dx)

f (x)q(y)N(x, dy)µH (dx) F×E0

f (x)E∗x [1{σF=∞} ◦ θt ; t < ζ ∧ R]µL (dx)+

f (x)P∗x (ζ > 0, σF = ∞)µL (dx)+

f (x)q(y)N(x, dy)µH (dx) F×E0

f (x)Py (σF = ∞)N(x, dy)µH (dx). F×E0

for any f ∈ B+ (E). This establishes (5.7.20).

232

CHAPTER FIVE

5.8. MORE EXAMPLES The first four examples of Section 5.3 concern the Dirichlet forms associated with time-changed processes of Brownian motion and reflecting Brownian motion. In this section, we give more examples of them with emphasis on the corresponding Feller measures. (1◦ ) Trace of RBM on the boundary Let D ⊂ Rn be a bounded domain with C3 -boundary ∂D for n ≥ 2. The surface measure on ∂D will be denoted by σ . Let X = (Xt , Px ) be the standard Brownian motion on Rn and X D be the absorbing Brownian motion on D as has been considered in Example 3.5.9. X D is the part process of X on D and it admits the transition density p0t (x, y) expressed as p0t (x, y) = gt (x − y) − Ex [gt−τ (Xτ − y); τ < t], where gt is given by (2.2.13) and τ is the exit time of X from D (the hitting time of Rn \ D). We let h(s, x, ξ ) =

1 ∂p0s (x, ξ ) , 2 ∂nξ

x ∈ D, ξ ∈ ∂D,

where nξ denotes the inward normal vector at ξ . We show that h is a density function of the joint distribution of (τ , Xτ ) under Px for x ∈ D: Px (τ ∈ ds, Xτ ∈ dξ ) = h(s, x, ξ )dsσ (dξ ).

(5.8.1)

To see this, let u be the harmonic function on D with a smooth boundary function f on ∂D, namely, u(x) = Ex [ f (Xτ ); τ < ∞], x ∈ D. Since p0t (x, y) is symmetric in x, y, and a fundamental solution of the heat equation ∂t∂ u(t, x) = 1 u(t, x) with the Dirichlet boundary condition on ∂D, Green’s formula 2 x yields 1 t ds h(s, x, ξ )f (ξ )dσ (ξ ) 2 0 ∂D 1 t ds (y p0s (x, y))u(y)dy = 2 0 D = u(x) − P0t u(x) = Ex [f (Xτ ); τ ≤ t], holding for x ∈ D, t > 0, proving (5.8.1). Therefore, the hitting distribution H(x, dξ ) and the α-order hitting distribution Hα (x, dξ ) admit density functions given for x ∈ D, ξ ∈ ∂D by ∞ ∞ h(s, x, ξ )ds, Kα (x, ξ ) = e−αs h(s, x, ξ )ds. K(x, ξ ) = 0

0

233


K is called the Poisson kernel for D. Define Uα (ξ , η) = α Kα (x, ξ )K(x, η)dx,

ξ , η ∈ ∂D,

D

which can be easily rewritten as Uα (ξ , η) =

1 4

∞

(1 − e−αt )

0

∂ 2 p0t (ξ , η) dt. ∂nξ ∂nη

Let U(ξ , η) = limα→∞ Uα (ξ , η). Then U(ξ , η) =

1 ∂K(ξ , η) , 2 ∂nξ

ξ , η ∈ ∂D.

(5.8.2)

This function U(ξ , η) is the density function of the Feller measure U with respect to σ × σ . Indeed we get from (5.5.14) U( f ⊗ g) = lim α(H α f , Hg)D α→∞ = lim Uα (ξ , η)f (ξ )g(η)σ (dξ )σ (dη) α→∞ ∂D×∂D

=

∂D×∂D

U(ξ , η)f (ξ )g(η)σ (dξ )σ (dη), f , g ∈ B+ (∂D).

We call U(ξ , η) defined by (5.8.2) the Feller kernel for the domain D. Consider the special case where D equals BR = {x ∈ Rn : |x| < R} and σ is the uniform spherical measure on R = {x ∈ Rn : |x| = R}. Expression (5.8.2) is then reduced to 1 (n/2) , ξ , η ∈ R , ξ = η, (5.8.3) U(ξ , η) = n/2 2π |ξ − η|n −|x| because the Poisson kernel for BR equals 1n R R|x−η| n , x ∈ BR , η ∈ R , with n/2 −1 n = 2π (n/2) being the area of 1 . When n = 2, (5.8.3) can be rewritten as 2

U(ξ , η) = [4π R2 (1 − cos(θ − θ ))]−1 ,

2

ξ = Reiθ , η = Reiθ ,

which coincides with the kernel (2.2.59) for R = 1. The expression of U(ξ , η) in (5.8.3) depends only on the distance of ξ and η, and it even coincides with the corresponding kernel (5.3.16) for the unbounded upper half-space Rn for n ≥ 2. Let us consider the reflecting Brownian motion X r = (Xtr , Prx ) on D = D ∪ ∂D, as has been considered in Example 3.5.9. X r is associated with the regular Dirichlet form ( 12 D, H 1 (D)) on L2 (D). The part process of X r on D is identical with the absorbing Brownian motion X D so that X r and the Brownian motion X have the same hitting distribution H, H α from D to ∂D as well as the Feller measure U and supplementally Feller measure V for the domain D.

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In particular, X r admits (5.8.2) as its Feller kernel. Since X r is irreducible recurrent, we see from (3.5.19) that H1(x) = 1, x ∈ D, and hence q(x) = 1 − H1(x) = 0, x ∈ D, and V is vanishing. Further, the energy measure for ( 21 D, H 1 (D)) is expressed as µu (B) = |∇u(x)|2 dx, u ∈ bH 1 (D), B ∈ B(D), B

which does not charge on ∂D. Moreover, the jumping and killing measures of X r are vanishing. Therefore, we get from Theorem 5.5.9 1 1 2 |∇Hf (x)| dx = ( f (ξ ) − f (η))2 U(ξ , η)σ (dξ )σ (dη) (5.8.4) 2 D 2 ∂D×∂D e1 (D) denotes the e1 (D)|∂D , where U(ξ , η) is given by (5.8.2) and H for any f ∈ H 1 collection of quasi continuous functions in He (D). This particularly recovers the expression of Eˇ in (5.3.9) when D is the planer unit disk. (2◦ ) Trace of Brownian motion on a hypersurface As in the preceding example, we consider the standard Brownian motion X = (Xt , Px ) on Rn with n ≥ 2. Let S be a C3 compact hypersurface so that E0 = Rn \ S is the union of the interior domain Di and exterior domain De . We denote by σ the surface measure on S. Further, ∂/∂niξ and ∂/∂neξ will denote the inward normal and outward normal derivative at ξ ∈ S from the view of Di , respectively. We consider the Poisson kernel K(x, ξ ), x ∈ E0 , ξ ∈ S, and the escape probability of X from S defined by q(x) = 1 − H1(x) = Px (σS = ∞),

x ∈ E0 .

Note that q(x) = 0 if n = 2 because X is then irreducible recurrent and (3.5.19) holds. If n ≥ 3, then q(x) is positive but only for x ∈ De . We shall show that the Feller measure U and the supplementary Feller measure V with respect to X and S have densities U(ξ , η) and v(ξ ) relative to σ × σ and σ , respectively, expressed as U(ξ , η) = U i (ξ , η) + U e (ξ , η), U i (ξ , η) = v(ξ ) =

1 ∂K(ξ , η) , 2 ∂niξ 1 ∂q(ξ ) , 2 ∂neξ

ξ , η ∈ S, ξ = η

U e (ξ , η) =

1 ∂K(ξ , η) , 2 ∂neξ

ξ ∈ S.

(5.8.5) (5.8.6)

In the special case of S = R , (5.8.5) is reduced to U(ξ , η) =

(n/2) 1 , n/2 π |ξ − η|n

ξ , η ∈ R , ξ = η,

(5.8.7)

235


because U i (ξ , η) equals (5.8.3), while the Poisson kernel for the exterior 2 −R2 domain {|x| > R} equals 1n R |x| . In this special case with n ≥ 3, (5.8.6) |x−η|n is also reduced to a constant function v(ξ ) =

n−2 , 2r

ξ ∈ R ,

(5.8.8)

n−2

R because q(x) = 1 − |x| n−2 for |x| > R. For the proof of (5.8.5) and (5.8.6), it suffices to show that 1 1 |∇Hf (x)|2 dx = ( f (ξ ) − f (η))2 U e (ξ , η)σ (dξ )σ (dη) 2 De 2 S×S + f (ξ )2 v(ξ )σ (dξ ) (5.8.9) S

e1 (Rn ) H

e1 (Rn ) denotes the holds for any f ∈ on account of (5.8.4), where H 1 n collection of quasi continuous functions in He (R ). To this end, take a sufficiently large N such that the ball BN = {x ∈ Rn : |x| < N} contains the surface S. The bounded domain De,N = De ∩ BN has the boundary consisting of two disjoint set S and N . We denote by De (u, v) and De,N (u, v) the integrals of ∇u(x) · ∇v(x) with respect to dx over De and De,N , respectively. By virtue of (5.8.4), we have, for f , g ∈ bC1 (Rn ), 1 e,N N D (H f , HN g) 2 1 ( f (ξ )−f (η))(g(ξ )−g(η))U e,N (ξ , η)σ (dξ )σ (dη), = 2 ∂De,N ×∂De,N

(5.8.10)

where HN denotes the hitting distribution of X from De,N to ∂De,N and U e,N denotes the Feller kernel (5.8.2) for the domain De,N . e1 (Rn )|De,N ⊂ As will be observed in the next example (3◦ ), we have H 1 1 e (De,N ). So we can derive from (5.8.10) that for any f ∈ H e (Rn ) vanishing H n on R \ BN , 1 1 1 e D ( f , f ) ≥ De,N ( f , f ) ≥ De,N (HN f , HN f ) 2 2 2 1 ( f (ξ ) − f (η))2 U e,N (ξ , η)σ (dξ )σ (dη) = 2 S×S + f (ξ )2 v N (ξ )σ (dξ ), S

where

v (ξ ) = N

N

U e,N (ξ , η)σ (dη),

ξ ∈ S.

(5.8.11)

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If we let qN (x) = Px (σS > σN ), then we get from (5.8.2) the expression v N (ξ ) =

1 ∂qN (ξ ) 2 ∂neξ

for ξ ∈ S.

(5.8.12)

We next prove that for ξ , η ∈ S with ξ = η, as N ↑ ∞, U e,N (ξ , η) ↑ U e (ξ , η),

(5.8.13)

v N (ξ ) ↓ v(ξ ).

(5.8.14)

For ξ , η ∈ S with ξ = η, choose a ball B centered at ξ with η ∈ / B and B ⊂ BN . Let C = ∂B ∩ De and h(x, y), x ∈ B ∩ De , y ∈ C, be the density function of the distribution Px (XσC ∈ dy, σC < σS ) with respect to the surface measure σ (dy) on C. For the Poisson kernel K N (x, η) of the domain De,N , it then holds that h(x, y)K N ( y, η)σ (dy), x ∈ B ∩ De , η ∈ S. K N (x, η) = C

Similarly, we have h(x, y)K(y, η)σ (dy),

K(x, η) =

x ∈ B ∩ De , η ∈ S.

C

By taking the outward normal derivatives in x at ξ ∈ S in the above two equations, we get (5.8.13) from (5.8.2) and (5.8.5) because K N (y, η) increases to K(y, η) as N → ∞ for y ∈ De , η ∈ S. To prove (5.8.14), we take N1 with S ⊂ BN1 and denote by σ1 the surface measure on N1 . Then we have K N1 (x, η)qN (η)σ1 (dη) for N > N1 qN (x) =

N1

and q(x) = N K N1 (x, η)q(η)σ1 (dη). By taking the outward normal derivative 1 in x at S on both sides of each of the above two equations, we arrive at (5.8.14) because qN (η) decreases to q(η) as N → ∞ for each η ∈ N1 . ˇ f , f ) the right hand side of (5.8.9). We can conclude Finally, denote by D( ˇ f,f ) from (5.8.11), (5.8.13), and (5.8.14) the followings. First, 12 De ( f , f ) ≥ D( 1 n for any f ∈ He (R ) and, by substituting H f in place of f , 1 e ˇ f , f ), D (H f , H f ) ≥ D( 2

e1 (Rn ). f ∈H

(5.8.15)

Second, lim

N→∞

1 e,N N ˇ f , f ) < ∞, D (H f , HN f ) = D( 2

e1 (Rn ). f ∈H

(5.8.16)

237


Since for every x ∈ De ,

HN f (x) = Ex f (XσS ); σS < σN ↑ H f (x)

and

as N ↑ ∞,

N N e,N0 0 De,N (HN f , HN f ) + 1 (H f , H f ) = D

(HN f (x))2 dx De,N0

are uniformly bounded in N for each N0 in view of (5.8.16), the Cesàro mean of a subsequence of {HN f } is D1e,N0 -strongly convergent to H f by Theorem A.4.1. Hence 1 e,N0 ˇ f,f) D (H f , H f ) ≤ D( 2 and, by letting N0 → ∞, we arrive at the converse inequality to (5.8.15). The proof of (5.8.9) is now complete. We remark that, in the trace Dirichlet form (5.3.17) of the reflecting Brownian motion X on the upper half-space of Rn , only the Feller kernel U defined by (5.3.16) is involved and the supplementary Feller measure V does not appear even when n ≥ 3 and X is transient. The reason is in that the vertical component of X is the one-dimensional reflecting Brownian motion so that the hitting probability H1(x) by X of the hyperplane always equals 1. (3◦ ) Trace of Brownian motion on a closed region We continue to consider the standard Brownian motion X = (Xt , Px ) on Rn (n ≥ 2) and a bounded domain D ⊂ Rn with C3 -boundary ∂D. The Dirichlet form (E, F) of X on L2 (Rn ) is given by 1 (5.8.17) D, H 1 (Rn ) . (E, F) = 2 e1 (Rn ) the family of all E-quasi-continuous functions in the We denote by H extended Sobolev space He1 (Rn ). Let us identify the trace Dirichlet form of (5.8.17) on the closed set D by (5.8.23), (5.8.24), and (5.8.25) below. To this end, we first compare X with the reflecting Brownian motion X r on D that appeared in Example 3.5.9. The Dirichlet form (E r , F r ) of X r on L2 (D)(= L2 (D)) is given by 1 r r 1 (5.8.18) D, H (D) . (E , F ) = 2 By Corollary 2.2.15, the extended Sobolev space He1 (D) coincides with the space BL(D). We shall denote by BL(D) the family of all E r -quasi-continuous functions in BL(D). The capacity Cap1 for (5.8.18) will be designated by Capr1 .

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As Capr1 (A ∩ D) ≤ Cap1 (A) for any open set A ⊂ Rn ,

f on Rn is E-quasi-continuous =⇒ f D is E r -quasi-continuous.

(5.8.19)

Moreover, N ⊂ D is E r -polar =⇒ N is E-polar.

(5.8.20)

Indeed, for any compact set K ⊂ D, it follows from [73, Lemma 2.2.7] that Capr1 (K) = inf{E1r (u, u) : u ∈ C∞ (D), u ≥ 1 on K}, Cap1 (K) = inf{E1 (u, u) : u ∈ C∞ (Rn ) ∩ H 1 (Rn ), u ≥ 1 on K}. On the other hand, by virtue of [142, VI:3], there exists a linear operator T called an extension operator from H 1 (D) to H 1 (Rn ) such that Tf |D = f ,

E1 (Tf , Tf ) ≤ ME1r ( f , f ),

f ∈ H 1 (D),

for some constant M > 0 independent of f ∈ H 1 (D). Moreover, T sends C∞ (D) into C∞ (Rn ) ∩ H 1 (Rn ). Therefore, Cap1 (K) ≤ MCapr1 (K) for any compact set K ⊂ D, which in particular means (5.8.20). The above considerations lead us to e1 (Rn )|D = BL(D). H

(5.8.21)

The inclusion ⊂ is immediate from (5.8.19). Conversely, any u ∈ BL(D) admits v ∈ BL(Rn ) with v|D = u (cf. [24, (2.21)]). Then v ∈ He1 (Rn ) by Theorem 2.2.13 if n = 2, while v = v0 + c for some v0 ∈ He1 (Rn ) and c ∈ R, by Theorem 2.2.12 if n ≥ 3. In the latter case, replace c by a function w ∈ Cc∞ (Rn ) with w = c on D so that v ∈ He1 (Rn ) with v|D = u. Take any E r -(resp. E-)quasi-continuous version u (resp. v) of u (resp. v). Then v|D is E r -quasir u = v|D E -q.e. on D, and consequently continuous on D by (5.8.19) so that E-q.e. on D, proving the inclusion ⊃ in (5.8.21). We now consider a measure µ on Rn defined by µ(dx) = g(x)dx for any bounded measurable function g which is strictly positive a.e. on D and vanishing on Rn \ D. µ charges no E-polar set and further ◦

µ ∈ SD ,

(5.8.22)

e1 (Rn ) and namely, D is an E-quasi-support of µ. To see this, take any f ∈ H r assume that f = 0 µ-a.e. f |D is then E -quasi-continuous by (5.8.19) and vanishing a.e. on D. Therefore, f |D = 0 E r -q.e. on D, and so f = 0 E-q.e. on D, which proves (5.8.22) in view of Theorem 3.3.5. t Let Y be the time-changed process of X by means of its PCAF At = 0 g(Xs )ds with Revuz measure µ. By virtue of Theorem 5.2.13, Y can be ˇ F) ˇ realized as a µ-symmetric Hunt process on D whose Dirichlet form (E, ˇ the extended Dirichlet on L2 (D; µ)(= L2 (D; µ)) is regular. Denote by (Fˇ e , E)


239

ˇ F). ˇ It then holds that space of (E, Fˇ = Fˇ e ∩ L2 (D; µ)

(5.8.23)

on account of Theorem 5.2.15. We claim that Fˇ e = BL(D) and, for f ∈ BL(D), ˇ f,f)= 1 E( 2 + +

(5.8.24)

|∇f |2 (x)dx D

1 2

∂D×∂D

∂D

( f (ξ ) − f (η))2 U e (ξ , η)σ (dξ )σ (dη)

f (ξ )2 v(ξ )σ (dξ ).

(5.8.25)

Identity (5.8.24) follows from Theorem 5.2.15 and (5.8.21). Since HD f = f · 1D + H∂D f · 1Rn \D , we get (5.8.25) from (5.2.4) and the identity (5.8.9) in e1 (Rn )|∂D = BL(D)| the preceding example holding for any f ∈ H ∂D .

Chapter Six REFLECTED DIRICHLET SPACES

Reflected Dirichlet space was introduced for a regular Dirichlet form by M. L. Silverstein in 1974 in [138, 139] and further investigated by the firstnamed author in [16]. Reflected Dirichlet space plays an important role for the boundary theory of symmetric Markov processes. While it is possible to introduce reflected Dirichlet form for a quasi-regular Dirichlet form directly, we choose to do it first in the regular Dirichlet form setting as this allows us to define the reflected Dirichlet space via terminal random variables and harmonic functions of finite energy. This approach sheds more insight into the probabilistic meaning and the structure of the reflected Dirichlet form. As illuminated by the relationship between the Dirichlet spaces for the absorbing Brownian motion and reflecting Brownian motion, given a Dirichlet form (E, F), heuristically, there are two ways to define its reflected Dirichlet space: (i) the linear span of F and all harmonic functions of finite “E-energy”; (ii) all functions that are “locally” in F and have finite “E-energy.” In this chapter, we will develop these two approaches simultaneously and show that they give the same object. In Section 6.1, we introduce the notion of terminal random variables and harmonic functions of finite energy for a Hunt process associated with a transient regular Dirichlet form. In Section 6.2, we will present and establish several equivalent notions of reflected Dirichlet space (E ref , F ref ) for a regular transient Dirichlet form (E, F). One of these equivalent notions can be used to define reflected Dirichlet space for a regular recurrent Dirichlet form in Section 6.3, where we will show that in this case F ref = Fe . In Section 6.4 we give yet another equivalent definition of reflected Dirichlet space that is invariant under quasi-homeomorphism of Dirichlet forms, which allows us to define reflected Dirichlet space for any general quasi-regular Dirichlet forms. Various concrete examples of reflected Dirichlet spaces will be exhibited in Section 6.5 for regular Dirichlet forms including most of those presented in Section 2.2. In Section 6.6, we first define a Silverstein extension of a quasi-regular Dirichlet form (E, F) on L2 (E; m). At the probabilistic level, a symmetric F) is a Silverstein extension of (E, F) if the symmetric Dirichlet form (E, F) extends the symmetric Markov Markov process X associated with (E,

REFLECTED DIRICHLET SPACES

241

, which spends zero process X associated with (E, F) to some state space E sojourn time at the “boundary” E \E. We then show that the active reflected space (E ref , Faref ) is the maximum among the Silverstein extensions of (E, F) with respect to a partial ordering defined among symmetric Dirichlet forms on L2 (E; m). The maximality of the reflected Dirichlet space is essentially due to Silverstein [138]. A probabilistic notion of a harmonic function for a symmetric Hunt process plays a role in the first two sections. In Section 6.7, the equivalence of analytic and probabilistic concepts of harmonicity is investigated. 6.1. TERMINAL RANDOM VARIABLES AND HARMONIC FUNCTIONS Let (E, F) be a transient regular Dirichlet form on L2 (E; m), and let X = {Xt , {Ft }t≥0 , Px , x ∈ E} be the m-symmetric Hunt process on E associated with it. The transition function and the resolvent kernel of X are denoted by {Pt ; t ≥ 0} and {Rα ; α ≥ 0}, respectively. We adopt the conventions that any element of the extended Dirichlet space Fe is quasi continuous already and that any numerical function f on E is extended to E∂ by setting f (∂) = 0. Recall the class S of smooth measures on E and the class S0(0) of positive Radon measures on E of finite 0-order energy integrals defined in Section 2.3. (0) Any µ ∈ S0(0) admits a unique 0-order potential Uµ ∈ Fe . We denote by S00 (0) the 0-order counterpart of the class S00 defined by (2.3.10) so that µ ∈ S00 if and only if µ ∈ S0(0) and µ(E) < ∞, Uµ∞ < ∞. L EMMA 6.1.1. For µ ∈ S, let Aµ be the PCAF of X with Revuz measure µ. Then µ (0) . Eν [Aζ ] = Uν(x)µ(dx) ≤ Uν∞ µ(E) for every ν ∈ S00 E

In particular,

µ Aζ

is Px -integrable for q.e. x ∈ E if µ(E) < ∞.

Proof. If µ ∈ S0(0) , then we have from (4.1.3) that for any h ∈ B+ (E) with (h, R0 h) < ∞, Eh·m [Aµζ ] = R0 h, µ = E(R0 h, Uµ) = (h, Uµ). µ

Consequently, Ex [Aζ ] as a function of x is a quasi continuous version of Uµ (0) , and for ν ∈ S00 µ

Eν [Aζ ] = ν, Uµ = E(Uν, Uµ) = Uν, µ ≤ Uν∞ µ(E). For a general µ ∈ S, there exists an E-nest {Fk , k ≥ 1} such that 1Fk µ ∈ S0(0) for every k ≥ 1 owing to the 0-order counterpart of Theorem 2.3.15. By replacing

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CHAPTER SIX

Aµ and µ in the above with 1Fk · Aµ and 1Fk · µ, respectively, and by letting k → ∞, we get the desired relation. The second statement of the lemma follows from it and the 0-order counterpart of Corollary 2.3.11. Let M be a MAF of X, namely, M is an additive functional of X satisfying (4.2.12). Denote by M and e(M) the predictable quadratic variation process and the energy of M, respectively. M is an element of A+ c and we denote by µM its Revuz measure. Then e(M) = 12 µM (E). M denotes the totality of ◦

MAFs of X and we let M = {M ∈ M : e(M) < ∞}. ◦

T HEOREM 6.1.2. Let M ∈ M and define M ∗ := supt≥0 |Mt |. Then Eν [(M ∗ )2 ] ≤ 8Uν∞ e(M) < ∞

(0) for every ν ∈ S00 .

In particular, Mt converges as t → ∞ to a random variable M∞ in L2 (Px ) as well as Px -a.s. for q.e. x ∈ E, and Eν [(M∞ )2 ] ≤ 8Uν∞ e(M) < ∞

(0) for every ν ∈ S00 .

(6.1.1)

Proof. By Doob’s maximal inequality (A.3.36), for T > 0, Eν sup (Mt )2 ≤ 4Eν [MT ] ≤ 4Eν [Mζ ] 0≤t≤T

≤ 8Uν∞ e(M) < ∞, where the last inequality follows from Lemma 6.1.1. Letting T → ∞, we get the desired estimate of M ∗ and hence {Mt : t ≥ 0} is a Pν -uniformly integrable (0) . We then get the desired conclusions with the help martingale for each ν ∈ S00 of the 0-order counterpart of Corollary 2.3.11. D EFINITION 6.1.3. A measurable function h is said to be harmonic on E if it is specified and finite up to quasi equivalence and if for every relatively compact open subset D ⊂ E, Ex [|h(XτD )|] < ∞ and h(x) = Ex [h(XτD )] for q.e. x ∈ E. Here we use the convention that X∞ = ∂ and h(∂) := 0. D EFINITION 6.1.4. A terminal random variable ϕ is a random variable on (, F∞ ) that is Px -integrable for q.e. x ∈ E and that for q.e. x ∈ E, Px -a.s. {ϕ = 0} ⊃ {Xζ − ∈ E, ζ < ∞} ∪ {ζ = 0} and ϕ ◦ θt = ϕ for every t < ζ . Here we notice that, due to the transience of X, lim Xt = ∂

t→∞

by Theorem 3.5.2.

Px -a.s. on {ζ = ∞} for q.e. x ∈ E,

(6.1.2)

243


L EMMA 6.1.5. If ϕ is a terminal random variable, then h(x) := Ex [ϕ] is harmonic. Conversely, a harmonic function h on E can be represented by h = Ex [ϕ] for some terminal random variable ϕ if and only if for q.e. x ∈ E, {h(XτDk ), FτDk } is a Px -uniformly integrable martingale, where {Dk , k ≥ 1} is an increasing sequence of relatively compact open subsets with ∪k≥1 Dk = E. In this case, lim h(XτDk ) = ϕ,

k→∞

Px − a.s. for q.e. x ∈ E.

(6.1.3)

Proof. Let ϕ be a terminal random variable. Note that for any relatively compact open subset D ⊂ E, {τD = ζ } ⊂ {Xζ − ∈ E, ζ < ∞} because of (6.1.2). Hence ϕ = 0 on {τD = ζ }, while ϕ ◦ θτD = ϕ on {τD < ζ }. Define h(x) := Ex [ϕ], which is finite q.e. on E. It follows that for q.e. x ∈ E, h(x) = Ex [ϕ ◦ θτD ; τD < ζ ] = Ex [h(XτD ); τD < ζ ] = Ex [h(XτD )]. So h is harmonic on E. Moreover, if {Dk , k ≥ 1} is an increasing sequence of relatively compact open subsets with ∪k≥1 Dk = E, then, Px -a.s. for q.e. x ∈ E, h(XτDk ) = 1{τDk k, since vj − vk = (gj − gk ) + (ψj − ψk ) is the unique decomposition of vj − vk ∈ bFloc into gj − gk ∈ Fe and ψj − ψk ∈ HN, we conclude from the second inequality of (6.2.11) that {ψk , k ≥ 1} is E ref -Cauchy. Thus by Lemmas 6.2.3 and 6.2.11, u) = lim E(v k , vk ) ≤ lim sup 2E ref (vk , vk ) E(u, k→∞

k→∞

= 2 lim sup E(gk , gk ) + E ref (ψk , ψk ) k→∞

= 2E( f , f ) + 2E ref (h, h) = 2E ref (u, u). The lemma is established.

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We are now ready to give the Proof of Theorem 6.2.5. Let G denote the linear space of functions u for which there is an E-Cauchy sequence of {un , n ≥ 1} ⊂ Floc that converges to u m-a.e. on E. For u ∈ G and its approximating sequence {un , n ≥ 1}, by Lemma 6.2.12, there are E-Cauchy sequence {fn ≥ 1} ⊂ Fe and E ref Cauchy sequence {hn , n ≥ 1} ⊂ HN such that un = fn + hn for every n ≥ 1. Therefore, fn is E-convergent to some f ∈ Fe . By taking a subsequence if necessary, we may assume that fn converges to f q.e. on E. Consequently, hn converges m-a.e. to h := u − f . By Lemma 6.2.3 and (6.2.11), h ∈ HN and u). It follows then u = f + h ∈ F ref ; E ref (h, h) = limn→∞ E ref (hn , hn ) ≤ 4E(u, in other words, we have shown G ⊂ F ref . On the other hand, evidently Fe ⊂ G and HN ⊂ G by Lemmas 6.2.7 and 6.2.8. It follows F ref ⊂ G and so G = F ref . n , un ) is well-defined on G. Sup u) := limn→∞ E(u We now show that E(u, pose that {un , n ≥ 1} ⊂ Floc is another E-Cauchy sequence that converges to u m-a.e. on E. Let fn ∈ Fe and hn ∈ HN be such that un = fn + hn . By the above reasoning, fn is E-convergent to some f ∈ Fe , hn is E ref -convergent to some h ∈ HN, and u = f + h . By the uniqueness of the decomposition for u ∈ F ref (see Remark 6.2.2), f = f and h = h. It follows that lim E ref (un − un , un − un )

n→∞

= lim E( fn − fn , fn − fn ) + lim E ref (hn − hn , hn − hn ) = 0. n→∞

n→∞

Thus by (6.2.11), n − un , un − un ) ≤ lim 2E ref (un − un , un − un ) = 0 lim E(u

n→∞

n→∞

n , un ) = limn→∞ E(u n , un ). and so limn→∞ E(u ref f , f ) < ∞, We finally prove that E = E on F ref . For f , g ∈ Floc with E( 1 1 E(g, g) < ∞, we define E( f , g) = 4 E( f + g, f + g) − 4 E( f − g, f − g) so that f , g) = 1 µcf ,g (E) + 1 (( f (x) − f (y))(g(x) − g(y))J(dx, dy) E( 2 2 E×E (6.2.12) + f (x)g(x)κ(dx), E

where µcf ,g = 14 µcf +g − 14 µcf −g . As E = E on Fe and, by Lemmas 6.2.7 f , h) = 0 for f ∈ Fe and and 6.2.8, E = E ref on HN, it suffices to show that E( f , h) = 0 for non-negative h ∈ HN. In fact, one needs only to show that E( functions f ∈ F ∩ Cc (E) and for h(x) = Ex [ϕ] with non-negative bounded functions ϕ ∈ N, since the linear spans of such functions are E-dense in Fe and in HN, respectively.

257


In this case, it holds that f , h) = e(M [f ] , M h ) + 1 E( 2

f (x)h(x)κ(dx) E

1 [f ] 1 = lim Em Mt Mth + t→0 2t 2

f (x)h(x)κ(dx),

(6.2.13)

E

◦

where M [f ] and M h ∈ M are martingale additive functionals of finite energy specified by Theorem 4.2.6 and (6.1.8), respectively, and e denotes the mutual energy defined by (4.2.2). In fact, by using the polarized expression of the quadratic variation in (4) of Section A.3.3, we can derive just as in the proof of Theorem 6.1.8 the equality 1 µM[f ],c ,Mh,c (E) 2

e(M [f ] , M h ) = +

1 2

( f (x) − f (y))(h(x) − h(y))J(dx, dy) + E×E

1 2

f (x)h(x)κ(dx). E

But, in view of the proof of Lemma 6.2.7, we have µMh,c = µch and so µM[f ],c ,Mh,c = µch,f . This combined with the above equality and (6.2.12) leads us to (6.2.13). From (6.2.13), we obtain

1 1 h f (x)h(x)κ(dx) E( f , h) = lim Em ( f (Xt ) − f (X0 ))Mt + t→0 2t 2 E

1 1 f (x)h(x)κ(dx) = lim Em f (Xt )Mth + t→0 2t 2 E 1 Em [f (Xt )(h(Xt ) − h(X0 )); t < ζ ] t→0 2t 1 + f (x)h(x)κ(dx), 2 E

= lim

(6.2.14) [f ]

where in the first equality we used Fukushima’s decomposition Mt = f (Xt ) − [f ] [f ] f (X0 ) − Nt and the fact that Nt has zero energy. Due to the symmetry of X and the expression (6.1.7) of the MAF M h , Em [f (Xt )(h(Xt ) − h(X0 )); t < ζ ] = Em [f (X0 )(h(X0 ) − h(Xt )); t < ζ ]

= Ef ·m ϕ1{t≥ζ } − h(X0 )1{t≥ζ }

= Ef ·m ϕ1{t≥ζ ,Xζ − =∂} − Pfh·m (t ≥ ζ ).

(6.2.15)

258

CHAPTER SIX

Since f ∈ F ∩ Cc (E), there exists some g ∈ F ∩ Cc (E) with f ≤ g2 on E. Then Ef ·m ϕ1{t≥ζ , Xζ − =∂} is dominated by ||ϕ||∞ · Pg2 ·m (t > ζ , Xζ − = ∂), which is o(t) on account of Proposition 4.2.3. Furthermore, by virtue of (4.2.8), 1 1 f (x)h(x)κ(dx). lim Pfh·m (t ≥ ζ ) = t→∞ 2t 2 E f , h) = 0. This completes the proof of Therefore, (6.2.14) and (6.2.15) yield E( the theorem. For k ≥ 1, define truncation operator τk by τk u := ((−k) ∨ u) ∧ k. T HEOREM 6.2.13. It holds that k u, τk u) < ∞ F ref = u : |u| < ∞ [m], τk u ∈ Floc , ∀k ≥ 1, sup E(τ k≥1

and

k u, τk u) E ref (u, u) = lim E(τ k→∞

for u ∈ F ref .

Proof. By Remark 6.2.6, k u, τk u) < ∞ . F ref ⊃ u : |u| < ∞ [m], τk u ∈ Floc , ∀k ≥ 1, sup E(τ k≥1

sequence {un , n ≥ 1} in For u ∈ F , by Theorem 6.2.5, there is an E-Cauchy Floc that converges to u m-a.e. on E. By (6.2.5) and (6.2.6), τk un ∈ F ref and for each k ≥ 1, ref

k un , τk un ) ≤ lim E(u n , un ) = E ref (u, u) < ∞. lim sup E(τ n→∞

n→∞

So by Remark 6.2.6, τk u ∈ F ref and E ref (τk u, τk u) ≤ E ref (u, u). As τk u is a bounded function in F ref , in view of Remark 6.2.2, there are bounded f ∈ Fe and bounded h ∈ HN so that τk u = f + h. It follows from Lemma 6.2.7 that τk u ∈ Floc . This shows that k u, τk u) < ∞ F ref ⊂ u : |u| < ∞ [m], τk u ∈ Floc , ∀k ≥ 1, sup E(τ k≥1

and so these two function spaces are the same. k u, τk u) is increasing in Note that for u ∈ F ref , by (6.2.5) and (6.2.6), E(τ k ≥ 1 and bounded by E(u, u). Since τk u is convergent pointwise to u as k → ∞, we have from Remark 6.2.6 k u, τk u) ≤ E(u, u) ≤ lim E(τ u). E ref (u, u) = E(u, k→∞

259


We are now ready to present the main theorem of this section. T HEOREM 6.2.14. Define Faref := F ref ∩ L2 (E; m). Then (E ref , Faref ) is a Dirichlet form on L2 (E; m). sequence {un , n ≥ 1} ⊂ Floc that Proof. For u ∈ Faref , there is an E-Cauchy converges to u m-a.e. on E. Let gn := (0 ∨ un ) ∧ 1. Then gn ∈ Floc , gn → g := n , un ) by (6.2.5) and (6.2.6). Thus n , gn ) ≤ E(u (0 ∨ u) ∧ 1 m-a.e. on E and E(g by Remark 6.2.6, g ∈ Faref with n , gn ) ≤ lim sup E(u n , un ) = E ref (u, u). E ref (g, g) ≤ lim sup E(g n→∞

n→∞

Since (Faref , E1ref ) is a Hilbert space by Theorem 6.2.4, it follows that (E ref , Faref ) is a Dirichlet form on L2 (E; m). The relation between F ref and the extended Dirichlet space Faref e of (E ref , Faref ) will be given in Theorem 6.6.10. On account of Theorem 6.2.13, (Eαref , Faref ) can be viewed as the reflected Dirichlet space of (Eα , F) for any α > 0. For u ∈ F ref , there are unique f ∈ Fe and ϕ ∈ N so that u = f + h where h(x) := Ex [ϕ]. In the following theorem, we write ϕu for ϕ. T HEOREM 6.2.15. Let α > 0 and u ∈ Faref . Then ϕu = 0 on {ζ = ∞}. Moreover, u − hα ∈ F, where hα (x) := Ex [e−αζ ϕu ]. Proof. Let Y be the subprocess of X killed at an independent exponential random time T of rate α > 0; that is, Xt (ω) if t < T(ω), Yt (ω) = ∂ otherwise. Since the transition function of Y equals {e−αt Pt ; t ≥ 0}, the Dirichlet space on L2 (E; m) and the extended Dirichlet space of Y are both (Eα , F). Let NY and HNY be the universal Dirichlet space and the harmonic function space defined by (6.2.1) and (6.2.2), respectively, but with Y in place of X. As u is in the hα ∈ F, reflected Dirichlet space of Y, there is a unique ψ ∈ NY so that u − where hα (x) := EYx [ψ]. Let {Dk , k ≥ 1} be an increasing sequence of relatively compact open sets with ∪k≥1 Dk = E. By noting that ϕu is a terminal random variable of X, it follows from Remark 6.2.2 that ψ = lim u(YτDY ) = lim u(XτDk )1{τDk 0 such that m(B(x, r)) ≥ C1 rn

for all x ∈ D and 0 < r ≤ 1,

then the space (6.5.14) can be identified with a certain Sobolev space W α/2,2 (D) of fractional order so that it becomes a regular Dirichlet form on L2 (D; m) possessing Cc∞ (Rn )|D as its core (cf. [14]). Its associated process X is called a reflecting α-stable process on D. By virtue of Theorem 3.3.9, the Dirichlet form (E 0 , F 0 ) defined above is the part form of (6.5.14) on D, and accordingly the censored stable process X 0 coincides with the part process of X on D. Note that every uniformly Lipschitz domain in Rn is an n-open set. But an n-set can

275


have very rough boundary. For example, any n-set with a closed set of zero Lebesgue measure removed is still an n-set. Let ((F 0 )ref , E 0,ref ) be the reflected Dirichlet space of the Dirichlet form 0 (E , F 0 ) defined above for an arbitrary open set D ⊂ Rn . Then it holds that 0 ref 0,ref (F ) , E = (W(D), B). (6.5.15) Recall that by Definition 6.4.4 ◦ 0 (F 0 )ref = u ∈Floc : B(u, u) < ∞ and E 0,ref = B on (F 0 )ref . The inclusion “⊂” in (6.5.15) follows as in Example (1◦ ). Conversely, for any non-negative u ∈ W(D), integer k ≥ 1 and any relatively compact open set U ⊂ D with smooth boundary ∂D, we can find v ∈ F 0 such that v = τk u on U by making use of the fact that τk u|U ∈ 0 . By W α/2,2 (U). (See [14, Theorem 2.2] for details.) This implies that τk u ∈ Floc the same argument as that in the paragraph preceding (6.5.3), we conclude that ◦

0 . Consequently, u ∈ (F 0 )ref u admits a quasi continuous version and u ∈Floc and (6.5.15) follows. When D is an n-set, we can readily show that

(F ref , E ref ) = (W(D), B)

(6.5.16)

for the reflected Dirichlet form of (E, F) given by (6.5.14) because F ref can be ◦

◦0

described as above but with F loc in place of F loc . As in Example (1◦ ), the identity (Fe , E) = (W(D), B)

(6.5.17)

holds if and only if (E, F ) is recurrent.

6.6. SILVERSTEIN EXTENSIONS Throughout this section, (E, F) is a quasi-regular symmetric Dirichlet form on L2 (E; m). F ) on L2 (E; m) (not necessarily D EFINITION 6.6.1. A Dirichlet form (E, quasi-regular) is said to be a Silverstein extension of (E, F ) if Fb is an ideal b and fg ∈ Fb for every f ∈ Fb and g ∈ F b ) and E = E b (that is, Fb ⊂ F in F on Fb × Fb . b and E = E on Fb imply that F ⊂ Remark 6.6.2. We note that Fb ⊂ F ◦ ◦ ⊂ F loc and E = E on F. To verify the inclusion F ⊂ F loc , we consider a F

276

CHAPTER SIX

F) on quasi-homeomorphism j from (E, F) to a regular Dirichlet form (E, 2 m). Let {Dn , n ≥ 1} be an increasing sequence of relatively compact L (E , b open subsets of E that increases to E. For each integer n ≥ 1, choose gn ∈ F −1 with gn = 1 on Dn . Then gn = j gn is an element of Fb taking values 1 on Dn . Note that {Dn } is a sequence of quasi open sets increasing to E Dn = j−1 b , since gn f ∈ Fb and gn f = f on Dn for every n ≥ 1, we q.e. For any f ∈ F ◦

have f ∈ F loc .

◦

we can use truncations τk f ∈F loc , k ≥ 1, to obtain For a general f ∈ F, ◦ f ∈ F loc by the same argument as in Section 6.5 (1◦ ). T HEOREM 6.6.3. Suppose that (E, F) is a quasi-regular Dirichlet form on L2 (E; m). Then its active reflected Dirichlet space (E ref , Faref ) is a Silverstein extension of (E, F). Proof. Recall that it follows immediately from Definition 6.4.4 that (E1ref , Faref ) is the reflected (and hence active reflected) Dirichlet space of the transient Dirichlet form (E1 , F). So in view of Theorem 1.4.3, without loss of generality we may and do assume that (E, F) is a regular transient Dirichlet form on a locally compact separable metric space E. By virtue of Theorem 6.2.13, it holds that Fb ⊂ bFaref

and

E = E ref

on Fb × Fb .

(6.6.1)

Let f ∈ Fb and g ∈ bFaref . By Theorem 6.2.13, g ∈ Floc . Choose a sequence { fk , k ≥ 1} ⊂ F ∩ Cc (E) that is E1 -convergent to f as well as q.e. convergent to f on E and with fk ∞ ≤ f ∞ . Then fk g ∈ Fb and obtain from Exercise 1.1.10 being applied to the active reflected Dirichlet form (E ref , Faref ) and (6.6.1) that fk gE1 = fk gE1ref ≤ fk ∞ gE ref + g∞ fk E1 + fk ∞ g2 . Therefore, supk≥1 fk gE1 < ∞. Since fk g converges to fg on E, it follows that fg ∈ Fb . This proves that Fb is an ideal in bFaref . Thus (E ref , Faref ) is a Silverstein extension of (E, F). In fact, we also have the following. P ROPOSITION 6.6.4. Suppose that (E, F) is a quasi-regular Dirichlet form on L2 (E; m). Then bFe is an ideal of bF ref . In particular, if 1 ∈ Fe , then (E, Fe ) = (E ref , F ref ). Proof. By making use of a quasi-homeomorphism as in the proof of Theorem 6.4.5, we may well assume that (E, F) is a regular Dirichlet form. According to Corollary 5.2.12 and Proposition 6.4.6, the extended Dirichlet form and the reflected Dirichlet form are unchanged under a full support time change. Hence replacing m by f (x)m(dx) for some strictly positive bounded


277

function g with E g(x)m(dx) < ∞ if necessary, we may and do assume that m(E) < ∞. In this case, bFe = bF and bF ref = bFaref . So the conclusion of this proposition follows directly from Theorem 6.6.3. The next theorem gives the probabilistic meaning of a Silverstein extension. T HEOREM 6.6.5. Suppose (E, F) is a quasi-regular Dirichlet form on F) is another symmetric Dirichlet form on L2 (E; m) (not L2 (E; m) and (E, necessarily quasi-regular). Then the following are equivalent. F ) is a Silverstein extension of (E, F ). (i) (E, (ii) There is a locally compact separable metric space E and a measurable map E; m) j: E → E so that, by letting m = m ◦ j−1 , j∗ is a unitary map from L2 ( F ) := j(E, F) is a regular onto L2 (E; m) and the image Dirichlet form (E, E, m). Moreover, there is an E-quasi-open Borel subset Dirichlet form on L2 ( E0 of E so that m(E \ E0 ) = 0 and j is the quasi-homeomorphism that maps the quasi-regular Dirichlet form (E, F ) on L2 (E; m) onto the quasi-regular F E ) is the part Dirichlet form F E ) on L2 ( E0 ; m). Here (E, Dirichlet form (E, 0 0 F) on the E-quasi-open of (E, subset E0 . E is an : u = 0 E-q.e. E = u ∈ F on E \ E0 , bF Proof. (ii) ⇒ (i): Since F 0 0 Consequently, bF is an ideal of bF and E = E on bF × bF; that ideal of bF. F) is a Silverstein extension of (E, F ). is, (E, ⊂ Floc by (i) ⇒ (ii): Since (E, F) is quasi-regular on L2 (E; m) and F has an E-quasi-continuous version on E. As Remark 6.6.2, every function in F L2 (E; m) is separable, there exists countable set B0 = {fn , n ≥ 1} of bounded that is dense in (F, E1 ) such that functions in F (1) B0 is an algebra over the rational numbers. (2) There is a function h ∈ F and an E-nest {Fk , k ≥ 1} consisting of compact sets such that 1/k ≤ h ≤ k on Fk , B0 ⊂ C({Fk }) and B0 separates points of ∪k≥1 Fk . ·∞

We let functions in B0 take value zero on E \ ∪k≥1 Fk . Define B := B0 , which is a commutative Banach algebra. We will use the Gelfand transform F) and the quasito construct a regularizing space E for Dirichlet form (E, homeomorphic map j which is very similar to the one given in the proof of Theorem 1.4.3. Notice, however, that we now use an E-nest {Fk } instead of an E-nest in taking quasi continuous versions of functions in an E1 -dense set in the above. For readers’ convenience, we provide the details of a B0 (⊂ F) proof here.

278

CHAPTER SIX

Step 1. Construct a locally compact separable metric space E. Let E be a collection of non-trivial real-valued functionals γ on B which satisfy for f , g ∈ B and for rational numbers a and b, |γ ( f )| ≤ f ∞ ,

γ ( fg) = γ ( f )γ (g),

γ (af + bg) = aγ ( f ) + bγ (g).

We equip E with the weakest topology so that the function f : γ → γ ( f ) is continuous for every f ∈ B. E is then a separable locally compact metrizable E). E is compact if and only if 1 ∈ B. space and {f , f ∈ B} = C∞ ( E such that Let j be the unique map from ∪k≥1 Fk into (jx)( f ) := f (x)

for f ∈ B and x ∈ ∪k≥1 Fk .

By (2) above, j is a continuous one-to-one map on each Fk . Hence Fk := j( Fk ) Fk is compact in E and j is a topological homeomorphism from Fk onto for every k ≥ 1. Note that j : ∪k≥1 Fk → E is Borel measurable and m(E \ Fk ) = 0. It follows m := m ◦ j−1 . Clearly m( E \ ∪k≥1 ∪k≥1 Fk ) = 0. Define m is a Radon measure and from the m-integrability of functions in B0 that E; m) further supp[ m] = E. Since B0 is dense in L2 (E; m), j∗ is unitary from L2 ( 2 onto L (E; m). F) is regular on L2 ( Step 2. The image Dirichlet form j(E, E; m). F) := j(E, F). Then (E, F) is clearly a Dirichlet form on L2 ( Let (E, E, m). ∩ C∞ ( E) ⊃ (B0 ) and the latter is uniformly dense in C∞ ( E) and Since F (E, F) is a regular Dirichlet form on L2 ( E1 -dense in F, E; m). Fk . There is some E-polar set N ⊂ E so that Step 3. Define E1 := ∪k≥1 E0 := subset of E and that j is a quasi-homeomorphism E1 \ j(N) is an E-quasi-open F E ) on L2 ( E0 ; m). from (E, F) on L2 (E; m) onto (E, 0 Note that for every k ≥ 1, hk := k(h ∧ (1/k)) ∈ Fb and hk = 1 on Fk for b with E = E on Fb , it follows that every k ≥ 1. Since Fb is an ideal in F bFFk = bFFk and so F = F Fk = FFk j∗ F k

with j∗ E = E on FFk .

(6.6.2)

Px ) be the symmetric Hunt process on E associated with the Let X = ( Xt , F) on L2 ( E; m) and let regular Dirichlet form (E, σE\E 1 Ex e−αt f ( Xt )dt , x ∈ E, f ∈ B+ ( E). REα1 f (x) = 0

F E1 ) by In view of Corollary 3.2.3, REα1 is related to the space (E, E , Eα ( REα1 f , v) = f (x)v(x) m(dx), REα1 f ∈ F 1 E1

(6.6.3)

279


E , where E; m) and v ∈ F for any f ∈ B( E) ∩ L2 ( 1 E = f ∈ F : f = 0 E-q.e. F on E \ E1 . 1 E , E1 ). Similarly, we define By (6.6.3), { REα1 f ; f ∈ L2 ( E; m)} is dense in (F 1 RFαk f (x) = Ex

σE\Fk

−αt

e

f (Xt )dt ,

x∈ E, f ∈ B+ ( E).

0

Fk = RFαk f (x) = E1 , limk→∞ REα1 f (x) E-q.e. on E for every f ∈ B+ ( E). As ∪k≥1 F is E1 -dense in F E . Since {Fk } is an E-nest, we It follows that ∪k F k 1 F E1 ) is the image Dirichlet form of (E, F) under the quasiconclude that (E, homeomorphism j from E to E1 . For emphasis, we denote this Dirichlet form F E ) by EE1 . (E, 1 On the other hand, since (E, F) is quasi-regular, it admits a properly associated special standard process X on E by virtue of Theorem 1.5.2. Denote by { R0α ; α > 0} the resolvent kernel of the image process jX. It follows from REα1 f , m-a.e. for every f ∈ L2 ( E1 ; m). For every f ∈ B0 , (6.6.2) that R0α f = 2 0 m). Rα (f ) is EE1 -quasi-continuous, and so is REα1 (f )|E1 f ∈ bB(E) ∩ L (E; because REα1 (f ) is E-quasi-continuous and the restriction to E1 of any E-nest E1 is an E -nest. Since they coincide m-a.e. on E1 and B0 is countable, there is an X-properly exceptional set N ⊂ E such that R0α (f )(x) REα1 (f )(x) =

for every x ∈ E1 \ j(N) and every f ∈ B0 . E), the above As the linear span of {f ; f ∈ B0 } is uniformly dense in C∞ ( E) in place of f . We thus have for every display holds for every g ∈ C∞ ( Xt , t < σE\E1 } under Px has the same distribution as that of x∈ E1 \ j(N), { the image process {jXt , t < ζ } starting from x. This in particular implies that E1 \ j(N), namely, the set E0 := E1 \ j(N) Px (τE1 \j(N) > 0) = 1 for every x ∈ is X -finely open and hence E-quasi-open by Theorem 3.3.3(i). Now the part X on the E-quasi-open E0 is associated with the Dirichlet form process X E0 of E1 E ; that is, FE0 = FE1 . Since N is E-polar, there is an E-nest {Kn , n ≥ 1} so that N ⊂ ∩n≥1 (E \ Kn ). By Theorem 1.2.13(iii), {Fn ∩ Kn , n ≥ 1} is an E-nest consisting of compact sets and clearly j is a topological homeomorphism from Fn ∩ Kn to j( Fn ∩ Kn ) for every n ≥ 1. It follows that j is a quasihomeomorphism between the quasi-regular Dirichlet form (E, F ) on L2 (E; m) F E ) on L2 ( E0 ; m). This completes the proof of and the part Dirichlet form (E, 0 the theorem. The following is an immediate consequence of Theorems 6.6.3 and 6.6.5.

280

CHAPTER SIX

C OROLLARY 6.6.6. Suppose that (E, F) is a quasi-regular Dirichlet form on L2 (E; m) such that its associated process is conservative. Then (E ref , Faref ) = (E, F). Remark 6.6.7. (i) Let X be an m-symmetric standard process on E properly associated with the quasi-regular Dirichlet form (E, F) on L2 (E; m). Identifying X E0 by the quasi-homeomorphism j, the above theorem E with E0 and X with says that (E, F) is a Silverstein extension of (E, F) if and only if there is an F) that m-symmetric Hunt process X associated with Dirichlet form (E, which contains E as an E-quasi-open extends X to some state space E subset up to an E-polar set, where the measure m is extended to E by setting of E \ E) = 0. The last condition amounts to saying that the process m(E X spends \ E. zero Lebesgue amount of time on the “boundary” E E0 in (ii) When (E, F) is a regular Dirichlet form on L2 (E; m), the set Theorem 6.6.5(ii) can be taken to be an open subset of E (see [138] and F E ) is a regular Dirichlet form on L2 ( E0 ; m) [73, Theorem A.4.4]) so that (E, 0 by Theorem 3.3.9. We define a partial ordering on Dirichlet forms on L2 (E; m). F) on L2 (E; m), D EFINITION 6.6.8. For two Dirichlet forms (E, F) and (E, we write (E, F) (E, F) if and F ⊂F

u) E(u, u) ≥ E(u,

for every u ∈ F.

We next show that (E ref , Faref ) is the maximum among the Silverstein extensions of (E, F) on L2 (E; m) with respect to the above partial ordering . T HEOREM 6.6.9. Suppose that (E, F) is a quasi-regular Dirichlet form on F) is a Dirichlet form on L2 (E; m) that is a Silverstein L2 (E; m) and that (E, F) (E ref , Faref ). extension of (E, F). Then (E, Proof. By Theorem 6.6.5 and Remark 6.6.7, there is a locally compact F) is a regular Dirichlet form, and on which (E, separable metric space E modulo a quasi-homeomorphism and an E-polar set, we can and do assume and the measure m is extended to space E that E is an E-quasi-open subset of E by letting m(E \ E) = 0. Let X be the symmetric Hunt process on E associated F) on L2 (E ; m) = L2 (E; m). Without loss of with the regular Dirichlet form (E, . Then the generality, we may assume that E is an X -finely open subset of E E part process X of X on E is a symmetric standard process properly associated with the quasi-regular Dirichlet form (E, F) on L2 (E; m). J and κ denote the energy measure of the continuous part of the Let µcf , [f ] , the jumping measure and killing measure of X . The corresponding MAF M

281


counterparts of X E are denoted as µcf , J, and κ. In view of Theorem 3.3.8, F ) as (E, F) can be identified with a subspace of (E, : \ E}, E = E on F × F . F = {f ∈ F f = 0 E-q.e. on E Therefore, we can readily deduce the Beurling-Deny decomposition F) yielding that (Theorem 4.3.3) for (E.F) from that for (E, \ E) . (6.6.4) J(dx, dy) = J (dx, dy) and κ(dx) = κ (dx) + J(dx, E E×E

E

E

◦

Further, µcf = µcf for f ∈ F. µcf can be extended to f ∈ F loc . But since ◦

⊂ F loc in view of Remark 6.6.2, we have by Theorem 4.3.10(i) F . µcf = µcf E for every f ∈ F

(6.6.5)

Let E be defined by (6.4.1) with respect to (E, F) in terms of µcf , J, and E( f,f) ≤ κ. Then on account of (6.6.4)–(6.6.5), we have for every f ∈ F, ref ref that E( f , f ) < ∞. This proves that F ⊂ F and E( f , f ) ≥ E ( f , f ) for f ∈ F; ref ref is, (E, F) (E , Fa ). It follows from the last statement in Theorem 6.4.5 that when a quasi-regular Dirichlet (E, F) is recurrent, F ref is the extended Dirichlet space of (E ref , Faref ). The following theorem deals with the transient case. T HEOREM 6.6.10. Suppose that (E, F) is a transient quasi-regular Dirichlet 2 form on L (E; m). Then (i) Faref e ⊂ F ref and the restriction of E ref on (Faref )e gives the extended Dirichlet form. (ii) When m(E) < ∞, (Faref )e = F ref . Proof. (i) Just as in the proof of Theorem 6.4.5, using a quasi-homeomorphism if necessary, we may and do assume that (E, F ) is a transient regular Dirichlet form on L2 (E; m). Suppose that u ∈ (Faref )e . Then there is an E ref -Cauchy sequence {uk , k ≥ 1} ⊂ Faref that converges to u m-a.e. on E. For each k ≥ 1, there are unique fk ∈ Fe and hk ∈ HN so that fk + hk = uk . It follows from (6.2.4) that {fk , k ≥ 1} is an E-Cauchy sequence in Fe and {hk , k ≥ 1} is an E ref Cauchy sequence in HN. Since (Fe , E) is a Hilbert space, fk is E-convergent to some f ∈ Fe . Moreover, by Theorem 2.1.5, there is a subsequence {fnk , k ≥ 1} that converges to f m-a.e. on E. Now by Lemma 6.2.3, hnk is E ref -convergent to u − f ∈ HN. Consequently, u = f + (u − f ) ∈ F ref and unk and hence uk is E ref -convergent to u. This proves that (Faref )e ⊂ F ref . (ii) When m(E) < ∞, then for every u ∈ F ref and integer k ≥ 1, uk := ((−k) ∨ u) ∧ k ∈ Faref and E ref (uk , uk ) ≤ E ref (u, u). As limk→∞ uk = u, we have

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by the Banach-Saks Theorem A.4.1) that u ∈ (Faref )e . So in this case (Theorem ref ref we have established F ⊂ Fa e . Remark 6.6.11. (i) Let (E, F) be a quasi-regular Dirichlet form on L2 (E; m) and (F ref , E ref ) be its reflected Dirichlet space. By Theorem 6.6.3, Theorem 6.6.5, and Remark 6.6.7, the active reflected Dirichlet form , m), where E (E ref , Faref ) can be represented as a regular Dirichlet form on L2 (E is some locally compact separable metric space containing E as an E ref -quasi by \ E) = 0. open set up to an E-polar set and m is the extension of m to E m(E Therefore, using Definition 6.4.4 again, we can further define the reflected Dirichlet space of (E ref , Faref ) being regarded as a regular Dirichlet form on , m). L 2 (E Actually this reflected Dirichlet space coincides with (F ref , E ref ). In fact, u) in Definition 6.4.4 can be viewed as the Beurling-Deny the expression E(u, , m), and accordingly formula for the regular Dirichlet form (E ref , Faref ) on L2 (E we get the stated coincidence by using the same definition again. In particular, when (E ref , Faref ) is recurrent, we have by Theorem 6.4.5 that (Faref )e = F ref . (ii) We keep the setting of (i). When (E ref , Faref ) is transient, we have by Theorem 6.4.12 that F ref is the linear span of (Faref )e and HNref , where HNref denotes the space of harmonic functions of (E ref , Faref ) having finite E ref -energy. Example (5◦ ) of Section 6.5 illustrates that the linear space HNref can be nontrivial. In fact, Remark 3.7 of [24] suggests that for any integer k ≥ 1, there exists a regular Dirichlet form (E, F) so that HNref has dimension k. Example 6.6.12. Let D ⊂ Bn be a bounded Lipschitz domain and X be the absorbing Brownian motion on D. Its associated Dirichlet form on L2 (D) is 1 1 (E, F), where F = H0 (D) and E(u, v) = 2 D ∇u(x) · ∇v(x) dx. The reflected Dirichlet form (E ref , F ref ) of (E, F) is given by (6.5.10). Its active reflected Dirichlet form (E ref , Faref ) = (E, H 1 (D)) is a regular Dirichlet form on L2 (D) = L2 (D; 1D dx) and its associated process is the normally reflecting Brownian motion X r on D. (i) Let µ be a positive smooth measure of X r whose quasi support is contained in ∂D (e.g., the surface measure σ on ∂D is a smooth measure of finite energy X be the reflecting Brownian motion killed by means of a integral of D). Let F) is given by PCAF with Revuz measure µ. Its associated Dirichlet form (E, 1 2 F = H (D) ∩ L (∂D; µ) and 1 2 |∇f (x)| dx + f (x)2 µ(dx) for f ∈ F. E( f , f ) = 2 D ∂D F) is a Silverstein extension of (E, F) and (E, F) (E ref , Faref ). Clearly (E, In Section 7.4, a lateral condition will be given to characterize the infinitesX of this example imal generator of the L2 -semigroup of such an extension as in a more general context.


283

X be the part process of (ii) Let F be a subset of ∂D that is not E ref -polar. Let the reflecting Brownian motion killed upon hitting F. Its associated Dirichlet F) is given by form (E, := u ∈ H 1 (D) : u = 0 E ref -q.e. on F F Clearly (E, F) is a Silverstein extension of (E, F) and and E = E ref on F. ref ref (E, F) (E , Fa ).

6.7. EQUIVALENT NOTIONS OF HARMONICITY It is well-known that a harmonic function u in a domain D ⊂ Rd can be defined or characterized by u = 0 in D in the distributional sense, that is, 1 2 2 (D) := {v ∈ Lloc (D) : ∇v ∈ Lloc (D)} so that u ∈ Hloc ∇u(x) · ∇v(x)dx = 0 for every v ∈ Cc∞ (D). Rd

It is equivalent to the following averaging property by running a Brownian motion X: for every relatively compact subset D1 of D,

u(XτD1 ) ∈ L1 (Px ) and u(x) = Ex u(XτD1 ) for every x ∈ D1 . Here τD1 := inf {t ≥ 0 : Xt ∈ / D1 }. It is a natural and fundamental question Whether the above two notions of harmonicity remain equivalent in a more general context or not, such as for discontinuous symmetric processes including symmetric Lévy processes and for symmetric diffusions on fractals. The probabilistic notion of harmonicity was briefly touched upon in Definition 6.1.3. It is a special case of the more general Definition 6.7.5 below. (See Theorems 6.7.9 and 6.7.13 for the connection.) In this section, we address the equivalence of the analytic and probabilistic notions of harmonicity in the context of symmetric Hunt processes on local compact separable metric spaces. In short, the answer is yes. See Theorem 6.7.13 below. The material in this section will not be needed for the rest of the book. Let X = (, F∞ , Ft , Xt , ζ , Px , x ∈ E) be an m-symmetric Hunt Markov process on a locally compact separable metric space E such that its associated Dirichlet form (E, F ) is regular on L2 (E; m). Here m is a positive Radon measure on E with full topological support. As before, denote by E∂ := E ∪ {∂} the one-point compactification of E. Let be the totality of right continuous, left-limited sample paths from [0, ∞[ to E∂ that hold the value ∂ once attaining it. For any ω ∈ , we set Xt (ω) := ω(t). Let ζ (ω) := inf{t ≥ 0 : Xt (ω) = ∂} be the lifetime of X. As usual, F∞ and Ft are the minimal augmented σ -algebras 0 := σ {Xs : 0 ≤ s < ∞} and Ft0 := σ {Xs : 0 ≤ s ≤ t} under obtained from F∞ / B} (the exit {Pν : ν ∈ P(E)}. For a Borel subset B of E, τB := inf{t > 0 : Xt ∈

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CHAPTER SIX

time of B) is an (Ft )-stopping time. Recall that we use the convention that X∞ := ∂ and every function u defined on E is extended to E∂ by setting u(∂) = 0. Moreover, any function in Fe is taken to be quasi continuous. Note that the Hunt process X is stochastically continuous: Px (Xt = Xt− ) = 1 for every t > 0 and x ∈ E. We emphasize here that to ensure a wide scope of applicability, unless otherwise stated, we do not assume that the process X (or equivalently, its associated Dirichlet form (E, F) is irreducible. Let us first prepare two general theorems. For t > 0, let rt denote the time-reversal operator defined on the path space of X as follows: For ω ∈ {t < ζ }, ω((t − s)−), if 0 ≤ s < t, rt (ω)(s) = ω(0), if s ≥ t. Exercise 6.7.1. Show that the restriction of the measure Pm to Ft is invariant under rt on ∩ {ζ > t}; that is, Em [ξ ; t < ζ ] = Em [ξ ◦ rt ; t < ζ ] for every non-negative random variable ξ ∈ Ft . T HEOREM 6.7.2. (Lyons-Zheng’s forward and backward martingale decomposition) For every u ∈ Fe and t > 0, Pm -a.e. on {t < ζ }, u(Xt ) − u(X0 ) =

1 [u] 1 [u] M − Mt ◦ rt , 2 t 2

(6.7.1)

where M [u] is the MAF of finite energy in the Fukushima’s decomposition (Theorem 4.2.6) of u(Xt ) − u(X0 ). Proof. Denote by R1 the 1-resolvent kernel of X. If u = R1 f for some nearly Borel function f ∈ L2 (E; m), then we have t Lu(Xs )ds, u(Xt ) − u(X0 ) = Mt[u] + 0

where Lu = u − f . Since X is stochastically continuous, applying time reversal operator rt to above display, we have Pm -a.e. on {t < ζ }, t [u] Lu(Xs )ds u(X0 ) − u(Xt ) = Mt ◦ rt + 0

Thus we have u(Xt ) − u(X0 ) =

1 [u] 1 [u] M − Mt ◦ rt 2 t 2

Pm -a.e. on {t < ζ }

(6.7.2)

285


For general u ∈ Fe , just as in the proof of Theorem 4.2.6, there is a sequence of {un = R1 fn , n ≥ 1} with fn ∈ L2 (E; m) so that limn→∞ E(un − u, un − u) = 0 and limn→∞ un = u m-a.e. on E. Note that limn→∞ e(M [un ] − M [u] ) ≤ limn→∞ E(un − u, un − u) = 0. Thus by Theorems 3.5.4 and 4.2.5, there is a subsequence {nk , k ≥ 1} so that unk (X) − unk (X0 ) and M [unk ] converges uniformly on each compact time interval to u(X) − u(X0 ) and M [u] , respectively, Px -a.s. for q.e. x ∈ E. Since (6.7.2) holds for every unk in place of u, we arrive at the conclusion of the theorem by passing k → ∞. The next lemma is a consequence of Theorem 6.7.2. L EMMA 6.7.3. If u ∈ Fe satisfies E(u, u) = 0, then Px (u(Xt ) = u(X0 ) for every t ≥ 0) = 1

for q.e. x ∈ E.

In other words, for q.e. x ∈ E, Ex := {y ∈ E : u(y) = u(x)} is an invariant set with respect to the process X in the sense that Px (X[0, ∞) ⊂ Ex ) = 1. Proof. Recall that in (4.3.15) µu is the energy measure of u, which is the Revuz measure of M [u] , and κ is the killing measure of (E, F). As µu (E) ≤ 2E(u, u) = 0, we have M [u] = 0 and thus, by Theorem 6.7.2, u(Xt ) = u(X0 ) Pm a.e. on {t < ζ } for every t > 0. This implies by Fukushima’s decomposition that N [u] = 0 on [0, ζ ) and hence on [0, ∞) Pm -a.e. Consequently, u(Xt ) − u(X0 ) = M [u] + N [u] = 0 for every t ≥ 0 Px -a.s. for q.e. x ∈ E, which proves the lemma. The next theorem accompanies a new notation and extends Theorem 2.1.12 as well as (2.3.18). Note that the process X is not assumed to be transient. T HEOREM 6.7.4. Suppose that ν is a smooth measure on E, whose corresponding PCAF of X is denoted by Aν . Define Uν(x) := Ex [Aνζ ]. Then Uν ∈ Fe if and only if E Uν(x)ν(dx) < ∞. In this case, E(Uν, u) =

u(x)ν(dx)

for every u ∈ Fe .

(6.7.3)

E

Proof. First assume that m(E) < ∞. It is easy to check directly that {x ∈ E : Ex [Aνζ ] > j} is finely open for every integer j ≥ 1. So Kj := {Uν ≤ j} is finely closed. Define νj := 1Kj ν. Clearly for x ∈ Kj , Uνj (x) ≤ Uν(x) ≤ j, while for x ∈ Kjc , ζ 1Kj (Xs )dAνs = Ex Uνj (XσKj ) ≤ j. Uνj (x) = Ex 0

So fj := Uνj ≤ j on E and hence fj is in L2 (E; m).

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Assume that Uν ∈ Fe . Then fj ≤ j ∧ Uν ∈ F. Since fj is excessive, we have by Lemma 1.2.3 that fj ∈ F and Eα ( fj , fj ) ≤ Eα (j ∧ Uν, j ∧ Uν)

for every α > 0.

Taking α → 0 yields that for every j ≥ 1, E( fj , fj ) ≤ E(j ∧ Uν, j ∧ Uν) ≤ E(Uν, Uν) < ∞. On the other hand, by Theorem 4.1.1, ν

1 1 E( fj , fj ) = lim ( fj − Pt fj , fj )m = lim Efj ·m At j t→0 t t→0 t fj (x)ν(dx). = fj (x)νj (dx) = E

Kj

Since Uν < ∞ q.e. on E, we have ν(E \ ∪∞ j=1 Kj ) = 0 and fj increases to Uν as j → ∞. Thus by the monotone convergence theorem, Uν(x)ν(dx) = lim fj (x)ν(dx) ≤ E(Uν, Uν) < ∞. E

j→∞ K j

Now assume that E Uν(x)ν(dx) < ∞. In this case, as Uν < ∞ ν-a.e. on E, we have ν(E \ ∪∞ j=1 Kj ) = 0. Since by Theorem 4.1.1 ν

1 1 lim ( fj − Pt fj , fj )m = lim Efj ·m At j = fj (x)νj (dx) t→0 t t→0 t E (6.7.4) ≤ Uν(x)ν(dx) < ∞,

E

we have fj ∈ F with E( fj , fj ) ≤ E Uν(x)ν(dx). The same calculation shows that 1K \K ·ν

for i > j, fi − fj = Ex Aζ i j and ( fi − fj )(x)ν(dx) ≤ Uν(x)ν(dx), E( fi − fj , fi − fj ) = Ki \Kj

Kl \Kj

which tends to zero as i, j → ∞; that is, {fj , j ≥ 1} is an E-Cauchy sequence in F. As limj→∞ fj = Uν on E, we conclude that Uν ∈ Fe . We have just proved that Uν ∈ Fe if and only if E Uν(x)ν(dx) < ∞ under the assumption that m(E) < ∞. In either cases, we deduce from (6.7.4) that (6.7.5) E(Uν, Uν) = lim E( fj , fj ) = Uν(x)ν(dx). j→∞

E

287


Moreover, for u ∈ bF+ , by Theorem 4.1.1 and monotone convergence theorem, we have 1 E(Uν, u) = lim E( fj , u) = lim lim ( fj − Pt fj , u) j→∞ j→∞ t→0 t = lim u(x)1Kj (x)ν(dx) = u(x)ν(dx). j→∞ E

E

Since the linear span of bF+ is E-dense in Fe , we have established (6.7.3). For a general σ -finite measure m, take a strictly positive m-integrable Borel measurable function g on E and define µ = g · m. Then µ is a finite measure on E. Let Y be the s time change of X via measure µ; that is, Yt = Xτt , where τt = inf{s > 0 : 0 g(Xs )ds > t}. The time-changed process Y is µ-symmetric. Let (E Y , F Y ) be the Dirichlet form of Y on L2 (E; µ). Then by Corollary 5.2.12, FeY = Fe and E Y = E on Fe . The measure ν is also a smooth measure with respect to process Y in view of Theorem 5.2.11. It is easy to verify that the PCAF AY,ν of Y corresponding to ν is related to corresponding PCAF Aν of X by = Aντt AY,ν t

for t ≥ 0.

In particular, we have U Y ν(x) = Uν on E. As we just proved that the theorem holds for Y, we conclude that the theorem also holds for X. D EFINITION 6.7.5. Let D be an open subset of E. We say a function u is harmonic in D (with respect to the process X) if for every relatively compact open subset D1 of D, t → u(Xt∧τD1 ) is a uniformly integrable Px -martingale for q.e. x ∈ D1 . Let (N(x, dy), H) be a Lévy system of X and let J(dx, dy) = N(x, dy)µH (dx)

and

κ(dx) = N(x, {∂})µH (dx),

where µH is the Revuz measure of the positive continuous additive functional H of X. Since (E, F ) is a regular Dirichlet form on L2 (E; m), for any relatively compact open sets D1 , D2 with D1 ⊂ D2 , there is φ ∈ F ∩ Cc (E) so that φ = 1 on D1 and φ = 0 on Dc2 . We then have from Theorem 4.3.3 c J(D1 , D2 ) = (φ(x) − φ(y))2 J(dx, dy) ≤ 2E(φ, φ) < ∞. (6.7.6) D1 ×Dc2

For open set D ⊂ E, let X D = (XtD , ζ D , Px ) denote the part process of X on the open set D. The Dirichlet form of X D is (E, F D ), where F D := {u ∈ F : u = 0 q.e. on E \ D}. ( For notational convenience, in this section we use notation F D instead of FD .) By Theorem 3.3.9, (E, F D ) is a regular Dirichlet form in L2 (D; m). A function f is said to be locally in F D , denoted

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CHAPTER SIX

D as f ∈ Floc , if for every relatively compact subset D1 of D, there is a function D g ∈ F such that f = g m-a.e. on D1 . By Theorem 4.3.10(i), µcu,v is well D defined on D for every u, v ∈ Floc . Further, for a smooth measure µ on D with µ D D respect to (E, F ), U µ will denote the function E· [Aζ D ] for the PCAF Aµ of X D with Revuz measure µ in accordance with Theorem 6.7.4. For an open set D ⊂ E, consider the following two conditions for a measurable function u on E. For any relatively compact open sets D1 , D2 with D1 ⊂ D2 ⊂ D2 ⊂ D, |u(y)|J(dx, dy) < ∞ (6.7.7) D1 ×(E\D2 )

and fu ∈ FeD1 . Here the function fu is defined by

fu (x) = 1D1 (x)Ex (1 − φD2 )|u| (XτD1 ) ,

(6.7.8)

x ∈ E,

(6.7.9)

φD2 ∈ Cc (D) ∩ F with 0 ≤ φD2 ≤ 1 and φD2 = 1 on D2 .

(6.7.10)

using a function φD2 satisfying

Note that both conditions (6.7.7) and (6.7.8) are automatically satisfied when X is a diffusion since in this case the jumping measure J vanishes and XτD1 ∈ ∂D1 on {τD1 < ζ }. In view of (6.7.6), every bounded function u satisfies the condition (6.7.7). In fact by the following lemma, every bounded function u also satisfies the condition (6.7.8). L EMMA 6.7.6. Suppose that u is a measurable function on E and D1 and D2 are two relatively compact open sets D1 , D2 such that D1 ⊂ D2 ⊂ D2 ⊂ D. Let φD2 be a function satisfying (6.7.10) and fu be defined by (6.7.9). Define (1 − φD2 (y))|u(y)|N(x, dy)µH (dx), (6.7.11) µu (dx) := 1D1 (x) E\D2

which is a smooth measure of X D1 . Then fu = U D1 µu . Furthermore, fu ∈ FeD1 if and only if fu (x)µu (dx) < ∞. (6.7.12) D1

Suppose u satisfies condition (6.7.7) and the condition that

sup Ex 1Dc2 |u| (XτD1 ) < ∞. x∈D1

Then (6.7.12) holds for u and so does (6.7.8).

(6.7.13)

289


Proof. Let φD2 be a function satisfying (6.7.10), and fu be defined by (6.7.9). Note that 1 − φD2 = 0 on D2 . Taking T = τD1 , g(s) = 1, and f (x, y) = |(1 − φD2 (y))|u|(y) − (1 − φD2 (x))|u|(x)| in the Lévy system formula (A.3.33), we get τD 1 (1 − φD2 (y))|u|(y)N(Xs , dy) dHs fu (x) = Ex 0

E\D2

for x ∈ E. Note that by Proposition 4.1.10, the Revuz measure µu on D1 of the PCAF t∧τD 1 (1 − φD2 (y))|u|(y)N(Xs , dy) dHs t → 0

E\D2

of X D1 is given by (6.7.11) and so fu = U D1 µ u . Applying Theorem 6.7.4 to X D1 , we conclude that fu ∈ FeD1 if and only if D1 fu (x)µu (dx) < ∞. Now assume that conditions (6.7.7) and (6.7.13) hold. Note that under condition (6.7.13), fu is a bounded function. On the other hand, under condition (6.7.7), (1 − φD2 (y))|u(y)|N(x, dy) µH (dx) µu (D1 ) = D1

E\D2

≤ So we have for u.

D1

|u(y)|N(x, dy) µH (dx) < ∞.

E\D2

U µu (x)µu (dx) ≤ fu ∞ µu (D1 ) < ∞; that is, (6.7.12) holds D1

D1

In many concrete cases such as in Examples 6.7.14–6.7.16 below, one can show that condition (6.7.7) implies condition (6.7.13). L EMMA 6.7.7. Let D be an open subset of E. Every u ∈ Fe that is locally bounded on D satisfies conditions (6.7.7) and (6.7.8). Proof. Let u ∈ Fe be locally bounded on D. For any relatively compact open sets D1 , D2 with D1 ⊂ D2 ⊂ D2 ⊂ D, take φ ∈ F ∩ Cc (D) such that φ = 1 on D1 and φ = 0 on Dc2 . Then uφ ∈ Fe and u(y)2 J(dx, dy) D1 ×(E\D2 )

=

D1 ×(E\D2 )

(((1 − φ)u)(x) − ((1 − φ)u)(y))2 J(dx, dy)

≤ 2E(u − uφ, u − uφ) < ∞. This together with (6.7.6) implies (6.7.7).

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Let fu be defined by (6.7.9). Note that |u| ∈ Fe is locally bounded on D and so (1 − φD2 )|u| = |u| − φD2 |u| ∈ Fe . Thus it follows from Theorem 3.4.8 that

fu (x) = Ex (1 − φD2 )|u| (XτD1 ) − (1 − φD2 (x))|u|(x) for q.e. x ∈ E and fu ∈ FeD1 .

L EMMA 6.7.8. Let D be a relatively compact open set of E. Suppose u is a D that is locally bounded on D and satisfies condition (6.7.7). function in Floc Then for every v ∈ Cc (D) ∩ F, the expression 1 c 1 (u(x) − u(y))(v(x) − v(y))J(dx, dy) µu,v (D) + 2 2 E×E + u(x)v(x)κ(dx) D

is well-defined and finite, and it will still be denoted as E(u, v). Proof. Clearly the first and the third terms are well defined and finite. To see that the second term is also well defined, let D1 be a relatively compact open subset of D such that supp[v] ⊂ D1 . By assumption, there is some f ∈ F so that f = u m-a.e. and hence q.e. on D1 . Then uφ ∈ F D for φ ∈ Cc (D) ∩ F with φ = 1 on D1 . Using property (6.7.6), |(u(x) − u(y))(v(x) − v(y))|J(dx, dy) E×E

=

D1 ×D1

|(u(x) − u(y))(v(x) − v(y))|J(dx, dy)

+2 ≤

D1 ×(E\D1 )

D1 ×D1

|u(x)v(x)|J(dx, dy) + 2

|v(x)| D1

|u(y)|J(dx, dy) E\D1

|(u(x)φ(x) − u(y)φ(y))(v(x) − v(y))|J(dx, dy)

+2uv∞ J(supp[v], Dc1 ) + 2v∞

supp[v]×Dc1

|u(y)|J(dx, dy)

< ∞.

This proves the lemma.

Recall Definition 1.3.17 of a special standard core C(⊂ F ∩ Cc (E)) of (E, F). F ∩ Cc (E) is a particular example of a special standard core. For a special standard core C of (E, F) and an open subset D ⊂ E, we let CD = {u ∈ C : supp[u] ⊂ D}.

291


D T HEOREM 6.7.9. Let D be an open subset of E. Suppose that u ∈ Floc is locally bounded on D satisfying conditions (6.7.7)–(6.7.8) and that

E(u, v) = 0

for every v ∈ CD

(6.7.14)

for some special standard core C of (E, F ). Then u is harmonic in D. If D1 is a relatively compact open subset of D such that Px (τD1 < ∞) > 0 for q.e. x ∈ D1 , then u(x) = Ex [u(XτD1 )] for q.e. x ∈ D1 . Proof. Take a relatively compact open set D2 with D1 ⊂ D2 ⊂ D2 ⊂ D and a function φ := φD2 as in (6.7.10). Then φu ∈ F D . So by Theorems 3.4.8 and 3.4.9, h1 (x) := Ex [(φu)(XτD1 )] ∈ Fe and φu − h1 ∈ FeD1 . Moreover, E(h1 , v) = 0

for every v ∈ FeD1 .

(6.7.15)

Let h2 (x) := Ex ((1 − φ)u)(XτD1 ) , which is well defined by condition (6.7.8). Define fu+ (resp. fu− ) by (6.7.9) with u+ = u ∨ 0 (resp. u− = (−u) ∨ 0) in place of u. Then 1D1 · h2 = fu+ − fu− . Using a Lévy system formula (A.3.33) as in the proof of Lemma 6.7.6, we have fu+ = U D1 µ+ and fu− = U D1 µ− , where µ+ (resp. µ− ) is defined by (6.7.11) with u+ (resp. u− ) in place of u. Thus we obtain 1D1 · h2 = U D1 µ+ − U D1 µ− . We claim that 1D1 h2 ∈ FeD1 and that for every v ∈ FeD1 ,

E(1D h2 , v) =

v(x)1D1 (x) E

Dc2

(1 − φ)u (z)N(x, dz) µH (dx).

(6.7.16)

Clearly U D1 µ+ ≤ U D1 µ. For j ≥ 1, let Fj := {x ∈ D1 : U D1 µ+ (x) ≤ j}, which is a finely closed subset of D1 . Define νj := 1Fj µ+ . Then for x ∈ Fj , U D1 νj (x) ≤ U D1 µ+ (x) ≤ j, while for x ∈ D1 \ Fj , U D1 νj (x) = Ex U D1 νj (XσFj ) ≤ j. In other words, we have U D1 νj ≤ j ∧ U D1 µ+ ≤ j ∧ f . As both U D1 νj and j ∧ f are excessive functions of X D1 and m(D1 ) < ∞, we have by Theorem 1.1.5 and Lemma 1.2.3 that {U D1 νj , j ∧ U D1 µ} ⊂ F D1 with E(U D1 νj , U D1 νj ) ≤ E( j ∧ f , j ∧ f ) ≤ E( f , f ) < ∞.

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Moreover, for each j ≥ 1, we have by Theorem 4.1.1 that E(U D1 νj , U D1 νj ) 1 D1 D1 1 U D1 (νj (x) − PD = lim t U νj (x))U νj (x)m(dx) t→0 t E 1 νj Ex At∧τ U D1 νj (x)m(dx) = lim D1 t→0 t E U D1 νj (x) 1Fj (x)µ1 (dx), = D1

which increases to

D1

U D1 µ+ (x)µ+ (dx). Consequently,

U D1 µ+ (x)µ+ (dx) ≤ E( f , f ) < ∞.

D1

So we have by Theorem 6.7.4 applied to X D1 that U D1 µ+ ∈ FeD1 D1 + + with E(U µ , v) = D1 v(x)µ (dx) for every v ∈ FeD1 . Similarly, we have U D1 µ− ∈ FeD1 with E(U D1 µ− , v) = D1 v(x)µ2 (dx) for every v ∈ FeD1 . It follows that 1D1 h2 = U D1 µ1 − U D1 µ2 ∈ FeD1 and claim (6.7.16) is established. As h2 = 1D1 h2 + (1 − φ)u and (1 − φ)u satisfies condition (6.7.7), we have by Lemma 6.7.8 and (6.7.16) that for every v ∈ Cc (D1 ) ∩ F, E(h2 , v) = E(1D1 h2 , v) + E((1 − φ)u, v) = v(x)(1 − φ(y))u(y)N(x, dy)µH (dx) E×E

v(x)(1 − φ(y))u(y)N(x, dy)µH (dx)

− E×E

= 0.

(6.7.17)

This combining with (6.7.15) and condition (6.7.14) proves that E(u − h1 − h2 , v) = 0

for every v ∈ CD1 .

(6.7.18)

Since u − (h1 + h2 ) = (φu − h1 ) − 1D1 h2 ∈ FeD1 and CD1 is E-dense in FeD1 by Theorem 3.3.9(iii), the above display holds for every v ∈ FeD1 . In particular, we have E(u − h1 − h2 , u − h1 − h2 ) = 0.

(6.7.19)

293


By Lemma 6.7.3, u(Xt ) − h1 (Xt ) − h2 (Xt ) is a bounded Px -martingale for q.e. x ∈ E. As

for x ∈ D1 , h1 (x) + h2 (x) = Ex u(XτD1 ) the above implies that t → u(Xt∧τD1 ) is a uniformly integrable Px -martingale for q.e. x ∈ D1 . If Px (τD1 < ∞) > 0 for q.e. x ∈ D1 , applying Lemma 6.7.3 to the Dirichlet form (E, F D1 ), we have u − h1 − h2 = 0 q.e. on D1 because u(XtD )−h1 (XtD )−h2 (XtD ) = 0 for t > τD1 . So u(x) = Ex [u(XτD1 )] for q.e. x ∈ D1 . This completes the proof of the theorem. Remark 6.7.10. (i) The principal difficulty in the above proof is establishing D satisfying condi(6.7.18) and that u − (h1 + h2 ) ∈ FeD1 for general u ∈ Floc tions (6.7.7) and (6.7.8). If u is assumed a priori to be in Fe , these facts and therefore the theorem itself are then much easier to establish. Note that when u ∈ Fe , it follows immediately from Theorem 3.4.8 that h1 + h2 = Ex [u(XτD1 )] ∈ Fe enjoys property (6.7.18) and u − (h1 + h2 ) ∈ FeD1 . Therefore, (6.7.19) holds and consequently u is harmonic in D. (ii) If we assume that the process X (or equivalently (E, F )) is m-irreducible and that Dc1 is not m-polar, then by Theorem 3.5.6, Px (τD1 < ∞) > 0 for q.e. x ∈ D. T HEOREM 6.7.11. Suppose D is an open set of E with m(D) < ∞ and u is a function on E satisfying the condition (6.7.7) such that u ∈ L∞ (D; m) and that {u(Xt∧τD ), t ≥ 0} is a uniformly integrable Px -martingale for q.e. x ∈ E. Then D u ∈ Floc

and

E(u, v) = 0 for every v ∈ Cc (D) ∩ F.

(6.7.20)

Proof. As for q.e. x ∈ E, {u(Xt∧τD ), t ≥ 0} is a uniformly integrable Px martingale, u(Xt∧τD ) converges in L1 (Px ) as well as Px -a.s. to some random variable ξ . By considering ξ + , ξ − and u+ (x) := Ex [ξ + ], u− (x) := Ex [ξ − ] separately, we may assume without loss of generality that u ≥ 0. Note that ξ 1{τD 0, by the Markov property of X D ,

PD t u2 (x) = Ex [u2 (Xt ); t < τD ] = Ex ξ 1{τD =∞} ◦ θt ; t < τD = u2 (x). Since u2 ∈ L2 (D; m), by (1.1.4)–(1.1.5), u2 ∈ F D On the other hand,

with E(u2 , u2 ) = 0.

PD t u(x) = Ex [u(Xt ); t < τD ] = Ex u(XτD ) : t < τD ≤ u(x).

(6.7.21)

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CHAPTER SIX

Let {Dn , n ≥ 1} be an increasing sequence of relatively compact open subsets of D with ∪n≥1 Dn = D and define σn := inf t ≥ 0 : XtD ∈ Dn . Let en (x) = Ex [e−σn ], x ∈ D, be the 1-equilibrium potential of Dn with respect to the part process X D . Clearly en ∈ F D is 1-excessive with respect to the process X D , en (x) = 1 q.e. on Dn . Let a := 1D u∞ . Then for every t > 0, e−t PD t ((aen ) ∧ u)(x) ≤ ((aen ) ∧ u)(x)

for q.e. x ∈ D.

By Lemma 1.2.3, we have (aen ) ∧ u ∈ F D for every n ≥ 1. Since (aen ) ∧ u = D . u m-a.e. on Dn , we have u ∈ Floc Let D1 be a relatively compact open subset of D. Let φ ∈ Cc (D) ∩ F so that 0 ≤ φ ≤ 1 and φ = 1 in an open neighborhood D2 of D1 . Define for x ∈ E

h1 (x) := Ex (φu)(XτD1 ) and h2 (x) := Ex ((1 − φ)u)(XτD1 ) . Then u1 = h1 + h2 on E. Since φu ∈ F, we know as in (6.7.15) that h1 ∈ Fe and E(h1 , v) = 0


By the same argument as that for (6.7.17), we have E(h2 , v) = 0


These together with (6.7.21) in particular imply that E(u, v) = E(h1 + h2 + u2 , v) = 0

for every v ∈ Cc (D1 ) ∩ F.

Since D1 is an arbitrary relatively compact subset of D, we have E(u, v) = 0

for every v ∈ Cc (D) ∩ F.

This completes the proof.

Remark 6.7.12. The principal difficulty for the proof of the above theorem is D establishing that a function u harmonic in D is in Floc with E(u, v) = 0 for every v ∈ F ∩ Cc (D). If a priori u is assumed to be in Fe , then Theorem 6.7.11 is easy to establish. In this case, it follows from Theorem 3.4.8 that u1 = h1 + h2 = Ex [u(XτD1 )] ∈ Fe and that u1 is E-orthogonal to F ∩ Cc (D). This together with (6.7.21) immediately implies that u enjoys (6.7.20). Combining Theorems 6.7.9 and 6.7.11, we have the following. T HEOREM 6.7.13. Let D be an open subset of E. Suppose that u is a function on E that is locally bounded on D and satisfies conditions (6.7.7)–(6.7.8).

295


D (i) u is harmonic in D if and only if u ∈ Floc and the condition (6.7.14) holds for some special standard core C of (E, F). (ii) Assume that for every relatively compact open subset D1 of D, Px (τD1 < ∞) > 0 for q.e. x ∈ D1 . (By Remark 6.7.10, this condition is satisfied if (E, F) is irreducible.) Then u is harmonic in D if and only if for every relatively compact subset D1 of D, u(XτD1 ) ∈ L1 (Px ) and u(x) = Ex [u(XτD1 )] for q.e. x ∈ D1 .

Example 6.7.14. (Stable-like process on Rn ) Consider the following Dirichlet form (E, F), where F := W α/2,2 (Rn ) := u ∈ L2 (Rn ) :

Rn ×Rn

and for u, v ∈ F , E(u, v) =

1 2

(u(x) − u(y))2 dxdy < ∞ |x − y|n+α

Rn ×Rn

(u(x) − u(y))(v(x) − v(y))

c(x, y) dxdy. |x − y|n+α

Here n ≥ 1, α ∈ (0, 2), and c(x, y) is a symmetric function in (x, y) that is bounded between two positive constants. In view of Section 2.2.2, (E, F) is a regular Dirichlet form on L2 (Rn ). Its associated symmetric Hunt process X is called a symmetric α-stable-like process on Rn , which is studied in [29]. Cc∞ (Rn ) is a special standard core of (E, F). The process X has a strictly positive jointly continuous transition density function p(t, x, y) and hence is irreducible. Moreover, there is constant c > 0 such that p(t, x, y) ≤ c t−n/α

for t > 0 and x, y ∈ Rn

(6.7.22)

and consequently by [33, Theorem 1], sup Ex [τD ] < ∞

(6.7.23)

x∈D

for any open set D having finite Lebesgue measure. When c(x, y) is constant, the process X is nothing but the rotationally symmetric α-stable process on Rn . In this example, the jumping measure J(dx, dy) =


Hence for any non-empty open set D ⊂ Rn , condition (6.7.7) is satisfied if and only if (1 ∧ |x|−n−α )u(x) ∈ L1 (Rn ). Moreover, for such a function u and relatively compact open sets D1 , D2 with D1 ⊂ D2 ⊂ D2 ⊂ D, we can use

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CHAPTER SIX

Lévy system formula (A.3.33) to obtain

sup Ex (1Dc2 |u|)(XτD1 ) x∈D1

τD1

= sup Ex x∈D1

0

Dc2

c(Xs , y) |u(y)| dy ds |Xs − y|n+α

−n−α ≤ c (1 ∧ |y| )|u(y)|dy sup Ex [τD1 ] < ∞. Rn

(6.7.24)

x∈D1

In other words, for this example, condition (6.7.13) and hence (6.7.8) is a consequence of (6.7.7). So Theorem 6.7.13 says that for an open set D and a function u on Rn that is locally bounded on D with (1 ∧ |x|−n−α )u(x) ∈ L1 (Rn ), the following are equivalent: (i) u is harmonic in D. (ii) For every relatively compact subset D1 of D, u(XτD1 ) ∈ L1 (Px ) and u(x) = Ex [u(XτD1 )] for q.e. x ∈ D1 . α/2,2 D = Wloc (D) and for every v ∈ Cc∞ (D), (iii) u ∈ Floc c(x, y) (u(x) − u(y))(v(x) − v(y)) dxdy = 0. |x − y|n+α Rn ×Rn It was shown in [29] that every bounded harmonic function in D admits a (locally) Hölder continuous version. Example 6.7.15. (Diffusion process on a locally compact separable metric space) Let (E, F) be a local regular Dirichlet form on L2 (E; m), where E is a locally compact separable metric space, and X is its associated Hunt process. In this case, X has continuous sample paths and so the jumping measure J is null in view of Theorem 4.3.4. Hence conditions (6.7.7)–(6.7.8) are automatically satisfied. Let D be an open subset of E and u be a function on E that is locally bounded in D. Then by Theorem 6.7.13, u is harmonic in D if and only if D and (6.7.14) holds for some special standard core C of (E, F). u ∈ Floc Now consider the following special case: E = Rn with n ≥ 1, m(dx) is the Lebesgue measure dx on Rn , F = H 1 (Rn ), and E(u, v) =

n ∂u(x) ∂v(x) 1 aij (x) dx 2 i,j=1 Rn ∂xi ∂xj

for u, v ∈ W 1,2 (Rn ),

where (aij (x))1≤i,j≤n is an n × n-matrix valued measurable function on Rn that is uniformly elliptic and bounded. Then (E, F) is a regular local Dirichlet form on L2 (Rn ) possessing Cc∞ (Rn ) as its special standard core and its

297


associated Hunt process X is a conservative diffusion on Rn having jointly continuous transition density function. Let D be an open set in Rn . Then by Theorem 6.7.13, the following are equivalent for a locally bounded function u on D: (i) u is harmonic in D. (ii) For every relatively compact open subset D1 of D, u(XτD1 ) ∈ L1 (Px ) and u(x) = Ex [u(XτD1 )] for every x ∈ D1 . 1 (D) and for every v ∈ Cc∞ (D), (iii) u ∈ Hloc n i,j=1

Rn

aij (x)

∂u(x) ∂v(x) dx = 0. ∂xi ∂xj

In fact, in this case, it can be shown (see, e.g., [83]) that every (locally bounded) function satisfying condition (iii) has a continuous version. Therefore, in condition (ii), the statement “for q.e. x ∈ D1 ” can be strengthened into “for every x ∈ D1 ” because Ex [u(XτD1 )] is a difference of two bounded excessive functions in x ∈ D1 relative to the part process of X on D1 which has an absolutely continuous transition function. Example 6.7.16. (Diffusions with jumps on Rn ) Consider the following Dirichlet form (E, F), where F = H 1 (Rn ) ∩ W α/2,2 (Rn ) and for u, v ∈ H 1 (Rn ) ∩ W α/2,2 (Rn ), n 1 ∂u(x) ∂v(x) E(u, v) = aij (x) dx 2 i,j=1 Rn ∂xi ∂xj

+

1 2

Rn ×Rn

(u(x) − u(y))(v(x) − v(y))


Here n ≥ 1, (aij (x))1≤i,j≤n is an n × n-matrix valued measurable function on Rn that is uniformly elliptic and bounded, α ∈ (0, 2), and c(x, y) is a symmetric function in (x, y) that is bounded between two positive constants. It is easy to check that (E, F) is a regular Dirichlet form on L2 (Rn ) possessing Cc∞ (Rn ) as a special standard core. Its associated symmetric Hunt process X has both the diffusion and jumping components. Such a process has recently been studied in [30]. It is shown there that the process X has strictly positive jointly continuous transition density function p(t, x, y) and hence is irreducible. Moreover, a sharp two-sided estimate is obtained in [30] for p(t, x, y). In particular, there is a constant c > 0 such that p(t, x, y) ≤ c t−n/α ∧ t−n/2

for t > 0 and x, y ∈ Rn .

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CHAPTER SIX

Note that when (aij )1≤i,j≤n is the identity matrix and c(x, y) is constant, the process X is nothing but the symmetric Lévy process that is the independent sum of a Brownian motion and a rotationally symmetric α-stable process on Rn . In this example, the jumping measure J(dx, dy) =


Hence for any non-empty open set D ⊂ Rn , condition (6.7.7) is satisfied if and only if (1 ∧ |x|−n−α )u(x) ∈ L1 (Rn ). By the same reasoning as that for (6.7.24), we see that for this example, condition (6.7.13) and hence (6.7.8) is implied by condition (6.7.7). So Theorem 6.7.13 says that for an open set D and a function u on Rn that is locally bounded on D with (1 ∧ |x|−n−α )u(x) ∈ L1 (Rn ), the following are equivalent: (i) u is harmonic in D with respect to X. (ii) For every relatively compact subset D1 of D, u(XτD1 ) ∈ L1 (Px ) and u(x) = Ex [u(XτD1 )] for q.e. x ∈ D1 . α/2,2 1 (D) ∩ Wloc (D) and for every v ∈ Cc∞ (D), (iii) u ∈ Hloc n n i,j=1 R

aij (x)

∂u(x) ∂v(x) dx ∂xi ∂xj

+

Rn ×Rn

(u(x) − u(y))(v(x) − v(y))

c(x, y) dxdy = 0. |x − y|n+α

It was shown in [30] that every bounded harmonic function in D admits a (locally) Hölder continuous version. Remark 6.7.17. It is possible to extend the results of this section to a general m-symmetric right process X on a Lusin space, where m is a positive σ -finite measure with full topological support on E. In this case, the Dirichlet (E, F) of X is a quasi-regular Dirichlet form on L2 (E; m). By Theorem 1.4.3, (E, F) is quasi-homeomorphic to a regular Dirichlet form on a locally compact separable metric space. So the results of this section can be extended to the quasiregular Dirichlet form setting, by using this quasi-homeomorphism. However, since the notion of open set is not invariant under quasi-homeomorphism, some modifications are needed. We need to replace open set D in Definition 6.7.5 by quasi open set D. Similar modifications are needed for conditions (6.7.7) and (6.7.8) as well. We say a function u is harmonic in a quasi open set D ⊂ E if for every quasi open subset D1 ⊂ D so that D1 ∩ Fk ⊂ D for every k ≥ 1, where {Fk , k ≥ 1} is an E-nest consisting of compact sets, t → u(Xt∧τD1 ∩Fk ) is a uniformly integrable Px -martingale for q.e. x ∈ D1 ∩ Fk , and for every k ≥ 1.

299

REFLECTED DIRICHLET SPACES ◦

D D needs to be replaced by Floc , which is defined The local Dirichlet space Floc D by (4.3.31) but with (E, F ) in place of (E, F ). Condition (6.7.20) should be replaced by ◦

D u ∈Floc

and

E(u, v) = 0 for every v ∈ F with E-supp[v] ⊂ D. (6.7.25)

Here E-supp[u] is the smallest quasi closed set outside which u = 0 m-a.e. We leave the details to interested readers.

Chapter Seven BOUNDARY THEORY FOR SYMMETRIC MARKOV PROCESSES

Let X 0 be an m0 -symmetric right process on a state space E0 . The boundary theory of symmetric Markov processes concerns all possible symmetric E containing E0 as an intrinsic open extensions of X 0 to some state space subset so that they admit no-sojourn (that is, spend zero Lebesgue amount of time) at E \ E0 . The no-sojorn condition ensures that the extension process is E by setting m0 ( E \ E0 ) = 0. m0 -symmetric after the measure m0 is extended to The initial “minimal” process X 0 can be taken as the part process on E0 of a symmetric Hunt process X on E killed upon leaving a quasi open subset E0 and this is the viewpoint we take in this chapter. Of course, one can just start with X 0 with no reference to X. As we see from (5.6.7), the Douglas integral described by the Feller measures determined by X 0 occupies a principal part of the trace Dirichlet form of X on E \ E0 . On the other hand, in view of Theorem 6.6.5, we may well expect that the reflected Dirichlet space (E 0,ref , (F 0 )ref ) of X 0 will play an important role in the boundary theory. One of the goals of this chapter is to demonstrate how these two tools can be used in the study of possible symmetric extensions of X 0 . On account of the quasi-homeomorphism theorem (Theorem 1.4.3), the three theorems in Section 1.5 and Theorem 3.1.13, without loss of generality, we may and do assume throughout this chapter that E is a locally compact separable metric space, m is a positive Radon measure on E with supp[m] = E, (E, F) is a regular irreducible symmetric Dirichlet form in L2 (E; m), and X = (, M, Xt , ζ , Px ) is an m-symmetric Hunt process associated with (E, F). Let F be a quasi closed subset of E that is not E-polar and X 0 be the part process of X on E0 := E \ F. The process X 0 is then symmetric with respect to the measure m0 := m|E0 . Let (E 0 , F 0 ) be the Dirichlet form of X 0 on L2 (E0 ; m0 ). We know that F 0 = {u ∈ F : u = 0 E-q.e. on F} and its extended Dirichlet space Fe0 equals {u ∈ Fe : u = 0 E-q.e. on F}. We shall consider the reflected Dirichlet space ((F 0 )ref , E 0,ref ) and the active reflected Dirichlet space 0 ref 2 0 0 (F 0 )ref a = (F ) ∩ L (E0 ; m0 ) of the quasi-regular Dirichlet form (E , F ) as defined in Section 6.4.

BOUNDARY THEORY FOR SYMMETRIC MARKOV PROCESSES

301

In Section 7.1, we investigate the relationship between the space (Fe , E) (re0,ref spectively, (F , Eα )) and the space ((F 0 )ref , E 0,ref ) (respectively, ((F 0 )ref a , Eα )). We will see that the former is always dominated by the latter. In Section 7.2 we focus our attention on the restricted spaces Fe F , F F and their descriptions in terms of the Feller measures U, V, and Uα introduced in Section 5.4 and the Douglas integrals defined by them. We give conditions to ensure that those spaces coincide with function spaces on F with finite Douglas integrals. We also use 1-order Feller measure U1 to introduce an intrinsic measure µ0 on F such that F F = Fe F ∩ L2 (F; µ0 ). These results are then applied to identifying the trace Dirichlet space of the reflecting extension X of a symmetric standard process X 0 with the function space on F of finite Douglas integrals. The process X will be called a reflecting extension of X 0 when the condition that (E, F) = (E 0,ref , (F 0 )ref a ) is fulfilled among others. It will then be shown that the active reflected Dirichlet space of any quasi-regular Dirichlet form admits such a representation on its regularizing boundary F. The infinitesimal generator A of the L2 -semigroup of X is a linear operator L, to be introduced in Section 7.3, satisfying a lateral (boundary) condition. The linear operator L is defined through the active reflected Dirich0 0 let form (E 0,ref , (F 0 )ref a ) of (E , F ). The aforementioned lateral (boundary) condition uses the notion of flux functional N that is also defined in terms of the reflected Dirichlet space (E 0,ref , (F 0 )ref ). The trace Dirichlet form on L2 (F; µ0 ) will play a role in the description of the lateral condition. In Section 7.4, we study the case where the set F consists of countably many points that are located in an invariant way under a quasi-homeomorphism. We shall show that not only the lateral condition but also the Dirichlet form and the resolvent of X can be described in quite tractable ways. In particular, when X admits no killing on F or jump from F to F, the trace Dirichlet form is the subspace of the space of finite Douglas integrals spanned by finitely supported functions, and accordingly the resolvent of X is uniquely determined by X 0 . The Dirichlet space F of X can be also characterized as a subspace of the active reflected Dirichlet space of X 0 spanned by F 0 and α-order hitting distributions to F. When F consists of only a single point a, X may be called a one-point extension of X 0 . In Section 7.5, the uniqueness of a one-point extension of a symmetric standard process will be established under a certain condition without requiring the regularity of the associated Dirichlet forms. A construction of a one-point extension of a symmetric standard process X 0 has been carried out in [75] and [28] by making use of a Poisson point process taking values in the space of excursions of X 0 around the point a. Consider a symmetric Hunt process X on a state space E, a closed set K ⊂ E, and the part process X 0 of X on E0 = E \ K. The above-mentioned probabilistic construction is robust enough to enable us to produce a one-point extension of X 0 by collapsing the set K into a single point a∗ , as will be shown in Section 7.5. While a

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CHAPTER SEVEN

one-point extension is always irreducible, the symmetric process X 0 we start with may not be irreducible, so X 0 may have more than one symmetrizing measure. Accordingly, X 0 may admit a variety of skew extensions as will be shown in Section 7.5. In Section 7.6, we collect various concrete examples of X 0 and identify their one-point extensions. In Section 7.7, we shall generalize the uniqueness theorem and existence theorem of the one-point extension established in Section 7.5 to the countably many points extension. The uniqueness is deduced from a theorem in Section 7.4 via quasi-homeomorphism. The construction will be carried out by repeating the one-point extensions of Section 7.5. Section 7.8 will present several concrete examples including a two-point reflecting extension in higher dimensions and darning of countably many holes. Recall that the following convention is in force: any numerical function f defined on a subset of E is extended to ∂ by setting f (∂) = 0.

7.1. REFLECTED DIRICHLET SPACE FOR PART PROCESSES From Section 7.1 to Section 7.3 except for the last part of Section 7.2, we shall adopt the same setting as in Section 5.5. Thus E is a locally compact separable metric space, m is a positive Radon measure on E with supp[m] = E, (E, F) is a regular irreducible symmetric Dirichlet form in L2 (E; m), and X = (, M, Xt , ζ , Px ) is an m-symmetric Hunt process associated with (E, F). The resolvent of X will be denoted by {Gα ; α > 0}. Let F be a quasi closed subset of E that is not E-polar. We denote by (N, H) a Lévy system of the Hunt process X on E. The jumping measure and the killing measure for the Dirichlet form (E, F) of X are given by J(dx, dy) := N(x, dy)µH (dx)

and

κ(dx) := N(x, {∂})µH (dx).

As before, each element of the extended Dirichlet space Fe will be represented by its quasi continuous version and we will assume without loss of generality that F is nearly Borel and finely closed. Then E0 = E \ F is finely open. Recall that τ0 := τE0 , defined by (5.5.2), is the first exit time from E0 = E \ F by X. The part process X 0 of X on E0 can then be realized as X 0 = {, M0 , Xt0 , ζ 0 , Px }x∈E0 , where

ζ = τ0 0

and

Xt0

=

Xt ∂

for 0 ≤ t < ζ 0 , for t ≥ ζ 0 ,

and M0 is the σ -field generated by Xt0 with a usual augmentation by null sets. For emphasis, the law of X 0 under Px and its expectation will be denoted as P0x

303


and E0x , respectively. The process X 0 is a standard process on E0 . {G0α ; α > 0} and G00 will denote the resolvent and the 0-order resolvent of X 0 , respectively. By Theorems 3.5.6 and 3.3.8 the Dirichlet form (E 0 , F 0 ) on L2 (E0 ; m0 ) defined by (5.5.4) is transient and quasi-regular with which X 0 is properly associated. In view of Theorem 3.3.8, X 0 can be considered as a special Borel standard process by restricting it to outside of an m-inessential set if necessary. In general (E 0 , F 0 ) is not regular on L2 (E0 ; m0 ). Let S(E0 ) (resp. S pur (E0 )) be the space of q.e. excessive functions (resp. q.e. purely excessive functions) of X 0 on E0 and L0 (f , g) be the energy functional of X 0 for f ∈ S pur (E0 ), g ∈ S(E0 ) defined by (5.4.4). As (E 0 , F 0 ) is a quasi-regular Dirichlet form on L2 (E0 ; m0 ), one can define its reflected Dirichlet space (E 0,ref , (F 0 )ref ) by Definition 6.4.4 but with X 0 in place of X. Let us define the notion of the terminal random variable by Definition 6.4.10 and put N = Φ : Φ is a terminal random variable of X 0 with E0x [Φ 2 ] < ∞ for q.e. x ∈ E0 and L0 E0· [Φ 2 ] − (E0· [Φ])2 , 1 < ∞ . By Theorem 6.4.12, it holds that (F 0 )ref = Fe0 + H0 N, where H0 N := h : h(x) = E0x [Φ] for q.e. x ∈ E0 withΦ ∈ N . Moreover, for f = f0 + h ∈ (F 0 )ref , where f0 ∈ Fe0 and h = E0· [Φ] with Φ ∈ N, 1 1 h(x)2 κ0 (dx), E 0,ref (f , f ) = E 0 (f0 , f0 ) + L0 (E0· [Φ 2 ] − h(·)2 , 1) + 2 2 E0 (7.1.1) where κ0 is the killing measure for X 0 . f = f0 + h as above is a unique E 0,ref -orthogonal decomposition of f ∈ (F 0 )ref . Exercise 7.1.1. Prove the identity κ0 (B) := κ(B) + J(B, F),

B ∈ B(E0 ).

(7.1.2)

0 0 The active reflected Dirichlet space (F 0 )ref a of (E , F ) is defined by 0 ref 2 (F 0 )ref a := (F ) ∩ L (E0 ; m0 ). 2 By Theorem 6.4.5, (E 0,ref , (F 0 )ref a ) is a Dirichlet form on L (E0 ; m0 ). In this 0,ref section, we first establish a unique Eα -orthogonal decomposition of f ∈ (F 0 )ref a . To this end, we put for α > 0 and a terminal random variable Φ of X 0

H0 Φ(x) := E0x [Φ]

and

H0α Φ(x) := Ex [e−αζ Φ], 0

x ∈ E0 ,

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CHAPTER SEVEN

whenever the expectations make sense. We also define for terminal random variables Φ, of X 0 Uα (Φ, ) := α(H0α Φ, H0 ). Here and in what follows, we denote by (u, v) the integral

E0

u(x) · v(x)m0 (dx).

L EMMA 7.1.2. If Φ is a non-negative terminal random variable of X 0 with H0 Φ(x) < ∞, then H0 Φ(x) − H0α Φ(x) = αG00 H0α Φ(x), H0β Φ(x) − H0α Φ(x) = (α − β)G0α H0β Φ(x)

for α, β > 0.

If Φ is a non-negative terminal random variable of X 0 , then Uα (Φ, Φ) is increasing in α > 0 and α1 Uα (Φ, Φ) is decreasing in α > 0. Proof. We have

H0 Φ(x) − H0α Φ(x) = αE0x

ζ0

e−α(ζ

−s)

e−αζ

(θs )

ds · Φ

0

= αE0x

∞ 0

= αE0x

0

∞ 0

E0X 0 s

0

Φ(θs ω)1{s α, Uβ (Φ, Φ) − Uα (Φ, Φ) = (β − α) (H0β Φ, H0 Φ) − (H0β Φ, αG0α H0 Φ) , which is non-negative because αG0α H0 Φ ≤ H0 Φ. The decreasing property of 1 U (Φ, Φ) is obvious. α α We introduce a subspace of N by N1 = {Φ ∈ N : U1 (|Φ|, |Φ|) < ∞}.

(7.1.3)

P ROPOSITION 7.1.3. A terminal random variable Φ of X 0 is in N1 if and only if H0α |Φ| ∈ (F 0 )ref a for some and hence for all α > 0. In this case, Φ = 1{ζ 0 0, 0 0 0 0 (F 0 )ref a = Hα N1 + F = {Hα Φ + f0 : Φ ∈ N1 , f0 ∈ F }.

(7.1.9)

The above decomposition is an Eα0,ref -orthogonal decomposition. Moreover, for 0 f = H0α Φ + f0 ∈ (F 0 )ref a with Φ ∈ N1 and f0 ∈ F , Eα0,ref (f , f ) = E 0,ref (H0 Φ, H0 Φ) + Uα (Φ, Φ) + Eα0 (f0 , f0 ).

(7.1.10)

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CHAPTER SEVEN

Proof. Again in the same way as the proof of Theorem 6.2.15, any non0 0 negative u ∈ (F 0 )ref a can be seen to be expressed as u = Hα Φ + f0 with Hα Φ ∈ 0 ref 0 (F )a , Φ ∈ N, Φ ≥ 0 and f0 ∈ F , which combined with the preceding 0 proposition proves that (F 0 )ref a ⊂ F + Hα N1 . The converse inclusion is obvious. The Eαref -orthogonality of the decomposition (7.1.9) can be also derived from (7.1.5) and (7.1.8): Eα0,ref (H0α Φ, f0 ) = −αE 0 (G00 H0α Φ, f0 ) + α(H0α Φ, f0 ) = 0 for Φ ∈ N1 and f0 ∈ F 0 . Expression (7.1.10) then follows from (7.1.6).

We now explore the relationship between (F 0 )ref and Fe . Throughout the remainder of this section, we shall assume that X admits no jumps from E0 to F; that is, J(E0 × F) = 0.

(7.1.11)

For a Borel measurable function ϕ on F and x ∈ E, we put Hϕ(x) := Ex ϕ(XσF ); σF < ∞ and Hα ϕ(x) := Ex e−ασF ϕ(XσF ); σF < ∞ , whenever they make sense. L EMMA 7.1.5. Assume that condition (7.1.11) holds. For ϕ ∈ B(F), we let Φ = 1{τ0 0 m-a.e. ψ(X ψdµ = E ); σ < ∞ belongs to the type of measure exhibited 0 H 1·m σ F 1 F F ◦

in the proof of Lemma 5.2.9. Therefore, µ0 ∈ SF .

T HEOREM 7.2.4. Assume that

It then holds that

m(F) = 0.

(7.2.18)

F F = Fe F ∩ L2 (F; µ0 ).

(7.2.19)

More specifically, let Xˇ be the time-changed process of X by means of the ˇ F) ˇ be the Dirichlet form of Xˇ on PCAF with Revuz measure µ0 and (E, L2 (F; µ0 ). Then Fˇ = F F and Eˇ1 (ϕ, ϕ) on Fˇ is equivalent to E1 (H1 ϕ, H1 ϕ) ˇ ϕ) + U1 (ϕ, ϕ)) on F . (= E(ϕ, F

Proof. We know by (5.2.5) that Fˇ = Fe F ∩ L2 (F; µ0 ). By (5.6.7), we have, ˇ for ϕ ∈ F, 1 (7.2.20) Eˇ1 (ϕ, ϕ) = C(ϕ, ϕ) + ϕ2L2 (F;µ0 ) + Mϕ Hϕ (F) 2 1 (ϕ(ξ ) − ϕ(η))2 J(dξ , dη) + ϕ(ξ )2 κ(dξ ). + 2 F×F F Let C be a special standard core of (E, F). By virtue of Theorem 5.2.8, C|F ˇ ˇ is √ E1 -dense in F. Since C is E1 -dense in F, C|F is dense in FF with metric E1 (H1 ϕ, H1 ϕ), ϕ ∈ FF . On account of (7.2.15), (7.2.16), (7.2.20), and the

315


assumption (7.2.18), we have for any ϕ ∈ CF E1 (H1 ϕ, H1 ϕ) ≤ Eˇ1 (ϕ, ϕ) ≤ 2E1 (H1 ϕ, H1 ϕ). This leads us to the desired identity (7.2.19). Therefore, we get Fˇ = FF and the metric equivalence. We can now formulate a counterpart of Theorem 7.2.1 for F in place of Fe . T HEOREM 7.2.5. (i) It holds that F F ⊂ G1 ⊂ N1

(7.2.21)

and Eα (Hα ϕ, Hα ϕ) ≥ C(ϕ, ϕ) + Uα (ϕ, ϕ), (ii) Suppose

then

ϕ ∈ F F .

F E0 = (F 0 )ref a ,

(7.2.23)

F F = G1 = N1 , H0α N1 = {Hα ϕ E0 : ϕ ∈ G1 },

(7.2.22)

(7.2.24) α > 0,

(7.2.25)

and Eα0,ref (Hα ϕ E0 , Hα ϕ E0 ) = C(ϕ, ϕ) + Uα (ϕ, ϕ),

ϕ ∈ G1 , α > 0, (7.2.26)

E 0,ref (Hϕ E0 , Hϕ E0 ) = C(ϕ, ϕ),

ϕ ∈ G1 . (7.2.27) Proof. (i) From (7.2.17), we get the inclusion F F ⊂ Fe F ∩ L2 (F; µ0 ). By Theorem 7.2.1(i), we have Fe F ⊂ G, from which we can deduce the first inclusion in (7.2.21) by taking intersections with L2 (F; µ0 ) and noting (7.2.12). Next take any ϕ ∈ G1 . Then ϕ ∈ G ⊂ N by (7.2.12) and Theorem 7.2.1(i). Since U1 (|ϕ|, |ϕ|) < ∞ by (7.2.13), we get ϕ ∈ N1 , proving the second inclusion in (7.2.21). The inequality (7.2.22) follows from (7.1.30). (ii) Combining the assumption (7.2.23) with the inclusion in (i) and the second inclusion in (7.1.27), we are led to the identities (7.2.24) and (7.2.25). Relations (7.2.26) and (7.2.27) then follow from (7.1.30) and (7.1.18), respectively.

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CHAPTER SEVEN

We remark that the condition (7.2.5) implies the condition (7.2.23), but the converse implication is not necessarily true as we saw for the reflecting Brownian motion in Section 6.5(5◦ ). So far in this chapter, we have employed the setting that we are given a symmetric Hunt process X on E whose Dirichlet form is regular irreducible and we have studied the properties of X in relation to the part process X 0 of X on a quasi open set E0 of E. We can regard X as an extension of X 0 from E0 to E. D EFINITION 7.2.6. A symmetric Hunt process X is said to be a reflecting extension of a symmetric standard process X 0 if the following holds: (RE.1) E is a locally compact separable metric space, m is an everywhere dense positive Radon measure on E, and X is an m-symmetric Hunt process on E whose Dirichlet form (E, F) on L2 (E; m) is regular. (RE.2) X 0 is the part process of X on a non-E-polar, E-quasi-open subset E0 of E whose Dirichlet form (E 0 , F 0 ) on L2 (E0 ; mE0 ) is irreducible. Further, F 0 is a proper subset of F. (RE.3) m(F) = 0 where F = E \ E0 . Dirichlet form of (RE.4) Denote by E 0,ref , (F 0 )ref a the active reflected (E 0 , F 0 ). Then (E, F) = E 0,ref , (F 0 )ref a . L EMMA 7.2.7. Suppose a symmetric Hunt process X is a reflecting extension of a symmetric standard process X 0 . Let µ0 be the intrinsic measure on F defined by (7.2.10). Then (i) X admits no jump from E0 to F in the sense of (7.1.11). (ii) The Dirichlet form (E, F) of X on L2 (E; m) is irreducible. (iii) The set F is non-E-polar. Proof. (i) Fix α > 0 and take any f ∈ F. Since Hα f is in the Eα -orthogonal complement of F 0 , we see from (RE.4) and Theorem 7.1.4 that Hα f = H0α Φ for some Φ ∈ N1 . Let {Ak } be an E 0 -nest consisting of compact subsets of E0 . Then τAk → σF ∧ ζ , k → ∞, Px -a.s. for q.e. x ∈ E0 in view of Theorem 3.1.13. By taking Proposition 7.1.3, its proof, Theorem 3.1.7, and the quasi-left-continuity of the Hunt process X into account, we can get Φ = Φ · 1{σF ∧ζ 0, F F ⊂ N1 and Hα N1 E0 ⊂ H0α N1 ⊂ (F 0 )ref (7.3.2) a . L EMMA 7.3.2. (i) For Φ ∈ N, let h(x) = E0x [Φ], x ∈ E. Then lim h(Xt0 ) = Φ P0x -a.s. on ζ 0 < ∞, Xζ00 − ∈ F t↑ζ 0

(7.3.3)

for q.e. x ∈ E0 . (ii) For ψ ∈ N1 , the function Hα ψ is X 0 -q.e. finely continuous and has the X 0 -fine limit function ψ on F for α > 0. Proof. (i) By virtue of Lemma 6.1.7, {Mth }t≥0 defined by (6.1.7) for X 0 is a 0 Px -square integrable martingale and hence h(Xt ) admits a left limit at ζ 0 P0x a.s. on {ζ 0 < ∞} for q.e. x ∈ E0 . Hence (7.3.3) follows from Lemma 6.4.11 and (7.3.1). (ii) By (7.1.5) and Lemma 7.1.5, Hα ψ − Hψ = −αG00+ Hα ψ ∈ F 0 . Hence we get from (7.3.3), Lemma 7.1.5, and Lemma 7.3.1(ii) that Hα ψ is X 0 -q.e. finely continuous and lim Hα ψ(Xt0 ) = ψ(Xζ00 − ), t↑ζ 0

P0x -a.s. on {ζ 0 < ∞, Xζ00 − ∈ F}

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CHAPTER SEVEN

for q.e. x ∈ E0 , namely, Hα ψ admits ψ as an X 0 -fine limit function on F.

We now consider the following condition: 0 0 If f ∈ (F 0 )ref a admits an X -fine limit function 0 on F, then f ∈ F .

(7.3.4)

Remark 7.3.3. (i) Let X 0 be the absorbing Brownian motion on the interval 1 E0 = (0, 1). Then F 0 = H01 (0, 1) and (F 0 )ref a = H (0, 1). Condition (7.3.4) is satisfied if F = {0, 1} (as in the case that X is the reflecting Brownian motion on E = [0, 1]) but it is not satisfied when F = {0} (as in the case that X is the Brownian motion on E = [0, 1) reflected at 0 and absorbed at 1). If E is the one-point compactification of E0 = (0, 1), then (7.3.4) is fulfilled by the one-point set F = E \ E0 . (ii) Assume that for q.e. x ∈ E0 , (7.3.5) P0x Xζ 0 − ∈ F ζ 0 < ∞ = 1 then (7.3.4) is fulfilled. To verify this, suppose f ∈ (F 0 )ref a and γ f = 0. By Theorem 7.1.4 and (7.1.5), f can be decomposed as f (x) = E0x [Φ] + f0 (x) for q.e. x ∈ E0 with Φ ∈ N1 and f0 ∈ F 0 . By Lemma 7.3.1 (ii), γ f0 = 0 and so Φ · 1{X 00 ∈F, ζ 0 0.

327


0,ref 0 Proof. By (7.3.7), u ∈ (F 0 )ref a and Eα (u, w) = 0 for any w ∈ F . Define 0 ref ψ := γ u and u0 := u − Hα ψ. Then u0 ∈ (F )a by (7.3.2) and γ (u0 ) = ψ − ψ = 0 by Lemma 7.3.2. Hence by assumption (7.3.4), u0 ∈ F 0 . Since by (7.3.2) Hα ψ ∈ H0α N1 , we have by Theorem 7.1.4 that

Eα0,ref (Hα ψ, w) = 0

for every w ∈ F 0 .

It follows then that Eαref (u0 , w) = Eα0,ref (u − Hα ψ, w) = 0

for every w ∈ F 0 .

Taking w = u0 we get u0 = 0 and therefore u = Hα ψ = Hα (γ u).

For f ∈ D(L) and ψ ∈ N1 , define N(f )(ψ) := E 0,ref (f , Hα ψ) + (Lf , Hα ψ),

α > 0.

(7.3.8)

Note that for α and β > 0, Hα ψ − Hβ ψ ∈ F 0 by Lemma 7.1.2 and Lemma 7.1.5. Hence N(f )(ψ) defined by (7.3.8) is independent of the choice of α > 0 in view of (7.3.7). We call N(f ) the flux functional of f being regarded as a linear functional on the space N1 . In the remainder of this section, we assume that m(F) = 0.

(7.3.9)

Denote by A the L2 -infinitesimal generator of X. That is, A is the self-adjoint operator on L2 (E; m) (= L2 (E0 ; m0 )) such that f ∈ D(A) with Af = g if and only if f ∈ F with E(f , v) = −(g, v) for every v ∈ F .

(7.3.10)

Recall the operator L is defined by (7.3.7). We see from Theorem 7.1.6 that L is an extension of A in the sense that D(A) ⊂ D(L)

and

Af = Lf for f ∈ D(A).

(7.3.11)

We are in a position to formulate a lateral (boundary) condition that gives a characterization of a function in D(L) to be in D(A). To this end, we

◦ consider the intrinsic measure µ0 (∈ SF ) on F defined by (7.2.10): F ψdµ0 = U1 (1, ψ), ψ ∈ B+ (F). We note that if B ∈ B(F) is µ0 -negligible, then (7.3.12) P0x Xζ00 − ∈ B ζ 0 < ∞, Xζ00 − ∈ F = 0 for q.e. x ∈ E0 , because U1 (1, 1B ) = 0 implies Px (σB < ∞, XσB ∈ B) = H1B (x) = 0 for m-a.e. x ∈ E0 and hence for q.e. x ∈ E0 , due to the X 0 -fine continuity of H1B . We then get (7.3.12) from Lemma 7.1.5.

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ˇ F) ˇ be the Dirichlet form on L2 (F; µ0 ) associated with the timeLet (E, changed process Xˇ of X by means of its PCAF with Revuz measure µ0 . By virtue of Theorem 7.2.4, we have (7.3.13) Fˇ = F F , which means that Fˇ consists of those µ0 -equivalence classes of µ0 -measurable functions on F admitting the restrictions to F of functions in F as their representatives. ˇ Suppose a function f ∈ (F 0 )ref a admits a µ0 -measurable function ϕ ∈ F as 0 its X -fine limit function on F. Then, on account of (7.3.12), f also has as its X 0 -fine limit function on F any function which is µ0 -equivalent to ϕ, so that ϕ can be taken from F F . T HEOREM 7.3.5. (i) Suppose f ∈ D(A). Then f ∈ D(L) and f satisfies the lateral conditions that ˇ (7.3.14) f admits an X 0 -fine limit function γ f ∈ F, ˇ and for every ψ ∈ F, 1 N(f )(ψ) + µc H(γ f ),Hψ (F) 2 1 ((γ f )(ξ ) − (γ f )(η))(ψ(ξ ) − ψ(η))J(dξ , dη) + 2 F×F (7.3.15) + (γ f )(ξ )ψ(ξ )κ(dξ ) = 0. F

(ii) Assume that the condition (7.3.4) holds. If f ∈ D(L) satisfies the lateral ˇ then f ∈ D(A). Here conditions (7.3.14) and (7.3.15) holding for every ψ ∈ D, ˇ ˇ ˇ D is any fixed E1 -dense subspace of F. Proof. (i) Suppose f ∈ D(A). Then for α > 0, f = Gα g with g = (α − A)f ∈ L2 (E; m0 ). Here Gα denotes the α-resolvent of X. Since f ∈ F, f admits the X 0 -fine limit function γ f = f |F ∈ Fˇ by Lemma 7.3.1(ii) and (7.3.13). ˇ For ψ ∈ F(= F F ), Hα ψ ∈ F. Since L is an extension of A, we have from (7.3.10) E(f , Hα ψ) + (Lf , Hα ψ) = 0, whose left hand side coincides with the left hand side of (7.3.15) in view of (7.3.8), (7.1.16), and (7.1.17). ˇ (ii) Suppose that f ∈ D(L) satisfies (7.3.14) and (7.3.15) for every ψ ∈ D. Let g = (α − L)f and w = f − Gα g E0 . Then by the preceding remark, (7.3.2), and Lemma 7.3.1(ii), w satisfies γ w ∈ F F ⊂ N1 and moreover (α − L)w = 0 on account of (7.3.11). Consequently, w = Hα (γ w) ∈ F by virtue of Lemma 7.3.4.


329

As w ∈ D(L) and (α − L)w = 0, N(w)(ψ) = E 0,ref (w, Hα ψ) + (Lw, Hα ψ) = Eα0,ref (w, Hα ψ) for every ψ ∈ N1 . On the other hand, we see by (i) that Gα g ∈ D(A) satisfies ˇ equation (7.3.15) and so does w. It follows then for every ϕ ∈ D, 1 Eα0,ref (Hα (γ w), Hα ϕ) + µc H(γ w),Hϕ (F) 2 1 (γ w(ξ ) − γ w(η))(ϕ(ξ ) − ϕ(η))J(dξ , dη) + 2 F×F + γ w(ξ )ϕ(ξ )κ(dξ ) = 0. F

Since w ∈ F, we see by (7.1.28) that the above identity is equivalent to ˇ which extends to every ϕ ∈ Fˇ because Eα (Hα (γ w), Hα ϕ) = 0 for every ϕ ∈ D, ˇ ˇ ˇ D is E1 -dense in F and Theorem 7.2.4 applies. Taking ϕ = γ w, we obtain w = Hα (γ w) = 0 and consequently, f = Gα g ∈ D(A). ˇ Remark 7.3.6. (i) Let H be any E1 -dense subspace of F . Then a set D ˇ ˇ ˇ ˇ satisfying H F ⊂ D ⊂ F is E1 -dense in F in view of Theorem 7.2.4. (ii) The function space Fˇ is involved in the lateral condition (7.3.14). In order to make Theorem 7.3.5 meaningful as a characterization of the domain D(A) ˇ of the L2 -generator of X, we need to have some explicit description of F. In regard to the second remark above, we recall some properties of the ˇ Fˇ ) on L2 (F; µ0 ) studied in Sections 5.2 and 5.6. By virtue Dirichlet form (E, ˇ F) ˇ is a quasi-regular Dirichlet form on L2 (F; µ0 ) and, in of Theorem 5.2.7, (E, view of (5.6.7), it admits the Beurling-Deny decomposition ˇ ϕ) = 1 µc (F) + 1 (ϕ(ξ ) − ϕ(η))2 (U + J)(dξ , dη) E(ϕ, 2 Hϕ

2 F×F ˇ (7.3.16) + ϕ(ξ )2 (V + κ)(dξ ), ϕ ∈ F. F

(Fˇ aref , Eˇ ref )

ˇ F). ˇ Explicit Denote by the active reflected Dirichlet space of (E, descriptions of reflected Dirichlet spaces can be given for various concrete examples as in Section 6.5 and also in the case where the underlying space is discretely countable as will be considered in the next section. L EMMA 7.3.7. (i) If we let Eˇ1ref (ϕ, ϕ) = Eˇref (ϕ, ϕ) + U1 (1, ϕ 2 ), then, for any ˇ Eˇ1 -dense subspace Dˇ of F, Fˇ = the Eˇ1ref -closure of Dˇ in Fˇ aref .

(7.3.17)

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CHAPTER SEVEN

(ii) Assume that X is recurrent. Then Fˇ = Fˇ aref and V = 0, κ = 0. Proof. (i) is obvious from Theorem 7.2.4. (ii) If X is recurrent, then the time-changed process Xˇ is also recurrent by ˇ E) ˇ = (Fˇ ref , Eˇ ref ) on account of Theorem 6.3.2, Theorem 5.2.5 and hence (F, a ˇ 1) cannot be zero. and further V + κ = 0 because otherwise E(1, For later use, we deduce here a direct characterization of the resolvent {Gα ; α > 0} of X. T HEOREM 7.3.8. For α > 0 and g ∈ L2 (E; m), let f = Gα g. Then ϕ = f F is the unique element of Fˇ satisfying the following equation: ˇ ψ) + Uα (ϕ, ψ) = (g, Hα ψ) for any ψ ∈ D, ˇ E(ϕ,

(7.3.18)

ˇ where Dˇ is any fixed Eˇ1 -dense subspace of F. Proof. Since (α − L)f = g, we have from (7.1.30) N(f )(ψ) = Eα0,ref (f , Hα ψ) − (g, Hα ψ) = C(ϕ, ψ) + Uα (ϕ, ψ) − (g, Hα ψ), ˇ which combined with the lateral condition (7.3.15) for f and for any ψ ∈ D, ˇ ψ) + Uα (ϕ, ψ) = 0 the identity (7.3.16) gives (7.3.18). If ϕ ∈ Fˇ satisfies E(ϕ, ˇ then ϕ = 0 on account of Theorem 7.2.4 and so the uniqueness for any ψ ∈ D, statement follows. Theorem 7.3.8 says that for α > 0 and g ∈ L2 (E; m), the resolvent Gα g can be recovered as follows: find the solution ϕ of (7.3.18) and then let Gα g = G0α g + Hα ϕ,

(7.3.19)

where {G0α ; α > 0} is the resolvent of X 0 . In the rest of this section, we exhibit the lateral condition appearing in Theorem 7.3.5 in special cases that X is a reflecting extension and its perturbation by a measure supported by F. Consider a reflecting extension X on E of a symmetric standard process X 0 on E0 in the sense of Definition 7.2.6. The crucial condition (RE.4) is (E, F) = (E 0,ref , (F 0 )ref a ).

(7.3.20)

Lemma 7.2.7 ensures that all the conditions on X and F = E \ E0 imposed in the first half of Section 7.2 as well as in Section 7.3 are satisfied. We can readily derive the following simple characterization of the domain of the generator A of the L2 -semigroup of X directly from the defining formulas (7.3.7) for L, (7.3.8) for N, and (7.3.10) for A: f ∈ D(A)

⇐⇒

f ∈ D(L),

N(f ) = 0.

(7.3.21)


331

Exercise 7.3.9. Derive (7.3.21) from (7.3.20). The right hand side of (7.3.21) does not involve the condition that γ f ∈ ˇ F(= F F ) as appeared in Theorem 7.3.5 because the condition f ∈ F is already implied by f ∈ D(L) under (7.3.20). We remark that (7.3.22) F F = G1 by virtue of Theorem 7.2.5, and that (C, G1 ) is a regular Dirichlet form on L2 (F ∗ ; µ0 ) which is associated with the time-changed process of X by means of the PCAF with Revuz measure µ0 in view of Corollary 7.2.8. On account of Theorem 5.2.8, a subset of F is E-polar if and only if it is C-polar. Let us denote by G˜1 the family of all C-quasi-continuous functions in G1 . We next take any positive Radon measure κ charging no E-polar set such that κ(E0 ) = 0.

(7.3.23)

The quasi support of κ is then contained in F. Let (E κ , F κ ) be the perturbed Dirichlet form of (E, F) defined by F κ = F ∩ L2 (E; κ), (7.3.24) E κ (u, v) = E(u, v) + (u, v)κ , u, v ∈ F κ . As we saw in Section 5.1, (E κ , F κ ) is a regular Dirichlet form on L2 (E; m) sharing the quasi notions in common with (E, F). The associated Hunt process X κ on E is obtained from X by killing by means of the PCAF with Revuz measure κ. In particular, X κ still admits no jump from E0 to F. The parts of (E κ , F κ ) and X κ on E0 are equal to (E 0 , F 0 ) and X 0 , respectively, because of the property (7.3.23). Owing to the irreducibility condition imposed in (RE.2) on (E 0 , F 0 ), we can see as in the proof of Lemma 7.2.7 that (E κ , F κ ) is irreducible. From (7.3.22) and (7.3.23) and the remarks made above, we get the identity of the spaces (7.3.25) F κ F = G˜1 ∩ L2 (F; κ). Furthermore, we have from (7.3.20) and (7.3.23) the equality E κ (u, u) = E 0,ref (u, u) + u2 dκ, u ∈ F κ ,

(7.3.26)

F

which particularly means that µc Hu (F) = 0 for u ∈ F κ , the restriction to F × F of the jumping measure of X κ vanishes and the restriction to F of the killing measure of X κ equals κ in view of Theorem 7.1.6.

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CHAPTER SEVEN

0 The condition in (RE.2) that F is a proper subspace of F and (7.3.22) κ imply that G1 = {0}. Then F F = {0} because (C, G1 ) is regular. Hence F 0 is again a proper subspace of F κ and we get the non-E κ -polarity of F as in the proof of Lemma 7.2.7. Thus all conditions imposed in the preceding section are fulfilled by X κ and F. Moreover, the condition (7.3.4) is met in view of (7.3.20) and Remark 7.3.3(iii). Therefore, we are led from Theorem 7.3.5, (7.3.25), (7.3.26), and Theorems 5.1.6, 5.2.8 to the following theorem.

T HEOREM 7.3.10. Let Aκ be the infinitesimal generator of the L2 -semigroup of X κ . f ∈ D(Aκ ) if and only if f ∈ D(L), f admits an X 0 -fine limit function γ f ∈ G˜1 ∩ L2 (F; κ), and

N(f )(ψ) +

γ f (ξ )ψ(ξ )κ(dξ ) = 0, F

for any ψ ∈ C F ,

(7.3.27)

(7.3.28)

where C is any fixed special standard core of (E, F). Example 7.3.11. We maintain the setting in Example 7.2.10. Thus D is a domain in Rn with n ≥ 1 satisfying (I) and (II). We consider the case that E = D, F = ∂D, E0 = D, m(dx) = 1D (x)dx, and (E, F) = ( 12 D, H 1 (D)). The part (E 0 , F 0 ) of (E, F) on D equals ( 21 D, H01 (D)). As we saw in Example 7.2.10, 1 0,ref 0 ref 1 D, H (D) (E , (F )a ) = 2 and the condition (7.3.20) is met. Hence the reflecting Brownian motion X r on D that is associated with (E, F) is a reflecting extension of its part X 0 on D, the absorbing Brownian motion. The linear operator L on L2 (D) = L2 (D; m) defined by (7.3.7) is now described as L=

1 , 2

D(L) = {f ∈ H 1 (D) : f ∈ L2 (D)},

(7.3.29)

where denotes the distribution derivative ni=1 ∂∂2 xi in the Schwartz distribution sense. In view of Theorem 7.2.5, N1 = G1 . So the flux functional N(f ), f ∈ D(L), defined by (7.3.8), is a linear functional on G1 specified by 2

N(f )(ψ) =

1 1 D(f , Hα ψ) + (f , Hα ψ), 2 2

ψ ∈ G1 ,

(7.3.30)

333


which is independent of α > 0. When ∂D, f , and Hα ψ are smooth enough, the right hand side of the above is equal to 1 2

∂D

∂f (ξ ) · ψ(ξ )σ (dξ ) ∂nξ

by the Gauss-Green formula, where nξ is the unit inward normal vector at ξ ∈ ∂D and σ (dξ ) is the surface measure on ∂D. Therefore, N(f ) is a general ∂f . substitute of the inward normal derivative 12 ∂n Let A be the infinitesimal generator of the semigroup on L2 (D) of X r . According to (7.3.21), f ∈ D(A)

⇐⇒

f ∈ D(L), N(f )(ψ) = 0 for all ψ ∈ Cc∞ (D), Af =

1 f , 2

f ∈ D(A).

(7.3.31)

(7.3.32)

Let κ be a positive Radon measure on D charging no E-polar set satisfying κ(D) = 0 and let X κ be the Hunt process on D being killed by means of the PCAF with Revuz measure κ. The part process of X κ on D is still the absorbing Brownian motion on D. The domain D(Aκ ) of the generator 2 of X κ is characterized as Theorem 7.3.10 with of Aκ of the L -semigroup ∞ F = ∂D, C F = Cc (D) ∂D , and L being given by (7.3.29). It holds that Aκ f = 1 f , f ∈ D(Aκ ). 2 As another example of the extension of the absorbing Brownian motion X 0 on D, consider a non-E-polar closed subset K of ∂D and the part process X r,K of the reflecting Brownian motion X r on D \ K. Its Dirichlet form (E K , F K ) on L2 (D \ K; m) = L2 (D) is given by F K = u ∈ H 1 (D) : u = 0 E-q.e. on K

and

EK =

1 D. 2

Its part on D is still given by (E 0 , F 0 ) = ( 12 D, H01 (D)) so that (E 0,ref , (F 0 )ref a )=

1 D, H 1 (D) . 2

In this case E = D \ K,

E0 = D,

F = ∂D \ K,

and the condition (7.3.4) is violated. The L2 -generator AK of X r,K cannot be characterized in a way of Theorem 7.3.5 using only the X 0 fine limit function on ∂D \ K.

334

CHAPTER SEVEN

7.4. COUNTABLE BOUNDARY In this section, we study the case where the set F is countable. The trace ˇ F) ˇ on L2 (F; µ0 ) then admits a simple explicit expression Dirichlet form (E, so that not only the lateral condition but also the Dirichlet form and resolvent of X can be described in quite tractable ways. Let E be a locally compact separable metric space, m be a positive Radon measure on X with full support, and X = (Xt , ζ , Px ) be an m-symmetric Hunt process on X whose Dirichlet form (E, F) on L2 (E; m) is regular and irreducible. Recall that any function in Fe is taken to be quasi continuous. Two subsets A and B of E are called quasi separated if there exist quasi open sets U, V such that A ⊂ U, B ⊂ V and U ∩ V = ∅. We assume that F = {a1 , a2 , . . . , ai , . . .} is a finite or countably infinite subset of E satisfying the following: (F.1) F is quasi closed, m(F) = 0, and {ai } is not E-polar for every i ≥ 1. (F.2) For each i ≥ 1, the one-point set {ai } and the set F \ {ai } are quasi separated. (F.3) There exists an E-nest {Kn , n ≥ 1} such that F ∩ Kn is a finite set for each n ≥ 1. The above three conditions are invariant under a quasi-homeomorphism. If a countable closed set F contains no accumulation point, then condition (F.2) is fulfilled. In this case, any sequence {Kn , n ≥ 1} of compact sets increasing to E satisfies (F.3). Let E0 := E \ F. We assume that the Hunt process X on E satisfies the condition (7.1.11). We let X 0 be the part process of X on E0 and U, V be the Feller measures on F relative to X 0 . For i, j ≥ 1, let U ij := U({ai }, {aj }) for i = j,

V i := V({ai })

ij

and Uα := Uα ({ai }, {aj }). Define for x ∈ E and i ≥ 1, ϕ (i) (x) := Px (σF < ∞, XσF = ai ) and

−ασF ; XσF = ai , u(i) α (x) := Ex e

which does not vanish m0 -a.e. on E0 because {ai } is also non-polar relative to the part of E on the set E0 ∪ {ai } in view of Theorem 3.3.8(iii). Since ϕ (i) = H1{ai } and u(i) α = Hα 1{ai } , we see that for i, j ≥ 1, (j) U ij = L0 (ϕ (i) , ϕ (j) ), i = j, V i = L0 (ϕ (i) , 1 − H1), Uαij = α(u(i) α , ϕ ).

335


Moreover, as X admits no jumps from E0 to F, we have from Lemma 7.1.5 ϕ (i) (x) = P0x ζ 0 < ∞ and Xζ00 − = ai and

0 −αζ 0 0 (x) = E ; X = a e . u(i) 0 i α x ζ −

The intrinsic measure µ0 on F defined by (7.2.10) is a measure assigning each point ai ∈ F a positive mass ! ij U1 , (7.4.1) µ0 ({ai }) = U1 1(i) = j≥1

which is extended to E by setting µ0 (E0 ) = 0. Clearly F is a quasi support of ˇ F) ˇ be the Dirichlet form on L2 (F; µ0 ) associated with the timeµ0 . Let (E, ˇ F) ˇ is changed process Xˇ of X by means of PCAF with Revuz measure µ0 . (E, quasi-regular by Theorem 5.2.7 admitting the expression (7.3.16). We denote ˇ F) ˇ and by Fˇ ref its active by (Fˇ ref , Eˇ ref ) the reflected Dirichlet space of (E, a reflected Dirichlet space. For a real-valued function ψ on F, we put C(ψ, ψ) =

! 1 ! (ψ(ai ) − ψ(aj ))2 U ij + ψ(ai )2 V i , 2 i,j≥1:i=j i≥1

B(ψ, ψ) = C(ψ, ψ) +

(7.4.2)

! 1 ! (ψ(ai ) − ψ(aj ))2 Jij + ψ(ai )2 κi , 2 i,j≥1:i=j i≥1 (7.4.3)

where Jij := J({ai }, {aj }) and κi := κ({ai }). Recall that the form C has already appeared as the Douglas integral in Section 7.2. Let B0 (F) be the space of functions on F vanishing except on finite many points. P ROPOSITION 7.4.1. (i) The strongly local term µc Hψ (F) in the expression ˇ (7.3.16) of E(ψ, ψ) vanishes for any ψ ∈ Fˇ and ˇ E(ψ, ψ) = B(ψ, ψ),

ˇ ψ ∈ F.

(7.4.4)

ˇ (ii) B0 (F) is an Eˇ1 -dense subspace of F. Proof. (i) The energy measure µc u for u ∈ bF does not charge any single point owing to the energy image density property formulated in Theorem 4.3.8 ˇ and in particular µc (ai ) = 0 for any i ≥ 1 and ψ ∈ F. Hψ

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CHAPTER SEVEN

(ii) Fix i ≥ 0 and consider the function

τU i −αs e f (Xs )ds , v(x) = Ex

x ∈ E,

0

for an m-integrable strictly positive bounded continuous function f on E, where Ui is a quasi neighborhood of ai appearing in condition (F.2). Then v ∈ F, v(ai ) > 0, and v(aj ) = 0 for any j = i. Since the one-point set {aj } is nonE-polar, aj is regular for itself for every j ≥ 1 in view of Theorem 3.1.10. ˇ Consequently, B0 (F) ⊂ F. ˇ Therefore, 1{ai } = v(a1 i ) Hv F ∈ F. Let {Kn } be an E-nest appearing in condition (F.3) and FKn = {u ∈ F : u = 0 m-a.e. on E \ Kn }. Wecan modify each function u ∈ FKn to vanish identically on E \ Kn so that FKn F ⊂ B0 (F) in view of the condition (F.3). Since ∪n FKn is E1 -dense in F, B0 (F) is Eˇ1 -dense in Fˇ in view of Remark 7.3.6. P ROPOSITION 7.4.2. (i) It holds that Fˇ ref = {ψ : B(ψ, ψ) < ∞},

Eˇ ref (ψ, ψ) = B(ψ, ψ), ψ ∈ Fˇ ref ,

and Fˇ aref = {ψ ∈ L2 (F; µ0 ) : B(ψ, ψ) < ∞}. (ii) For ψ ∈ Fˇ aref , let B1 (ψ, ψ) := B(ψ, ψ) +

i≥1

ψ(ai )2 U1 1(i). Then

Fˇ = the B1 -closure of B0 (F) in Fˇ aref . (iii) For each i ≥ 1,

!

(7.4.5)

(U ij + Jij ) < ∞.

(7.4.6)

(7.4.7)

j≥1, j=i

Proof. (i) On account of Proposition 7.4.1, condition (F.3), and Theorem 5.2.6, the local Dirichlet space defined by (4.3.31) for the space Fˇ consists of all real-valued functions on F. Hence (7.4.5) follows from Definition 6.4.4. (ii) This follows from (i), Lemma 7.3.7(ii), and Proposition 7.4.1(ii). ˇ (iii) We get this from (7.4.2) with ψ = 1{ai } ∈ F. We can define for f ∈ D(L), the flux of f at ai by (i) N(f )(ai ) := N(f )(1{ai } ) = E 0,ref (f , u(i) α ) + (Lf , uα ).

(7.4.8)

Theorem 7.3.5 and Lemma 7.3.7 now read as follows: T HEOREM 7.4.3. (i) Assume that condition (7.3.4) is fulfilled. f ∈ D(A) if and only if f ∈ D(L) and f satisfies the lateral conditions that ˇ f admits an X 0 -fine limit function γ f ∈ F,

(7.4.9)

337


for the space Fˇ specified by (7.4.6), and, for every i ≥ 1, N(f )(ai ) +

1 ! ((γ f )(ai ) − (γ f )(aj ))Jij + (γ f )(ai )κi = 0. 2 j≥1,j=i

(7.4.10)

(ii) If X is recurrent, then V = κ = 0 and     ! ! 1 (ψ(ai ) − ψ(aj ))2 (U ij +Jij )+ ψ(ai )2 U1 1(i) < ∞ . Fˇ = ψ :   2 i=j i≥1 (7.4.11) Taking Proposition 7.4.1(ii) into account, we can rewrite Theorem 7.3.8 and (7.3.19) as follows: 7.4.4. For α > 0 and g ∈ L2 (E; m), let f = Gα g. T HEOREM ψ = f F is then the unique element of Fˇ such that ! Uαij ψ(aj ) = (u(i) i ≥ 1. B(ψ, 1ai ) + α , g),

(7.4.12)

j≥1

Moreover, Gα g admits the representation ! u(i) Gα g(x) = G0α g(x) + α (x)ψ(ai ),

x ∈ E.

(7.4.13)

i≥1

In the remainder of this section, we assume that the symmetric Hunt process X on E admits no killing on F or jumps from F to F in the following sense: κi = 0

and

Jij = 0

for every i, j ≥ 1.

(7.4.14)

This condition is equivalent to B = C. Recall the space G1 of functions on F with finite Douglas integrals introduced by (7.2.11): G1 = {ψ ∈ L2 (F; µ0 ) : C(ψ, ϕ) < ∞}. T HEOREM 7.4.5. Assume that X satisfies condition (7.4.14). Then the following hold true: (i) Fˇ aref = G1 and Eˇ ref (ψ, ψ) = C(ψ, ψ) for ψ ∈ Fˇ aref . (ii) Let C1 (ψ, ψ) = C(ψ, ψ) + i≥1 ψ(ai )2 µ0 ({ai }). Then Fˇ = the C1 -closure of B0 (F) in G1 , ˇ ˇ E(ψ, ψ) = C(ϕ, ψ) for ψ ∈ F. (iii)

j≥1, j=i

U ij < ∞ for every i ≥ 1.

(7.4.15)

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CHAPTER SEVEN

(iv) For α > 0 and g ∈ L2 (E; m), let f = Gα g. Then ψ = f F is the unique element of Fˇ such that ! C(ψ, 1ai ) + Uαij ψ(aj ) = (u(i) i ≥ 1. (7.4.16) α , g), j≥1

Moreover, Gα g admits the representation (7.4.13). 0,ref (v) F ⊂ (F 0 )ref a , Eα (u, v) = Eα (u, v) for u, v ∈ F, and F = F 0 ⊕ Hα ,

(7.4.17)

where Hα is the Eα0,ref -closure of the linear span of {u(i) α ; i ≥ 1} and (7.4.17) is an Eα0,ref -orthogonal decomposition. (vi) Fe ⊂ (F 0 )ref , E(u, v) = E 0,ref (u, v) for u, v ∈ Fe . Furthermore, Fe0 and the linear span of {ϕ (i) : i ≥ 1} are subsets of Fe which are E 0,ref -orthogonal to each other. (vii) When F consists of finite number of points {a1 , a2 , . . . , aN }, Fe is the linear subspace of (F 0 )ref spanned by Fe0 and {ϕ (i) , 1 ≤ i ≤ N}, while F is the linear 0 (i) subspace of (F 0 )ref a spanned by F and {uα , 1 ≤ i ≤ N}. (viii) Assume that condition (7.3.4) holds. Then f ∈ D(A) if and only if f ∈ D(L), (7.4.9) is satisfied for the space Fˇ specified by (7.4.15), and N(f )(ai ) = 0

for every i ≥ 1.

(7.4.18)

Proof. (v) follows from (ii) and Theorem 7.1.8 by noting Lemma 7.2.3. (vi) uses Theorem 7.1.6. Other assertions are restatements of the preceding propositions and theorems under the assumption (7.4.14). Keeping the assumption (7.4.14), we consider the special case where the set F consists of finite number of points: F = {a1 , . . . , aN }. Let Aˇ be the N × N matrix with entry Aˇ ij being given by Aˇ ij = U ij ,

1 ≤ i, j ≤ N, i = j, ! U ik − V i , 1 ≤ i ≤ N. Aˇ ii = −

(7.4.19)

k≥1, k=i ij

Uα denotes the N × N matrix with entry Uα . We then let Rα (x, y) = −uα (x) · (Aˇ − Uα )−1 t uα (y),

x, y ∈ E,

(7.4.20)

(N) t where uα (x) = (u(1) α (x), . . . , uα (x)) and uα (y) is the transpose of uα (y).

339


The next decomposition formula of the resolvent follows from (7.4.16) and (7.4.13): Gα g(x) =

G0α g(x)

+

Rα (x, y)g(y)m(dy),

x ∈ E.

(7.4.21)

E0

We finally consider the case where the set F consists of only one point. T HEOREM 7.4.6. killing at a.

Suppose F consists of one point a and X admits no

(i) The resolvent {Gα ; α > 0} of X admits the expression for g ∈ L2 (E; m): Gα g(a) =

(uα , g) α(uα , ϕ) + L0 (ϕ, 1 − ϕ)

Gα g(x) = G0α g(x) + uα (x)Gα g(a),

(7.4.22)

x ∈ E,

where uα (x) = Ex e−ασa ; σa < ∞ ,

ϕ(x) = Px (σa < ∞),

x ∈ E.

(ii) Fe is the linear subspace of (F 0 )ref spanned by Fe0 and ϕ. 0 F is the linear subspace of (F 0 )ref a spanned by F and uα . 0 For f = f0 + cϕ, f0 ∈ Fe , c ∈ R,

E(f , f ) = E 0,ref (f , f ) = E 0 (f0 , f0 ) + c2 L0 (ϕ, 1 − ϕ).

(7.4.23)

(iii) Assume that condition (7.3.4) is fulfilled. Let A be the L2 -generator of X. f ∈ D(A) if and only if f ∈ D(L), f admits an X 0 -fine limit value γ f (a) at a and N(f )(a) = 0.

(7.4.24)

Here N(f )(a) is the flux of f ∈ D(L) at a defined by N(f )(a) = E 0,ref (f , uα ) + (Lf , uα ).

(7.4.25)

Proof. The formulas (7.4.20) and (7.4.21) for N = 1 reduce to (7.4.22). (ii) follows from Theorem 7.1.6 and Theorem 7.1.8 by noting that V({a}) = L0 (ϕ, 1 − ϕ). (iii) is a special case of Theorem 7.4.5.

340

CHAPTER SEVEN

7.5. ONE-POINT EXTENSIONS In this section, every sample path of a right process X on a state space E will be assumed to possess left limit Xt− in E for all t ∈ (0, ζ ), where ζ is the lifetime of X. A right process X = (Xt , Px , ζ ) is said to be of no killing inside if Px (ζ < ∞, Xζ − ∈ E) = 0 for any x ∈ E, where {ζ < ∞, Xζ − ∈ E} means the event that the left limit of Xt at t = ζ < ∞ exists and belongs to E. Let E be a locally compact separable metric space and m be a positive Radon measure on E with supp[m] = E. We fix a non-isolated point a ∈ E with m({a}) = 0. Put E0 = E \ {a}. E∂ = E ∪ {∂} denotes the one-point compactification of E. (E0 )∂ = E0 ∪ {∂} is regarded as a topological subspace of E∂ . Let X 0 = (Xt0 , P0x , ζ 0 ) be an m-symmetric Borel standard process on E0 satisfying the following condition: (A.1) X 0 admits no killing inside and P0x ζ 0 < ∞, Xζ00 − = a > 0 for every x ∈ E0 . (7.5.1) Then for x ∈ E0 and α > 0, we let ϕ(x) = P0x ζ 0 < ∞, Xζ00 − = a ,

0 uα (x) = E0x e−αζ ; Xζ00 − = a .

(7.5.2)

D EFINITION 7.5.1. A right process X = (Xt , Px , ζ ) on E is called a onepoint extension of X 0 if X is m-symmetric and of no killings on {a}, and the part process of X on E0 is X 0 . Exercise 7.5.2. Suppose a one-point extension X = (Xt , Px ) of X 0 is a Borel standard process. Show that X satisfies the following properties. Denote by σa the hitting time of the set {a} by X. (i) For any x ∈ E0 and α > 0, ϕ(x) = Px (σa < ∞),

uα (x) = Ex e−ασa ; σa < ∞ .

(7.5.3)

(ii) X admits no jump from E0 to {a}: Px (Xt− ∈ E0 , Xt = a for some t > 0) = 0, x ∈ E.

(7.5.4)

L EMMA 7.5.3. Suppose a one-point extension X of X 0 is a Hunt process whose Dirichlet form (E, F) is regular. Then the point a is non-E-polar and regular for itself with respect to X in the sense that Pa (σa = 0) = 1. Further, X is irreducible.

341


Proof. By (7.5.1) and (7.5.3), {a} is not m-polar with respect to X and hence non-E-polar. On account of Theorem 3.1.10, {a} is then regular for itself. Denote by {Gα ; α > 0} and {G0α ; α > 0} the resolvents of X and X 0 , respectively. Take any non-negative g ∈ Cc (E) and let w = Gα g. Then w ∈ F admits two types of decomposition w = G0α g + cuα = f0 + cϕ

with c = w(a), f0 = w − cϕ,

as a sum of Eα -orthogonal elements in F and a sum of E-orthogonal elements in Fe . On account of Exercise 2.1.13 and Lemma 2.1.15, we can find a uniformly bounded sequence vn of functions in F that is E-convergent as well as m-a.e. convergent to ϕ. Letting n → ∞ in the equation (g, vn ) = Eα (w, vn ) = E(f0 + cϕ, vn ) + α(G0α g + cuα , vn ), we get c [E(ϕ, ϕ) + α(uα , ϕ)] = (g, ϕ) − α(G0α g, ϕ) = (uα , g), and accordingly 0 < E(ϕ, ϕ) + α(uα , ϕ) < ∞,

Gα g(a) =

(uα , g) , E(ϕ, ϕ) + α(uα , ϕ)

which extends to g ∈ B(E). Therefore, for any A, B ∈ B(E) with positive m-measures, we have (1A , Gα 1B ) ≥ (1A , uα )Gα 1B (a) > 0, the irreducibility of X. T HEOREM 7.5.4. Let X 0 be an m-symmetric Borel standard process on E0 satisfying condition (A.1). A one-point extension of X 0 is then unique in law. More specifically, let X = (Xt , Px , ζ ) be a one-point extension of X 0 . Then X enjoys the following properties. Denote by {G0α ; α > 0} and (E 0 , F 0 ) (resp. {Gα ; α > 0} and (E, F )) the resolvent and the Dirichlet form of X 0 (resp. X) on L2 (E0 ; m)(= L2 (E; m)). (i) The point {a} is non-m-polar and regular for itself with respect to X. The process X is irreducible. (ii) For any bounded g ∈ L2 (E; m), Gα g(a) =

(uα , g) α(uα , ϕ) + L0 (ϕ, 1 − ϕ)

Gα g(x) = G0α g(x) + uα (x)Gα g(a),

(7.5.5) x ∈ E,

(7.5.6)

where L is the energy functional for X . (iii) Fe is the linear subspace of (F 0 )ref spanned by Fe0 and ϕ. F is the linear 0 subspace of (F 0 )ref a spanned by F and uα . 0

0

E(f , f ) = E 0,ref (f , f ),

f ∈ Fe .

(7.5.7)

Further, for f = f0 + cϕ, f0 ∈ Fe0 , c ∈ R, E 0,ref (f , f ) = E 0 (f0 , f0 ) + c2 L0 (ϕ, 1 − ϕ).

(7.5.8)

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CHAPTER SEVEN

(iv) Assume that condition (7.3.4) is fulfilled. Let A be the L2 -generator of X. f ∈ D(A) if and only if f ∈ D(L), f admits an X 0 -fine limit value at a, and N(f )(a) = 0,

(7.5.9)

where N(f )(a) is the flux of f at a defined by (7.4.25). It then holds that Af = Lf ,

f ∈ D(A).

(7.5.10)

We remark that since X 0 is an m-symmetric standard process, (E 0 , F 0 ) is quasi-regular by Theorem 1.5.3 and so its reflected Dirichlet space ((F 0 )ref , E 0,ref ) is well defined. Proof. The uniqueness follows from (ii). We prove (i)∼(iv) by a transfer method. Since X is an m-symmetric right process on a locally compact separable metric space E, (E, F) is quasi-regular with which X is properly associated by Theorem 1.5.3. According to Theorem 1.4.3, (E, F) is quasi F) on L2 ( E; m) for some locally homeomorphic to a regular Dirichlet form (E, compact separable metric space E by a quasi-homeomorphism j. In view of Theorem 3.1.13, X is equivalent in law to a pull back X˜ by j of a Hunt process F) outside some Borel properly Px , ζ ) on E associated with (E, X = ( Xt , exceptional set. To be more precise, j is a quasi-homeomorphism between an E-nest {Fk } and an E-nest { Fk } both consisting of compact sets so that j is a one-toFk . There exists then an E1 = ∪k X -properly one map from E1 = ∪k Fk onto Px (limk→∞ τFk = ζ) = 1 exceptional Borel set N containing E \ E1 such that E1 ∩ N ), then XE\N is for all x∈ E\ N , and if we define N ⊂ E by E \ N = j−1 ( X E\ ). In particular, equivalent to the Borel special standard process X˜ = j−1 ( N N is properly exceptional for X in the sense of Theorem 3.1.13. Since X extends X 0 satisfying (A.1), the set {a} must be located in E \ N. We can then apply Exercise 7.5.2 to the standard process X E\N extending X 0 E0 \N to conclude that a is not m-polar for X and X E\N admits no jump from E0 \ N to a. X E Therefore, a := ja ∈ E\ N is non- m-polar for the Hunt process X := on E := E\ N . Obviously m({ a}) = 0. Let X 0 be the part process of X on E\ N \ { a}, which is an m-symmetric standard process satisfying (A.1). It is X0 admitting no jump from E \ { a} now clear that X is a one-point extension of F) is regular, { X to { a}. Since (E, a} is regular for itself with respect to X and is irreducible in view of Lemma 7.5.3. In particular, X and { a} satisfy all the conditions imposed in Section 7.4 so that Theorem 7.4.6 applies. a}, and X 0 . It Consequently, (i), (iii), (iv) as well as (7.5.5) hold for X , { is not hard to verify that those properties are honestly inherited by X, a, and


343

X 0 through the quasi-homeomorphic map j−1 . For instance, if we denote by G0α ; α > 0} the resolvents of X and X 0 , respectively, and by { Gα ; α > 0} and { ϕ the α-order and 0-order hitting probabilities of X for a, then uα and g( a) = Gα

g) ( uα , m , 0 α( uα , ϕ ) ϕ, 1 − ϕ) m + L (

0 j∗ g = Gα j∗ g, Gα where L0 is the energy functional relative

∗to X . Since ∗ 0 0 ∗ ∗ ∗ f dm = E f d m, we can remove g = Gα j g, j uα = uα , j ϕ = ϕ, and E j j Gα the sign from both sides of the above identity to get the desired equality a with respect (7.5.5). a is regular for itself with respect to X , so is a = j−1 As to X E\N , and accordingly with respect to X. Hence the identity (7.5.6) is valid for every x ∈ E because X is an extension of X 0 . Similarly, we can conclude just as in the proof of Theorem 7.2.9 that the reflected Dirichlet form (E 0,ref , (F 0 )ref ) of X 0 is the image by j−1 of the reflected Dirichlet form of X0. A one-point extension X of an m-symmetric Hunt process X 0 can be constructed under some of the following additional conditions for X 0 : (A.2)

E

uα (x)m(dx) < ∞ for α > 0, and for every x ∈ E0 , P0x ζ 0 < ∞, Xζ00 − ∈ {a, ∂} = P0x (ζ 0 < ∞).

(7.5.11)

(A.2) P0x (Xζ00 − ∈ {a, ∂}) = 1 for every x ∈ E0 (regardless the length of the lifetime ζ 0 ∈ (0, ∞]). (A.3) The exists a neighborhood U of a such that infx∈V G01 ϕ(x) > 0 for any compact set V ⊂ U \ {a}. (A.4) Either E \ U is compact for any neighborhood U of a, or for any open neighborhood U1 of a in E, there exists an open neighborhood U2 of a in E with U 2 ⊂ U1 such that J0 (U2 \ {a}, E0 \ U1 ) < ∞. Here J0 denotes the jumping measure of X 0 . We do not need condition (A.4) when X 0 is a diffusion. L EMMA 7.5.5. Assume that X 0 is an m-symmetric Hunt process on E0 satisfying the condition (A.1). (i) If X 0 is a diffusion, then it satisfies the condition (7.5.11). (ii) If X 0 is a diffusion and the Dirichlet form of X 0 is regular, then condition (A.2) is fulfilled for q.e. x ∈ E0 . (iii) If X 0 admits a one-point extension X on E, then X 0 satisfies the integrability condition in (A.2). (iv) If either G01 f or G00+ f is lower semicontinuous on E0 for any f ∈ B+ (E0 ), then X 0 satisfies condition (A.3).

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CHAPTER SEVEN

Proof. (i) Since X 0 admits no killing inside E0 , the quasi-left-continuity 0 of X implies that Xζ00 − = ∂ 0 Px -a.s. for every x ∈ E0 , where ∂ 0 denotes the point at infinity of E0 for the one-point compactification of E0 . By the continuity of the sample paths of X 0 , this means that for any relatively compact neighborhood U of a and any compact set K ⊃ U, the entire portion {Xt0 : t ∈ [T, ζ 0 )} of the path is included either in U \ {a} or E \ K for some T ∈ (0, ζ 0 ) Px -a.s. for x ∈ E0 . (ii) By (A.1) and Proposition 2.1.10, X 0 is transient. Theorem 3.5.2 then implies that limt→ζ 0 Xt0 = ∂ 0 regardless the length of ζ 0 Px -a.s. for q.e. x ∈ E0 . By the same reasoning as that in (i), one can conclude that (A.2) holds for q.e. x ∈ E0 . (iii) Denote by {Gα ; α > 0} the resolvent of X and choose a non-negative f ∈ Cc (E) with f (a) > 0. Then Gα f (a) > 0 and uα (x)Gα f (a) ≤ Gα f (x) for x ∈ E0 , which yields the m-integrability of uα . (iv) It suffices to note the identity G01 ϕ = G00+ u1 .

T HEOREM 7.5.6. Let X 0 be an m-symmetric Hunt process on E0 satisfying conditions (A.1) and (A.3) as well as (A.4) in a non-diffusion case. Assume also that either (A.2) or (A.2) is satisfied by X 0 . Then there exists a one-point extension X = (Xt , Px , ζ ) of X 0 from E0 to E such that X admits no jumps to or from a and Px (Xζ − = ∂, ζ < ∞) = Px (ζ < ∞),

x ∈ E.

(7.5.12)

When X 0 is a diffusion, so is its one-point extension X.

Sketch of proof. A construction of a one-point extension X of X 0 can be carried out probabilistically by using a Poisson point process of excursions of X 0 around the point a, as will now be explained briefly. We first assume that X 0 satisfies the conditions (A.1), (A.2), and (A.3) as well as (A.4) in non-diffusion cases. Let {P0t ; t ≥ 0} be the transition function of X 0 . Recall that a system {νt ; t > 0} of σ -finite measures on E0 is said to be X 0 -entrance law if νs P0t = νs+t for every s, t > 0. By (A.1), X 0 is transient and, by virtue of Lemma 5.7.1, there exists a unique X 0 -entrance law {νt ; t > 0} such that ∞ νt dt = ϕ · m. (7.5.13) 0

Due to the integrability of uα in the assumption (A.2) and (5.7.4), each νt is a finite measure. Denote by W the space of cadlàg paths w defined on a time interval (0, ζ (w)) taking values in E0 with w0+ = a and wζ − ∈ {a, ∂} whenever


345

ζ (w) < ∞. We can define a σ -finite measure n on W so that f1 (wt1 )f2 (wt2 ) · · · fn (wtn )n(dw) W

= νt1 f1 P0t2 −t1 f2 · · · P0tn−1 −tn−2 fn−1 P0tn −tn−1 fn

(7.5.14)

for 0 < t1 < t2 < · · · < tn and f1 , f2 , . . . , fn ∈ B+ (E0 ). Put W + = {w ∈ W : ζ (w) < ∞, wζ − = a}, W − = W \ W + and denote by n+ , n− the restrictions of n to W + , W − , respectively. Owing to the assumptions (A.3), (A.4), it can be shown that n(σE0 \U < ζ ) < ∞

for any neighborhood U of a,

(7.5.15)

from which it follows that n(W − ) < ∞. Let p = {pt , t ≥ 0} be the Poisson point process taking values in W with characteristic measure n on an appropriate probability space (, P). Clearly, p is a sum of independent Poisson point processes p+ and p− with characteristic measures n+ and n− , respectively. We then create a path Xa starting at a by piecing together the returning excursions p+ until the first occurrence time T of a non-returning excursion p− and then adjoining p− T to finish the construction of Xa . We note that P(T > t) = e−n(W

−

)t

− −1 − and the distribution of p− n . T is n(W )

(7.5.16)

The resulting path Xa is not only cadlàg but also continuous at those moments t when Xa (t) = a on account of the property (7.5.15) of n. The one-point extension X can be eventually constructed by joining X 0 to Xa . We refer the readers to [75] (when X 0 is a diffusion) and [28] (when X 0 is a general symmetric Markov process) as well as [21, Remark 3.2(ii)] for more details of the proof sketched above. The assumption (A.2) can be replaced by (A.2) using a method of a time change (cf. [28]). Indeed, consider a continuous function γ (x) on E0 such that

0 < γ ≤ 1, E γ dm < ∞ with γ = 1 on a neighborhood of U of a and let Y 0 = (Xτ0t , P0x ) be the time-changed process of X 0 where τt is the inverse of

t the PCAF At = 0 γ (Xs0 )ds of X 0 . This time-changed process Y 0 is a γ · msymmetric Hunt process on E0 and it is easy to see that properties (A.1) and (A.3) remain valid for Y 0 . Furthermore, the jumping measure J0 is invariant under a time change by a strictly increasing PCAF At and so (A.4) also remains valid for Y 0 . To see this, let (N, H) be a Lévy system of X 0 and µH be the Revuz measure of H with respect to m. Substitute T = τt , h(s) = 1 in (A.3.33) for X 0 and make a change of variable s → τs to obtain (A.3.31) with Yt0 , Hτs in place of Xt0 , Hs . This means that Y 0 admits (N, Hτ· ) as its Lévy system. Since the Revuz measure of the PCAF Hτ· of Y 0 with respect to γ · m coincides with µH (cf. [73, Lemma 6.2.9]), the jumping measure for Y 0 equals N(x, dy)µH (dx) = J0 .

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CHAPTER SEVEN

Y 0 trivially satisfies the integrability in (A.2), whose second condition also follows from (A.2) . Therefore, a one-point extension Y of Y 0 exists by the method described above. We then do a time change of Y by means of its PCAF

t −1 γ (Y ) ds, which is evidently a one-point extension of X 0 . s 0 Remark 7.5.7. Let X 0 be an m-symmetric Hunt process on E0 satisfying (A.1). Suppose we are given a one-point extension X = (Xt , Px ) of X 0 such that X is a Hunt process whose Dirichlet form is regular and X admits no jumps from or to a. By applying Theorem 5.7.5 to X and F = {a}, we readily see that the entrance law {Q∗t } induced by the exit system for (X, {a}) is necessarily characterized by equation (7.5.13). In fact, one can say more ([28]). Since {a} is not m-polar with respect to X, there exists a PCAF t of X with Revuz measure δ{a} . Let τt be its right continuous inverse and (p, Pa ) be a point process taking values in W defined by Dp(ω) = {s ∈ (0, ∞) : τs− ω < τs ω}, ps (ω) = iτs− ω for s ∈ Dp(ω) . Here the sample space = {ω} is taken to be of function space type and it = kσa ◦ θt for the killing operator kt specified by Xs (kt ω) being equal to Xs (ω) if s < t and to if s ≥ t. Define n by (7.5.13) and (7.5.14). Then (p, Pa ) is a Poisson point process with characteristic measure n absorbed at the random time T as is described above. Furthermore, n = kσa P∗a where (P∗a , ) is an exit system for (X, {a}). The above-mentioned proof of Theorem 7.5.6 asserts that the converse procedure starting from X 0 and constructing X is possible. We call the X 0 -entrance law {νt ; t > 0} characterized by (7.5.13) the entrance law for the one-point extension X. According to (5.7.5), it admits the following explicit expression: for t > 0, B ∈ B+ (E0 ), t νs (B)ds = P0x ζ 0 ≤ t, Xζ 0 − = a m(dx). (7.5.17) 0

B

Kiyosi Itô [94] introduced the notion of the Poisson point process of excursions around one point a in the state space of a standard Markov process X just in a way of Remark 7.5.7. He was motivated by giving systematic constructions of Markovian extensions of the absorbing diffusion process X 0 on the half-line (0, ∞) subjected to Feller’s general boundary conditions [96]. Itô had constructed the most general jump-in process from the exit boundary 0 by using the Poisson point process in his unpublished lecture notes [93] that preceded [94]. In that case, a is just the point 0. However, recent work ([75], [21], [28]) reveals that Itô’s program works equally well in constructing a onepoint extension by conceiving a certain set K as a single point a∗ .


347

To be precise, let E be a locally compact separable metric space and m be an everywhere dense positive Radon measure E. Consider a closed subset K of E and put E0 = E \ K. We assume that either K is compact or E0 is relatively compact in E. Let us extend the topological space E0 to E∗ = E0 ∪ {a∗ } by adding an extra point a∗ to E0 whose topology is prescribed as follows: a subset U of E∗ containing the point a∗ is an open neighborhood of a∗ if there is an open set U1 ⊂ E containing K such that U1 ∩ E0 = U \ {a∗ }. In other words, E∗ is obtained from E0 by identifying K into one point a∗ . Notice that in the special case that E0 is compact in E, E∗ = E ∪ {a∗ } is nothing but the one-point compactification of E0 . The restriction of the measure m to E0 will be denoted by m0 , which is then extended to E∗ by setting m0 ({a∗ }) = 0. (f , g) will denote the integral of f · g on E0 against the measure m0 . Let X = (Xt , Px , ζ ) be an m-symmetric Hunt process on E whose Dirichlet form (E, F) on L2 (E; m) is regular. Let X 0 = (Xt0 , P0x , ζ 0 ) be the part process of X on E0 . X 0 is an m0 -symmetric Hunt process on E0 (cf. Exercise 3.3.7). The resolvent of X 0 is denoted by {G0α ; α > 0}. The energy functional for X 0 is denoted by L0 again. We aim at creating a q.e. one-point extension of X 0 from E0 to E∗ . We shall assume once and for all that X satisfies the following conditions: (B.1) X is irreducible. (B.2) m0 (U ∩ E0 ) is finite for some neighborhood U of K. (B.3) X admits no killings inside or jumps from E0 to K. Define the functions {ϕ K , uKα , α > 0} on E0 by 0 ϕ K (x) = P0x ζ 0 < ∞, Xζ00 − ∈ K , uKα (x) = E0x e−αζ ; Xζ00 − ∈ K . (7.5.18) Just as in Exercise 7.5.2, we can then verify that for x ∈ E0 ϕ K (x) = Px (σK < ∞) and uKα (x) = Ex e−ασK

(7.5.19)

Recall the conditions (A.1)–(A.4) imposed on X 0 in Theorem 7.5.6. Let us denote by (A◦ .1), (A◦ .3), and (A◦ .4) the conditions on the present X 0 obtained by replacing a with K in (7.5.1), (A.3), and (A.4), respectively. D EFINITION 7.5.8. A Borel right process X ∗ = (Xt∗ , P∗x , ζ ∗ ) on E∗ is called a q.e. one-point extension of X 0 if X ∗ is m0 -symmetric and has no killings inside, and the part process of X ∗ on E0 coincides with X 0 q.e., namely, outside some m-polar set for X 0 . T HEOREM 7.5.9. Assume that conditions (B.1)–(B.3) hold for X and that X 0 satisfies (A◦ .1), (A◦ .3) as well as (A◦ .4) in the non-diffusion case. Then the following hold.

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CHAPTER SEVEN

(i) There exists a q.e. one-point extension X ∗ of X 0 from E0 to E∗ . Such an extension process is unique in law. (ii) Denote by {G∗α ; α > 0} the resolvent of X ∗ . Then (7.5.5) holds with Gα , ϕ and uα being replaced by G∗α , ϕ K and uKα , respectively. With this replacement, (7.5.6) holds for q.e. x ∈ E0 . (iii) Denote by (E ∗ , F ∗ ) the Dirichlet form of X ∗ on L2 (E∗ ; m0 ). Then the statement (iii) of Theorem 7.5.4 holds with F ∗ , E ∗ , ϕ K and uKα in place of F, E, ϕ, and uα , respectively. (iv) Denote by A∗ the generator of the L2 -semigroup of X ∗ . Define the flux N (f )(a∗ ) at a∗ by (7.4.25) with uKα in place of uα . Then the statement (iv) of Theorem 7.5.4 holds true with X ∗ , A∗ and a∗ in place of X, A and a, respectively. If in addition X satisfies the condition (AC) in Definition A.2.16, then “q.e.” in statements (i) and (ii) above can be dropped. Proof. (i) It follows from (B.3) that X 0 admits no killing inside E0 . This together with (A◦ .1) implies that X 0 satisfies (A.1) with a∗ in place of a. The properties (A.3) and (A.4) for X 0 with a∗ in place of a follow from (A◦ .3) and (A◦ .4), respectively. By virtue of Theorem 7.5.6, it now suffices to prove that X 0 satisfies the condition (A.2) with a∗ in place of a holding for q.e. x ∈ E0 . We note that Px (XσK − ∈ K, σK < ∞) = Px (σK < ∞) for x ∈ E, as X admits no jumps from E0 to K by (B.3). Moreover, the function ϕ K defined by (7.5.18) coincides with the hitting probability of K by X so that K is non-m-polar with respect to X by the assumption (A◦ .1). By the irreducibility assumption (B.1), X is either recurrent or transient. In the recurrent case, Px (σK < ∞) = 1 for q.e. x ∈ E, by Theorem 3.5.6(ii). Consequently, Px (XσK − ∈ K, σK < ∞) = 1 and P0x (Xζ 0 − ∈ K, ζ 0 < ∞) = 1 for q.e. x ∈ E. When X is transient, XσK − ∈ K, Px -a.s. on {σK < ∞}, while Xζ − = ∂, Px -a.s. for q.e. x ∈ E, in view of Theorem 3.5.2 and the condition (B.3). Therefore, P0x (Xζ 0 − ∈ K ∪ {∂}) = 1 for q.e. x ∈ E0 , as was to be proved. Notice that if in addition X satisfies the absolute continuity condition (AC), then so does X 0 and we can then replace “for q.e. x ∈ E0 ” by “for every x ∈ E0 ” in the above conclusion. That the law of X is unique follows directly from Theorem 7.5.4. (ii), (iii), and (iv) are consequences of Theorem 7.5.4. We shall call such a procedure of obtaining X ∗ from X (or from X 0 ) darning a hole K or collapsing a hole K. The entrance law {νtK } of X 0 taking part in darning a hole K is characterized by 0

∞

νtK dt = ϕ K · m0 .

(7.5.20)

349


In view of (5.7.5), it admits the following explicit expression: for t > 0, B ∈ B+ (E0 ), t νsK (B)ds = Px (σK ≤ t) m0 (dx), (7.5.21) B

0

in terms of the original process X = (Xt , Px ). Assume that the originally given Hunt process X on E as above satisfies an additional condition that it admits no jumps from K to E0 = E \ K. We assume that the part process X 0 on E0 satisfies (A◦ .1). Then the closed set K is not m-polar with respect to X and the exit system (P∗x , L) for (X, K) is well defined by Definition 5.7.4. Note that the jump component J in the exit system vanishes due to the above-mentioned assumption on X. Let Q∗t (x, ·), t > 0, x ∈ K, be the associated entrance law on E0 defined by (5.7.13). By putting f = 1K in equation (5.7.14) of Theorem 5.7.5, we are immediately led to the following theorem from (7.5.20): T HEOREM 7.5.10. The entrance law {νtK (·)} for the darning of the hole K is related to the entrance law {Q∗t (x, ·)} induced by the exit system (P∗ , L) for (X, K) as K Q∗t (x, B)µL (dx), t > 0, B ∈ B(E0 ), (7.5.22) νt (B) = K

where µL is the Revuz measure of the PCAF L. We notice that the relation (7.5.22) holds for any process X on a state space E as above whose part process on E0 ⊂ E equals X 0 . For instance, consider a compact set K ⊂ Rn with continuous boundary and the absorbing Brownian X either the Brownian motion motion X 0 on D = Rn \ K. Then we can take as on Rn or the RBM on D. A q.e. one-point extension X ∗ of an m0 -symmetric standard process X 0 is irreducible in view of Theorem 7.5.4. However, X 0 may not be irreducible so that if we change m0 by multiplying an arbitrary positive constant on each irreducible component, X 0 is still symmetric with respect to the changed measure and may admit a different one-point extension. Let us consider the assumption that E0 = E01 ∪ · · · ∪ E0k for some disjoint open sets E0i , 1 ≤ i ≤ k, and each E0i is X 0 -invariant. Assumption (7.5.23) means that Px (E0i ) = 1

for every x ∈ E0i and every i = 1, . . . , k,

(7.5.23)

350 where

CHAPTER SEVEN

0 E0i := ω ∈ : {Xt0 (ω), Xt− (ω)} ⊂ E0i for every t ∈ [0, ζ 0 ) .

Condition (7.5.23) is equivalent to saying that X 0 does not travel over two different sets E0i and E0j , i = j, 1 ≤ i, j ≤ k. If X 0 is a diffusion, then the second condition in (7.5.23) is automatically satisfied. The restrictions of functions and measures on E0 to E0i will be designated by the superscript i for 1 ≤ i ≤ k. Choosing any k-vector p with positive entries: p = (p1 , . . . , pk )

with p1 , . . . , pk > 0,

we define a new measure m0 on E0 by mi0 = pi · mi0 ,

1 ≤ i ≤ k.

(7.5.24)

p

The measure m0 will be also designated by m0 to indicate its dependence on p. m0 -symmetric and we extend m0 to E∗ = E ∪ {a∗ } by setting Clearly X 0 is ∗ m0 ({a }) = 0. P ROPOSITION 7.5.11. Assume that the same conditions as in Theorem 7.5.9 as well as the condition (7.5.23) are satisfied. Let m0 be the measure defined by (7.5.24). X ∗ on E∗ that is a q.e. (i) There exists an m0 -symmetric Borel right process 0 m0 -symmetric Hunt one-point extension of the process X being regarded as an process on E0 . X ∗ and {νt , t > 0} for X ∗ are related by (ii) The entrance laws { νt , t > 0} for νti = pi · νti

for 1 ≤ i ≤ k.

(iii) The q.e. one-point extensions X ∗ and X ∗ corresponding to two different k-vectors p and p are equivalent in law if and only if p = λ p

for some λ > 0.

Proof. (i) Follows from Theorem 7.5.9 with m0 in place of m0 there. (ii) is immediate from the characterization (7.5.13) of the entrance law. By p p substituting m0 and m0 in (7.5.5), the corresponding one-point extensions can be seen to have the same resolvents if and only if the condition in (iii) is valid, Under the assumption (7.5.23), the Dirichlet form (E 0 , F 0 ) of X 0 on L (E0 , m0 ) and its reflected Dirichlet space (E 0,ref , (F 0 )ref ) can be described 0i 0i 0 0 as follows. For each 1 ≤ i ≤ k, define the restriction (E , F ) of (E , F ) to 0i 0 E0i by F = F E0i and 2

E 0i (u|E0i , v|E0i ) = E 0 (u1E0i , v1E0i )

for u, v ∈ F 0 .

351


This is a transient Dirichlet form on L2 (E0i ; mi0 ), whose reflected Dirichlet space will be denoted by ((F 0i )ref , E ref,i ). It holds then that F 0 = {u : u|E0i ∈ F 0i E 0 (u, v) =

k !

for 1 ≤ i ≤ k},

E 0i (u|E0i , v|E0i )

for u, v ∈ F 0 ,

i=1

(F 0 )ref = {u : u|E0i ∈ (F 0i )ref E 0,ref (u, v) =

k !

for 1 ≤ i ≤ k},

and

E ref,i (u|E0i , u|E0i ) for u, v ∈ (F 0 )ref .

(7.5.25)

i=1

0 ) the Now, for measure m0 defined by (7.5.24), we denote by (E0 , F 0 2 0 ref 0,ref m0 ) and by ((F ) , E ) its reflected Dirichlet Dirichlet form of X on L (E0 , space. We then readily see that

0 = F 0 F

and E0 (u, v) =

k !

pi E 0i (ui , v i ) for u, v ∈ F 0 ,

(7.5.26)

i=1

0 )ref = (F 0 )ref (F

and E0,ref (u, v) =

k !

pi E ref,i (ui , v i )

(7.5.27)

i=1

for u, v ∈ (F 0 )ref . X ∗ and taking (7.5.26) and (7.5.27) Applying Theorem 7.5.9(iii) to m0 and into account, we arrive at the next theorem. T HEOREM 7.5.12. Under the same assumptions as in Proposition 7.5.11, let X ∗ be the m0 -symmetric Borel right m0 be the measure defined by (7.5.24) and ∗ process on E as appeared in Proposition 7.5.11(i). ∗ ) be the Dirichlet form of X ∗ on L2 (E∗ ; m0 ). Then (i) Let (E∗ , F ∗ 0 ref Fe is a linear subspace of (F ) spanned by Fe0 and ϕ, 0 ∗ is a linear subspace of (F 0 )ref F a spanned by F and uα , and E∗ (u, v) =

k !

pi E ref,i (u|E0i , u|E0i )

∗ . for u, v ∈ F

(7.5.28)

i=1

∗ be the L2 (E∗ ; (ii) Assume further that condition (7.3.4) is satisfied. Let A m0 )∗ ∗ infinitesimal generator of X . Then, f ∈ D(A ) if and only if f |E0i ∈ D(Li ) for 1 ≤ i ≤ k,

f admits an X 0 -fine limit at a∗

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CHAPTER SEVEN

and k !

pi Ni (f |E0i )(a∗ ) = 0,

i=1

where, for 1 ≤ i ≤ k, Li is the linear operator defined by (7.3.7) with L2 (E0 , m0 ), F 0 , (E 0,ref , (F 0 )ref ) being replaced by L2 (E0i , mi0 ), F 0i and (E ref,i , (F 0i )ref ), respectively, and Ni denotes a flux at a∗ defined by (7.4.25) with L, E 0,ref and uα being replaced by Li , E ref,i and uK,i α , respectively. As compared to X ∗ , we may call X ∗ a skew extension of X 0 .

7.6. EXAMPLES OF ONE-POINT EXTENSIONS In this section, we shall consider various concrete examples of X 0 and exhibit their one-point extensions. (1◦ ) One-dimensional absorbing Brownian motion Let R be the real line, I be its open subset and m be the Lebesgue measure on it. The restriction of m to I is denoted by m0 . We introduce function spaces by 2 BL(I) := u : absolutely continuous on I with (u ) dx < ∞ , I 1 (I) := {u H0e

∈ BL(I) : u = 0 at the finite boundary points of I} ,

1 (I) ∩ L2 (I; m0 ), H 1 (I) := BL(I) ∩ L2 (I; m0 ) and H01 (I) := H0e DI (u, v) := u (x)v (x)m0 (dx) and (u, v) := u(x)v(x)m0 (dx). I

I

Let X 0 (I) be the absorbing Brownian motion on I and (E 0 , F 0 ) be its Dirichlet form on L2 (I; m0 ). It holds then that (E 0 , F 0 ) = ( 21 DI , H01 (I)),

1 Fe0 = H0e (I),

and

(F 0 )ref = BL(I). (7.6.1)

The linear operator LI on L2 (I; m0 ) introduced by (7.3.7) reads as follows: D(LI ) = f ∈ H 1 (I) : f has an absolutely continuous version (7.6.2) with f ∈ L2 (I; m0 ) , 1 LI f = f 2

for f ∈ D(L).

353


We let NI (f ) =

1 I D (f , uα ) + (LI f , uα ) 2

for f ∈ D(LI ) and uα (x) = E0x [e−αζ ; ζ 0 < ∞], x ∈ I. An integration by parts gives    f (b−) − f (a+) when I = (a, b); 0

NI (f ) =

f (a−)   −f (a+)

when I = (−∞, a);

(7.6.3)

when I = (a, ∞).

In fact, the existence of the limit of f at finite end points of I is clear from (7.6.1). When I = (a, ∞), NI (f ) = lim f (ξ )uα (ξ ) − f (a+) ξ →∞

and, if the first term of the right hand side does not vanish, then f ∈ / BL(I). We shall examine the one-point extension of X 0 (I) in several cases of I. (i) Reflecting and circular Brownian motions Let I = (0, ∞),

E0 = I,

E = [0, ∞) = I ∪ {0}.

Note that the point 0 is approachable in finite time by X 0 (I) with probability 1; that is, ϕ(x) := Px (σ0 < ∞) = 1 for every x ∈ E0 . So X 0 (I) satisfies (A.1) and (A.3). It also satisfies (A.2) on account of Lemma 7.5.5(ii) and the fact that q.e. is a synonym of “everywhere” for X 0 (I) (see Example 3.5.7). So by Theorem 7.5.6, X 0 (I) has a one-point extension X to E. Denote by (E, F) the Dirichlet form of X on L2 (E; m0 )(= L2 (E0 ; m0 )). We conclude from Theorem 7.5.4(iii) and (7.6.1) that F = H 1 (I)

and

E=

1 I D, 2

namely, X is the reflected Brownian motion of E = [0, ∞). This follows also from the uniqueness statement of Theorem 7.5.4 because the reflecting Brownian motion on [0, ∞) is obviously a one-point extension of X 0 (I). Since F 0 = {u ∈ (F 0 )ref a : u(0+) = 0}, the condition (7.3.4) is satisfied for F = {0}. By Theorem 7.5.4(iv) and (7.6.3), the L2 -generator A of X can be described as D(A) = {f ∈ D(LI ) : f (0+) = 0}

and

Af (x) =

1 f (x), x ∈ I, 2

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CHAPTER SEVEN

where D(LI ) is defined by (7.6.2). We can also get the Skorohod equation for X = (Xt , Px ): Xt = X0 + Bt + t ,

t > 0,

Px -a.s.,

x ∈ [0, ∞),

where Bt is a Brownian motion with B0 = 0 and t is the positive continuous additive functional of X with Revuz measure δ{0} . Next let I = (0, 1), E0 = I and E = I ∪ {a} be the one-point compactification of I. As in the preceding case, we can apply Theorem 7.5.6 to X 0 (I) to get its one-point extension X to E. Let (E, F) and A be the Dirichlet form and the L2 -generator of X. Since F 0 = {u ∈ (F 0 )ref a : u(0+) = u(1−) = 0}, condition (7.3.4) is satisfied for F = {0, 1}. In the same way as above, we can conclude that F = H01 (I) ∪ {constant functions} D(A) = {f ∈ D(LI ) : f (0+) = f (1−), 1 Af (x) = f (x) 2

and

E=

1 I D, 2

f (0+) = f (1−)},

and

for f ∈ D(A) and x ∈ I.

Consequently, X is the Brownian motion on the circle E, which can also be obtained by wrapping the Brownian motion on R to [0, 1) (more precisely, by modulo 1). (ii) Skew Brownian motion Let I = (−∞, 0) ∪ (0, ∞),

E = R,

E0 = I,

K = {0},

E∗ = R.

We now apply Theorem 7.5.9 and Proposition 7.5.11 to Brownian motion X on R and absorbing Brownian motion X 0 (I) to get one-point extensions of the latter to E∗ = R by darning the hole K = {0}. Obviously X satisfies the conditions (B.1), (B.2), and (B.3) and X 0 (I) satisfies the conditions (A◦ .1) and (A◦ .3), so Theorem 7.5.9 and Proposition 7.5.11 are applicable. Note that R+ = (0, ∞) and R− = (−∞, 0) are two invariant sets for X 0 . The restrictions of functions and measures on R to R+ , R− will be denoted by putting superscript + and −, respectively. We can then rewrite expression (7.6.1) as 1 (R± ) , Fe0 = u : u± ∈ H0e (7.6.4) 1 + 1 − E 0 (u, v) = DR (u+ , v + ) + DR (u− , v − ) 2 2

355


for u, v ∈ Fe0 , and we get

(F 0 )ref = u : u± ∈ BL(R± ) ,

E 0,ref (u, v) =

1 R+ + + 1 − D (u , v ) + DR (u− , v − ) 2 2

(7.6.5)

for u, v ∈ (F 0 )ref . Note that the functions in (F 0 )ref may not be continuous at 0. For any p+ > 0 and p− > 0, let m0 be the measure on R defined by m0 (dx) = m0 (dx) = p− dx on (−∞, 0) and m0 ({0}) = 0. Then X 0 (I) can p+ dx on (0, ∞), 0 ) be regarded as an m0 -symmetric diffusion on R0 whose Dirichlet form (E0 , F 2 m0 ) is described as on L (R0 , e0 = Fe0 , E0 (u, v) = p+ DR+ (u+ , v + ) + p− DR− (u− , v − ), u, v ∈ F e0 , F 2 2 and accordingly, 0 )ref = (F 0 )ref , (F p+ R+ + + p− R− − − 0 )ref . E0,ref (u, v) = D (u , v ) + D (u , v ) for u, v ∈ (F 2 2 − + 0 = {u ∈ (F 0 )ref Since F a : u (0−) = u (0+) = 0}, condition (7.3.4) is satisfied for K = {0}. Let m0 be the Lebesgue measure and X ∗ be the m0 -symmetric extension of 0 X (I) to E∗ = R by Theorem 7.5.9, namely, X ∗ is constructed based on the ∞ entrance law {µt , t > 0} for X 0 (I) specified by 0 µt dt = m0 . By (7.5.21), it is given by ± (7.6.6) µt (B) = Px (σ0 ∈ dt)dx, B ∈ B(R± ), B

in terms of the Brownian motion X = (Xt , Px ) on R. By Theorem 7.5.9, the Dirichlet form (E ∗ , F ∗ ) of X ∗ on L2 (R; m0 )(= L2 (E0 ; m0 )) is given by Fe∗ = {f = f0 + c : f0 ∈ F 0 , c ∈ R}

and

E ∗ (u, v) = E ref (u, v)

(7.6.7)

for u, v ∈ Fe∗ . In particular, we see from (7.6.7) that every u ∈ F ∗ is continuous at 0. We can conclude from (7.6.4), (7.6.5), (7.6.7), and (7.6.8) that (Fe∗ , E ∗ ) = (BL(R), 12 D). Hence X ∗ is nothing but the Brownian motion on R.

(7.6.8)

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CHAPTER SEVEN

On the other hand, let Proposition 7.5.11: namely, for X 0 given by µ± t (B) = p±

X ∗ be the m-symmetric extension of X 0 in ∗ µt X is constructed based on the entrance law Px (σ0 ∈ dt)dx,

B ∈ B(R± ).

B

∗ ) of X ∗ on L2 (R; m0 ) is By Theorem 7.5.12, the Dirichlet form (E∗ , F given by ∗ = F ∗ = H 1 (R), F E∗ (u, v) = Eref (u, v)

(7.6.9)

1 1 + − = p+ DR (u+ , v + ) + p− DR (u− , v − ) 2 2 for u, v ∈ H 1 (R). X ∗ on L2 (R; m0 ). We then see from Let A∗ be the infinitesimal generator of ∗ ) if and only if Theorem 7.5.12(ii) and (7.6.3) that f ∈ D(A f ± ∈ D(LR± ) with f (0−) = f (0+) and p− f (0−) = p+ f (0+) (7.6.10) and A∗ f (x) = 12 f (x), x ∈ E0 . Here D(LR± ) is defined by (7.6.2). In accordance with Harrison and Shepp [84], we call a real-valued process Y a skew Brownian motion on R with parameter β ∈ (−1, 1) if Yt = Y0 + Bt + βLt ,

t ≥ 0,

(7.6.11)

where B is Brownian motion on R and L is the symmetric local time of Y at 0, that is, 1 Lt = lim ε→0 2ε

t

1{|Ys |≤ε} ds. 0

Actually the process X ∗ can be shown to be a skew Brownian mop+ −p− tion with parameter p+ +p− in this sense. Furthermore, the local time L in equation (7.6.11) can be verified to be the positive continuous additive functional of Y having Revuz measure (p+ + p− )δ0 . We refer the readers to [21, §5] for the proof.

357


(iii) One-point skew extensions of X0 obtained by identifying multi-points Choose any k − 1 points −∞ < a1 < a2 < · · · < ak−1 < ∞ and let I = R \ {a1 , a2 , . . . , ak−1 } =

k %

Ii ,

i=1

where I1 := (−∞, a1 ),

Ij := (aj , aj+1 ), 1 ≤ j ≤ k − 1,

Ik := (ak−1 , ∞).

We consider the case that E0 = I,

E = R,

K=

k−1 %

{ai },

E∗ = I ∪ {a∗ },

i=1

where E∗ is obtained from I by identifying the compact set K as one point a∗ in the way described in Section 7.5. Measure m0 is the restriction of the Lebesgue measure to I. The restrictions of functions and measures on I to the interval Ii will be designated by using the superscript i . As in the preceding case, the Brownian motion X on R and the absorbing Brownian motion X 0 (I) satisfy the conditions imposed in Theorem 7.5.9. Let p = (p1 , p2 , . . . , pk ) be a k-vector with positive entries. The absorbing Brownian motion X 0 (I) is then symmetric with respect to the measure m0 =

k !

pi mi0 ,

i=1

so that we can construct its m0 -symmetric extension X ∗ to E∗ according to ∗ ∗ m0 ) of X ∗ then Proposition 7.5.11. The Dirichlet form (E , F ) on L2 (E∗ ; admits the description ∗ = {f ∈ H 1 (R) : f (a1 ) = f (a2 ) = · · · = f (ak−1 )}, F E∗ (f , g) =

k ! 1 i=1

2

pi DIi (f i , gi )

∗ . for f , g ∈ F

and

(7.6.12)

Let A∗ be the L2 -infinitesimal generator of X ∗ on L2 (R; m0 ). As in previous ∗ ) if and only if cases, condition (7.3.4) is satisfied. Accordingly, f ∈ D(A

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CHAPTER SEVEN

f |Ii ∈ D(LIi )

for 1 ≤ i ≤ k

with

f (a1 ±) = f (a2 ±) = · · · = f (ak−1 ±)

p1 f (a1 +)+

k−1 !

and

(7.6.13)

pj f (aj −)−f (aj−1 +) − pk f (ak−1−) = 0.

j=2

Both (7.6.12) and (7.6.13) can be shown in the same way as in the previous cases by using Theorem 7.5.12 and (7.6.3). (2◦ ) Absorbing diffusion on a half-line We consider an open half-line I = (0, ∞), a pair (s, m) of a canonical scale and a canonical measure on diffusions studied in

I,dvand associated (s) ds and F be the space of those Example 3.5.7. Let E (s) (u, v) = I du ds ds functions u on I which are absolutely continuous with respect to s and E (s) (u, u) < ∞. We assume that the left boundary 0 of I is regular but the right boundary ∞ is non-regular. Then (E, F) = (E (s) , F (s) ∩ L2 (I; m))

(7.6.14)

is a strongly local regular Dirichlet form on L2 ([0, ∞); m) and the associated diffusion X = (Xt , Px ) on [0, ∞) is by definition the reflecting diffusion, which satisfies Px (σ0 < ∞) > 0 for any x ∈ I. The part process X 0 = (Xt0 , P0x , ζ 0 ) of X on I is the absorbing diffusion satisfying P0x Xζ 0 − = 0, ζ 0 < ∞ > 0 for any x ∈ I. (7.6.15) The Dirichlet form (E 0 , F 0 ) of X 0 is given by F 0 = {u ∈ F : u(0+) = 0},

E 0 = E (s) .

(7.6.16)

X is a one-point extension of X 0 . In particular, X 0 satisfies the integrability condition in (A.2) on account of Lemma 7.5.5(iii). In view of (6.5.8), 0,ref (s) ∩ L2 (I; m). (7.6.17) E , (F 0 )ref = E (s) , F (s) , (F 0 )ref a =F Characterization (7.3.7) now reads df D(L) = f ∈ F (s) ∩ L2 (I; m) : d is absolutely continuous with respect ds (7.6.18) to m, Lf ∈ L2 (I; m) ,

359


Lf =

d d f dm ds

for f ∈ D(L).

(7.6.19)

Furthermore, from (7.6.14), (7.6.16), and (7.6.17), we get F 0 = {(F 0 )ref a : f (0+) = 0}, which means that the condition (7.3.4) is satisfied for F = {0}. Let us compute the flux N(f )(0) for f ∈ D(L) at 0 defined by (7.4.25). L EMMA 7.6.1. We have N(f )(0) = −

df (0+) ds

for f ∈ D(L).

Proof. For f ∈ D(L), the integration by parts gives x x df duα df df df d uα = (x)uα (x) − (0+). ds + ds ds ds 0 ds ds 0 Since the left hand side converges to N(f )(0) as x → ∞, the finite limit c = lim uα (x) · x→∞

df (x) ds

exists and hence it suffices to prove c = 0. Since ∞ is assumed to be non-regular, either m or s diverges near ∞. Suppose m diverges near ∞. Then limx→∞ uα (x) = 0 because uα is nonincreasing in x and m-integrable by the preceding observation. If c were not 0, diverges near ∞, violating the property that f ∈ F (s) . Next suppose s then df ds diverges near ∞. Then the same property of f implies limx→∞ df (x) = 0 and ds we get c = 0. Let A be the infinitesimal generator of the L2 -semigroup of the reflecting diffusion X. By virtue of Theorem 7.5.4, we conclude that f ∈ D(A) if and only if f ∈ D(L), the limit f (0+) exists, and df (0+) = 0. ds

(7.6.20)

d d f , f ∈ D(A). In this case we have Af = dm ds In view of (7.6.14) and (7.6.17), X is a reflecting extension of X 0 in the sense of Definition 7.2.6, so the above characterization of A also follows from (7.3.21).

(3◦ ) Diffusions on half-lines merging at one point We consider a finite number of disjoint rays i , i = 1, . . . , k, on R2 merging at a point a ∈ R2 . Each ray i is homeomorphic to the open half-line (0, ∞)

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CHAPTER SEVEN

and the point a ∈ R2 is the boundary of each ray at 0-side. We put E0 = ∪ki=1 i ,

E = E0 ∪ {a}.

E is endowed with the induced topology as a subset of R2 . Let m be a positive Radon measure on E such that supp[m] = E and m({a}) = 0. The restriction of m to i is denoted by mi . For any function g on E0 , its restriction to i will be denoted by gi . We consider a diffusion process X 0 = {Xt0 , ζ 0 , P0x } on E0 such that its restriction X 0,i to each open halfline i ∼ (0, ∞) is the absorbing diffusion governed by the speed measure mi and a canonical scale, say si , which is assumed to satisfy si (0+) > −∞,

1 ≤ i ≤ k.

Since mi ((0, 1)) < ∞, 1 ≤ i ≤ k, 0 is a regular boundary for each X 0,i , 1 ≤ i ≤ k. We shall also assume that ∞ is non-regular for each X 0,i , 1 ≤ i ≤ k. In view of the observation made in the preceding example, each X 0,i satisfies 2 i (si ) , (7.6.15) and the integrability condition in (A.2). Since R0,i 1 (L (i ; m )) ⊂ F 0,i i R1 f is lower semicontinuous for any non-negative Borel function f on E0 . Therefore, by taking Lemma 7.5.5(i) into account, we conclude that the diffusion X 0 meets all conditions (A.1), (A.2), and (A.3) in Theorem 7.5.6 yielding a one-point diffusion extension X of X 0 from E0 to E. For each i ≤ i ≤ k, denote by (E 0,i , F 0,i ) the Dirichlet form of X 0,i on L2 (i ; mi ) and by ((F 0,i )ref , E ref,i ) the reflected Dirichlet space. We use i i (E (s ) , F (s ) ) to denote the function space considered in (i) but with i and si in place of I = [0, ∞) and s there. It follows from (i) that ref,i i i E , (F 0,i )ref = E (s ) , F (s ) , (7.6.21) i F 0,i = v ∈ F (s ) ∩ L2 ((0, ∞); mi ) : v(0+) = 0 ,

(7.6.22)

and i

E 0,i (v1 , v2 ) = E (s ) (v1 , v2 )

for v1 , v2 ∈ F 0,i .

By (7.6.21) and (7.6.22), we see that the condition (7.3.4) is fulfilled again. Now let (E, F) be the Dirichlet form on L2 (E : m) of the one-extension X of X 0 to E as given by Theorem 7.5.6. Since X 0 has the property (7.5.23) with E0i = i , 1 ≤ i ≤ k, we have from (7.5.25) and Theorem 7.5.4(iii) F = {f = f0 + c uα : f0i ∈ F 0,i , 1 ≤ i ≤ k, c ∈ R}(⊂ (F 0 )ref a ) E(u, v) =

k !

i

E s (ui , v i )

i=1

where F 0,i are specified by (7.6.22).

for u, v ∈ F,

361


The entrance law {µt , t > 0} from a for X is the sum of its restriction µit to i , which is describable as 0,i 0,i µit (f )dt = P0,i = 0 and ζ ∈ dt . (7.6.23) X i 0,i f ·m ζ − We next choose any k-vector p = (p1 , . . . , pk ) with positive entries and define a new measure m on E0 by mi = pi mi ,

1 ≤ i ≤ k,

which is extended to E by setting m({a}) = 0. Since X 0 is also m-symmetric, we can construct by Proposition 7.5.11 a unique m-symmetric diffusion X on E with no sojorn or killing at a that extends X 0 . By virtue of Theorem 7.5.12(i), F) on L2 (E; m) of X can be described as follows: the Dirichlet form (E, = F = f = f0 + c uα : f0i ∈ F 0,i for 1 ≤ i ≤ k, and c ∈ R , F e = Fe = f = f0 + c ϕ : f0i ∈ Fe0,i for 1 ≤ i ≤ k, and c ∈ R , F v) = E(u,

k !

e . for u, v ∈ F

i

pi E s (ui , v i )

i=1

be the L2 (E; m0 )-infinitesimal generator of X . Combining Theorem Let A 7.5.12(ii) with (7.6.18), (7.6.19), and Lemma 7.6.1, we can see that f ∈ D(A) if and only if the following conditions are satisfied: i

f i ∈ F (s ) ∩ L2 (i ; mi ) df i is absolutely continuous with respect to mi , dsi d d i f ∈ L2 (i ; mi ) for 1 ≤ i ≤ k, Li f i = dmi dsi −∞ < f 1 (0+) = · · · = f k (0+) < ∞,

k ! i=1

pi

df i (0+) = 0. dsi

We have in this case Af (x) = Li f i (x)

if x ∈ i

for 1 ≤ i ≤ k.

The entrance law { µt , t > 0} from a for X is given by µit = pi µit , where

{µit , t

> 0} is given by (7.6.23).

1 ≤ i ≤ k,

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CHAPTER SEVEN

Clearly Example (1◦ ) (ii) of this section may be considered as a special case of the present one with k = 2 and E = R. X When mi (dx) = dx and si (x) = x for all i = 1, . . . , k, it is easy to see that is a Walsh Brownian motion [7] with k number of random markers. We remark that in this example, one can also replace diffusions on some half-lines by discontinuous Markov processes that can approach 0 such as censored α-stable processes on (0, ∞) with α > 1. The latter will be described in (5◦ ) below. (4◦ ) Brownian motion on Rn Let X = (Xt , Px ) be the Brownian motion on the Euclidean space Rn for n ≥ 2, K be a non-polar compact subset of Rn , and X 0 = (Xt0 , P0x , ζ 0 ) be the absorbing Brownian motion on E0 = Rn \ K. Conditions (B.1), (B.2), and (B.3) are trivially satisfied by X. The irreducibility of X and the non-polarity of K imply that Px (σ < ∞) > 0 for a.e. x ∈ Rn and consequently for every x because the transition function of X is absolutely continuous. Hence X 0 satisfies (A◦ .1). Since the resolvent G0α of X 0 is related to the resolvent Gα of X by G0α f (x) = Gα f (x) − Ex e−ασK Gα f (XσK ) and the second term of the right hand side is α-harmonic in x ∈ E0 , X 0 is strong Feller in the sense that G0α (bB(E0 )) ⊂ bC(E0 ) and so the condition (A◦ .3) is fulfilled by X 0 . By Theorem 7.5.9, a unique m0 -symmetric diffusion X ∗ extending X 0 to ∗ E = E0 ∪ {a∗ } can be constructed by darning the hole K. Here E∗ is the onepoint extension of E0 by regarding the set K as one point a∗ and m0 is the Lebesgue measure on E0 extended to E by setting m0 ({a∗ }) = 0. By (7.5.21), the entrance law for X 0 taking part in the darning is given by

t 0

νsK (B)ds

=

Px (σK ≤ t)dx,

t > 0, B ∈ B(E0 ),

(7.6.24)

B

in terms of the Brownian motion X = (Xt , Px ). When K is a ball centered at the origin, νtK is therefore a spherically symmetric measure on E0 . For the open set D ⊂ Rn , we consider the function spaces BL(D), 1 1 (D) as well as the Dirichlet integral D introduced in H (D), H01 (D) and H0,e Section 2.2.4. The Dirichlet form (E 0 , F 0 ) of X 0 on L2 (E0 ) equals ( 21 D, H01 (E0 )). In view of Example (4◦ ) of Section 6.5, the reflected Dirichlet space of (E 0 , F 0 ) is equal to (BL(E0 ), 12 D).

363


The linear operator L on L2 (E0 ) specified by (7.3.7) and the flux N(f )(a∗ ) specified by Theorem 7.5.9(iv) are with D(L) = f ∈ H 1 (E0 ) : f ∈ L2 (E0 ) , L = 12 1 N(f )(a∗ ) = 12 D(f , uKα ) + (f , uKα ) 2

for f ∈ D(L).

By Theorem 7.5.9, the Dirichlet form (E ∗ , F ∗ ) of X ∗ on L2 (E∗ ; m0 ) and its extended Dirichlet space Fe∗ can be expressed as follows: F ∗ = {f = f0 + cuKα : f0 ∈ H01 (E0 ), c ∈ R}, 1 Fe∗ = {f = f0 + cϕ K : f0 ∈ H0,e (E0 ), c ∈ R},

E ∗ (u, v) =

1 D(u, v) 2

for u, v ∈ Fe∗ .

(7.6.25)

(7.6.26)

Note that F ∗ ⊂ H 1 (E0 ) and Fe∗ ⊂ BL(E0 ). We know that uα (resp. ϕ) is 1 (E0 )). Eα∗ -orthogonal (resp. E ∗ -orthogonal) to the space H01 (E0 ) (resp. H0,e ∗ The process X is an irreducible diffusion by Theorem 7.5.4(i) and Theorem 7.5.6. When n = 2, X ∗ is recurrent by Theorem 2.1.8 because ϕ K = 1 on E0 . When n ≥ 3, X ∗ is transient but still conservative. Indeed, 1 − ϕ K (x) = P0x (ζ 0 = ∞) is P0t -invariant and so for every t, s > 0, νt+s , 1 − ϕ = νt , P0s (1 − ϕ) = νt , 1 − ϕ ; that is, νt , 1 − ϕ is a constant function in t > 0. Thus by (ii) and (iii) of Lemma 5.7.1, for each fixed α > 0, L0 (ϕ K , 1 − ϕ K ) = lim νtK , 1 − ϕ K = α ¯ναK , 1 − ϕ K = α(uKα , 1 − ϕ K ). t↓0

Now it follows from Theorem 7.5.9(ii) that αG∗α 1(a∗ ) = 1 and, consequently, αG∗α 1 = 1 on E∗ . This shows that X ∗ is conservative. Remark 7.6.2. The above proof shows that under the condition of Theorem 7.5.9, if X is conservative, then so is X ∗ . Clearly X 0 satisfies

P0x Xζ00 − ∈ K ζ 0 < ∞ = 1,

x ∈ E0 ,

and condition (7.3.4) is fulfilled on account of Remark 7.3.3. By Theorem 7.5.9(iv), the generator A∗ of X ∗ on L2 (E∗ ; m0 ) can be characterized as f ∈ D(A∗ ) ⇐⇒ f ∈ D(L), f admits X 0 -fine limit at a∗ and N(f )(a∗ ) = 0 and A∗ f =

1 2

for f ∈ D(A∗ ).

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When ∂K and f are smooth enough and f was of compact support, then the Gauss-Green formula yields ∂f 1 (ξ )σ (dξ ), N(f )(a∗ ) = − 2 ∂K ∂n where n denotes the outward normal for ∂K and σ is the surface measure on ∂K. In this sense, N(f )(a∗ ) may be interpreted as the flux of the vector field 1 ∇f at a∗ or into K. 2 Note that (E ∗ , F ∗ ) is a quasi-regular Dirichlet form on L2 (E∗ ; m0 ) but may not be a regular Dirichlet form unless every point of ∂K is a regular boundary point of E0 with respect to the Dirichlet problem for (α − 12 ) on E0 . Therefore, we cannot construct X ∗ by using the theory of the regular Dirichlet form in general. Like planar reflecting Brownian motion (cf. Example (2◦ ) of Section 5.3), X ∗ enjoys the following conformal invariance when n = 2. Note that the compact sets K and K in the next theorem can be disconnected. K are two non-polar compact T HEOREM 7.6.3. Suppose n = 2, K and subsets of R2 . Let X ∗ be the diffusion process on E∗ := (R2 \ K) ∪ {a∗ } obtained from X by darning the hole K into a single point a∗ . Suppose that K that maps ∞ to ∞. Identify φ is a conformal map from R2 \ K onto R2 \ E∗ := (R2 \ K ) ∪ { a∗ } the the compact set K with a single point a∗ and equip 2 ∗ ∗ ∗ a . Define φ(a ) = a . Then topology induced from R by collapsing K into E∗ , and φ(X ∗ ) is, up to a φ is a topological homeomorphism from E∗ onto time change, the diffusion process obtained from a Brownian motion on R2 by darning the hole K. K by E0 and E0 , respectively. Since φ maps Proof. Denote R2 \ K and R2 \ E0 ∞ to ∞, it maps the E0 -portion of any neighborhood of K into the portion of a neighborhood of K , and vice versa. Hence φ is a topological E∗ = E0 ∪ { a∗ }. The Lebesgue homeomorphism from E∗ = E0 ∪ {a∗ } onto 2 measure on R will be denoted by λ. As is noted in the above, X ∗ = ∗ (Xt∗ , P∗z )z∈E∗ is an irreducible recurrent diffusion on E . Further, its transition function is absolutely continuous with respect to λ D because so is its resolvent kernel by Theorem 7.5.9(ii) and consequently Corollary 3.1.14 applies. We let t |φ (Xs∗ )|2 1E0 (Xs∗ )ds. (7.6.27) m(dz) = |φ (z)|2 1E0 (z)dz, At = 0

◦

Just as in Example (2 ) of Section 5.3, we see that for every z ∈ E∗ , P∗z -a.s. At is strictly increasing to ∞, as t → ∞, and so is its inverse τt . The time-changed process Xˇ ∗ = (Xτ∗t , P∗z ) is an m-symmetric conservative diffusion on E∗ . In fact, X ∗ is recurrent in view of Theorem 5.2.5.


365

Define a process Y ∗ = ( Yt∗ , P∗w )w∈E∗ by Yt∗ = φ(Xτ∗t ),

P∗w = P∗φ −1 (w) , w ∈ E∗ .

(7.6.28)

Y ∗ is then a conservative diffusion on E∗ and, as in the above cited example, it E0∗ . is symmetric with respect to the zero extension of λ on E0 to ∗ We claim that Y has the same law as the diffusion process obtained from K . In view of Theorem 7.5.4 Brownian motion on R2 by darning the hole on the uniqueness one-point extension process, it suffices to show that the Y ∗ in E0 is an absorbing Brownian motion in E0 . Notice part process Y ∗,0 of ∗,0 that Y is the image by φ of the part process on E0 of the time-changed process (Xτ∗t , P∗z ). The latter equals the part process of X ∗ on E0 , namely, the absorbing Brownian motion X 0 on E0 , being time-changed by the inverse of

t∧ζ its PCAF A0t = 0 0 |φ (Xs0 )|2 ds. The desired property of Y ∗,0 is now a direct consequence of Theorem 5.3.1. Remark 7.6.4. (i) When n = 2 and K is a disk in R2 , let {νtK ; t > 0} be the entrance law for the absorbing Brownian motion X 0 on E0 := R2 \ K that takes part in darning the hole K for the Brownian motion X on R2 . As is noted right after Theorem 7.5.10, we can then apply this theorem to the RBM X on E0 so that Q∗t (x, B)µL (dx), t > 0, B ∈ B(E0 ) νtK (B) = ∂K

{Q∗t (x, ·), t

Here > 0, x ∈ ∂K} is the entrance law induced by an exit system X on the boundary set ∂K and µL is the Revuz measure (P∗ , L) for the RBM of a PCAF (boundary local time) L of X on ∂K. While exit system (P∗ , L) ∗ of X on ∂K (and hence the pair ({Qt (x, ·), t > 0, x ∈ ∂K}, L)) is not unique, {Q∗t (x, ·)µL (dx), t > 0} is unique by (5.7.11). Since ∂K is a circle, we can and do take µL as the uniform measure on ∂K. The corresponding {Q∗t (x, ·), t > 0, x ∈ ∂K} is then the commonly used entrance law of reflecting Brownian motion X on ∂K in literature (cf. [90, 115]). Hence the process X ∗ can also be obtained in the following heuristic way. Run a reflecting Brownian motion Y on R2 \ K. When Y hits the boundary ∂K, rotate the next excursion of Y away from ∂K by a random angle uniform over [0, 2π), and then continue this process. Collapsing the hole K into a single point {a} then results in a continuous process on E∗ that has the same distribution as X ∗ . It is now easy to see that the process X ∗ obtained from X by darning the hole K as given in Theorem 7.5.9 can be identified with the excursion-reflected Brownian motion coined in Lawler [115]. The latter arose in the study of SLE in multiply-connected planar domains. Theorem 7.6.3 extends such an identification to any compact set K for which R2 \ K is conformally equivalent to the complement of a closed ball in R2 .

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(ii) Same consideration can be made with the absorbing Brownian motion on a domain D ⊂ Rn in place of Brownian motion on Rn . In this case, K is a compact subset of the domain D. When n = 2, Theorem 7.6.3 (conformal invariance of X ∗ ) remains true for those conformal maps φ that map the (D \ K)-portion of any neighborhood of K into the (D \ K )-portion of a neighborhood of K . See Example (1◦ ) of Section 7.8 for darning multiple numbers of holes. We next consider the case where the closed set K is the complement of a bounded open set E0 ⊂ Rn . In this case, E∗ = E0 ∪ {a∗ } is just the one-point compactification of E0 . The symmetric diffusion X ∗ extending the absorbing Brownian motion X 0 = (Xt0 , ζ 0 , P0x ) on E0 to E∗ has the Dirichlet form (E ∗ , F ∗ ) on L2 (E∗ ; m0 ) expressible as F ∗ = H01 (E0 ) + {constant functions on E∗ }, E ∗ (w1 , w2 ) =

1 D(f1 , f2 ) 2

for wi = fi + ci with fi ∈ H01 (E0 ) and ci ∈ R, i = 1, 2,

which is a regular, strongly local and irreducible recurrent Dirichlet form as has been studied in [75, §3]. Hence we can construct the symmetric diffusion X ∗ on E∗ by a direct use of the Dirichlet form theory in this case. The L2 -generator of X ∗ can be characterized exactly in the same way as the preceding case. Finally, let n ≥ 3 and m(dx) = m(x)dx be a measure on Rn with density m being strictly positive, bounded continuous, and integrable on Rn . Let Rn Y = (Yt , Px , ζ Y ) be the time change

t of the Brownian motion X = (Xt , Px ) on ◦ by means of its PCAF At = 0 m(Xs )ds. As we saw in Example (1 ) of Section 5.3, Y is m-symmetric and its Dirichlet form (E Y , F Y ) on L2 (Rn ; m) is given by Y 1 e1 (Rn ) ∩ L2 (Rn ; m) . E , FY = (7.6.29) D, H 2 The process Y is transient and its 0-order resolvent RY f for f ∈ bB(Rn ) has the expression RY f = 2I2 ∗ (fm) with the Newtonian convolution kernel 2I2 defined by (2.2.26). We notice that I2 ∗ g ∈ C∞ (Rn ) for any g ∈ bL1 (Rn ). Since Xt converges to the point ∂ at infinity of Rn as t → ∞ Px -a.s. for any x ∈ Rn and RY 1(x) < ∞ for any x ∈ Rn , we have ϕ(x) = Px ζ Y < ∞, Yζ Y − = ∂ = 1 for every x ∈ Rn . (7.6.30) We can now apply Theorem 7.5.6 to E0 = Rn , X 0 = Y, a = ∂, E = Rn ∪ {∂}


367

in getting a one-point extension X of Y from Rn to Rn ∪ {∂} because conditions (A.1), (A.2) are clearly satisfied and (A.3) follows from the lower semicontinuity of RY f for f ∈ B+ (Rn ) and Lemma 7.5.5(iv). By setting m({∂}) = 0, X is an m-symmetric recurrent diffusion on Rn ∪ {∂} and its Dirichlet form (E, F ) on L2 (Rn ; m) is given by 1 n 2 n (7.6.31) D, BL(R ) ∩ L (R ; m) (E, F ) = 2 in view of (7.6.30), Theorem 7.5.4(iii), and Theorem 2.2.12. On account of Example (5◦ ) of Section 6.5, the active reflected Dirichlet form of (7.6.29) equals (7.6.31). We can draw from this the following two conclusions. First, X is a reflecting extension of Y in the sense of Definition 7.2.6. Therefore, X is not only a one-point extension but also the maximal Silverstein extension of Y. Based on this fact, X has been proved to be the unique genuine symmetric extension of Y in [24]. Second, the associated operator L specified by (7.3.7) is given by  1 n 2 n 2 n  D(L) = {f ∈ BL(R ) ∩ L (R ; m) : m f ∈ L (R ; m)}, (7.6.32) 1  Lf (x) = f (x), x ∈ Rn , f ∈ D(L). 2m(x) The function uα (x) = Ex [e−αζ ; Yζ Y − ], x ∈ Rn , satisfies uα = 1 − αRY uα . Hence limx→∂ uα (x) = 1. Accordingly, the flux N(f )(∂), f ∈ D(L), specified by 1 N(f )(∂) = D(f , uα ) + (Lf , uα )L2 (Rn ;m) 2 reads via the Gauss-Green formula as 1 fr (rξ )σ (dξ ), N(f )(∂) = − lim rn−1 2 r→∞ 1 Y

provided that f ∈ D(L) ∩ C1 (Rn ). Here 1 = {x ∈ Rn : |x| = 1} and σ is the surface measure on 1 . The infinitesimal generator of the L2 -semigroup of X can be characterized as Theorem 7.5.4(iv) in terms of L and N(f )(∂) as above. The entrance law {νt ; t > 0} taking part in the above construction of X is, in view of (7.5.17), given by t νs (B)ds = Px (ζ Y ≤ t)m(x)dx, t > 0, B ∈ B(Rd ). 0

B

If m is spherically symmetric, then so is νt . An alternative construction of X by means a Dirichlet form is possible as in the case of the one-point extension of the absorbing Brownian motion on a bounded open set.

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(5◦ ) Censored stable process We consider a case where X 0 is of pure jump type and admits no killings inside E0 . A typical example of such a process is a censored stable process on a Euclidean open set, as previously seen in Example (6◦ ) of Section 6.5 and Example 7.2.11. Let D be an open n-set in Rn as is defined in Example (6◦ ) of Section 6.5. D can be disconnected. Fix 0 < α < 2 and c > 0, and consider the function space (W(D), B) defined by (6.5.13). We let W α/2,2 (D) = W(D) ∩ L2 (D; m), where m denotes the Lebesgue measure. Then the bilinear form defined by (E, F) = (B, W α/2, 2 (D)) is a regular irreducible Dirichlet form on L2 (D; m) and the associated Hunt process X on D is a reflecting α-stable process by definition. It is shown in [29] that X has Hölder continuous transition density functions with respect to the Lebesgue measure m on D and therefore X can be refined to start from every point in D. X admits no killing inside D. Moreover, X admits no jump from D to ∂D or from ∂D to ∂D. The part process X 0 = (Xt0 , P0x , ζ 0 ) of X on D is identical with the censored α-stable process on D. Its Dirichlet form on L2 (D; m) is given by α/2, 2 α/2, 2 (D)), where W0 (D) is the closure of Cc∞ (D) in F with respect (E, W0 to E1 := E + (·, ·)L2 (D;m) . Note that the censored stable process X 0 has no α/2, 2 (D)) is denoted killings inside D. The extended Dirichlet space of (E, W0 α/2, 2 by W0,e (D). Let us assume that D ⊂ Rn is a proper open n-set, ∂D is a compact set of a positive d-dimensional Hausdorff measure with α > n − d when n ≥ 2 and α > 1 when n = 1. Let D∗ = D ∪ {a} be the topological space obtained from D = D ∪ ∂D by regarding ∂D as one point {a} in the way prescribed in Section 7.5. We consider the extension of the censored stable process X 0 to D∗ . When D is bounded, D∗ is just the one-point compactification of D. We now apply Theorem 7.5.9 to the case that E = D, K = ∂D. By the above-mentioned properties of the reflecting stable process X = (Xt , Px ) on D, it clearly satisfies conditions (B.1), (B.2), (B.3). Note that ϕ(x) := Px (σ∂D < ∞) = 1 for x ∈ D, when D is bounded, and 0 < ϕ < 1 on D when D is unbounded with compact boundary in view of [14]. Hence the condition (A◦ .1) for X 0 is also satisfied. Any bounded measurable function f on D is extended to D by defining ∞ f (x) = 0 on ∂D. By [29], Gα f (x) := Ex 0 e−βt f (Xt )dt is a continuous / D}. Then we have for G0β f (x) := function on D. Let τD := inf{t > 0 : Xt ∈ τD −βt Ex 0 e f (Xt )dt , for x ∈ D. G0β f (x) = Gβ f (x) − Ex e−βτD Gβ f (XτD )


369

Since x → Ex e−βτD Gβ f (XτD ) is a β-harmonic function of X 0 and so it is continuous on D (see [14, (3.8)]), we conclude that G0β f is continuous on D. Hence the condition (A◦ .3) is always satisfied for censored α-stable process X 0 in any open n-set D. A Lévy system of X 0 is given by (N(x, dy), dt) with N(x, dy) = c |x − y|−(n+α) dy and the condition (A◦ .4) is clearly satisfied. By Theorem 7.5.9, we can thus construct a unique symmetric extension X ∗ on D∗ of X 0 by darning the hole ∂D. The entrance law {νt ; t > 0} taking part in the darning is given by 0

t

νs (B)ds =

Px (σ∂D ≤ t)m(dx),

t > 0, B ∈ B(D),

B

in terms of the reflecting stable process X on D. Let u1 := Ex e−τD . By virtue of (6.5.15), the reflected Dirichlet space of (E 0 , F 0 ) equals (W(D), B) and it follows from Theorem 7.5.9 that the Dirichlet F) and its extended Dirichlet form (E, F e ) is given by form (E, = f = f0 + cu1 : f0 ∈ W α/2, 2 (D) and c ∈ R , F 0 e = f = f0 + c : f0 ∈ W α/2, 2 (D) and c ∈ R , F 0,e , g) = B(f , g), E(f

e . f,g ∈ F

7.7. MANY-POINT EXTENSIONS We now generalize the uniqueness theorem and existence theorem of the one-point extension established in Section 7.5 to a countably many points extension. As in Section 7.5, every sample path of a right process X on a state space E will be assumed to possess the left limit Xt− in E for all t ∈ (0, ζ ), where ζ is the lifetime of X. Let E be a locally compact separable metric space and m be a positive Radon measure on E with supp[m] = E. We fix a closed set F = {a1 , a2 , . . . , ai , . . . } consisting of finite or countably many non-isolated points of E such that F possesses no accumulating point and m(F) = 0. Put E0 = E \ F. As usual, let E∂ = E ∪ {∂} denote the one-point compactification of E. (E0 )∂ = E0 ∪ {∂} is regarded as a topological subspace of E∂ . Let X 0 = (Xt0 , P0x , ζ 0 ) be an m-symmetric Borel standard process on E0 satisfying the following condition:

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(M.1) X 0 admits no killing inside and P0x ζ 0 < ∞, Xζ00 − = ai > 0 for every x ∈ E0 and i ≥ 1.

(7.7.1)

We then define, for x ∈ E0 , i ≥ 1, and α > 0, 0 −αζ 0 ; Xζ00 − = ai . (7.7.2) ϕ (i) (x) = P0x ζ 0 < ∞, Xζ00 − = ai , u(i) α (x) = Ex e D EFINITION 7.7.1. A right process X = (Xt , Px , ζ ) on E is called an extension of X 0 if X is m-symmetric, having no killings on F, admitting no jumps from F to F, and with X 0 the part process of X on E0 . Suppose an extension X = (Xt , Px ) of X 0 is a Borel standard process. Then it can be verified exactly as in Exercise 7.5.2 that the following holds. For any x ∈ E0 , i ≥ 1, and α > 0, −ασF ; XσF = ai , (7.7.3) ϕ (i) (x) = Px σF < ∞, XσF = ai , u(i) α (x) = Ex e and X admits no jump from E0 to F: Px (Xt− ∈ E0 , Xt ∈ F for some t > 0) = 0 for x ∈ E.

(7.7.4)

L EMMA 7.7.2. Suppose an extension X of X 0 is a Hunt process whose Dirichlet form (E, F) is regular. Then each point ai is non-E-polar and regular for itself with respect to X in the sense that Pai (σai = 0) = 1. Moreover, X is irreducible. Proof. Fix i ≥ 1 and let X (i) be the part process of X on the set E0 ∪ {ai }. Then X (i) is a one-point extension of X 0 . By Theorem 3.3.9, X (i) is a Hunt process with the associated Dirichlet form being regular. Therefore, X (i) is irreducible by Lemma 7.5.3. Since this holds for every i ≥ 1, it follows that X is irreducible. The proof of other assertions is the same as the one for Lemma 7.5.3. T HEOREM 7.7.3. Let X 0 be an m-symmetric Borel standard process on E0 satisfying condition (M.1). An extension of X 0 is then unique in law. More specifically, let X = (Xt , Px , ζ ) be an extension of X 0 . Denote by {G0α ; α > 0} and (E 0 , F 0 ) (resp. {Gα ; α > 0} and (E, F)) the resolvent and the Dirichlet form of X 0 (resp. X) on L2 (E0 ; m) = L2 (E; m). Let B0 (F) be the space of functions on F vanishing except on finite many points. The energy functional relative to X 0 is denoted by L0 . (i) Each point {ai } is non-m-polar and regular for itself with respect to X. The process X is irreducible.

371


(ii) Define for i, j ≥ 1, ij

U ij = L0 (ϕ (i) , ϕ (j) ) for i = j, & ' ! ϕ (k) , V i = L0 ϕ (i) , 1 −

(j) Uα = α(u(i) α , ϕ ),

µ0 ({ai }) =

k≥1

!

(7.7.5) U1ik ,

k≥1

and define a function space on F by  C(ψ, ψ) =

1 2

i,j≥1,i=j (ψ(ai )

− ψ(aj ))2 U ij +

i≥1

ψ(ai )2 V i ,

 G1 = {ψ ∈ L2 (F; µ0 ) : C(ψ, ψ) < ∞}.

(7.7.6)

It holds that B0 (F) ⊂ F F ,

F F = the C1 -closure of B0 (F) in G1 ,

(7.7.7)

where C1 (ψ, ψ) = C(ψ, ψ) + ψ2L2 (F;µ0 ) . 2 (iii) For any α > 0 and g ∈ L (E; m), let ψ = Gα g F . Then ψ is an element of F F such that ! C(ψ, 1ai ) + Uαij ψ(aj ) = (u(i) i ≥ 1. (7.7.8) α , g), j≥1

Moreover, Gα g admits the representation Gα g(x) = G0α g(x) +

!

u(i) α (x)ψ(ai ),

x ∈ E.

(7.7.9)

i≥1 0,ref (iv) F ⊂ (F 0 )ref a , Eα (u, v) = Eα (u, v) for u, v ∈ F, and

F = F 0 ⊕ Hα , where Hα is Eα0,ref -closure of the linear span of {u(i) α ; i ≥ 1} and the above is an Eα0,ref -orthogonal decomposition. (v) Fe ⊂ (F 0 )ref , E(u, v) = E 0,ref (u, v) for u, v ∈ Fe . Fe0 and the linear span of {ϕ (i) : i ≥ 1} are subsets of Fe , which are E 0,ref -orthogonal to each other. (vi) When F consists of a finite number of points {a1 , a2 , . . . , aN }, Fe is the linear subspace of (F 0 )ref spanned by Fe0 and ϕ (i) , 1 ≤ i ≤ N. F is the linear 0 (i) subspace of (F 0 )ref a spanned by F and uα , 1 ≤ i ≤ N. (vii) Assume that the condition (7.3.4) is fulfilled. f ∈ D(A) if and only if f ∈ D(L), (7.4.9) is satisfied for the space Fˇ specified by (7.4.15), and N(f )(ai ) = 0

for every i ≥ 1.

(7.7.10)

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Proof. The uniqueness follows from (ii) and (iii). To see this, suppose two functions on F satisfy equation (7.7.8) and denote by ψ their difference. Then C(ψ, η) + Uα (ψ, η) = 0 for any η ∈ B0 (F), which implies C1 (ψ, ψ) = 0 and ψ = 0 by virtue of (7.7.7) and Lemma 7.2.3. Therefore, the solution of (7.7.8) is unique and so is the resolvent {Gα ; α > 0} of X in view of (7.7.9). The assertions (i), (ii), and (iii) can be deduced from Theorem 7.4.5 with the help of a quasi-homeomorphism j that has been used in the proof of Theorem 7.5.4. In fact, since X is an m-symmetric right process on the locally compact separable metric space E, we can start with the setting of the first two paragraphs in the proof of Theorem 7.5.4 using the same notations appearing there. Since X extends X 0 satisfying (M.1), each set {ai } must be located outside the properly exceptional (7.7.3) and (7.7.4) to set N for X. Applying the Borel standard process X E\N extending X 0 E0 \N , we conclude that each ai is not m-polar for X and X E\N admits no jump from E0 \ N to F. E\ N is non- m-polar for the Hunt process X = each ai := jai ∈ Therefore, a1 , a2 , . . . }. Evidently m(F ) = 0. Moreover, F X E on E = E \ N . Let F = { satisfies the conditions (F.1), (F.2), and (F.3) of Section 7.4 with respect to the F), because these properties are invariant under the quasiDirichlet form (E, homeomorphism j and the set F trivially satisfies these conditions with respect X on E \ F , which is an m-symmetric to (E, F). Let X 0 be the part process of X 0 admitting standard process satisfying (M.1). Clearly X is an extension of ai is regular for itself with no jumps from E \ F . Since (E, F) is regular, each F), X , (E, respect to X and X is irreducible in view of Lemma 7.7.2. Now 0 F and X satisfy all the conditions imposed in Section 7.4, and accordingly Theorem 7.4.5 holds true for them. X 0 will be designated All quantities defined by (7.7.5) and (7.7.6) for X and by the superscript. Then by Theorem 7.4.5(ii), , F = the 1 . B0 ( F) ⊂ F C1 -closure of B0 ( F ) in G (7.7.11) F F The restriction to F of the quasi-homeomorphism j is denoted by ˇj. By noting (i) ∗ (i) f dm = that j∗ g = G0α j∗ g, j∗ u(i) ϕ = ϕ (i) , and the relation E j∗ G0α α = uα , j

ij ij ij ij i i , ψ ) m, we have U = U , Uα = Uα , V = V so that C1 (ψ, ψ) = C1 (ψ E f d ∗ ∗ ∗ 1 . Since F = ˇj (F ), (7.7.7) follows from and G1 = ˇj G for ψ = ˇj ψ F F (7.7.11). E; m), the function We see also from Theorem 7.4.5(i) that for g ∈ L2 ( = g F satisfies ψ Gα ! ( , 1ai ) + αij ψ U u(i) g) i ≥ 1. (7.7.12) aj ) = ( C(ψ m, α , j≥1

for ψ = Gα g g, Gα g = j∗ g so that ψ = ˇj∗ ψ Gα For g ∈ L (E; m) with g = j F = and ψ Gα gF . By noting that (u(i) u(i) g) m , we are led from (7.7.12) α , g) = ( α , to (7.7.8). 2

∗

373


X , so is ai = j−1 ai with respect As each ai is regular for itself with respect to to X E\N , and accordingly with respect to X. Hence the identity (7.7.9) is valid because X is an extension of X 0 . (iv), (v), (vi), and (vii) also follow from Theorem 7.4.5(v), (vi), (vii), and (viii), respectively, by using the quasi-homeomorphism map j. Many-point extension X of an m-symmetric Hunt process X 0 can be constructed under some of the following additional conditions for X 0 : (M.2)

E

u(i) α (x)m(dx) < ∞ for every i ≥ 1 and P0x (ζ 0 < ∞, Xζ00 − ∈ F ∪ {∂}) = P0x (ζ 0 < ∞),

x ∈ E0 .

(7.7.13)

(M.2) P0x (Xζ00 − ∈ F ∪ {∂}) = 1 for every x ∈ E0 (regardless the length of the lifetime ζ 0 ∈ (0, ∞]). (M.3) For every i ≥ 1, there exists a neighborhood of Ui of ai such that infx∈V G01 ϕ (i) (x) > 0 for any compact set V ⊂ Ui ∩ E0 . (M.4) For each i ≥ 1 and every open neighborhood U1 of ai in E, there exists an open neighborhood U2 of ai in E with U 2 ⊂ U1 such that J0 (E0 ∩ U2 \ {ai }, E0 \ U1 ) < ∞. Here J0 denotes the jumping measure of X 0 . We do not need condition (M.4) when X 0 is a diffusion. T HEOREM 7.7.4. Let X 0 be an m-symmetric Hunt process on E0 satisfying conditions (M.1) and (M.3) as well as (M.4) in a non-diffusion case. We also assume that either (M.2) or (M.2) is satisfied by X 0 . Then there exists an extension X of X 0 from E0 to E. X admits no jump to or from the set F. When X 0 is a diffusion, so is X. Proof. For n ≥ 1, we let En = E0 ∪ {a1 , a2 , . . . , an } and F (n) = E \ En = F \ {a1 , a2 , . . . , an }. First we can construct a one-point extension of X 0 from E0 to E1 . Indeed, under the stated conditions on X 0 , the corresponding conditions imposed in Theorem 7.5.6 are satisfied for {a1 } and F (1) ∪ {∂} in place of {a} and ∂, respectively. Therefore, by defining the spaces of excursion paths with this replacement, the proof of Theorem 7.5.6 works to produce a one-point extension X 1 = (Xt1 , P1x , ζ 1 ) of X 0 from E0 to E1 such that P1x (ζ 1 < ∞, Xζ11 − ∈ F (1) ∪ {∂}) = P1x (ζ 1 < ∞),

x ∈ E1 .

(7.7.14)

Lemma 7.5.5(iii) then guarantees the integrability E u(1) α (x)m(dx) < ∞. In the is ensured for every i ≥ 1. same way, the integrability of u(i) α

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Next we verify that the process X 1 on E1 satisfies the conditions (A.1), (A.2), (A.3), and (A.4) relative to the point a2 . Define ϕ 12 (x) = P1x (ζ 1 < ∞, Xζ11 − = a2 ),

x ∈ E2 .

Since the part process of X 1 on E0 equals X 0 , we have ϕ 12 (x) ≥ ϕ (2) (x) > 0 for x ∈ E0 . We also have ϕ 12 (a1 ) > 0.

(7.7.15)

To see this, we use the same notations νt , W, W + , W − , n as in the proof of Theorem 7.5.6 for {a1 } and F (1) ∪ {∂} in place of {a} and ∂. By (7.5.14), νt (dx)ϕ (2) (x), n− {w ∈ W − : t < ζ , wζ − = a2 } = E0

which is strictly positive for some t > 0 in view of (7.5.13) ( with ϕ (1) in place of ϕ) and the assumption (M.1). By letting t ↓ 0, we get n− {wζ − = a2 } > 0. This together with (7.5.16) leads us to (7.7.15). Condition (A.1) is verified. The integrability condition in (A.2) is verifiable because for x ∈ E0 , 1 E1x e−αζ ; Xζ11 − = a2 1 1 = E1x e−αζ ; Xζ11 − = a2 , ζ 1 < σa1 + E1x e−αζ ; Xζ11 − = a2 , ζ 1 > σa1 1 −ασa1 (1) ≤ u(2) = u(2) α (x) + Ex e α (x) + uα (x). The second condition (7.5.11) of (A.2) is also met in view of (7.7.14). Since the resolvent of X 1 dominates that of X 0 and ϕ 12 ≥ ϕ (2) , the condition (A.3) for X 1 and a2 follows from (M.1), (M.3). As X 1 admits no jump from E0 to a1 , the condition (A.4) for X 1 and a2 follows from (M.4). We can now apply Theorem 7.5.6 to X 1 and a2 in constructing a one-point extension X 2 of X 1 from E1 to E2 . In the same way, we can construct a onepoint extension X 3 of X 2 from E2 to E3 . For instance, the integrability condition in (A.2) for X 2 = (Xt2 , P2x , ζ 2 ) relative to a3 can be verified as for x ∈ E0 , 2 2 E2x e−αζ ; Xζ2− = a3 = E2x e−αζ ; Xζ2− = a3 , ζ 2 < σa1 ∧ σa2 2 +E2x e−αζ ; Xζ2− = a3 , ζ 2 > σa1 ∧ σa2 −ασa ∧σ a2 1 1 ≤ u(3) α (x) + Ex e (1) (2) ≤ u(3) α (x) + uα (x) + uα (x).

By continuing this procedure, we get a sequence of symmetric right processes X n = (Xtn , Pnx , ζ n ) on En such that X n is a one-point extension of X n−1


375

from En−1 to En , n ≥ 1. Each X n admits no jump from or to {a1 , a2 , . . . , an }. We can define the sequence on a common probability space. Now define Xt := Xtn if t < ζ n and Xt := ∂ for t ≥ ζ := limn→∞ ζ n . Note that for every j k ≥ j and x ∈ Ej , Pkx = Px on Fζ j . Therefore, for every x ∈ E, there is a unique probability measure Px on F∞ so that Px = Pnx on Fζ n for every n ≥ j, where x ∈ Ej . It is easy to see that {Xt , Px , x ∈ E} is an extension of X 0 from E0 to E that admits no jumps from E to F or from F to E. To formulate a counterpart of Theorem 7.5.9, let E be a locally compact separable metric space and m be an everywhere dense positive Radon measure E. We consider a closed subset K of E such that either

(K.1) K = ∪i Ki , where {Ki } are finite or countably infinite disjoint compact sets which are locally finite in the sense that any compact set intersects only with finite many of Ki ’s; or (K.2) K = K1 ∪ · · · ∪ KN , where {Ki }1≤i≤N , are disjoint, K1 , . . . , KN−1 are compact and E \ KN is relatively compact.

( We put E0 = E \ K, F ∗ := i {a∗i } and let E∗ := E0 ∪ F ∗ be the topological Hausdorff space obtained by adding to E0 extra points {a∗1 , a∗2 , . . . }, whose topology is prescribed as follows: for each i ∈ , a subset U of E∗ containing ⊂ E containing the point a∗i is a neighborhood of a∗i if there is an open set U ∗ ∗ Ki such that U ∩ E0 = U \ {ai }. In other words, E is obtained from E by identifying each closed set Ki with the point {a∗i } for every i ∈ . We denote by m0 the restriction of the measure m on E to E0 . The measure m0 is then extended to E∗ by setting m0 (F ∗ ) = 0. In the remainder of this section, (f , g) will denote the integral of f · g on E0 against the measure m0 . Let X = (Xt , Px , ζ ) be an m-symmetric Hunt process on E whose Dirichlet form (E, F) on L2 (E; m) is regular. Let X 0 = (Xt0 , P0x , ζ 0 ) be the part process of X on E0 , which is an m0 -symmetric Hunt process on E0 (cf. Exercise 3.3.7). / E0 }. The Denote by τ0 the lifetime of X 0 ; that is, τ0 = inf{t > 0 : Xt ∈ resolvent of X 0 is denoted by {G0α ; α > 0}. We aim at constructing a q.e. manypoint extension of X 0 from E0 to E∗ , which can be intuitively viewed as the process obtained from X by collapsing holes Ki ’s into a∗i . We impose the following conditions on X: (C.1) X is irreducible. (C.2) m0 (U ∩ E0 ) is finite for some neighborhood U of K. (C.3) X admits no killings inside or jumps from E \ Ki to Ki for each i ∈ .

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We define, for α > 0 and i ∈ , the functions ϕ (i) , u(i) α on E0 by ϕ (i) (x) = P0x ζ 0 < ∞ and Xζ00 − ∈ Ki , 0 −αζ 0 ; Xζ00 − ∈ Ki . u(i) α (x) = Ex e Due to the condition (C.3), these two functions can also be expressed as ϕ (i) (x) = Px σK < ∞, XσK ∈ Ki and −ασK ; XσK ∈ Ki . u(i) α (x) = Ex e Let us consider the following conditions on X 0 : (M◦ .1) For each i ∈ , ϕ (i) (x) > 0 for every x ∈ E0 . (M◦ .3) For each i ∈ , there is some neighborhood Ui of Ki such that supx∈V G01 ϕ (i) (x) < ∞ for every compact set V ⊂ Ui ∩ E0 . ◦ (M .4) Either E \ U is compact for any neighborhood U of K in E, or, for any open neighborhood U1 of K in E, there exists an open neighborhood U2 of K in E with U 2 ⊂ U1 such that J0 (U2 \ K, E0 \ U1 ) < ∞, where J0 denotes the jumping measures of X 0 . These three conditions are the counterparts of (7.7.1), (M.3), and (M.4), respectively, with ai being replaced by Ki . A right process X ∗ on E∗ = E0 ∪ F ∗ is called a q.e. extension of X 0 if X ∗ is m0 -symmetric, having no killings on F admitting no jumps from F ∗ to F ∗ , and the part process of X ∗ on E0 coincides with X 0 q.e., namely, outside some m0 -polar set for X 0 . T HEOREM 7.7.5. Assume that X satisfies the conditions (C.1), (C.2) and (C.3). Assume further that X 0 satisfies conditions (M◦ .1) and (M◦ .3) as well as condition (M◦ .4) when X 0 is not a diffusion. Then there exists an m0 -symmetric Borel right process X ∗ on E∗ which is a q.e. extension of X 0 on E0 . Such extension is unique in law. The process X ∗ admits no jumps to or from the set F ∗ . If X 0 is a diffusion, then so is X ∗ . If X satisfies the condition (AC) of Definition A.2.16 additionally, “q.e.” can be removed from the above conclusion. Proof. We shall give the proof only for the case where K is of the form (K.1) because the second case (K.2) can be treated in a similar way. It follows from (C.3) that X 0 admits no killing inside E0 , which together with (M◦ .1) means that X 0 satisfies (M.1) with a∗i in place of ai . The properties


377

(M.3), (M.4) for X 0 with a∗i in place of ai follow from (M◦ .3), (M◦ .4), respectively. Furthermore, exactly in the same way as the proof of Theorem 7.5.9, we can obtain the following property of X 0 : (M◦ .2) P0x (Xζ 0 − ∈ K ∪ {∂}) = 1 for q.e. x ∈ E0 . ( This implies that X 0 also satisfies the condition (M.2) with F ∗ = i {a∗i } in place of F holding for q.e. x ∈ E0 . If, in addition, X satisfies the condition (AC), then so does X 0 and we can replace “for q.e.” by “for every” in the above assertion. We can now deduce the desired conclusion from Theorem 7.7.4. As an extension of X 0 from E0 to E∗ in the sense of Definition 7.7.1, X ∗ is unique in law by virtue of Theorem 7.7.3. We call the above procedure of obtaining X ∗ from X (or from X 0 ) darning each hole Ki into one point a∗i , i ≥ 1. X ∗ enjoys those properties listed in Theorem 7.7.3 but with {a∗1 , a∗2 , · · · } in place of {a1 , a2 , · · · }. 7.8. EXAMPLES OF MANY-POINT EXTENSIONS We present several concrete examples of many-point extensions. (1◦ ) Darning multiple holes for multidimensional Brownian motion We(consider the Brownian motion X = (Xt , Px ) on Rn with n ≥ 2. Let K = i≥1 Ki where {Ki } is a collection of finite or countably infinite number of disjoint non-polar compact subsets of Rn , which is locally finite. Let X 0 = (Xt0 , P0x , ζ 0 ) be the absorbing Brownian motion on E0 = Rn \ K. Conditions (C.1), (C.2), (C.3) are trivially satisfied by X. By Theorem ( 3.3.8, each set Ki is non-polar relative to the part process X 0i of X on Rn \ j=i Kj . As n ≥ 2, X 0i is irreducible and it has a transition density with respect to the Lebesgue measure. Therefore, the condition (M◦ .1) is fulfilled by X 0 . The condition (M◦ .3) for X 0 can be verified just as in Example (4◦ ) of Section 7.6. Thus we can use Theorem 7.7.5 to obtain a diffusion extension X ∗ of X 0 from E0 to E∗ = E0 ∪ F ∗ with F ∗ = ∪i≥1 {a∗i } by darning each hole Ki into a point a∗i . Assume that n = 2 and consider the case that the number of holes Ki is finite: K = ∪Ni=1 Ki for disjoint non-polar compact subsets {K1 , K2 , . . . , KN } of R2 . X ∗ then enjoys the conformal invariance property analogous to those in Theorem 7.6.3 and Remark 7.6.4(ii), as will be formulated below. Ki , where { T HEOREM 7.8.1. Let K = ∪Ni=1 Ki ; 1 ≤ i ≤ N} is a second set of disjoint non-polar compact subsets of R2 . Suppose that φ is a conformal map K that maps ∞ to ∞ and, for each i ≥ 1, φ maps from R2 \ K onto R2 \ K -portion of a the R2 \ K-portion of any neighborhood of Ki into the R2 \ Ki with a single neighborhood of Ki , and vice versa. Identify the compact set

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point a∗i and equip E∗ := (R2 \ K ) ∪ { a∗1 , a∗2 , . . . , a∗N } the topology induced 2 a∗i . Define φ(a∗i ) = a∗i , 1 ≤ from R by collapsing each set Ki into one point ∗ ∗ i ≤ N. Then φ is a topological homeomorphism from E onto E . Moreover, φ(X ∗ ) is, up to a time change, the diffusion process obtained from a Brownian Ki∗ into a∗i with 1 ≤ i ≤ N. motion on R2 by darning each hole Proof. Let m0 be the Lebesgue measure on E0 = R2 \ K extended to E∗ by E∗ is denoted setting m0 ({a∗i }) = 0, 1 ≤ i ≤ N. The corresponding quantity on ∗ ∗ ∗ by m0 . X = (Xt , Pz ) is an extension of the absorbing Brownian motion in E0 to E∗ and m0 -symmetric. By Theorem 7.7.3(vi), the extended Dirichlet space (Fe∗ , E ∗ ) of X ∗ = (Xt∗ , P∗z ) is given by

Fe∗ = {f +

N (i) i=1 ci ϕ

E ∗ (u, v) = 12 DE0 (u, v),

1 : f ∈ H0,e (E0 ), ci ∈ R} ⊂ BL(E0 ),

u, v ∈ Fe∗ ,

where ϕ (i) (x) := Px (XσK − ∈ Ki ) for x ∈ R2 \ K. By the recurrence of X, we N (i) have i=1 ϕ (x) = Px (σK < ∞) = 1 for every x ∈ E0 . This in particular implies that 1 ∈ Fe∗ with E ∗ (1, 1) = 0 and therefore, by Theorem 2.1.8, X ∗ is recurrent. The process X ∗ is irreducible by Theorem 7.7.3(i). Its resolvent kernel is absolutely continuous with respect to m0 on account of Theorem 7.7.3(iii) and so is its transition function by virtue of Corollary 3.1.14. We can now proceed exactly along the same line as in the proof of Theorem 7.6.3. The PCAF At of X ∗ defined by (7.6.27) is strictly increasing to ∞ as t → ∞ P∗z -a.s. for any z ∈ E∗ . Let Xˇ ∗ be the time change of X ∗ by means of Y ∗ be the image of Xˇ ∗ by φ defined as (7.6.28). Since φ the inverse of At and E∗ , Y ∗ is a conservative (in fact, is a topological homeomorphism from E∗ to ∗ m0 -symmetric. Furthermore, as in the recurrent) diffusion process on E . It is last paragraph in the proof of Theorem 7.6.3, we see by using Theorem 5.3.1 E0 is an absorbing Brownian motion in E0 . Thus that the part process of Y ∗ in ∗ E0 to E∗ in the sense Y is an extension of the absorbing Brownian motion in of Definition 7.7.1 and so we can apply Theorem 7.7.3 on uniqueness to obtain the desired conclusion. Remark 7.8.2. Theorem 7.8.1 remains valid in the general case where X is the absorbing Brownian motion on a domain D ⊂ Rn and {Ki } is a collection of countably infinite number of disjoint non-polar compact subsets of D, which is locally finite. In this general case, the PCAF At of X ∗ defined by (7.6.27) is strictly increasing in t ∈ [0, ζ ∗ ) P∗z -a.s. for any z ∈ E∗ , where ζ ∗ is the lifetime Y ∗ defined by (7.6.28) is an m0 -symmetric diffusion of X ∗ . The image process ∗ ai for every i ≥ 1. process on E with lifetime Aζ ∗ admitting no killing at E0 in the Therefore, Y ∗ is an extension of the absorbing Brownian motion on sense of Definition 7.7.1 and the uniqueness theorem applies.


379

(2◦ ) Darning multiple holes for a reflecting stable process Suppose that D is an open n-set in Rn and m denotes the Lebesgue measure on D. Let X = (Xt , Px , ζ ) be the reflecting α-stable process ( on D considered in Example (5◦ ) of Section 7.6. But this time we let K = i∈ Ki be the union of a finite or countably infinite number of disjoint non-trivial compact subsets of ∂D which are locally finite. In view of the properties of X explained in Example (5◦ ) of Section 7.6, we readily see that conditions (C.1)–(C.3) are satisfied by X for E = D and K = ∪i∈ Ki . Suppose each compact set Ki ⊂ ∂D has finite and strictly positive di -dimensional Hausdorff measure when n ≥ 2 and is non-empty when n = 1. Let σK := inf{t ≥ 0 : Xt ∈ K} ∧ ζ. It follows from [14, Theorem 2.5 and Remark 2.2(i)] that ϕ (i) (x) := Px (σK < ∞ and XσK − ∈ Ki ) > 0

(7.8.1)

for every x ∈ D \ K if and only if α > n − di when n ≥ 2 and α > 1 when n = 1. Now we assume that α > n − di for each i ≥ 1 when n ≥ 2 and α > 1 when n = 1. Let X 0 = (Xt0 , P0x , ζ 0 ) of X killed upon hitting K. X 0 then satisfies the condition (M◦ .1). It is easy to see that X 0 has a symmetric transition density function p0 (t, x, y), which can be represented using the Hölder continuous transition density p(t, x, y) of X as p0 (t, x, y) = p(t, x, y) − Ex p t − τD , XτD , y ; τD < t for t > 0 and x, y ∈ D \ K. Moreover, the density function p0 (t, x, y) is continuous on (0, ∞) × (D \ K) × (D \ K). Thus X 0 satisfies also the condition (M◦ .3). The property (M◦ .4) of X 0 can be readily verified as in Example (5◦ ) of Section 7.6. Hence we can apply Theorem 7.7.5 to get the symmetric extension X ∗ of X 0 by darning the holes {Ki , i ∈ } into single points {a∗i , i ∈ }. In the above two examples, the uniqueness of the extension X ∗ , the description of its Dirichlet form, and the characterization of its L2 -generator by the zero flux condition can be derived from Theorem 7.7.3 by replacing points ai with a∗i . We say that a symmetric right process X admits a reflecting extension if F Faref , Faref

(7.8.2)

is its active reflected Dirichlet where F is the Dirichlet space of X and space. If X admits a reflecting extension, then X is non-conservative by

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Corollary 6.6.6. In the following two examples, we consider such a situation together with all possible symmetric extensions with no killing or jump on the boundary. (3◦ ) One-dimensional minimal diffusion Let I = (r1 , r2 ) be a one-dimensional interval, X 0 be a minimal diffusion on I with no killing inside I in the sense that it satisfies the properties (d.1), (d.2), (d.3), and (d.4) of Example 3.5.7, and (s, m) be a pair of a canonical scale and a canonical measure corresponding to X 0 . X 0 is then m-symmetric and, by Theorem 3.5.8, the Dirichlet form (E 0 , F 0 ) of X 0 on L2 (I; m) is given by F 0 = {u ∈ F (s) ∩ L2 (I; m) : u(ri ) = 0 if ri is regular} E 0 = E (s) 0 0 . F ×F

In view of (6.5.8), (s) ∩ L2 (I; m), (F 0 )ref a =F

E 0,ref = E (s) ,

and consequently X 0 admits a reflecting extension if and only if either r1 or r2 is regular. In this case, let I ∗ be the interval obtained from I by adding the boundary ri whenever ri is regular. Then (E, F) = (E (s) , F (s) ∩ L2 (I; m)) is a regular strongly local Dirichlet form on L2 (I ∗ ; 1I · m) and the associated diffusion process X on I ∗ is by definition the reflecting diffusion (Example 3.5.7). X is a symmetric extension of X 0 and actually a reflecting extension in the sense of Definition 7.2.6. When both r1 and r2 are regular, then the reflecting extension X is a symmetric two-point extension of X 0 from I to I ∗ = [r1 , r2 ]. X 0 admits three other kinds of symmetric (one-point) extensions: the extension to [r1 , r2 ) reflecting only at r1 , the extension to (r1 , r2 ] reflecting only at r2 , and an extension X˙ to the one-point compactification I˙ = I ∪ {∂} of I. Extend m to ˙ F) ˙ of X˙ on L2 (I; ˙ m) ˙ m˙ on I˙ by setting m({∂}) ˙ = 0. Then the Dirichlet form (E, is regular and it can be described as E˙ = E (s)

and

F˙ = {u ∈ F (s) : u(r1 ) = u(r2 )}.

Without using Dirichlet forms, these three one-point extensions of X 0 can be constructed by means of Theorem 7.5.6, while the two-point extension to [r1 , r2 ] can be done by means of Theorem 7.7.4. (4◦ ) Time-changed transient reflecting Brownian motion Let D be a domain of Rn with a continuous boundary and Z = (Zt , Qx ) be the reflecting Brownian motion on D. The Dirichlet form of Z on L2 (D) is ( 21 D, H 1 (D)). In what follows, we assume the transience of Z so that n must be


381

more than 3 and D must be unbounded. If further D contains an infinite cone, then Z is transient as has been seen in Example 3.5.9. More specifically, we assume that D is an unbounded Lipschitz domain in Rn with n ≥ 3 in the following sense: there are constants δ > 0, M > 0 and a locally finite open covering {Uj }j∈J of ∂D such that, for each j ∈ J, the set D ∩ Uj is expressed as (2.2.55) in some coordinate system by some Lipschitz ( continuous function Fj with Lipschitz constant bounded by M, and ∂D ⊂ j∈J {x ∈ Uj : dist(x, ∂Uj ) > δ}. By [76], there exists then a reflecting Brownian motion (RBM) Z = (Zt , Qx ) on D whose resolvent {GZα ; α > 0} has the strong Feller property in the sense that GZα (bL1 (D)) ⊂ bC(D).

(7.8.3)

In particular, Z satisfies the absolute continuity condition (AC) in Definition A.2.16. Such a process Z had been constructed in [5] for a bounded Lipschitz domain. The process Z is always conservative but we see from Theorem 3.5.2 that Z escapes to infinity as t → ∞: (7.8.4) Qx lim Zt = ∂ = 1 for every x ∈ D, t→∞

where ∂ is the point at infinity of D. Any function in the space BL(D) is represented by its quasi continuous version. By virtue of Corollary 3.5.3, any u ∈ He1 (D) then satisfies Qx lim u(Zt ) = 0 = 1 for every x ∈ D. (7.8.5) t→∞

The above two statements hold “for q.e. x ∈ D”, which are now strengthened to “for every x ∈ D” owing to the property (AC) of Z. By Section 6.5(5◦ ), the reflected Dirichlet space of the Dirichlet form of 1 ( 2 D, H 1 (D)) is identical with the space ( 21 D, BL(D)) of BL functions. The extended Sobolev space He1 (D) is a subspace of BL(D), and indeed a proper subspace because the former does not contain a non-zero constant function due to the transience assumption while the latter does. Let H (D) be the space of functions in BL(D) which are D-orthogonal to He1 (D). Then H (D) ⊂ H(D), where H(D) denotes the space of harmonic functions on D with finite Dirichlet integral. We will be concerned with the condition that H (D) consists of constant functions on D.

(7.8.6)

Condition (7.8.6) holds true when D = R , n ≥ 3, in view of Theorem 2.2.12. This property remains valid for any unbounded uniform domain. A domain D is called a uniform domain if there exists C > 1 such that for every x, y ∈ D, there is a rectifiable curve γ in D connecting x and y with length(γ ) ≤ C|x − y| n

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and moreover min{|x − z|, |z − y|} ≤ C dist(z, Dc )

for every z ∈ γ .

An infinite cone is a special unbounded uniform domain. We refer the readers to [24, Proposition 3.6] for a proof of the following implication: D \ Br (0) is a unbounded uniform domain =⇒

(7.8.6) holds.

(7.8.7)

Here Br (0) denotes the open ball with center 0 and radius r > 0. Since the active reflected Dirichlet space of Z coincides with BL(D) ∩ L2 (D) = H 1 (D), condition (7.8.2) fails and the reflecting Brownian motion Z does not admit a reflecting extension. But if we make a time change of Z, then the situation may change radically. Let m be a positive Radon measure on D charging no polar set possessing full quasi support with respect to the Dirichlet form ( 12 D, H 1 (D)). For instance, 1 (D) has these properties. Let m(dx) = f (x)dx for a strictly positive f ∈ Lloc X = (Xt , ζ , Px ) be the time-changed process of the reflecting Brownian motion Z on D by means of its PCAF A with Revuz measure m. X is then m-symmetric, and the Dirichlet form of X on L2 (D; m) and its active reflected Dirichlet space are given by 1 2

D, He1 (D) ∩ L2 (D; m) ,

1 2

D, BL(D) ∩ L2 (D; m) ,

(7.8.8)

respectively, as was seen in Section 6.5(5◦ ). We see that X admits a reflecting extension if m(D) < ∞, because then / the two Dirichlet forms in (7.8.8) obviously differ. If m(D) = ∞, then 1 ∈ L2 (D; m) and they coincide under (7.8.6). Thus P ROPOSITION 7.8.3. Assume that a domain satisfies condition (7.8.6). Then X admits a reflecting extension if and only if m(D) < ∞. We now take a strictly

t positive continuous and integrable function f on D and let dm = fdx. At = 0 f (Zs )ds is the associated PCAF of Z in the strict sense with full support D. The time-changed process X = (Xt , Px ) with Xt = Zτt , τ = A−1 , ζ = A∞ is a diffusion process on D whose 0-order resolvent is given by GX0+ u(x) = GZ0+ (uf )(x),

x ∈ D,

u ∈ B+ (D).

This in particular implies that X also satisfies the condition (AC) and GX0+ u is lower semicontinuous for any u ∈ B+ (D) in view of (7.8.3).

383


As was observed in Section 6.5(5◦ ), the finiteness of m(D) implies that ζ is finite a.s. and so we get from (7.8.4) and the above observation Px (ζ < ∞, Xζ − = ∂) = 1 for every x ∈ D.

(7.8.9)

Therefore, the conditions (A.1), (A.2), (A.3) of Section 7.5 with X and ∂ in place of X 0 and a, respectively, are fulfilled so that X admits a unique onepoint symmetric extension X to the one-point compactification D ∪ {∂} of D by Theorems 7.5.4 and 7.5.6. By Theorem 7.5.4, its extended Dirichlet space (Fe , E ) can be described as 1 D. 2 In particular, X is recurrent. When condition (7.8.6) is satisfied, this extension is a reflecting extension of X. The special case that D = Rn , n ≥ 3, was considered in Example (4◦ ) of Section 7.6. If an unbounded domain D is not a uniform domain, the dimension of the space H∗ (D) may exceed 2 as we shall see in the following example. Let Fe = He1 (D) ⊕ {c : c ∈ R} ⊂ BL(D),

D = B1 (0) ∪ x = (x1 , x2 , . . . , xn ) :

xd2

>

n−1 !

E =

) xk2

, n ≥ 3.

(7.8.10)

k=1

D contains the upper cone C+ and lower cone C− where   & n−1 '1/2  & n−1 '1/2      ! ! , C− = B1 (0)c ∩ xn < − xk2 xk2 C+ = B1 (0)c ∩ xn >     k=1

k=1

so that D is transient as it contains an infinite cone C+ but D cannot be a uniform domain because it has a bottleneck B1 (0). Note that D is a Lipschitz domain. The point at infinity of D at the upper end (lower end) is denoted by ∂+ (∂− ). Let Z = (Zt , Qx ) be the RBM on D and ϕ + (x) = Qx lim Zt = ∂+ , ϕ − (x) = Qx lim Zt = ∂− . t→∞

t→∞

We then see from (7.8.4) that ϕ + (x) + ϕ − (x) = 1

for every x ∈ D.

(7.8.11)

Furthermore, ϕ + (x) > 0, +

ϕ − (x) > 0

for every x ∈ D,

(7.8.12)

because ϕ (x) is either identically positive or identically zero due to the irreducibility, so the above follows from the symmetry of the domain D.

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CHAPTER SEVEN

Choose again a strictly positive integrable function f on D and let dm = fdx. Consider the time-changed diffusion process X = (Xt , Px , ζ ) on D obtained t from Z by the time change with respect to the PCAF At = 0 f (Zs )ds. X is m-symmetric and has the property (7.8.9). In particular, it holds for x ∈ D that ϕ + (x) = Px (ζ < ∞, Xζ − = ∂+ ), ϕ − (x) = Px (ζ < ∞, Xζ − = ∂− ). (7.8.13) We put for α > 0, x ∈ D −αζ −αζ u+ ; Xζ − = ∂+ , u− ; Xζ − = ∂− . α (x) = Ex e α (x) = Ex e

P ROPOSITION 7.8.4. (i) There exists a unique m-symmetric two-point extension X of X from D to D ∪ {∂+ } ∪ {∂− }. X is recurrent. Denote by (E , F ) and Fe the Dirichlet form of X on L2 (D ∪ {∂+ } ∪ {∂− }; m)(= L2 (D; m)) and its extended Dirichlet space, respectively. Then − 2 F = H 1 (D) ⊕ {c+ u+ α + c− uα : c+ , c− ∈ R} ⊂ BL(D) ∩ L (D; m),

Fe = He1 (D) ⊕ {c+ ϕ + +c− ϕ − : c+ , c− ∈ R} ⊂ BL(D), 1 D(u, v), u, v ∈ Fe . 2 (ii) For any u ∈ BL(D), there exist constants c+ , c− such that E (u, v) =

 Qx (Z∞− = ∂+ , limt→∞ u(Zt ) = c+ ) = Qx (Z∞− = ∂+ ), Q (Z x ∞− = ∂− , limt→∞ u(Zt ) = c− ) = Qx (Z∞− = ∂− ).

(7.8.14) (7.8.15)

(7.8.16)

If c+ = c− = 0, then u ∈ H1e (D). Proof. (i) We check conditions (M.1), (M.2), (M.3) for X on D and ∂+ , ∂− in place of X 0 on E0 and a1 , a2 , respectively. (M.1) is ensured by (7.8.12) and (7.8.13). (M.2) is trivially true because m(D) < ∞. (M.3) follows from the lower semicontinuity of GX0+ u for u ∈ B+ (D) proved in the paragraph below Proposition 7.8.3 and an obvious counterpart of Lemma 7.5.5(iv). Therefore, X admits a two-point extension X by virtue of Theorem 7.7.4. By Theorem 7.7.3, we get the uniqueness of X and the properties (7.8.14) and (7.8.15) of its Dirichlet form. Properties (7.8.11) and (7.8.14) imply the recurrence of X because 1 ∈ Fe and E (1, 1) = 0. (ii) Define, for n ≥ 1, Dn = B1 (0) ∪ C+ ∪ (C− ∩ Bn (0)),

n = C− ∩ {|x| = n}.

Then Dn is a uniform Lipschitz domain and Dn increases to D as n → ∞.

385


For any u ∈ BL(D), uDn ∈ BL(Dn ) which is a sum of a function in He1 (Dn ) and some constant c+ in view of (7.8.7). Let Z n = (Ztn , Qnx ) be the reflecting Brownian motion on Dn , namely, a diffusion associated with the Dirichlet form ( 21 D, H 1 (Dn )) on L2 (Dn ). On account of (7.8.4) and (7.8.5), we have n Qnx lim u(Ztn ) = c+ , Z∞− = ∂+ = 1, t→∞

x ∈ Dn ,

and we see that c+ is independent of n. Since the part processes of Z and Z n on Dn \ n are identical in law, we further obtain Qx Z∞− = ∂+ , lim u(Zt ) = c+

t→∞

= lim Qx σn = ∞, Z∞− = ∂+ , lim u(Zt ) = c+ n→∞

t→∞

n = ∂+ , lim u(Ztn ) = c+ = lim Qnx σn = ∞, Z∞− n→∞

=

t→∞

lim Qn (σn n→∞ x

= ∞) = lim Qx (σn = ∞) = Qx (Z∞− = ∂+ ), n→∞

completing the proof of the first identity of (7.8.16). The second one can be shown similarly. If c+ = c− = 0 for u ∈ BL(D), then the restriction of u to B1 (0) ∪ C+ belongs to the space He1 (B1 (0) ∪ C+ ) in view of the above proof. Similarly, the restriction of u to B1 (0) ∪ C− is an element of He1 (B1 (0) ∪ C− ). Hence u ∈ He1 (D).

P ROPOSITION 7.8.5. It holds that H (D) = {c+ ϕ + + c− ϕ − : c+ , c− ∈ R} and the extension X of Proposition 7.8.4 is a reflecting extension of the timechanged RBM X on D in the sense of Definition 7.2.6. Proof. By (7.8.14), ϕ+ , ϕ− ∈ BL(D) and so each of them has an associated pair of constants (c+ , c− ) according to (ii) of Proposition 7.8.4. Let τn be the exit time of Z from the set D ∩ Bn (0), n ≥ 1. Then {ϕ+ (Zτn )} is a bounded Qx -martingale and possesses an a.s. limit with ϕ+ (x) = EQx []. Then 1{Z∞− =∂+ } = c+ 1{Z∞− =∂+ } , 1{Z∞− =∂− } = c− 1{Z∞− =∂− } .

386

CHAPTER SEVEN

But c+ 1{Z∞− =∂+ } = lim ϕ+ (Zτn )1{Z∞− ◦θτn =∂+ } n→∞

= lim Qx Z∞− ◦ θτn = ∂+ , Z∞− ◦ θτn = ∂+ Fτn n→∞

= lim Qx Z∞− ◦ θτn = ∂+ Fτn = lim ϕ+ (Zτn ) = . n→∞

n→∞

By taking the Qx -expectation, we get c+ = 1. In the same way, we get c− = 0. We have shown that the associated pair with ϕ+ is (1, 0). Similarly, we see that the associated pair with ϕ− is (0, 1). Take now any u ∈ BL(D) and let (c+ , c− ) be the associated pair with u. Define u0 = u − (c+ ϕ+ + c− ϕ− ). Then, by the above observation, u0 ∈ BL(D) and the associated pair with u0 equals (0, 0). Therefore, u0 ∈ He1 (D) by virtue of the second assertion of Proposition 7.8.4. The second statement follows from the first one and (7.8.14). Just as in the preceding example of the one-dimensional diffusion with two regular boundaries, the time-changed reflecting Brownian motion X on D admits three other kinds of one-point symmetric extensions: the extension to D ∪ {∂+ } reflecting only at ∂+ , the extension to D ∪ {∂− } reflecting only at ∂− , and the extension to the one-point compactification D ∪ {∂}. The last one was constructed in the second paragraph after Proposition 7.8.3. Other two can be constructed in similar ways. (5◦ ) One-dimensional Brownian motion with countable boundary We investigate the uniqueness and existence of the extension of a onedimensional absorbing Brownian motion with a countable boundary. Let a0 = 0 and {an }n≥1 be a sequence of positive numbers strictly decreasing to 0. Set F := {an }n≥0 ,

I0 := (−∞, 0),

I1 := (a1 , ∞),

In := (an , an−1 )

for n ≥ 2 and E0 := R \ F = ∪∞ n=0 In . The Lebesgue measure on R is denoted by λ. Let X 0 be the absorbing Brownian motion on E0 , namely, the Brownian motion being killed upon hitting the set F. Since a0 = 0 is an accumulation point in F, we cannot use Theorem 7.7.3 in characterizing the extension of X 0 to R. But we can combine Theorem 7.1.8 with three other general theorems in getting its unique existence. In accordance with Example 3.5.7, a Markov process on an open interval is called a minimal diffusion with no killing inside if it satisfies the properties (d.1), (d.2), (d.3), and (d.4) stated in Example 3.5.7.


387

P ROPOSITION 7.8.6. Let X be a minimal diffusion on R with no killing inside such that X is symmetric with respect to λ and the part process of X on E0 coincides with X 0 . Then X is the Brownian motion on R. Proof. Let (s, m) be a pair of a canonical scale and a canonical measure associated with X. Then X is symmetric with respect to m so that m must be a constant time of λ by a uniqueness theorem of a symmetrizing measure due to Ying-Zhao [153] and a strong irreducibility (d.4) of X. We may take m = λ. It then suffices to show ds = 2λ.

(7.8.17)

By virtue of Theorem 3.5.8, the Dirichlet form (E, F) of X on L2 (R; λ) is identical with (3.5.9) expressed in terms of the current pair (s, λ). Especially (E, F) is a regular strongly local Dirichlet form. Accordingly, J = 0,

κ = 0,

(7.8.18)

in its Beurling-Deny representation. Furthermore, the energy measure µc u for u ∈ bF does not charge any single point owing to the energy image density property formulated in Theorem 4.3.8, and in particular µc Hu ({ai }) = 0

for every i ≥ 0 and for any u ∈ F.

(7.8.19)

0 2 Denote by (E 0 , F 0 ) and (E 0,ref , (F 0 )ref a ) the Dirichlet form of X on L (E0 ; λ) and its active reflected Dirichlet space, respectively. For a function f on R and n ≥ 0, we let fn := f |In . Using the notations in Example (1◦ ) of Section 7.6, we then have F 0 = f ∈ L2 (R; λ) : fn ∈ H01 (In ) for every n ≥ 0 , 2 1 (F 0 )ref a = f ∈ L (R; λ) : fn ∈ H (In ) for every n ≥ 0 ,

E 0,ref (f , f ) =

∞ ! 1 n=0

2

DIn (fn , fn )

for f ∈ (F 0 )ref a .

(7.8.20)

We now apply Theorem 7.1.8 to the current X 0 and X. On account of (7.8.18) and (7.8.19), we get (7.8.21) F 0 ⊂ F E0 ⊂ (F 0 )ref a , E(u, u) = E 0,ref uE0 , uE0

for u ∈ F.

(7.8.22)

Taking the expression (3.5.9) of (E, F) into account, we conclude from (7.8.21) ds = f > 0, that ds and λ = dx is mutually absolutely continuous. If we let dx

388

CHAPTER SEVEN

then we get from (7.8.22)

1 u (s(x)) f (x)dx = 2 In

u (s(x))2 f (x)2 dx

2

In

for any u ∈ C01 (s(In )) and n ≥ 0, which means f = 2 and the desired (7.8.17). Next let us take positive numbers {pn }n≥0 such that α ≤ pn ≤ β,

n = 0, 1, 2, . . .

for some positive constants α, β, and we let m(dx) :=

∞ !

pn 1In (x)dx.

(7.8.23)

n=0

The absorbing Brownian motion X 0 is symmetric with respect to the Lebesgue measure λ but it can also be viewed as an m-symmetric diffusion on E0 , 0 ) and (E0,ref , (F 0 )ref as observed in Section 7.5 already. Let (E0 , F a ) be the 0 2 m) and its active reflected Dirichlet space, Dirichlet form of X on L (E0 ; respectively. They are given by 0 = f ∈ L2 (R; m) : fn ∈ H01 (In ) for every n ≥ 0 , F 2 0 )ref m) : fn ∈ H 1 (In ) for every n ≥ 0 , (F a = f ∈ L (R; E0,ref (f , f ) =

∞ ! pn n=0

2

DIn (fn , fn )

0 )ref for f ∈ (F a .

In exactly the same way as the proof of the preceding proposition, we can prove the following: P ROPOSITION 7.8.7. Let X be a minimal diffusion on R with no killing inside such that X is symmetric with respect to m and the part process of X on E0 coincides with X 0 . Then X is the diffusion associated with the canonical scale s(dx) = 2

∞ ! n=0

and the canonical measure m.

p−1 n 1In (x)dx

(7.8.24)


389

By repeating the one-point skew extensions formulated in Theorem 7.5.9, the process X of Proposition 7.8.7 can be constructed from the Brownian motion directly as follows. Let B− and B+ be the absorbing Brownian motions on R− = (−∞, 0) and R+ = (0, ∞), respectively. Let X 01 be the subprocess of B+ on R+ \ {a1 } killed upon hitting a1 . The process X 01 is symmetric with respect to the measure m1 (dx) = p2 1(0,a1 ) (x)dx + p1 1(a1 ,∞) (x)dx, and we can apply Theorem 7.5.9 to construct a unique m1 -symmetric diffusion 01 1 X 1 on R+ extending

∞ 1 X by darning the hole a1 with entrance law µt determined by 0 µt dt = m1 . We next consider the subprocess X 02 of X 1 on R+ \ {a2 } being killed upon hitting the point a2 . X 02 is symmetric with respect to the measure m2 (dx) =

p3 1(0,a2 ) m1 (dx) + 1(a2 ,∞) m1 (dx) p2

= p3 1(0,a2 ) dx + p2 1(a2 ,a1 ) dx + p1 1(a1 ,∞) dx, and we can construct a unique m2 -symmetric diffusion X 2 on R+ extending X 02 just as above. Repeating this procedure and taking the limit as in [22, §3], we get a diffusion X + on R+ satisfying the following: X + is symmetric with respect to the measure m+ (dx) = 1R+ (x) m(dx) =

∞ !

pn 1In (x)dx

n=1

and it is actually an m+ -symmetric extension of the subprocess X 0,+ of X 0 on + E0 ∩ (0, ∞) = ∪∞ n=1 In . The process X has a finite lifetime and approaches 0 almost surely. We finally piece X + together with B− at 0 via Proposition 7.5.11 to get a desired diffusion X on R which is symmetric with respect to m(dx) = 1R− (x)dx + m+ (dx) and actually an m-symmetric extension of X 0 . Thus we may call a diffusion associated with the pair ( s, m) of (7.8.24) and (7.8.23) a skew Brownian motion on R. Even if the boundary set F has no accumulation point, the absorbing Brownian motion does not satisfy the condition (7.7.1) so that Theorem 7.7.3 is not applicable. Nevertheless, a skew Brownian motion can be characterized and constructed as above.

390

CHAPTER SEVEN

For instance, for any sequence q = {qn : n ≥ 0} of positive numbers, the diffusion process X q on R associated with the pair  ∞ !   −1 q  = 2q 1 (x)dx + 2 q−1 ds  (−∞,0) n 1(n−1,n) (x)dx, 0  n=1

∞ !   q  = q 1 (x)dx + qn 1(n−1,n) (x)dx dm  0 (−∞,0)  n=1

is a skew Brownian motion characterized and constructed in a similar manner to the above. When sq (0, ∞) is finite, +∞ is approachable and X q becomes transient. But X q is always conservative because n n k ! q q m (0, x)s (dx) ≥ mq (k − 1, x)sq (dx) = n → ∞, n → ∞, 0

k=1

k−1

namely, +∞ is non-exit. Thus X q admits no reflecting extension.

Appendix A ESSENTIALS OF MARKOV PROCESSES

A.1. MARKOV PROCESSES In this section, we introduce various concepts of Markov processes, such as the Markov process, right process, standard process, and Hunt process. A.1.1. Preliminaries In this subsection, we collect some preliminaries for Sections A.1, A.2, and A.3. Let be a set. A class M of subsets of is called a σ -field if ∈ M and it is closed under the operations of taking a countable union and complement. The couple (, M) is said to be a measurable space in this case. Given a measurable space (, M) and a subset of , the trace of M on is defined by { ∩ : ∈ M} and denoted by M| . (, M| ) is then a measurable space. For a family C of subsets of , σ (C) will denote the smallest σ -field containing C. A probability measure P on a measurable space (, M) is a map P : M → [0, 1] such that P() = 1 and P is countably additive in the sense that ∞ ∞ n = P(n ) P n=1

n=1

for n ∈ M such that j ∩ j = ∅ for i = j. In this case, the triple (, M, P) is called a probability space. Denote by P() the family of all probability measures on a measurable space (, M). For P ∈ P(), we put N P := {N ⊂ : ∃ ∈ M such that P() = 0 and N ⊂ }, MP := {N : ∈ M, N ∈ N P },

P(N) := P().

The σ -field MP so defined is called the completion of M with respect to P. N ∈ N P is called a null set in MP . The σ -field M∗ defined by M∗ = ∩P∈P() MP is called the universal completion of M. Let R+ = [0, ∞) and B be the family of Borel subsets of R+ , namely, B is the σ -field generated by all open subsets of R+ . Given a measurable

392

APPENDIX A

space (, M), we denote by B × M the product σ -field of the product space R+ × , namely, the σ -field generated by sets of the type B × with B ∈ B and ∈ M. P ROPOSITION A.1.1. Let A ∈ B × M (t, ω) ∈ A}. Then π(A) ∈ M∗ .

and

π (A) = {ω ∈ : ∃t ≥ 0,

This proposition follows from the observations that the notion of analytic sets is stable under projections ([37, III:13]) and any M-analytic subset of belongs to MP for any P ∈ P() ([37, III:33]). We next state monotone class theorems. For a set , a class C of its subsets is said to be a π-system if it is closed under finite intersections. A class D of its subsets is called a Dynkin class if (i) ∈ D, (ii) if A ⊂ B are both from D, then so is B \ A, (iii) if {Aj , j ≥ 1} is a countable disjoint sets from D, then so is ∪∞ j=1 Aj . P ROPOSITION A.1.2. Let C be a π-system on a set . (i) If D is a Dynkin class containing C, then σ (C) ⊂ D. (ii) Suppose H is a linear space consisting of bounded real functions on such that 1 ∈ H, 1A ∈ H for every A ∈ C, and H is closed under the operation of taking uniformly bounded increasing limits. Then H contains any bounded σ (C)-measurable function. Proof. (i) See [13, Chapter 0, (2.2)]. (ii) follows from (i) because D = {A ⊂ : 1A ∈ H} is a Dynkin class. Let E be a topological space. The Borel σ -field B(E) of E is by definition the σ -field generated by all open subsets of E. Denote by P(E) the family of all probability measures on the measurable space (E, B(E)). We let B∗ (E) = ∩µ∈P(E) Bµ (E) where Bµ (E) is the µ-completion of B(E). Each element of B(E) (resp. B∗ (E)) is called a Borel (resp. universally measurable) subset of E. P ROPOSITION A.1.3. Let (F, d) be a compact metric space equipped with a metric d and E be an arbitrary subset of F. We endow E with the relative topology and let B(E) be the Borel σ -field of E with respect to this topology. We denote by Cu (E) the collection of all d-uniformly continuous real functions on E. Suppose H is a linear space consisting of bounded real functions on E such that 1 ∈ H, Cu (E) ⊂ H and H is closed under the operation of taking uniformly bounded increasing limits. Then H contains any bounded B(E)-measurable function. Proof. For the family O of all open subsets of F, its trace O|E on E is the family of all open subsets of E (with respect to the relative topology on E).

ESSENTIALS OF MARKOV PROCESSES

Accordingly, B(E) is the trace of B(F) on E: B(E) = B(F)E .

393

(A.1.1)

Notice that Cu (E) = C(F)|E and in particular Cu (E) ⊂ bC(E). Let H(F) = { f ∈ bB(F) : f |E ∈ H}. Then 1 ∈ H(F), C(F) ⊂ H(F) and H(F) is closed under uniformly bounded increasing limits. For the distance function d(x, y) on F and A ⊂ F, we put d(x, A) := infy∈A d(x, y). If we let, for any U ∈ O, f (x) = d(x, F \ U) ∧ 1, x ∈ F, then f ∈ C(F), f 1/n (∈ C(F)) ↑ 1U , n → ∞, and hence 1U ∈ H(F). As O is a π-system, we have by Proposition A.1.2 that H(F) coincides with the space of all bounded B(F)-measurable functions. Let (E, B(E)) be a measurable space and denote by T either R+ = [0, ∞) or [0, ∞]. A quadruplet X = (, M, {Xt }t∈T , P) is called a stochastic process with time parameter set T and state space (E, B(E)) if (, M, P) is a probability space and, for each t ∈ T, Xt is a measurable map from (, M) to (E, B(E)). In this book, the last property is designated by Xt ∈ M/B(E). We call a sample space, ω ∈ a sample, ∈ M an event and P() its probability, respectively, of X. For each t ∈ T, Xt = Xt (ω) is a function of ω ∈ with value in E. For each fixed ω ∈ , the function X· (ω) : T → E is called the sample path of ω. A family {Mt ; t ≥ 0} of sub-σ -fields of M with parameter R+ is said to be a filtration if Ms ⊂ Mt for 0 ≤ s < t. In this case, we always extend the parameter of the filtration to [0, ∞] by setting M∞ = σ {Mt , t ≥ 0}. For a stochastic process X and a filtration {Mt ; t ≥ 0}, the former (resp. latter) is called adapted to (resp. admissible for) the latter (resp. former) if Xt ∈ Mt /B(E) for every t ≥ 0. For a filtration {Mt ; t ≥ 0}, we put Mt+ = ∩t >t Mt , t ≥ 0. A filtration is called right continuous if Mt = Mt+ for every t ≥ 0. D EFINITION A.1.4. Let {Mt }t≥0 be a filtration. A function σ : → [0, ∞] is called an {Mt }-stopping time if {σ ≤ t} ∈ Mt for every t ≥ 0. Obviously, a non-random constant time σ (ω) = r(≥ 0) for every ω ∈ is an {Mt }-stopping time. Exercise A.1.5. Prove the following. (i) A function σ : → [0, ∞] is an {Mt+ }-stopping time if and only if {σ < t} ∈ Mt for every t ≥ 0. (ii) If σ is an {Mt }-stopping time and c ≥ 0, then σ + c is also an {Mt }stopping time. (iii) Let σn , n = 1, 2, . . . , be a sequence of {Mt }-stopping times. If σn (ω) ↑ σ (ω) as n → ∞ for w ∈ , then σ is an {Mt }-stopping time.

394

APPENDIX A

If σn (ω) ↓ σ (ω) as n → ∞ for w ∈ , then σ is an {Mt+ }-stopping time. (iv) For an {Mt+ }-stopping time σ and for n ∈ N, put ∞ −n k=1 k2 1{(k−1)2−n ≤σ (ω) 0, we define a map Xs(n,ε) (ω) from [0, t] × to E∂ by (n) (ω) for 0 ≤ s < t Xsn,ε (ω) = Xs∧(t−ε)

and Xtn,ε (ω) = Xt (ω).

Since {Xt } is adapted to {Mt }, this map can be seen to be Bt × Mt -measurable for sufficiently large n. By the right continuity of the sample path (X.6)r (iii), Xs∧(t−ε) (ω), 0 ≤ s < t, n,ε X (ω) = lim n→∞ s Xt (ω), s = t. Since ε > 0 is arbitrary, we get the desired measurability of the map t . (ii) Take any t ≥ 0. As a composition of two maps ω ∈ → (σ (ω) ∧ t, ω) and (s, ω) ∈ [0, t] × → Xs (ω) ∈ E∂ , we get Xσ ∧t ∈ Mt /E∂ . Therefore, {Xσ ∈ B} ∩ {σ ≤ t} = {Xσ ∧t ∈ B} ∩ {σ ≤ t} ∈ Mt for B ∈ B(E∂ ). Remark A.1.14. For an admissible filtration {Mt } of X and an {Mt }-stopping time σ , define the σ -fields Mσ − of events strictly prior to time σ by Mσ − = σ { ∩ {s < σ } : ∈ Ms , s ≥ 0}.

(A.1.13)

If a real-valued {Mt }-adapted stochastic process {Zt } is predictable, then Zσ · 1{σ 0 : Xt (ω) ∈ B},

(A.1.19)

σ˙ B (ω) = inf{t ≥ 0 : Xt (ω) ∈ B}.

(A.1.20)

When B is an open set, we have by the right continuity of the sample path {σB (ω) < t} = {σ˙ B (ω) < t} =

{Xr ∈ B} ∈ Ft0 ,

t > 0,

r∈Q∩(0,t)

0 0 }-stopping times. Since Ft+ ⊂ Ft+ = Ft by and hence both σB and σ˙ B are {Ft+ the preceding theorem, σB , σ˙ B are {Ft }-stopping times. The following theorem asserts that the same is true for a general Borel set B.

T HEOREM A.1.19. Let X be a Borel right process. For any Borel set B ∈ B(E∂ ), both the hitting time σB and the entrance time σ˙ B are {Ft }-stopping times. Proof. Take any µ ∈ P(E∂ ). It suffices to show that σB and σ˙ B are both µ {Ft }-stopping times. For a fixed t > 0, let us consider a subset A = {(s, ω) ∈ (0, t) × : Xs (ω) ∈ B}

403


of the product space [0, ∞) × . Since {Ft0 } is an admissible filtration of X, we have by Lemma A.1.13 ∞ −1 (B) \ (B) ∈ Bt × Ft0 ⊂ B∞ × Ft0 . −1 A= t−(1/k) 1/k k=[1/t]+1

Here for a ∈ R, [a] denotes the largest integer that does not exceed a. Let be the projection of A on : = {ω ∈ : ∃s ∈ [0, t), (s, ω) ∈ A}. On account of Proposition A.1.1, belongs to the Pµ -completion (Ft0 )Pµ of the σ -field Ft0 . Since (Ft0 )Pµ ⊂ Ftµ , we get ∈ Ftµ . On the other hand, we have clearly {σB (ω) < t} = and hence {σB < t} ∈ µ µ Ft . Since this is true for every t > 0 and {Ft } is right continuous by Theorem µ A.1.18, σB is an {Ft }-stopping time. The proof for σ˙ B is the same. According to Theorem A.1.18 and (A.1.18), a Borel right process X enjoys 0 , and µ ∈ the following property: for any {Ft }-stopping time σ , F ∈ bF∞ P(E∂ ),

F ◦ θσ ∈ bF∞ and Eµ F ◦ θσ Fσ = EX [F] Pµ -a.s. (A.1.21) σ

If F ∈ bFt0 for some t ≥ 0, then F ◦ θσ ∈ bFσ +t . The relation (A.1.21) can be further extended to F ∈ bF∞ . Exercise A.1.20. Show the following: (i) If F ∈ bF∞ , then Ex [F] is B∗ (E∂ )-measurable as a function of x ∈ E∂ . (ii) If σ is an {Ft }-stopping time, then (Fσ )Pµ = Fσ , Xσ ∈ Fσ /B∗ (E∂ ), (A.1.22) µ∈P(E∂ )

where (Fσ )Pµ denotes the Pµ -completion of Fσ . T HEOREM A.1.21. Let X be a Borel right process and σ be an {Ft }-stopping time. For any F ∈ bF∞ and any µ ∈ P(E∂ ),

F ◦ θσ ∈ bF∞ , Eµ F ◦ θσ Fσ = EXσ [F], Pµ −a.s. (A.1.23) If F ∈ bFt , t ≥ 0, then F ◦ θσ ∈ bFσ +t . Proof. For F ∈ bF∞ , we have by Exercise A.1.20, EXσ [F] ∈ Fσ .

(A.1.24)

Take any µ ∈ P(E∂ ) and define ν ∈ P(E∂ ) by ν(B) = Pµ (Xσ ∈ B) for 0 , we take the Pµ -expectation on B ∈ B(E∂ ). Then, particularly for F ∈ bF∞ both sides of (A.1.21) to obtain

Eµ [F ◦ θσ ] = Eµ EXσ (F ) = Eν [F ].

404

APPENDIX A µ

ν Since F ∈ bF∞ ⊂ bF∞ , this identity means that F ◦ θσ ∈ b(F∞ )Pµ ⊂ bF∞ . We can also derive from (A.1.21) the equality

Eµ [F ◦ θσ ; ] = Eµ EXσ (F); for ∈ Fσ ,

which, together with (A.1.24), implies the latter half of (A.1.23). The proof of the last assertion is similar. Since any {Ft }-stopping time σ is obviously Fσ -measurable, we multiply both sides of (A.1.23) by 1{σ 0, ∞ ∞ −αt −αt e Ex ( f (Xt ))dt = Ex e f (Xt )dt . Rα f (x) = 0

0

405


Here we are using a consequence of Lemma A.1.13 that f (Xt (ω)) is a 0 -measurable function of (t, ω). B([0, ∞)) × F∞ ∗ Let f ∈ B (E). For any µ ∈ P(E), put ν(B) = µ, Rα 1B , B ∈ B(E). There exist f1 , f2 ∈ bB(E) such that f1 ≤ f ≤ f2 , ν, f2 − f1 = 0. Then both f1 , f2 satisfy equation (A.1.26) and µ, Rα f1 = µ, Rα f2 . Hence Rα f ∈ bBµ (E), P µ F ∈ F∞µ ⊂ F∞ , and f also satisfies (A.1.26). Since this holds for every F ∈ F∞ . µ ∈ P(E), Rα f ∈ bB∗ (E) and We next apply the strong ∞ Markov property (A.1.23) to F and any F ◦ θσ , the right hand side {Ft }-stopping time σ . Since σ e−αt f (Xt )dt = e−ασ of (A.1.27) equals

F ◦ θσ = Eµ Eµ e−ασ F ◦ θσ Fσ = Eµ e−ασ EXσ [ F] , Eµ e−ασ which coincides with the left hand side of (A.1.27). ∗ (E), we first apply (i) to f ∧ n, then let α ↓ 0 and n → ∞ (ii) For f ∈ B+ successively to get (ii). Now let us denote by (X.6)s , (X.6)h the conditions obtained from (X.6)r by replacing its third requirement (iii) with stronger ones (iii) and (iii) , respectively: (iii) For each ω ∈ , sample path t → Xt (ω) is right continuous on [0, ∞) and has left limits on (0, ζ (ω)) in E. (iii) For each ω ∈ , its sample path t → Xt (ω) is right continuous on [0, ∞) and has left limits on (0, ∞) in E∂ . A Markov process X is called quasi-left-continuous on (0, ζ ) (resp. (0, ∞)) if it possesses the following property (A.1.30) (resp. (A.1.31)): there exists a right continuous admissible filtration {Mt } for X such as, for any increasing {Mt }-stopping times {σn } with σ = limn→∞ σn , Pµ

n→∞

Pµ

lim Xσn = Xσ , σ < ζ = Pµ (σ < ζ ) for µ ∈ P(E∂ ),

lim Xσn = Xσ , σ < ∞ = Pµ (σ < ∞)

n→∞

for µ ∈ P(E∂ ).

(A.1.30)

(A.1.31)

We introduce two subfamilies of Borel right processes. D EFINITION A.1.23. (i) X is called a Borel standard process if X is a Markov process on a Lusin space (E, B(E)) and satisfies (X.6)s as well as the strong Markov property and the quasi-left-continuity on (0, ζ ) with respect to an admissible filtration {Mt }.

406

APPENDIX A

(ii) X is called a Hunt process if X is a Markov process on a Lusin space (E, B(E)) and satisfies (X.6)h as well as the strong Markov property and the quasi-left-continuity on (0, ∞) with respect to an admissible filtration {Mt }. The next theorem can be proved in exactly the same way as the proof of Theorem A.1.18. T HEOREM A.1.24. Let (E, B(E)) be a Lusin space. For a Markov process X = (, M, Xt , Px ) on (E, B(E)) satisfying condition (X.6)s (resp. (X.6)h ) the following three conditions are mutually equivalent: (α) X is a standard (resp. Hunt) process on E. (β) The minimum augmented admissible filtration {Ft } of X is right continuous and X is strong Markov and quasi-left-continuous on (0, ζ ) (resp. (0, ∞)) with respect to it. µ (γ ) For each µ ∈ P(E∂ ), the filtration {Ft } is right continuous and the identity µ (A.1.15) as well as (A.1.30) (resp. (A.1.31)) hold for {Ft }-stopping times and for the probability measure Pµ . Remark A.1.25 (A generalization of state space E). In defining a Borel right process, a Borel standard process, and a Hunt process in Definitions A.1.17 and A.1.23, respectively, the state space (E, B(E)) is assumed to be a Lusin space so that E is regarded as a Borel subset of a compact metric space F and the Borel σ -field B(E) is the trace of B(F) on E. By dropping the Borel measurability assumption on E in these definitions, we can take an arbitrary subset E of a compact metric space F, endow E with the relative topology, and take as B(E) the Borel σ -field with respect to this topology. In what follows, we shall admit this generalization of the state space (E, B(E)) for a Borel right process, a Borel standard process, and a Hunt process, respectively, with no specific mentions. With this generalization, nothing is changed because B(E) is still the trace of B(F) on E and the basic monotone class theorem Proposition A.1.3 continues to work.

Exercise A.1.26. Let X be a Hunt process on a Lusin space (E, B(E)). (i) Show that {Xt− (ω)}t>0 is progressive with respect to any admissible filtration {Mt } for X. (ii) For B ∈ B(E), show that σB defined below is an {Ft }-stopping time. σB (ω) = inf{t > 0 : Xt− (ω) ∈ B},

inf ∅ = ∞.

(A.1.32)

As an application of Theorem A.1.18 (resp. Theorem A.1.24), we can formulate a restriction of a Borel right process (resp. Hunt process) to its

407


invariant set, which is one of the most trivial and yet important transformations of a Markov process. Let X = (, M, {Xt }t∈[0,∞] , {Px }x∈E∂ ) be a Borel right process (resp. Hunt process) on a Lusin space (E, B(E)). A Borel set A ⊂ E is said to be X-invariant if Px (σE\A < ∞) = 0 for every x ∈ A,

resp. Px (σE\A ∧ σE\A < ∞) = 0

(A.1.33)

for every x ∈ A .

(A.1.34)

Since σE\A (resp. σE\A ∧ σE\A ) is an {Ft }-stopping time by Theorem A.1.19 (resp. Theorem A.1.19 and Exercise A.1.26), we have ∈ F∞ , where = {ω ∈ : σE\A (ω) = ∞} σE\A = ∞}). (resp. = {ω ∈ : σE\A (ω) = ∞ and Let A ⊂ E(⊂ F) be a Borel X-invariant set endowed with the relative topology. Put A∂ = A ∪ {∂} and regard it as a topological subspace of E∂ . Then B(A∂ ) = B(A) ∪ {B ∪ {∂} : B ∈ B(A)}. We then call X|A = (, M| , {Xt }t∈[0,∞] , {Px (· ∩ )}x∈A∂ )

(A.1.35)

the restriction of X to its invariant set (A, B(A)). L EMMA A.1.27. For a Borel right process (resp. Hunt process) on (E, B(E)), X|A is a Borel right process (resp. Hunt process) on (A, B(A)). Proof. We give a proof only for a Borel right process X because the same proof as below works for a Hunt process X by using Theorem A.1.24 instead of Theorem A.1.18. First X|A is a Markov process on A, namely, all the five conditions of Definition A.1.8 are fulfilled with (E∂ , B(E∂ )) being replaced by (A∂ , B(A∂ )). In fact, the measurability condition (X.2) for X|A follows from that for X and Px ({Xt ∈ B} ∩ ) = Px (Xt ∈ B)

for x ∈ A∂ , B ∈ B(A∂ ).

Four other conditions for X|A are clear. The property (X.6)r for the pair (, Xt ) follows readily from that for the pair (, Xt ). We now examine the property (γ ) of Theorem A.1.18 for X|A . Take any µ ∈ P(A∂ ) and regard it as an element of P(E∂ ) by taking µ(E \ A) = 0. The σ -fields generated by X|A will be designated by putting . It then follows µ µ µ µ F t = F t ∩ ⊂ Ft . from Ft0 = Ft0 ∩ , ∈ F∞ , and Pµ () = 1 that Consequently, the property (γ ) for X immediately implies that for X|A , yielding that X|A is a Borel right process by virtue of Theorem A.1.18.

408

APPENDIX A

Suppose that X = (, M, {Xt }t∈[0,∞] , {Px }x∈E∂ ) is a Borel right process on a Lusin space (E, B(E)). It will be useful to introduce for X a σ -field Bn (E∂ ) slightly larger than B(E∂ ) as follows: D EFINITION A.1.28. A set B ⊂ E∂ is said to be nearly Borel measurable if, for any µ ∈ P(E∂ ), there exist sets B1 , B2 ∈ B(E∂ ) such that B1 ⊂ B ⊂ B2 , Pµ (Xt ∈ B2 \ B1 for some t ≥ 0) = 0. The totality of nearly Borel measurable subsets of E∂ is denoted by Bn (E∂ ). Exercise A.1.29. (i) Show that Bn (E∂ ) is a σ -field contained in B∗ (E∂ ). (ii) Show that for every µ ∈ P(E) and for a non-negative nearly Borel measurable function f , there are non-negative Borel measurable functions g and h such that g ≤ f ≤ h and Pµ (g(Xs ) < h(Xs ) for some s ≥ 0) = 0. The hitting time σB and the entrance time σ˙ B of B ∈ B(E∂ ) are {Ft }-stopping times by Theorem A.1.19. This property remains valid for B ∈ Bn (E∂ ), because, for any µ ∈ P(E∂ ), we take the set B1 ∈ B(E∂ ) in the above definition to see that for each t ≥ 0, {σB ≤ t} differs from {σB1 ≤ t} ∈ Ft only by a Pµ -null set and consequently {σB ≤ t} ∈ Ft . Remark A.1.30 (Restriction to a nearly Borel invariant set). In the above, we assume that an X-invariant set A for a Borel right process (resp. Hunt process) X is a Borel subset of E. But more generally we can take as A a nearly Borel measurable subset of E defined by Definition A.1.28 above. A nearly Borel measurable set is universally measurable but not necessarily Borel measurable. The X-invariance of a nearly Borel measurable set A ⊂ E is σE\A ) is still defined by (A.1.33) (resp. (A.1.34)). Note that σE\A (resp. σE\A ∧ an {Ft }-stopping time. Hence, for a nearly Borel measurable X-invariant set A ⊂ E, the restriction X|A of X to A is well defined by (A.1.35). On the other hand, since E is regarded as a subset of a compact metric space F, we can endow A with the relative topology and define B(A) as the Borel σ -field with respect to this topology. Then, for any B ∈ B(A), there is B1 ∈ B(E) so that B = B1 ∩ A and Px (Xt ∈ B) = Px (Xt ∈ B1 ), x ∈ A, yielding the measurability (X.2). Thus X|A is a Borel right process (resp. Hunt process) on (A, B(A)) in the extended sense of Remark A.1.25. Exercise A.1.31. Let X be a Borel right process on a Lusin space (E; B(E)) and A be a nearly Borel measurable X-invariant subset of E. Show that for any nearly Borel measurable function f on E, f |A is nearly Borel measurable with respect to the restricted Borel right process X|A .


409

L EMMA A.1.32. Let X be a Hunt process on a Lusin space (E, B(E)) and A be a nearly Borel measurable subset of E. If (A.1.33) holds for A, then so does (A.1.34). In other words, A is invariant with respect to the Hunt process X whenever it is so with respect to X regarded as a Borel right process. Proof. Assume first that A ∈ B(E) satisfies (A.1.33). Due to the quasileft-continuity on [0, ∞) of the Hunt process X, it is known (cf. [73, Theorem σE\A ) = 1 for any x ∈ E, from which the property A.2.3]) that Px (σE\A ≤ (A.1.34) for A follows. If A is a nearly Borel measurable subset of E satisfying (A.1.33), there exists, for any x ∈ A, a Borel set A1 such that x ∈ A1 ⊂ A and Px (σE\A1 < ∞) = 0 in view of Definition A.1.28. This combined with the above consideration yields (A.1.34) for A. A topological space E is called a Radon space if it is homeomorphic to a universally measurable subset of a compact metric space F. By identifying E with its image under this homeomorphism, we can then regard E as a subspace of F with relative topology and as an element of B∗ (F). Any Lusin space is a Radon space. We equip a Radon space E with the universally measurable σ -field B∗ (E). Exercise A.1.33. Show that for a Radon space E, B∗ (E) is the trace of B∗ (F) on E. Even if we start with a Borel right process X on a Lusin space (E, B(E)), we are led to Markov processes on Radon spaces by non-trivial transformations of X such as time changes, Feynman-Kac transforms, and killing upon leaving a nearly Borel finely open set, since the transition functions of the transformed Markov processes can hardly make the space of bounded B(E)-measurable functions invariant. Let (, M, Xt , Px ) be a Markov process on a Radon space (E, B∗ (E)). We 0∗ = σ {Xs−1 (B) : B ∈ B∗ (E), s < ∞} may then consider two σ -fields of : F∞ 0 −1 and F∞ = σ {Xs (B) : B ∈ B(E), s < ∞}. Similarly, we have two σ -fields 0∗ , Ft0∗ , t ≥ 0, and following Ft0∗ and Ft0 for each t ≥ 0. Starting with F∞ the procedure described in Section A.1.1, we can construct the minimum augmented admissible filtration {Ft }t∈[0,∞] for X. Exercise A.1.34. Show that the filtration {Ft }t∈[0,∞] constructed as above 0∗ and {Ft0∗ } is the same as the minimum augmented admissible from F∞ 0 and {Ft0 , t ≥ 0}. filtration constructed in Section A.1.1 from the filtration F∞ The transition function {Pt ; t ≥ 0} of X is now a transition function on (E, B∗ (E)) (instead of (E, B(E))) in the sense of Definition 1.1.13.

410

APPENDIX A

D EFINITION A.1.35. For a Markov process on a Radon space (E, B∗ (E)) with a transition function {Pt ; t ≥ 0} and for α ≥ 0, a [0, ∞]-valued function u on E is said to be α-excessive if u is B∗ (E)-measurable and e−αt Pt u(x) ↑ u(x) as t ↓ 0

for every x ∈ E.

A 0-excessive function will be simply called excessive. D EFINITION A.1.36. X = (Xt , Px ) is called a right process if it is a Markov process on a Radon space (E, B∗ (E)) satisfying condition (X.6)r as well as the property that for every α-excessive function g for X with α ≥ 0 and for every µ ∈ P(E), ! g(Xt ) = g(Xt ) = 1. (A.1.36) Pµ lim t ↓t

The next theorem gives useful criteria for a Markov process on a Radon space to be a right process. It also shows that the notion of a right process on a Radon space is a genuine extension of the notion of a Borel right process on a Lusin space. T HEOREM A.1.37. (i) Every Borel right process on a Lusin space is a right process; if X = (Xt , Px ) is a Borel right process on a Lusin space (E, B(E)), then X possesses the property (A.1.36). (ii) The following conditions are mutually equivalent for a Markov process X = (Xt , Px ) on a Radon space (E, B∗ (E)) satisfying condition (X.6)r . The resolvent kernel for X is denoted by {Rα ; α > 0}. (a) X is a right process. (b) X is strong Markov in the sense of Definition A.1.15, and {g(Xt )}t≥0 is Pµ -indistinguishable from an {Ft }-adapted optional process for any g ∈ Rα (Cu+ (E)), α > 0 and µ ∈ P(E). (c) X has the property (A.1.36) for any g ∈ Rα (Cu+ (E)), α > 0 and every µ ∈ P(E). Proof. (i) This will be proved in Theorem A.2.2 of Section A.2.1. (ii) Since g ∈ Rα (bC+ (E)) is α-excessive, the implication (a) ⇒ (c) is trivial. (c) ⇒ (b): Condition (c) obviously implies the second condition of (b). We shall derive from condition (c) the strong Markov property of X with respect to 0 }, or more specifically (A.1.16) holding for the right continuous filtration {Ft+ B ∈ B(E) and ∈ Fσ0 + .

411


0 Let σ be an {Ft+ }-stopping time and σn , n ≥ 1, be its discrete approximations from the above defined by (A.1.2). For ∈ Fσ0 + and for f ∈ bC+ (E), α > 0, we have ∞ ∞

−αt −αt e Ex f (Xσ +t ); dt = lim Ex e f (Xσn +t )dt; n→∞

0

= lim

∞

n→∞

k=1

Ex

∞

0

e−αt f (Xt+k2−n )dt; σn = k2−n , .

0

0 Notice that {σn = k2−n } ∩ = {(k − 1)2−n ≤ σ < k2−n } ∩ ∈ Fk2 −n . 0 Using the Markov property of X relative to the filtration {Ft } and the assumption (c), we see that the above expression equals ∞ ∞ −αt −n Ex EXk2−n e f (Xt )dt ; σn = k2 , lim n→∞

k=1

0

= lim Ex Rα f (Xσn ); = n→∞

∞ 0

e−αt Ex EXσ [f (Xt )] ; dt.

Since both Ex f (Xσ +t ); and Ex EXσ [ f (Xt )] ; are right continuous in t, they must coincide owing to the uniqueness of the Laplace transformation 0 } by using and we arrive at the desired identity (A.1.16) for the filtration {Ft+ Proposition A.1.3. (b) ⇒ (a): Assume (b). Then by virtue of Theorem A.1.18 and Exercise A.1.34, the minimum augmented admissible filtration {Ft } for X is right continuous and X has the strong Markov property with respect to {Ft }, which can be further extended to (A.1.23). For fixed α > 0, let H be the collection of f ∈ bB(E) such that the process {Rα f (Xt )}t≥0 is Pµ -indistinguishable from an {Ft }-adapted optional process for each µ ∈ P(E). Then Cu (E) ⊂ H and H is a linear space closed under uniformly bounded increasing limits. Accordingly, H = bB(E) by Proposition A.1.3. For α ≥ 0, we can then make use of the strong Markov property of X with respect to {Ft } to show that (A.1.36) holds for any α-excessive function g of X in exactly the same way as in the proof of Theorem A.2.2 below. Remark A.1.38. We shall prove in Theorem A.2.2 that any α-excessive function of a Borel right process X on a Lusin space satisfies not only the property (A.1.36) but is also nearly Borel measurable. But for a general right process X on a Radon space, the latter property may not hold. In Definition A.1.23, we defined a Borel standard process and a Hunt process as specific Borel right processes on a Lusin space with additional properties on

412

APPENDIX A

the left limits of sample paths. A general standard process will now be defined as a specific right process X on a Radon space. D EFINITION A.1.39. A right process X on a Radon space (E, B∗ (E)) is called a standard process if X satisfies additionally (X.6)s and the quasileft-continuity on (0, ζ ). We end this subsection by quoting two fundamental theorems from Blumenthal-Getoor [13]. A filtration {Mt }t∈[0,∞] is called quasi-left-continuous if, for any increasing sequence {σn } of {Mt }-stopping times with limit σ , ∞ "

Mσn = Mσ .

(A.1.37)

n=1

Here ∨∞ n=1 Mσn denotes the σ -field generated by {Mσn , n ≥ 1}. The minimum augmented T HEOREM A.1.40. {Ft }t∈[0,∞] of a Hunt process is quasi-left-continuous.

admissible

filtration

Proof. See [13, IV, (4.2)].

This theorem indicates an advantage of a Hunt process over a Borel standard process especially in dealing with its martingale additive functionals. If X is a Hunt process, then for any sequence {σn } of {Ft }-stopping times increasing to σ , ∞ " Xσ ∈ Fσn /B(E∂ ). (A.1.38) n=1

# This is because {σ < ∞} = ∪k ∩n {σn ≤ k} ∈ ∞ n=1 Fσn and X∞ (ω) = ∂ for every ω ∈ , and accordingly, the quasi-left-continuity (A.1.31) of X on [0, ∞) implies (A.1.38). Actually the property (A.1.38) is shown in [13, IV, (4.2)] to be equivalent to the quasi-left-continuity of {Ft }. However, a standard process # may fail to satisfy (A.1.38) because {σ < ζ } = {Xσ ∈ E} may not be in ∞ n=1 Fσn . D EFINITION A.1.41. A standard process X is called special if its minimum augmented admissible filtration {Ft }[0,∞] is quasi-left-continuous. We have verified that the transition function {Pt ; t ≥ 0} of a Borel right process X defined by (A.1.4) is a transition function on a Lusin space (E, B(E)) as an analytic quantity satisfying conditions (t.1)–(t.4) of Definition 1.1.13. Conversely, given a transition function {Pt ; t ≥ 0} in this analytic sense, it is important to construct an associated nice Markov process X. A well-know theorem of this sort is provided by a Feller transition function on a locally compact separable metric space E.

413


Let E be a locally compact separable metric space and E∂ = E ∪ {∂} be the one-point compactification of E. We put C∞ (E) = { f ∈ C(E) : lim f (x) = 0}. x→∂

If a family of Markovian kernels {Pt ; t ≥ 0} on (E, B(E)) satisfies (t.1), (t.3) and Pt (C∞ (E)) ⊂ C∞ (E), t ≥ 0, lim Pt f (x) = f (x), x ∈ E, f ∈ C∞ (E), (A.1.39) t↓0

then it is called a Feller transition function or Feller semigroup regarded as a semigroup of linear operators on C∞ (E). Exercise A.1.42. Verify that a Feller transition function is a transition function on (E, B(E)) in the sense of Definition 1.1.13. T HEOREM A.1.43. For any Feller transition function {Pt ; t ≥ 0} on a locally compact separable metric space E, there exists a Hunt process on E having {Pt ; t ≥ 0} as its transition function. Such a process will be called a Feller process.

Proof. See (9.4) in Chapter 1 of [13].

We remark that, once a Markov process X on E with property (X.6)r is constructed having Feller transition function, it is clear then that X satisfies the property (A.1.36) for u ∈ Rα (Cu+ (E)) and consequently, by Theorem A.1.37, the strong Markov property.

A.2. BASIC PROPERTIES OF BOREL RIGHT PROCESSES A.2.1. Excessive Functions Let X = (, M, {Xt }t∈[0,∞] , {Px }x∈E∂ ) be a Borel right process on a Lusin space E with transition function {Pt , t ≥ 0}. Recall that the notions of an α-excessive function and an excessive function were defined in Definition A.1.35 for a general right process. Suppose a non-negative universally measurable function u on E satisfies the inequality e−αt Pt u(x) ≤ u(x)

for every t > 0 and x ∈ E.

(A.2.1)

An application of the operator e−αs Ps to both sides of (A.2.1) yields that e−α(s+t) Ps+t f (x) ≤ e−αs Ps u(x);

414

APPENDIX A

namely, e−αt Pt u(x) increases as t decreases. Denote by u(x) its limit as t ↓ 0. Then u can be easily seen to be an α-excessive function satisfying u(x) ≤ u(x) for every x ∈ E. The function u will be called the α-excessive regularization of u. A non-negative constant function on E is excessive because of Pt 1(x) = Pt (x, E) ≤ 1 for every t ≥ 0 and x ∈ E, and properties (t.3)–(t.4) of {Pt ; t ≥ 0}. For any non-negative universally measurable function f on E, the resolvent kernel Rα f is α-excessive for every α ≥ 0 in view of Theorem A.1.22. L EMMA A.2.1. Let α > 0. (i) For any α-excessive function u, there exists a sequence { fn } of non-negative universally measurable functions such that for every x ∈ E, Rα fn (x) ↑ u(x) as n ↑ ∞. (ii) The limit of an increasing sequence of α-excessive functions is again α-excessive. Proof. (i) We put un = u ∧ n. By the resolvent equation (1.1.25), nRn+α un = Rα (n(un − nRn+α un )) . Further,

nRn+α un (x) =

∞

e−s g(s, n, x)ds,

0

where g(s, n, x) = e−α n P ns un (x). Since un satisfies (A.2.1), g(s, n, x) increases to some a(x) ≤ ∞ as n → ∞. Clearly e−αt Pt u(x) ≤ a(x) ≤ u(x) for every t > 0, and we get a(x) = u(x) by letting t ↓ 0. Hence nRn+α un (x) ↑ u(x) as n ↑ ∞, and fn (x) := n(un (x) − nRn+α un (x)) for x ∈ E is the desired non-negative function. (ii) Let {un } be a sequence of α-excessive functions increasing to a function u. Then u is universally measurable and satisfies the inequality (A.2.1). Since e−αt Pt un (x) increases as t decreases or as n increases, we obtain by exchanging the order of limits that for x ∈ E, s

lim e−αt Pt u(x) = lim lim e−αt Pt un (x) = lim lim e−αt Pt un (x) t↓0

t↓0 n→∞

n→∞ t↓0

= lim un (x) = u(x). n→∞

Recall the convention that any numerical function u on E is extended to E∂ by setting u(∂) = 0.


415

T HEOREM A.2.2. Let α ≥ 0 and u be an α-excessive function. (i) u is right continuous along the sample path of X in the sense of (A.1.36). (ii) u is nearly Borel measurable. Proof. Since any (0−) excessive function is α-excessive for any α > 0, we may assume that α > 0. We start with the case that u = Rα f with f ∈ bB+ (E). Since u is Borel measurable and Xt is right continuous on E adapted to the filtration {Ft }, u(Xt ) is optional in the sense of Section A.1.1. Indeed, for any v ∈ Cu (E), {v(Xt )}t≥0 is a right continuous real valued process adapted to {Ft } and hence an optional process. Therefore, by Proposition A.1.3, u(Xt ) is also optional. Let σn be any sequence of uniformly bounded and decreasing {Ft }-stopping times with limit σ . Since {Ft } is right continuous, σ is again an {Ft }-stopping time by Exercise A.1.5. By virtue of Theorem A.1.22, we have the identity (A.1.27) with σ being replaced by σn . Letting n → ∞, we get

lim Eµ e−ασn u(Xσn ) = Eµ e−ασ u(Xσ ) for every µ ∈ P(E). n→∞

Therefore, u satisfies (A.1.36) by Theorem A.1.6. ∗ Next we let u = Rα f with f ∈ bB+ (E). For any µ ∈ P(E), we define a finite measure ν by ν(B) = µ, Rα 1B , B ∈ B(E). There exist f1 , f2 ∈ bB(E) such that f1 ≤ f ≤ f2 and ν, f2 − f1 = 0. Let ui := Rα fi , i = 1, 2, which are Borel measurable functions with u1 ≤ u ≤ u2 . The Markov property (A.1.3) yields that for any t ≥ 0, Eµ [u2 (Xt ) − u1 (Xt )] = µ, Pt Rα ( f2 − f1 ) ≤ eαt ν, f2 − f1 = 0. Since ui (Xt ), i = 1, 2, are right continuous, we conclude that Pµ (u1 (Xt ) = u2 (Xt ) for every t ≥ 0) = 1. Consequently, u satisfies (A.1.36) and u is also nearly Borel measurable. We further let Yt = e−αt u(Xt ). Then {Yt }t≥0 is an ({Ft }, Pµ )-supermartingale for any µ ∈ P(E), because, by the Markov property of X, especially the formula (A.1.23) with σ = s, we have for t > s ≥ 0 Eµ [Yt Fs ] = e−αt EXs [u(Xt−s )] = e−αs e−α(t−s) Pt−s u(Xs ) ≤ e−αs u(Xs ) = Ys . When u is an arbitrary α-excessive function, we choose a sequence { fn } satisfying the condition of Lemma A.2.1(i). Since un = Rα fn satisfies (A.1.36), so does its limit function u by virtue of Theorem A.1.7. Since un is nearly Borel measurable, so is u. Exercise A.2.3. Show that if u and v are α-excessive, then so is u ∧ v.

416

APPENDIX A

Let B ⊂ E be nearly Borel measurable. As we noted in Section A.1, the hitting time σB of B is an {Ft }-stopping time. We define the hitting distribution and α-order hitting distribution of B by HB (x, A) := Px XσB ∈ A, σB < ∞ ,

(A.2.2) HαB (x, A) := Ex e−ασB ; XσB ∈ A , respectively, where x ∈ E and A ∈ B∗ (E). Both of them are kernels on (E, B∗ (E)) (cf. Exercise A.1.20). We further define

(A.2.3) pB (x) = Px (σB < ∞), pαB (x) = Ex e−ασB , x ∈ E, the hitting probability and α-order hitting probability of B, respectively. Clearly pB = HB 1 and pαB = HαB 1.

L EMMA A.2.4. (i) If u is α-excessive for α > 0 and σ is an {Ft }-stopping time, then

Ex e−ασ u(Xσ ) ≤ u(x), x ∈ E. (ii) Let B ⊂ E be nearly Borel measurable. If u is α-excessive for α > 0, then so is HαB u. If u is excessive, then so is HB u. In particular, pαB is α-excessive and pB is excessive. Proof. (i) follows immediately from Lemma A.2.1 and the identity (A.1.27). When B ⊂ E is nearly Borel measurable, we see by the identity (A.1.27) and ∗ (E) and α > 0, the Markov property that for u = Rα f with f ∈ B+ ∞ α −αt −αt −αs e Pt HB u (x) = e Ex e f (Xs ◦ θt )ds = Ex

σB ◦θt

∞

−αs

e t+σB ◦θt

f (Xs )ds ,

x ∈ E.

Since t + σB ◦ θt ↓ σB as t ↓ 0, HαB u is α-excessive. The same conclusion holds for a general α-excessive function in view of Lemma A.2.1. If u is excessive, then u is α-excessive for any α > 0 and so is HαB u. If we let α ↓ 0, then HαB u ↑ HB u and hence Pt (HB u) ≤ HB u for every t ≥ 0. Moreover, interchanging the order of taking monotone limits, we have lim Pt (HB u)(x) = lim lim e−αt Pt (HαB u)(x) = HB u(x), x ∈ E. t↓0

α↓0 t↓0


417

A.2.2. Fine Topology, Excessive Measures, and Exceptional Sets Let X be a Borel right process on a Lusin space (E, B(E)) with transition function {Pt ; t > 0}. L EMMA A.2.5 (Blumenthal’s 0-1 law). For any ∈ F0 and any x ∈ E, either Px () = 0 or Px () = 1. Proof. By (X.6)r (ii), θ0−1 = for every ⊂ . When ∈ F00 , we have, from the Markov property (A.1.11) of X and (X.5),

Px () = Px ( ∩ θ0−1 ) = Ex PX0 (); = (Px ())2 . Thus Px () = 0 or Px () = 1. For ∈ F0 and x ∈ E, we then choose ∈ F00 with Px ( ) = 0 to get Px () = Px ( ) ∈ {0, 1}. For any nearly Borel measurable set B ⊂ E, σB is an {Ft }-stopping time and in particular {σB = 0} ∈ F0 . Hence, for each x ∈ E, the probability Px (σB = 0) is either 1 or 0 by the above lemma. A point x ∈ E is called regular (resp. irregular) for B if this value equals 1 (resp. 0). We denote by Br the totality of regular points of B. Since Br = {x ∈ E : pαB (x) = 1} for α > 0 and pαB is nearly Borel measurable in x in view of Theorem A.2.2 and Lemma A.2.4, we see that Br ∈ Bn (E). We now introduce several notions of smallness of subsets of E relative to the Borel right process X. D EFINITION A.2.6. (i) B ∈ B∗ (E) is called of potential zero if R(x, B) = 0 for every x ∈ E. (ii) B ⊂ E is called polar if there is a set D ∈ Bn (E) such that B ⊂ D and Px (σD < ∞) = 0 for every x ∈ E. (iii) B ⊂ E is called thin if there is a set D ∈ Bn (E) such that B ⊂ D and Dr is empty, namely, if Px (σD = 0) = 0 for every x ∈ E. (iv) A set B ⊂ E contained in a countable union of thin sets is called semipolar. Obviously any polar set is thin and any thin set is semipolar. D EFINITION A.2.7. (i) A set B ⊂ E is said to be thin at a point x ∈ E if there is a set D with B ⊂ D, D ∈ Bn (E) such that x is irregular for D, namely, Px (σD = 0) = 0. (ii) B ⊂ E is said to be finely open if Bc = E \ B is thin at each point x ∈ B, namely, there is for each x ∈ B a set D(x) with Bc ⊂ D(x), D(x) ∈ Bn (E) such that Px (σD(x) > 0) = 1.

418

APPENDIX A

By the right continuity of the sample path, any open set D ⊂ E in the original topology of E is finely open because Px (σDc > 0) = 1 for every x ∈ D. Exercise A.2.8. Denote by O the totality of finely open subsets of E. Show that O satisfies the following axiom of a family of open subsets to determine a topology of E: (i) E, ∅ ∈ O. (ii) O1 , O2 ∈ O ⇒ O1 ∩ O2 ∈ O. (iii) {Oλ ; λ ∈ } ⊂ O ⇒ ∪λ∈ Oλ ∈ O. The topology of E determined by O is called the fine topology with respect to X. A function on E is called finely continuous if it is continuous with respect to the fine topology. The fine topology is finer that the original one and any continuous function on E with respect to the original topology is finely continuous. In what follows, we say that an event ∈ M occurs almost surely or a.s. in abbreviation if Px () = 1 for every x ∈ E. T HEOREM A.2.9. A nearly Borel measurable function u on E is finely continuous if and only if the real-valued composite process t ∈ [0, ∞) → u(Xt ) is right continuous a.s. Here we set u(∂) = 0 as usual. Proof. See [13, II.(4.8)] or [73, Theorem A.2.7].

In particular, any α-excessive function on E is finely continuous (α ≥ 0) in view of Theorem A.2.2. L EMMA A.2.10. Let B ∈ Bn (E). For any µ ∈ P(E), Pµ XσB ∈ B ∪ Br , σB < ∞ = Pµ (σB < ∞).

(A.2.4)

In particular, for any x ∈ E, the α-order hitting distribution HαB (x, ·) and the hitting distribution HB (x, ·) of B are concentrated on the set B ∪ Br . Proof. See [13, I, (11.4)] or [73, Lemma A.2.7].

If B ∈ Bn (E) is finely closed, then any point x ∈ Bc cannot be regular for B and so (A.2.5) Pµ XσB ∈ B, σB < ∞ = Pµ (σB < ∞), µ ∈ P(E). For a kernel κ on (E, B(E)) and a measure µ on E, we denote by µκ the measure defined by E µ(dx)κ(x, B), B ∈ B(E). Let {Pt ; t ≥ 0} be the transition function of a Borel right process X. A measure m on E is called excessive with respect to {Pt } if m is σ -finite and satisfies mPt ≤ m for every t > 0.

419


L EMMA A.2.11. If m is an excessive measure, then for any B ∈ B(E), mPt (B) ↑ m(B)

as t ↓ 0.

(A.2.6)

Proof. For B ∈ B(E), we denote the increasing limit limt↓0 mPt (B) by nu(B). , n ≥ 1} ⊂ B(E), we Then ν(B) ≤ m(B) and, for mutually disjoint $∞ sets {Bn ∞ interchange increasing limits to obtain ν n=1 Bn = n=1 ν(Bn ), thus ν is a σ -finite measure. We next show that ν = m. Fix an α > 0. If A ∈ B(E) satisfies ν(A) < ∞, then, for any ε > 0, it follows from ανRα (A) ≤ ν(A) that ν{x ∈ E : Rα (x, A) > ε} ≤ (αε)−1 ν(A) < ∞. Take a sequence {A to E with ν(An ) < ∞ for every n } of Borel sets increasing n ≥ 1. Let Bn = x ∈ E : Rα (x, An ) > 1n . Then {Bn } is a sequence of Borel finely open sets increasing to E with ν(Bn ) < ∞ for every n ≥ 1. For any non-negative continuous function f on E and integer n ≥ 1, we have by Fatou’s lemma lim inf Pt ( f 1Bn ) ≥ f 1Bn . t↓0

By Fatou’s lemma again, ν, f 1Bn = limmPt , f 1Bn = limm, Pt ( f 1Bn ) ≥ m, f 1Bn . t↓0

t↓0

Thus ν = m on Bn and consequently ν = m on ∪∞ n=1 Bn = E.

In the remainder of this section, we shall fix an excessive measure m for a Borel right process X. D EFINITION A.2.12. (i) A set N ⊂ E is called m-polar if there is a nearly Borel measurable set N ⊃ N such that Pm (σ N < ∞) = 0.

(A.2.7)

(ii) “Quasi everywhere” or “q.e.” in abbreviation means “except for an m-polar set.” (iii) A subset N ⊂ E is called an m-inessential set for the Borel right process X if N is an m-negligible nearly Borel measurable set such that E \ N is X-invariant. Clearly any m-inessential set is m-polar. T HEOREM A.2.13. (i) If f is nearly Borel measurable on E, then there exist Borel measurable functions g, h such that g ≤ f ≤ h and g = h q.e. Any m-polar set is contained in a Borel m-polar set.

420

APPENDIX A

(ii) If B ⊂ E is m-polar, then m(B) = 0. If B ∈ Bm (E) satisfies mRα (B) = 0 for some α > 0, then m(B) = 0. (iii) If B ⊂ E is nearly Borel, finely open, and m-negligible, then B is m-polar. (iv) If functions f , g on E are nearly Borel measurable, finely continuous, and satisfy f ≥ g [m], then f ≥ g q.e. (v) Let α ≥ 0. If f is α-excessive and f < ∞ m-a.e., then f < ∞ q.e. Proof. (i) By Exercise A.1.29(ii), there are Borel measurable functions g, h such that g ≤ f ≤ h and Pm (g(Xs ) < h(Xs ) ∃s ≥ 0) = 0, which means g = h q.e. If in particular B is a nearly Borel measurable m-polar set, then there are Borel sets B1 ,B2 such that B1 ⊂ B ⊂ B2 and B2 \ B1 is m-polar, and hence B is contained in the Borel m-polar set B2 = B ∪ (B2 \ B). (ii) If B is a Borel m-polar set, then mPt (B) = Pm (Xt ∈ B) = 0 for every t > 0. By Lemma A.2.11, we have m(B) = 0. Next note that a set B ∈ Bm (E) belongs to the completions of B(E) with respect to the measures mPt and mRα for any t > 0 and α > 0. From ∞ e−αt mPt (B)dt = mRα (B) = 0, 0

we have mPt (B) = 0 for almost every t > 0 and so m(B) = 0 by Lemma A.2.11 again. (iii) By the Fubini theorem and the excessiveness of m, ∞ ∞ 1B (Xs )ds = mPt (B)dt = 0. Em 0

0

Since B is finely open, it follows from Lemma A.2.10 and the strong Markov property of X that   {Xt ∈ B, ∀t ∈ (r1 , r2 )} = 0. Pm (σB < ∞) ≤ Pm  r1 0. Choose q1 , q2 ∈ bB+ (E) such that

q1 ≤ q ≤ q2

and

mRα (q1 < q2 ) = 0.

Then Rα q1 ≤ Rα q ≤ Rα q2 and Rα q1 = Rα q2 [m]. Since by Theorem A.2.13, Rα qi , i = 1, 2, are Borel measurable α-excessive functions, we have Rα q1 = Rα q2 q.e. on E. Next, for a general α-excessive function f , by Lemma A.2.1 there is a ∗ (E) such that Rα fn ↑ f as n ↑ ∞. Then for each n ≥ 1, sequence { fn } ⊂ bB+ we can find Borel measurable α-excessive functions gn and hn such that gn ≤ Rα fn ≤ hn and gn = hn q.e. Put g¯ := lim infn→∞ gn and h¯ := lim infn→∞ hn . ¯ ¯ g¯ = h¯ q.e. and for every t > 0, e−αt Pt g¯ ≤ g¯ and e−αt Pt h¯ ≤ h. Clearly g¯ ≤ f ≤ h, ¯ respectively, which Let g and h be the α-excessive regularization of g¯ and h, ¯ are Borel measurable α-excessive functions satisfying g ≤ g¯ ≤ f ≤ h ≤ h. But for each x ∈ E, e−αt Pt g¯ (x) is a decreasing function of t and hence possesses at most a countable number of discontinuous points. It is easy to see that Pt g(x) = Pt g¯ (x) for every continuity point t of Pt g¯ (x). Hence Rα g = Rα g¯ and consequently m(g < g¯ ) = 0 by Theorem A.2.13(ii). Furthermore, m(¯g < h) = 0 by Theorem A.2.13(ii). Thus m(g < h) = 0, which implies g = h q.e. by Theorem A.2.13(iv). Finally, when f is a (0-)excessive function, f is 1n -excessive for each n ≥ 1, and by the preceding result, there exist Borel measurable 1n -excessive functions gn , hn with gn ≤ f ≤ hn

and

gn = hn q.e.

Put g¯ := lim infn→∞ gn and h¯ := lim infn→∞ hn . Then ¯ g¯ ≤ f ≤ h,

g¯ = h¯ q.e.,

Pt g¯ ≤ g¯ ,

Pt h¯ ≤ h¯ for t > 0.

In the same way as above, the excessive regularization g and h of g¯ and h¯ are the desired functions. T HEOREM A.2.15. Any m-polar set is contained in an m-inessential Borel set. Proof. Let B be an m-polar set. By Theorem A.2.13(i), we may assume that B ∈ B(E). If we put ϕ(x) = Px (σB < ∞), x ∈ E, then ϕ = 0 [m]. Since ϕ is excessive, by Lemma A.2.14 there exists a Borel measurable function g with ϕ ≤ g = 0 [m]. Then B := B ∪ C

where C = {x ∈ E : g(x) > 0}

is a desired set. In fact, B is a Borel set with m( B) = 0 by Theorem A.2.13(ii). It remains to show that Bc = Bc ∩ {g = 0} is X-invariant. Take

422

APPENDIX A

any x ∈ Bc . For x ∈ E, since g(x) = 0, we have ϕ(x) = Px (σB < ∞) = 0. Recall our convention g(∂) = 0. Since Ex (g(Xt )) = Pt g(x) ≤ g(x) = 0, the right continuity of g(Xt ) yields Px (g(Xt ) = 0 for every t ≥ 0) = 1, which implies Px (σC < ∞) = 0 and consequently Px (σB < ∞) = 0. D EFINITION A.2.16. The following (AC) and (AC) are called the absolute continuity condition for the transition function {Pt ; t ≥ 0} and the resolvent kernel {Rα ; α > 0}, respectively. (AC) for any t > 0, x ∈ E, the measure Pt (x, ·) is absolutely continuous with respect to m. (AC) For some fixed α > 0 and for any x ∈ E, the measure Rα (x, ·) is absolutely continuous with respect to m. T HEOREM A.2.17. (i) (AC) implies (AC) . (ii) Condition (AC) holds if and only if every m-polar set is polar. (iii) Assume condition (AC) . If f and g are α-excessive for some α ≥ 0 and f ≥ g [m], then f ≥ g. Proof. (i) is trivial. (ii) For any nearly Borel measurable m-polar set B and x ∈ E, let ϕ(x) := Px (σB < ∞). Then ϕ = 0 [m] and so, under the condition (AC) , we have for any x ∈ E ϕ(x) = lim β Rβ (x, dy)ϕ(y) = 0, β→∞

E

namely, B is polar. Conversely, assume that every m-polar set is polar. If B ∈ B(E) satisfies m(B) = 0, then by the excessiveness of m we have, for α > 0, mRα (B) = 0 and hence Rα (·, B) = 0 [m]. On account of Theorem A.2.13, the set N = {x ∈ E : Rα (x, B) > 0} is m-polar and hence polar. So for any x ∈ E, Rα (x, B) = / N] = 0. So condition (AC) holds. limt↓0 Ex [Rα (Xt , B); Xt ∈ (iii) Under the stated conditions, we have for any x ∈ E f (x) = lim β Rα+β (x, dy)f (y) β→∞

E

≥ lim β β→∞

Rα+β (x, dy)g(y) = g(x). E

L EMMA A.2.18. (i) For any B ∈ Bn (E), B \ Br is semipolar. (ii) If B ⊂ E is semipolar, then Xt ∈ B occurs for at most countable many t ≥ 0 a.s. Proof. See [13, II, (3.3)–(3.4)].

423


C OROLLARY A.2.19. Let α ≥ 0 and B be a semipolar set. If f , g are α-excessive functions on E and f (x) = g(x) for every x ∈ E \ B, then they are equal identically. Proof. Since B is of potential zero by Lemma A.2.18, βRβ+α f and βRβ+α g are equal identically for any β > 0. It then suffices to let β → ∞.

A.3. ADDITIVE FUNCTIONALS OF RIGHT PROCESSES A.3.1. Revuz Measures of Positive Continuous Additive Functionals We continue to consider a Borel right process X = (, M, Xt , ζ , Px ) on a Lusin space (E, B(E)) with transition function {Pt ; t ≥ 0}. Let us fix a σ -finite excessive measure m of {Pt ; t ≥ 0}. Any numerical function f on E is extended to E∂ by setting f (∂) = 0. D EFINITION A.3.1. A numerical function At (ω) of two variables t ≥ 0, ω ∈ is called an additive functional of X if there exist ∈ F∞ and an m-inessential set N ⊂ E with Px () = 1 for x ∈ E \ N

and

θt ⊂

for t > 0,

(A.3.1)

and the following conditions are satisfied: (A.1) For each t ≥ 0, At | is Ft | -measurable. (A.2) For any ω ∈ , A· (ω) is right continuous on [0, ∞) has the left limits on (0, ζ (ω)), A0 (ω) = 0, |At (ω)| < ∞ for t < ζ (ω), and At (ω) = Aζ (ω) (ω) for t ≥ ζ (ω). Further, the additivity At+s (ω) = At (ω) + As (θt ω) for every t, s ≥ 0,

(A.3.2)

is satisfied. We call and N in the above definition a defining set and an exceptional set of At (ω), respectively. If in particular the exceptional set N can be taken to be empty, in other words, if a defining set satisfies Px () = 1 for every x ∈ E, then the additive functional At (ω) is said to be an additive functional in the strict sense. Two additive functionals A, B are called m-equivalent if Pm (At = Bt ) = 0 and we write A ∼ B in this case.

for every t > 0,

(A.3.3)

424

APPENDIX A

L EMMA A.3.2. If additive functionals A and B are m-equivalent, then there are a common defining set and a common Borel exceptional set N such that At (ω) = Bt (ω)

for every t ≥ 0 and ω ∈ .

(A.3.4)

Proof. Denote the defining sets and exceptional sets of additive functionals A and B by A , NA , B , and NB , respectively. Put N0 := NA ∪ NB , 0 := A ∩ B , 1 := {At = Bt for every t > 0}, and := 0 ∩ 1 . Then N0 is m-inessential and it is easy to see that θt () ⊂ for every t > 0. Moreover, Px (c0 ) = 0 for x ∈ E \ N0 . Hence Px (c ) ≤ Px (c0 ) + Px (0 \ 1 ) = Px (0 \ 1 ). On account of the right continuity and the m-equivalence of A and B, g(x) = Px (0 \ 1 ) vanishes m-a.e. on E. Further, we can easily see that g|E\N0 is an excessive function of the restricted right process XE\N0 . Applying Theorem A.2.9 and Theorem A.2.13 to the right process XE\N0 , we get g = 0 q.e. on E \ N0 , and accordingly, g(x) = 0 for every x ∈ E \ N1 for some m-polar set N1 ⊃ N0 . By virtue of Theorem A.2.15, there exists a Borel m-inessential set N containing N1 . and N have the desired properties of this lemma. An additive functional A is called finite càdlàg if for every ω in its defining set , t → At (ω) is a real-valued right continuous function of t ∈ [0, ∞) possessing the left limits on (0, ∞). In Section A.3.3, we shall deal with finite càdlàg additive functionals in the strict sense. An additive functional A is called a positive continuous additive functional (PCAF in abbreviation) if for every ω in its defining set , t → At (ω) is a [0, ∞]-valued continuous function of t ≥ 0. In the rest of this section, we shall be exclusively concerned with PCAFs. Denote by A+ c the totality of PCAFs of + X. What we are interested in is the family of equivalence classes A+ c / ∼ of Ac . A typical example of a PCAF is given by t f (Xs (ω))ds, t ≥ 0, ω ∈ , (A.3.5) At (ω) = 0

where f ∈ bB+ (E). In this case, At (ω) is a PCAF in the strict t+ssense with f (Xu )du = as its defining set and its additivity (A.3.2) follows from t s 0 f (Xu (θt ω))du. If we let f equal 1 identically on E, then we have the simplest PCAF, At = t ∧ ζ . ∗ For A ∈ A+ c and f ∈ B+ (E), we define f · A by ( f · A)t (ω) =

t

f (Xs (ω))dAs (ω), 0

t ≥ 0, ω ∈ .

425

ESSENTIALS OF MARKOV PROCESSES + Exercise A.3.3. Show that if A ∈ A+ c and f ∈ bB+ (E), then f · A ∈ Ac .

L EMMA A.3.4. (i) For A ∈ A+ c , let ϕ(t) = Em [At ], t ≥ 0. If ϕ(t) is finite for some t > 0, then it is finite for all t > 0. In this case, ϕ(t) is a continuous concave function of t ∈ [0, ∞). (ii) For A ∈ A+ c , f ∈ B+ (E), 1 Em [( f · A)t ] is increasing as t decreases to 0. t Proof. (i) If we put ct (x) = Ex [At ], then by (A.3.2) ϕ(t + s) = ϕ(t) + mPt , cs ,

t, s ≥ 0.

(A.3.6)

(A.3.7)

Since m is excessive, ϕ(t + s) ≤ ϕ(t) + ϕ(s), t, s ≥ 0. Combining this with the fact that ϕ(t) is non-decreasing in t, we get the first conclusion of (i). Next suppose ϕ(t) < ∞ for every t ≥ 0. Then by the dominated convergence theorem, ϕ(t) is continuous in t ≥ 0. Due to the excessiveness of m, the measure mPt decreases in t and hence, for 0 < t < t and s > 0, we have from (A.3.7) ϕ(t + s) − ϕ(t ) ≤ ϕ(t + s) − ϕ(t). By taking t = t + s, 1 (ϕ(t + 2s) + ϕ(t)) ≤ ϕ(t + s) for s, t > 0. 2 Thus, ϕ is midpoint concave and, since it is continuous, ϕ is concave. + (ii) Let A ∈ A+ c . If f ∈ bB+ (E), then f · A ∈ Ac by Exercise A.3.3. We put ϕ(t) = Em [( f · A)t ]. If ϕ(t) < ∞ for some t > 0, then ϕ is concave on [0, ∞) by (i) and ϕ(0) = 0. Hence, for 0 < s < t, we have ϕ(s) ≥ st ϕ(t), namely, (A.3.6) holds. The monotonicity is trivially true when ϕ(t) = ∞ for every t > 0. For a general f ∈ B+ (E), we put fn (x) = f (x) ∧ n, x ∈ E. Then Em [( fn · A)t ] ↑ Em [( f · A)t ] as n ↑ ∞, and consequently (A.3.6) for fn implies the same for f . T HEOREM A.3.5. (i) For A ∈ A+ c , there exists a unique measure µA on (E, B(E)) satisfying t 1 f (x)µA (dx) = lim Em f (Xs )dAs , ∀f ∈ B+ (E). (A.3.8) t↓0 t 0 E + (ii) If A, B ∈ A+ c , A ∼ B, then µA = µB . For A ∈ Ac , µA charges no semipolar set or m-polar set. + (iii) For A ∈ A+ c , f ∈ bB+ (E), the measure corresponding to f · A ∈ Ac in the sense of (i) equals f · µA . (iv) It holds for A ∈ A+ c , f ∈ B+ (E) that ∞ −αt f (x)µA (dx) = lim αEm e f (Xt )dAt . (A.3.9) E

α→∞

0

426

APPENDIX A

Proof. (i) By virtue of Lemma A.3.4, the limit on the right hand of (A.3.8) exists. For B ∈ B(E), we define µA (B) to be this value with f = 1B . By interchanging the order of taking increasing limits, we get the complete additivity of µA as well as the identity (A.3.8). (ii) is clear from Lemma A.3.2, property of a semipolar set in Lemma A.2.18 and Definition A.2.12 of an m-polar set. (iii) follows from (i). (iv) We may assume that f ∈ bB+ (E). We put ϕ(t) = Em [( f · A)t ]. Using integration by parts and taking expectation, we get t t e−αs f (Xs )dAs = Em e−αs d( f · A)s Em 0

0

= e−αt ϕ(t) + α

t

e−αs ϕ(s)ds.

0

On account of Lemma A.3.4, if ϕ(t) is finite for some t > 0, then so it is for every t > 0 and ϕ(t) ≤ ϕ(1) · t for t ≥ 1. Therefore, we have in this case ∞ ∞ −αs 2 e f (Xs )dAs = α e−αs ϕ(s)ds < ∞. (A.3.10) αEm 0

0

As α → ∞, the right hand side increases to ∞ for every t > 0, (A.3.9) is trivial.

E

fdµA by (A.3.8). When ϕ(t) =

The measure µA determined by (A.3.8) is called the Revuz measure of A ∈ A+ c . This measure was introduced by D. Revuz [128] in 1970. The Revuz measure of the simplest PCAF At = t ∧ ζ equals m by Lemma A.2.11. Hence, for f ∈ bB+ (E), the Revuz measure of the PCAF given by (A.3.5) is equal to f · m. However, the Revuz measure of A ∈ A+ c is not generally absolutely continuous with respect to the reference excessive measure m. A.3.2. Time Change and Killing of a Borel Right Process Let X = (, M, Xt , ζ , Px ) be a Borel right process on a Lusin space (E, B(E)). In this subsection, we consider a positive continuous additive functional in the strict sense At (ω) of X and formulate the time change Xτt of Xt with respect to the right continuous inverse τt (ω) of A which lives on the support F of A defined below. We do not consider a reference excessive measure m for X or an exceptional set N for A. In the case that we admit an exceptional set N for A, we can apply the consideration of the present subsection to the restricted right process X|E\N for which A can be regarded as a PCAF in the strict sense. At the end of this subsection, we shall also give a definition of the canonical subprocess of X with respect to the multiplicative functional e−At for a PCAF A in the strict sense.

427


Thus we assume that At (ω) satisfies conditions of Definition A.3.1 with N = ∅ and that At (ω) is a [0, ∞]-valued continuous function in t ≥ 0 for every ω in the defining set ⊂ . Further, by replacing (, M) with (, M| ) and by restricting (Xt , ζ , Px ) to this measurable space, we may and do assume that the defining set of At (ω) is itself. Define R(ω) = inf{t > 0 : At (ω) > 0},

F = {x ∈ E : Px (R = 0) = 1}.

(A.3.11)

Observe that {R = 0} ∈ F0+ = F0 so, by Blumenthal’s 0-1 law (Lemma A.2.5), Px (R = 0) is either 0 or 1 for x ∈ E. The set F is called the

support of the PCAF A. If we introduce a function ϕA by ϕA (x) = Ex e−R , x ∈ E, then F = {x ∈ E : ϕA (x) = 1}. Since t + R(θt ω) ↓ R(ω), t ↓ 0, ϕA is 1-excessive and consequently F is a nearly Borel finely closed set. Denote by σF the hitting time of F and by p1F the 1-order hitting probability of F : p1F (x) = Ex [e−σF ], x ∈ E. For each ω ∈ , we introduce the time sets I(ω) = {t : At+ε (ω) − At (ω) > 0, for every ε > 0} and Z(ω) = {t : Xt (ω) ∈ F}. I(ω) is the set of right increasing points of A· (ω). P ROPOSITION A.3.6. For every x ∈ E, Px (R = σF ) = 1,

(A.3.12)

Px (I ⊂ Z) = 1,

(A.3.13)

Px (At = (1F · A)t

for every t > 0) = 1.

(A.3.14)

Further, each point of F is regular for F. Proof. Due to the right continuity of ϕA (Xt ), it holds for any x ∈ E that Px (XσF ∈ F) = 1 and hence Px (σF < R) = Px (σF < R, R ◦ θσF > 0)

= Ex PXσF (R > 0); σF < R = 0. We next show Px (R < σF ) = 0 for x ∈ F r ∪ (E \ F).

428

APPENDIX A

This is trivially true for x ∈ F r . If x ∈ E \ F, then for any t > 0

Px (R < σF ) = Px (AR+t > 0, R < σF ) = Ex PXR (At > 0); R < σF

≤ Ex PXR (R < t); XR ∈ E \ F , which tends to zero as t ↓ 0. Hence we have Px (σF ≥ R) = 1 for all x ∈ E and p1F (x) = ϕA (x) for x∈ / F \ F r . Since both ϕA and p1F are 1-excessive and the set F \ F r is semipolar, we see from Lemma A.2.18 and Corollary A.2.19 that the last identity holds for all x ∈ E and consequently (A.3.12) is valid. In particular, p1F (x) = ϕA (x) = 1 for every x ∈ F, and so each x ∈ F is regular for F. Next, the continuity of t → At and (A.3.12) imply that for every x ∈ E, {σF ◦ θt > 0} Px (I ⊂ Z) ≤ Px t∈I

 ≤ Px 

 {∃ q ∈ Q+ with q > r, Aq − Ar > 0, r + σF ◦ θr > q}

r∈Q+

≤

Ex PXr (R < σF ) = 0,

r∈Q+

yielding (A.3.13). Since the set I(ω) of right increasing times of At (ω) differs from the set of its increasing times by at most a countable set and At (ω) is continuous, we can get from (A.3.13) that, Px -a.s. for any x ∈ E, At (ω) = dAs (ω) [0,t]∩I(ω)

=

1F (Xs (ω))dAs (ω) = (1F · A)t (ω). [0,t]∩I(ω)

This completes the proof of Proposition A.3.6.

Here we quote a change of variable formula from [13, V.(2.2)]. Let a(t) be a right continuous, non-decreasing function from [0, ∞] to [0, ∞] with a(0) = 0, a(∞) = limt→∞ a(t). We define τ (t) = inf{s : a(s) > t}, t ≥ 0,

inf ∅ = ∞.

(A.3.15)

The function τ : [0, ∞) → [0, ∞] will be called the inverse of a. It is right continuous and non-decreasing.

429


L EMMA A.3.7. (i) For any non-negative Borel measurable function f on [0, ∞] with f (∞) = 0, ∞ ∞ f (t)da(t) = f (τ (t))dt. (A.3.16) 0

0

(ii) Suppose a is continuous, then  max{s : a(s) = t} for t ∈ [0, a(∞)), τ (t) = ∞ for t ≥ a(∞). In particular, τ is strictly increasing on [0, a(∞)) and a(τ (t)) = t for t ∈ [0, a(∞)). Now, for the PCAF At (ω) of the Borel right process X = (, M, Xt , θt , ζ , Px ), its inverse is defined for each ω ∈ by  inf{s : As (ω) > t} for t < Aζ (ω)− (ω), τt (ω) = ∞ for t ≥ Aζ (ω)− (ω).

(A.3.17)

We then let Xˇ t (ω) = Xτt (ω) (ω),

ζˇ (ω) = Aζ (ω)− (ω),

t ≥ 0, ω ∈ .

(A.3.18)

Recall that X∞ (ω) is defined to be ∂ so Xˇ t (ω) = ∂ for t ≥ ζˇ (ω). The support F of A defined by (A.3.11) is nearly Borel measurable and hence F ∈ B∗ (E). We denote the set F ∪ {∂} by F∂ , which is regarded as a topological subspace of E∂ . P ROPOSITION A.3.8. (i) For each s ≥ 0, τs is an {Ft }-stopping time. (ii) τs+t (ω) = τs (ω) + τt (θτs (ω) (ω)), s ≥ 0, t ≥ 0, ω ∈ . (iii) Define Fˇ t = Fτt ,

t ≥ 0.

(A.3.19)

{Fˇ t } is then a right continuous filtration and Xˇ t ∈ Fˇ t /B∗ (E∂ ) for every t ≥ 0. ˇ (iv) Px Xt ∈ F∂ for every t ≥ 0 = 1 for every x ∈ E. (v) If η is an {Fˇ t }-stopping time, then τη is an {Ft }-stopping time. (vi) If B ⊂ E is nearly Borel measurable with respect to X, then its hitting time σˇ B = inf{t > 0 : Xˇ t ∈ B} by the process {Xˇ t } is an {Fˇ t }-stopping time.

430

APPENDIX A

Proof. (i) For any u > 0, {τs < u} = ∪∞ n=1 {Au− 1n > s} ∈ Fu . (ii) τs + τt ◦ θτs = τs + inf{u : Au ◦ θτs > t}. If τs < ∞, then Au ◦ θτs = Au+τs − Aτs = Au+τs − s in view of Lemma A.3.7(ii). Hence τs + τt ◦ θτs = inf{u + τs : u ≥ 0, Au+τs > s + t} = τs+t . (iii) Since {τt , t ≥ 0} is a non-decreasing sequence of {Ft }-stopping times, {Fˇ t } are clearly non-decreasing. If ∈ ∩n Fˇ t+ 1 , by the right continuity of τt , n we have for any u > 0 { ∩ {τt+ 1 < u} ∈ Fu , ∩ {τt < u} = n

n

which means that ∈ Fˇ t , yielding the right continuity of {Fˇ t }. The second assertion follows from Exercise A.1.20. (iv) On account of Lemma A.3.7(ii), we see for every ω ∈ that the time set {t < ∞ : τu (ω) = t for some u} coincides with the time set I(ω) of right increase of At (ω). Therefore, the assertion follows from (A.3.13). (v) Since {η < t} ∈ Fˇ t for any t > 0, we have {η < t} ∩ {τt < s} ∈ Fs for any s > 0 and {ω : η(ω) < r} ∩ {ω : τr (ω) < s} ∈ Fs . {ω : τη(ω) (ω) < s} = r∈Q+

(vi) Since {Xˇ t } is a right continuous {Fˇ t }-adapted process and Fˇ t is equal to ˇ Pµ according to Exercise A.1.20(ii), the proof of Theorem A.1.19 µ∈P(E∂ ) (Ft ) shows that σˇ B is an {Fˇ t }-stopping time provided that B ∈ B(E). For B ∈ Bn (E) and µ ∈ P(E), the sets B1 , B2 ∈ B(E) in Definition A.1.28 satisfy Pµ (Xˇ t ∈ B2 \ B1 for some t ≥ 0) = 0. This means that for each t ≥ 0, t = {σˇ B ≤ t} differs from {σˇ B1 ≤ t} ∈ Fˇ t only by Pµ -null set and t ∈ (Fˇ t )Pµ . Hence t ∈ Fˇ t . We finally let 0 = {ω ∈ : Xˇ t (ω) ∈ F∂ for every t ≥ 0}.

(A.3.20)

Since the set E \ F is nearly Borel measurable with respect to X and 0 = {ω ∈ : σˇ E\F (ω) = ∞}, we see that 0 ∈ F∞ by Proposition A.3.8(vi) and Px (0 ) = 1 for any x ∈ E by Proposition A.3.8(iv). The restrictions to 0 of those functions on will be denoted by the same notations. The traces on 0 of the σ -fields M, Ft , t ∈ [0, ∞], will also be denoted by the same notations. We then redefine Fˇ t , t ≥ 0, by (A.3.19). T HEOREM A.3.9. Xˇ = 0 , M, Xˇ t , θτt , ζˇ , {Px }x∈F∂ is a strong Markov process on ( F∂ , B∗ ( F∂ )) satisfying condition (X.6)r . Proof. Since τ0 = R, it holds for x ∈ F∂ that Px (Xˇ 0 = x) = Px (X0 = x) = 1. For B ∈ B∗ ( F∂ ), Px (Xˇ t ∈ B) is B∗ ( F∂ )-measurable in x ∈ F∂ by virtue of Proposition A.3.8(iii) and Exercise A.1.20(i). Thus Xˇ satisfies conditions (X.1), (X.2), (X.4), and (X.5) in Definition A.1.8.

431


Obviously Xˇ t , ζˇ , θτt satisfy the condition (X.6)r of Section A.1.3. {Fˇ t } is a right continuous admissible filtration for Xˇ t by Proposition A.3.8(iii). Let η be an {Fˇ t }-stopping time. By Proposition A.3.8(ii), we have τη+t = τη + τt ◦ θτη and Xˇ η+t = Xτt ◦ θτη . On account of Proposition A.3.8(v) and (A.1.23), it then holds for any f ∈ bB ∗ ( F∂ ) and µ ∈ P( F∂ ) that ,

Eµ f (Xˇ η+t )Fˇ η = Eµ f (Xτt ◦ θτη )Fτη

= EXτη f (Xτt ) = EXˇ η [ f (Xˇ t )],

Pµ -a.s.,

ˇ proving the strong Markov property of X.

Xˇ is called the time-changed process of the Borel right process X by its PCAF A. The transition function {Pˇ t ; t ≥ 0} and the resolvent kernel {Rˇ p ; p ≥ 0} of Xˇ are given for f ∈ bB ∗ ( F), x ∈ F, by ∞

Pˇ t f (x) = Ex f (Xˇ t ) , Rˇ p f (x) = Ex e−pt f (Xˇ t )dt . (A.3.21) 0

Here any numerical function f on F is extended to F∂ by setting f (∂) = 0 as usual. They are a transition function and a resolvent kernel on ( F, B∗ ( F)) in the sense of Definition 1.1.13, respectively. α , RAα on (E, B∗ (E)) by For α ≥ 0, p ≥ 0, we define kernels Up,A ∞ e−αt−pAt f (Xt )dAt (A.3.22) Up,α A f (x) = Ex 0

and

RAα f (x) = Ex

∞

e−αt−At f (Xt )dt

(A.3.23)

0

∗ for x ∈ E and f ∈ B+ (E).

L EMMA A.3.10. The following identities hold for p > 0 and α > 0. Rˇ p f (x) = Up,0 A f (x),

x ∈ F, f ∈ bB ∗ ( F).

(A.3.24)

For x ∈ E and f ∈ bB ∗ (E), 0 Up,0 A f (x) − Up,α A f (x) − αRpA α (Up, A f )(x) = 0,

where

UAα

:=

(A.3.25)

UAα f (x) − Up,α A f (x) − pUAα Up,α A f (x) = 0,

(A.3.26)

RAα f (x) − Rα f (x) + UAα RAα f (x) = 0,

(A.3.27)

α U0,A .

432

APPENDIX A

Proof. Identity (A.3.24) follows from Lemma A.3.7. Equation (A.3.25) folt lows from 1 − e−αt = α 0 e−αs ds. Equation (A.3.26) can be derived from the t ∞

expression Up,α A f (x) = Ex 0 e−ατt −pt f (Xˇ t )dt and 1 − e−pt = pe−pt 0 eps ds. Equation (A.3.27) appears in Exercise 4.1.2. T HEOREM A.3.11. ( F, B∗ ( F)).

The time-changed process Xˇ is a right process on

Proof. In Theorem A.3.9, we proved that Xˇ = (Xˇ t , {Px }x∈F ) is a strong Markov process on ( F, B∗ ( F)) with the property (X.6)r . In view of Theorem A.1.37, it suffices to show that the process Rˇ p f (Xˇ t ) is Pµ -indistinguishable from an optional process for any f ∈ Cu+ ( F) and any µ ∈ P( F) ⊂ P(E). For any α > 0 and f ∈ bB+ (E), let g = RAα f . By (A.3.27), UAα g is then finite on E. Combining (A.3.25), (A.3.26), and (A.3.27), we see that Up,0 A g can be α g3 for g1 , g2 , g3 ∈ bB+ (E), a linear expressed as UAα g + Rα g1 − UAα g2 − UpA combination of four finite α-excessive functions, and consequently Up,0 A g(Xt ) is right continuous in t ≥ 0 Pµ -a.s. by virtue of Theorem A.2.2. Therefore, so is the process Rˇ p g(Xˇ t ) = Rˇ p g(Xτt ) on account of (A.3.24). Now take any f ∈ Cu+ (E). We have seen that the process Rˇ p gα (Xˇ t ) is Pµ -indistinguishable from an optional process for gα = αRAα f . If we let α → ∞, the process converges to Rˇ p f (Xˇ t ), which must enjoy the desired property accordingly. Finally, we take from [13, III.3] a definition of the canonical subprocess of a Borel right process X = (, M, Mt , ζ , Xt , θt , Px ) on a Lusin space (E, B(E)) with respect to the multiplicative functional e−At for a positive continuous additive functional A of X in the strict sense. Without loss of generality, we assume that the defining set of A is itself. We also make a convention that A∞ (ω) = 0 for every ω ∈ . = × [0, ∞] and write . Let R Let ω = (ω, λ) for the generic point in = M × R. Define be the Borel σ -field of [0, ∞] and set M Xt (ω) if t < λ and ζ ( ω) = ζ (ω) ∧ λ. (A.3.28) ω) = Xt ( ∂ if t ≥ λ θt ω = (θt ω, (λ − t) ∨ 0) so Note that ζ = inf{t > 0 : Xt = ∂}. Further, define t be the space t = × (t, ∞] and M θh = Xt+h for t, h ≥ 0. Let that Xt ◦ for which there exists a ∈ Mt such that ∈M that consists of all sets ∩ t = × (t, ∞]. For each ω ∈ , there is a unique probability measure αω on ([0, ∞], R) denote by ∈ M, ω = {λ : (ω, λ) ∈ } such that αω ([t, ∞]) = e−At (ω) . For

433


its ω-section and define

) = Ex αω ( ω ) . Px (

x ∈ E.

(A.3.29)

ω = (ω, λ) : Xt ( ω) ∈ B} = {ω : Xt (ω) ∈ B} × In particular, for B ∈ B∗ (E), { (t, ∞] and we get from (A.3.29)

Px ( Xt ∈ B) = Ex e−At ; Xt ∈ B . Pt (x, B) = We refer the reader to [13, III. (3.3)] for a proof of the next lemma. M t, , M, X = ( Xt , ζ , θt , Px ) is a Markov process on L EMMA A.3.12. (E, B∗ (E)) with transition function determined by

Pt f (x) = Ex f (Xt ) e−At , t ≥ 0, x ∈ E, f ∈ bB(E). (A.3.30) M t, , M, T HEOREM A.3.13. (i) X = ( Xt , ζ , θt , Px ) is a right process on ∗ (E, B (E)). (ii) If X is a Hunt process on (E, B(E)), then X is a Hunt process on E with the B(E)-measurability of the transition function being weakened to B∗ (E)measurability. Proof. (i) By Lemma A.3.12, X is a Markov process on (E, B∗ (E)) whose ∞ −αt Pt f (x)dt, x ∈ E, coincides with the function resolvent kernel Rα f (x) = 0 e A Rα f (x), x ∈ E, defined by (A.3.23). Obviously X satisfies the property (X.6)r . On account of Theorem A.1.37, it only remains to show that u( Xt ) is right continuous in t ≥ 0 Pµ -a.s. for u = RAα f , f ∈ bB+ (E). But in view of (A.3.27), u is then a difference of two bounded excessive functions with respect to X and hence the process u(Xt (ω)) is right continuous in t ≥ 0 Pµ -a.s. by virtue of Theorem A.2.2. Choose 1 ∈ M such that Pµ (1 ) = 1 and this statement is true for all ω ∈ 1 . ω)) is equal to u(Xt (ω)) when t < ζ ( ω) and to 0 otherwise, it is Since u( Xt ( Pµ -measure 1. right continuous in t ≥ 0 for any ω ∈ 1 × [0, ∞] which is of (ii) It can be shown that the quasi-left-continuity of X is inherited by X (cf. [13, III, (3.13)]). X is called the canonical subprocess of X with respect to the multiplicative functional e−At . It is also called the process obtained from X by killing with respect to the positive continuous additive functional A. A.3.3. Review of Martingale Additive Functionals In this subsection, we list some basic facts on martingale additive functionals in the strict sense and related additive functionals in the strict sense that are utilized in the main text. They can be deduced from the corresponding facts

434

APPENDIX A

in the general theory of square integrable martingales and this deduction is summarized in [73, A.3]. An excellent reference that contains more details is [34]. Let X = (, M, {Xt }, θt , ζ , {Px }x∈E∂ ) be a Hunt process on a Lusin space (E, B(E)) and {Ft } be its minimum augmented admissible filtration. {Ft } is quasi-left-continuous by Theorem A.1.40. A statement concerning ω ∈ is said to hold a.s. if it is true Px -a.s. for every x ∈ E. By a stochastic process, we mean in this subsection a real-valued {Ft }-adapted process whose sample paths are right continuous on [0, ∞) and have left limits on (0, ∞) a.s. Two stochastic processes are called equivalent if they are Px -indistinguishable for every x ∈ E. In this case, we regard them as identical. We call a stochastic process {At } an additive functional (AF in abbreviation) if almost surely As+t = As + At ◦ θs for every s, t ≥ 0, and At = Aζ for every t ≥ ζ . Thus an additive functional in the present sense is a finite càdlàg additive functional in the strict sense in the terminology of Section A.3.1. Let us introduce several classes of additive functionals: V+ := {AF Z : Zt ≥ 0 for every t ≥ 0 a.s.} , V := V+ − V+ = {A − B : A, B ∈ V+ }, . A = A ∈ V : Ex

,

t

/

-

|dA|s < ∞ for every t > 0 and x ∈ E ,

0

PA = {A ∈ A : A is predictable}, CA = {A ∈ A : A is continuous}, M = AF M : Ex [Mt2 ] < ∞, Ex [Mt ] = 0 for all t ≥ 0 and x ∈ E , Mc = {M ∈ M : M is continuous}. See Section A.2.1 for the definition of a stochastic process being predictable. Obviously CA ⊂ PA ⊂ A. M ∈ M is called a martingale additive functional (MAF in abbreviation) because it is easily seen to be a martingale with respect to ({Ft }, Px ) for every x ∈ E. The space M is equipped with the seminorm ηt,x = Ex [Mt2 ], t > 0, x ∈ E. (1) For any M ∈ M, there exists a unique M ∈ CA+ such that Ex [Mt ] = Ex [Mt2 ],

∀t > 0, ∀x ∈ E,

435


or, equivalently, Mt2 − Mt is a Px -martingale for every x ∈ E. M is called the predictable quadratic variation of M. For M, N ∈ M, we define their covariation by M, N :=

1 (M + N − M − N) ∈ CA. 2

(2) For any A ∈ A, there exists a unique Ap ∈ PA such that A − Ap is a Px martingale for every x ∈ E. Ap is called the dual predictable projection of A. (3) We let Md = {M ∈ M : M, N = 0 for every N ∈ Mc }. Any M ∈ M admits a unique decomposition M = Mc + Md ,

M c ∈ Mc ,

M d ∈ Md .

M c is called the continuous part of M, while M d is called the purely discontinuous part of M. It is known (see [85]) that M d admits the expression p Mtd = lim Bεt − (Bε )t , ε→0

t ≥ 0,

where Bεt := 0<s≤t (Ms − Ms− )1{|Ms −Ms− |>ε} , and the convergence is in probability as well as in L2 with respect to Px for every x ∈ E. By a stopping time, we mean an {Ft }-stopping time. A stopping time T is said to be predictable if there exists an increasing sequence of stopping times {Tn , n ≥ 1} such that Tn < T and limn→∞ Tn = T a.s. on {T > 0}. A stopping time T is called totally inaccessible if, for any predictable stopping time S, Px (T = S < ∞) = 0 for every x ∈ E. If T is a stopping time such that XT− = XT a.s. on {T < ∞}, then T is totally inaccessible owing to the quasi-left-continuity of X. Moreover, we can find a sequence of totally inaccessible stopping times {Tn } such that {(t, ω) ∈ (0, ∞) × : Xt = 0} =

∞

{(Tn (ω), ω) : ω ∈ },

n=1

the right hand side being a disjoint union. For A ∈ A, Ap ∈ CA if and only if Ex [AT ] = Ex [AT− ] for every x ∈ E and every predictable stopping time T. In particular, if A ∈ A jumps only when X jumps, namely, the set {(t, ω) : At (ω) = 0} is contained in the left hand side of the above identity, then Ap ∈ CA.

436

APPENDIX A

(4) For M ∈ M, we define [M] ∈ A+ by (Ms )2 , [M]t = M c t +

t ≥ 0,

s≤t

which is called the quadratic variation of M. Here Ms := Ms − Ms− . For M, N ∈ M, we let [M, N] = 12 ([M + N] − [M] − [N]). It then holds that [M]p = M,

M ∈ M.

Furthermore, for M, N ∈ M, the approximation 2 n

[M, N]t = lim

n→∞

n Mtjn − Mtj−1

n Ntjn − Ntj−1 ,

tjn = jt/2n ,

j=1

takes place in L1 (, Px ) for every x ∈ E. Moreover, the above convergence is uniform in probability with respect to Px on every compact time interval for every x ∈ E (cf. [85]). (5) Let M ∈ M. For any stopping time T, define MtT = MT · 1{T≤t} − (MT · 1{T≤t} )p ,

t ≥ 0.

Then M T ∈ Md . stopping times appearing Let {Tn } be the sequence of totally inaccessible Tn M converges to the purely in (3). Then M Tn , M T = 0, n = , and ∞ n=1 discontinuous part M d of M in the topology of M. A.3.4. Lévy system of a Hunt Process We keep the same setting as in the preceding subsection. Let X = (, {Ft }, {Xt }, {θt }, ζ , {Px }x∈E∂ ) be a Hunt process on a Lusin space (E, B(E)), where {Ft }t≥0 is the minimum augmented admissible filtration. In this subsection, we introduce the notion of a Lévy system for X that describes the jump behaviors of the sample path Xt completely. The pair (N, H) of a kernel N(x, dy) on (E∂ , B(E∂ )) and a positive continuous additive functional H in the strict sense of X in the terminology of Section A.3.1 is called a Lévy system of X if the following equation holds for any f ∈ B+ (E∂ × E∂ ) with f (x, x) = 0, x ∈ E∂ , and for every t > 0 and x ∈ E: 1 0 ! t Ex f (Xs− , Xs ) = Ex f (Xs , y)N(Xs , dy) dHs . (A.3.31) 0<s≤t

0

E∂

When H is a specific PCAF arising in relation to a boundary problem of a Markov process, M. Motoo [123] had called such a pair (N, H) a Lévy system,

437


but it was S. Watanabe [149] who first formulated a Lévy system in the above fashion and proved its existence for a general Hunt process X. A kind of the absolute continuity condition called (L) for X was imposed in [149], and was later removed by A. Benveniste and J. Jacod [9] by using a method of Ray compactification. There are many choices of a Lévy system (N, H). For example, if (N, H) is ) for any strictly positive bounded Borel a Lévy system of X, then so is ( N, H function f on E, where t −1 f (Xs )dHs . N (x, dy) := f (x) N(x, dy) and Ht := 0

Nevertheless, according to (A.3.31), the measure N(x, dy)µH (dx) on E × E∂ is uniquely determined by X and its excessive measure m, where µH is the Revuz measure of the PCAF of H relative to m. In this book, we take and fix one Lévy system (N, H) of X and we take its existence for granted. The defining formula (A.3.31) of a Lévy system is known to be equivalent to the following more general one (cf. [137, p. 346]): for any non-negative predictable process {Ys }, any f ∈ B+ (E∂ × E∂ ) with f (x, x) = 0, x ∈ E∂ , and for every x ∈ E∂ , 1 0 ! ∞ Ex Ys f (Xs− , Xs ) = Ex Ys f (Xs , y)N(Xs , dy) dHs . 0

s>0

E∂

(A.3.32) In particular, for any stopping time T and for any non-negative Borel function g(s) on (0, ∞), we can substitute Ys = 1(0,T] (s) · g(s), s ≥ 0, in the above formula to obtain 1 0 g(s)f (Xs− , Xs ) Ex 0<s≤T

T

= Ex

g(s) 0

! f (Xs , y)N(Xs , dy) dHs .

(A.3.33)

E∂

This identity is frequently utilized in the main text of the present book. In fact, the formula (A.3.32) is proved in [137] for a general right process X but by interpreting Xt− as the left limit of X on E relative to a Ray topology induced on E. When X is a Hunt process, however, this limit coincides with the left limit of Xt relative to the original topology (cf. [137, (47.10)]). More generally, (A.3.32) is known to be true relative to the original topology (with a certain interpretation of Xζ − ) for any special standard process X (see [27, §2.4]). If X is a Hunt process, its transition function sends Borel measurable functions into Borel measurable functions, and consequently the kernel N appearing in the formula (A.3.32) can be taken to be a kernel on (E∂ , B(E∂ )) (cf. [137, (17.12) and (73.5)]).

438

APPENDIX A

We mention an important viewpoint on the identity (A.3.31). If the left hand f side of (A.3.31) is finite for every t > 0 and x ∈ E, then At = s≤t f (Xs− , Xs ), t ≥ 0, can be viewed as an element of the class A+ of additive functionals for X introduced in the preceding subsection. Moreover, equation (A.3.31) implies that the positive continuous additive functional (Af )p in the strict sense defined by ! t p f (Xs , y)N(Xs , dy) dHs , t > 0. (A.3.34) (Af )t = 0

E∂

is the dual predictable projection of Af in the sense of (3) of Section A.3.3. A.3.5. Itô’s Formula In this section, we present Itô’s formula for semimartingales. We will restrict our presentation to a subclass of semimartingales that is used in this book. A good reference on this subject is [85]. Let (, F, P) be a probability space with filtration {Ft }t≥0 satisfying the usual condition. In the following, all the processes are assumed to be adapted to the filtration {Ft }t≥0 . Martingale property and dual predictable projections are related to this filtration as well. Let M be a square integrable martingale. Its predictable quadratic variation process M is the unique predictable process A with A0 = 0 so that M 2 − A is a martingale. M can be uniquely decomposed as M = M c + M d , where M c is a continuous square integrable martingale and p t ≥ 0, Mtd = lim Bεt − (Bε )t , Bεt

ε→0

where := 0<s≤t (Ms − Ms− )1{|Ms −Ms− |>ε} , and the convergence is in probability as well as in L2 with respect to P. M c and M d are called the continuous part and purely discontinuous part of M, respectively. For two semimartingales, X = M + A and Y = N + C, where M and N are square integrable martingales and A and C are processes of finite variations with A0 = C0 = 0, we define [X, Y]t = M c , N c t + (Xs − Xs− )(Ys − Ys− ), t ≥ 0, (A.3.35) 0<s≤t

with M c , N c := (M c + N c − M c − N c ). For simplicity, we will define X c := M c and Y c := N c , as the semimartingale decomposition of X and Y into such M + A and N + C is unique. For a square integrable martingale M, both [M] and M are non-decreasing processes and M is the dual predictable projection of [M] ([85, Theorem 6.26]). Thus we have E[Mt2 ] = E[M]t = E[Mt ] for every t ≥ 0. If the square integrable martingale M has Mt = MT for every t ≥ T, where T is an {Ft }t≥0 -stopping time, then Mt = MT and [M]t = [M]T for every t ≥ T (see [85, Theorem 6.31]). 1 4

439


The following Doob’s maximal inequality holds for every square integrable martingale M: , 2 E sup |Ms | (A.3.36) ≤ 4 E[Mt2 ], t ≥ 0. s≤t

Moreover, for every stopping time T and t > 0, [M]t∧T = lim

n

n→∞

M kt ∧T − M (k−1)t ∧T n

2

n

,

(A.3.37)

k=1

where the convergence is in probability with respect to P (see the Remark after Theorem 9.33 in [85]). For a square integrable martingale M and a stopping time T, Kt := MT · 1{t≥T} − (MT · 1{t≥T} )p is a purely discontinuous square integrable martingale. So by (A.3.35), we have in particular [M − K, K]t = 0 for t ≥ 0. (A.3.38) t A stochastic integral Kt := 0 Hs dMs can be defined for a predictable t 2 integrand H ·and for a square integrable martingale M. If E[ 0 Hs dMs ] < ∞, then K = 0 Hs dMs is also a square integrable martingale. For a predictable process H, a semimartingale X = M + A, where M is a square integrable t martingale and A is a process of finite variation, the stochastic integral 0 Hs dXs t t t is defined to be 0 Hs dMs + 0 Hs dAs . Here 0 Hs dAs is the Lebesgue-Stieltjes integral. For f ∈ C2 (R2 ) and two semimartingales X (i) = M + A and Y (2) = N + C as above, the following Itô’s formula holds (see [85, Theorem 9.35]): P-a.s. for t ≥ 0, 2 t ∂ (1) (2) f (Xs− , Xs− )dXs(k) f (Xt(1) , Xt(2) ) = f (X0(1) , X0(2) ) + ∂x k 0 k=1 +

+

2 1 t ∂2 (1) (2) f (Xs− , Xs− )dX (k),c , X (j),c s 2 k,j=1 0 ∂xk ∂xj

(1) (2) , Xs− ) f (Xs(1) , Xs(2) ) − f (Xs−

0<s≤t

2 ∂ (1) (2) (k) (k) f (Xs− , Xs− )(Xs − Xs− ) . − ∂xk k=1

(A.3.39)

440

APPENDIX A

In particular, we have the following product rule: t t Xt Yt = X0 Y0 + Ys− dXs + Xs− dYs + [X, Y]t , 0

t ≥ 0.

(A.3.40)

0

A.4. REVIEW OF SYMMETRIC FORMS Let H be a real Hilbert space with inner product (·, ·). The domain of a linear operator S on H is denoted by D(S). A linear operator S on H is said to be symmetric if D(S) = H, (Sf , g) = ( f , Sg), f , g ∈ H. A symmetric form (E, D(E)) on H and its closedness are defined at the beginning of Section 1.1. The notions of a strongly continuous contraction semigroup {Tt ; t > 0} of symmetric linear operators on H and a strongly continuous contraction resolvent {Gα ; α > 0} on H are also introduced there. The latter is obtained from the former by taking the Laplace transform (1.1.1), which is called the resolvent of a semigroup {Tt : t > 0}. A self-adjoint linear operator A on H is called non-negative definite if (Af , f ) ≥ 0 for every f ∈ D(A). The generator of a strongly continuous contraction resolvent {Gα ; α > 0} is defined by Af = α f − G−1 α f,

D(A) = Gα (H),

and −A is easily seen to be a non-negative definite self-adjoint operator independent of α > 0. The generator A of a strongly continuous contraction semigroup {Tt ; t > 0} is defined by Af = lim t↓0

Tt f − f , D(A) = { f ∈ H : Af exists as a strong limit}, t

which can be verified to be identical with the generator of the resolvent of {Tt ; t > 0}. A symmetric linear operator S on H satisfying S2 = S is called a projection operator. A family {Eλ ; −∞ < λ < ∞} of projection operators on H is said to be spectral family if it satisfies the following conditions: Eλ Eµ = Eλ , λ ≤ µ, limλ ↓λ Eλ f = Eλ f , limλ→−∞ Eλ f = 0, limλ→∞ Eλ f = f , f ∈ H. It holds then that 0 ≤ (Eλ f , f ) ↑ ( f , f ), λ ↑ ∞, f ∈ H. Further, (Eλ f , g) is of bounded variation in λ for f , g ∈ H. function Given a spectral family {Eλ ; λ ∈ R} and a continuous real-valued ∞ ϕ on R, there exists a unique self-adjoint operator A = −∞ ϕ(λ)dEλ on H characterized by  ∞   ϕ(λ)d(Eλ f , g), ∀g ∈ H (A f , g) =   −∞ (A.4.1) ∞   2  = {f ∈ H : ϕ(λ) d(Eλ f , f ) < ∞}. D(A) −∞

441


Any non-negative definite self-adjoint operator −A on H admits a unique spectral family {Eλ ; λ ∈ R} with Eλ = 0 for every λ < 0, such that −A = ∞ λdE (cf. [154]). In this case, for a non-negative continuous real function ϕ λ 0 ∞ on [0, ∞), the self-adjoint operator determined by 0 ϕ(λ)dEλ will be denoted by ϕ(−A), which is also non-negative definite. We now state mutual correspondences among the four objects below: (a) The totality of closed symmetric forms E on H. (b) The totality of self-adjoint operators on H with −A being non-negative definite. (c) The totality of strongly continuous contraction semigroups {Tt ; t > 0} of symmetric operators on H. (d) The totality of strongly continuous contraction resolvents {Gα ; α > 0} of symmetric operators on H. A more detailed proof can be found in [73, §1.3]. (i) (b), (c) and (d) are mutually in one-to-one correspondence. The correspondences (b) ⇒ (c), (b) ⇒ (d) are defined by Tt = exp(tA), Gα = (α − A)−1 , respectively. The correspondences (c) ⇒ (b), (d) ⇒ (b) are defined by the operations of taking generators, respectively. The correspondences (c) ⇒ (d), (d) ⇒ (c) are defined by (1.1.1), (1.1.2), respectively. (ii) (a) and (b) are in one-to-one correspondence. The correspondence (b) ⇒ (a) is defined by √ √ √ D(E) = D( −A), E( f , g) = ( −Af , −Ag), f , g ∈ D(E). (A.4.2) The correspondence (a) ⇒ (b) is characterized by a direct relation D(A) ⊂ D(E),

E( f , g) = −(Af , g),

∀f ∈ D(A), ∀g ∈ D(E).

(A.4.3)

Expression (A.4.2) can be rewritten using (A.4.1) as / .  ∞   λd(Eλ f , f ) < ∞ ,  D(E) = f ∈ H :    EE( f , g) =

0

∞

(A.4.4)

λd(Eλ f , g), f , g ∈ D(E).

0

(iii) The one-to-one correspondence of (a), (d) is characterized by a direct relation (1.1.6). (iv) The direct correspondence (c) ⇒ (a) is described by (1.1.4), (1.1.5), where E (t) is an approximating form defined by (1.1.3) for {Tt ; t > 0}. For f ∈ H, E (t) ( f , f ) increases as t decreases.

442

APPENDIX A

The direct correspondence (d) ⇒ (a) is described by D(E) = { f ∈ H : lim E (β) ( f , f ) < ∞}, E( f , g) = lim E (β) ( f , g), β→∞

β→∞

where E ( f , g) = β( f − βGβ f , g) for f , g ∈ H. For f ∈ H, E (β) ( f , f ) increases as β increases. (v) Let {Tt ; t > 0} and {Gα ; α > 0} be related to E as above. Then Tt (H) ⊂ D(E), Tt g2E ≤ 2t1 g2 for every g ∈ H, and furthermore limt↓0 Tt f − f E1 = 0, limα→∞ αGα f − f E1 = 0 for every f ∈ D(E). Using the spectral family expressing E as (A.4.4), the first assertion of ∞ (iv) follows from E (t) ( f , g) = 1t 0 (1 − e−tλ )d(Eλ f , g), f , g ∈ H. The proof of the second is similar. The first assertion of (v) is obtained by integrating the inequality e−2tλ λ ≤ 2t1 by d(Eλ g, g). The second follows from the expression ∞ λ 2 (λ + 1)d(Eλ f , f ). The next theorem is frequently αGα f − f 2E1 = 0 α+λ used in the main text. (β)

T HEOREM A.4.1 (Banach-Saks). (i) Let H be a real Hilbert space with inner product (·, ·) and norm · . If, for fn ∈ H, n ≥ 1, supn fn = M is finite, then the Cesàro mean sequence of a suitable subsequence of { fn } converges strongly to an element of H. (ii) Let H be a real linear space and C( f , g), f , g ∈ H, be a non-negative definite symmetric bilinear form on H. If fn ∈ H, n ≥ 1, satisfies supn C( fn , fn ) < ∞, then the Cesàro mean sequence of a suitable subsequence of { fn } is C-Cauchy. Proof. (i) Under the present assumption, a suitable subsequence of { fn } converges weakly to some element f ∈ H. Denoting the difference of the subsequence and f by { fn } again, { fn } converges to zero and hence we can choose its subsequence nk as follows. Let n1 = 1. If n1 , . . . , nN are chosen, we select nN+1 > nN such that |( fn1 , fnN+1 )| ≤ N1 , . . . , |( fnN , fnN+1 )| < N1 . Then the Cesàro mean gN = N1 Nk=1 fnk of { fnk } satisfies gN 2 = ≤

N 1 2 fnk 2 + 2 2 N k=1 N

( fni , fnk )

1≤i τ (ω) − s = τ (θs0 ω), Bs = As and Bt (θs0 ω) = At∧τ (θs0 ω) (θs0 ω) = Aτ (θs0 ω) (θs0 ω) = Aτ (θs ω) (θs ω) so that Bs + Bt (θs0 ω) = As + Aτ (θs ω) (θs ω) = Aτ , Bs+t = A(s+t)∧τ = Aτ . When τ (ω) ≤ s, Bs = As∧τ = Aτ , Bs+t = A(s+t)∧τ = Aτ , while θs0 ω = ω∂ and Bt (θs0 ω) = At∧τ (ω∂ ) (ω∂ ) = A0 (ω∂ ) = 0. When A is continuous, it is predictable so that Bt ∈ F(τ0 ∧t)− ⊂ Gt by (A.1.14) and (ii). 4.3.12: Since E (c) (u, v) = 12 µcu,v (E) for u, v ∈ Fe , (4.3.34) follows from Lemma 4.3.6 immediately. 5.1.2: Note that

Ps+t f (x) = Ex e−As e−At ◦θs f (Xt ◦ θs )

= Ex e−As EXs e−At f (Xt ) = Ps Pt f (x). (t.3) and (t.4) are obvious. 5.4.1: From (5.4.5), we see that e(t) is non-decreasing as t ↑ and further the subadditivity e(t + s) ≤ e(t) + e(s) holds. Therefore, if e(t) is finite for some t > 0, then so it is for any t > 0. Suppose e(t) < ∞ for any t > 0. Since 0 ≤ f − Pt f decreases to zero as t ↓ 0, we get by the dominated convergence theorem that e(t) ↓ 0, t ↓ 0. For t < t , we have from (5.4.5) e(t ) = e(t) + (f − Pt −t f , Pt g). The second term of the right hand side is dominated by (f − Pt −t f , g) = e(t − t), which decreases to zero as t ↓ t or t ↑ t , so that e(t) is continuous in t > 0. We obtain also from (5.4.5) the inequality e(t + s) − e(t ) ≤ e(t + s) − e(t) for t < t . Substitution t = t + s then yields 12 {e(t) + e(t + 2s)} ≤ e(t + s), namely, e is midpoint concave. Since it is continuous on [0, ∞) with ↑ as t ↓ 0. e(0) = 0, we have e(s) ≥ st e(t) for s < t, that is to say, e(t) t

447

SOLUTIONS TO EXERCISES

5.5.1: We denote τ0 by τ . 1 (Hβ f (x) − Hα f (x)) α−β τ −(α−β)t −βτ = Ex e e f (Xτ )dt 0

= Ex

∞

e−αt 1{t 0 = 1. Consequently, O1 ∩ O2 ∈ O. Other properties are obvious. A.3.3: Let and N be a defining set and an exceptional set, respectively, for A ∈ A+ c . For any f ∈ bB+ (E), the process Y(t, ω) = f (Xt (ω)) is progressive with respect to {Ft , t ≥ 0} in view of Lemma A.1.13, and so is the restriction of Y to [0, ∞) × relative to {Ft , t ≥ 0}. t 0 f (Xs (ω))dAs (ω), ω ∈ , is therefore well defined and is Ft -measurable for each t ≥ 0. It is then easily seen to be a PCAF with defining set and exceptional set N.

Notes

NOTES ON CHAPTERS 1 TO 4 Many materials in Sections 1.1 and 2.1, Chapter 3, and Chapter 4 are based on a recently published Japanese book by Fukushima-Takeda [74]. The notion of h-capacity and quasi-regular Dirichlet form were first introduced in Albeverio and Ma [2] for symmetric Dirichlet forms. It was later extended to non-symmetric Dirichlet forms by Albeverio, Ma, and Röckner in [4]. The presentation in Sections 1.2 and 1.3 are based on the lecture notes of a graduate course the first author taught in the Autumn semester of 1995 at Cornell University. Capacitary characterization of an E-nest was also presented in the book of Ma-Röckner [119] under a more general non-symmetric setting. Section 1.4 is based on Chen-Ma-Röckner [31]. The analytic potential theory presented in Section 2.3 goes back to J. Deny [39] and its content is not much different from that of Fukushima-Oshima-Takeda [73, §2.1, §2.2]. The first half of Section 3.5 on probabilistic features of transience is mostly taken from Chen-Fukushima [21, §2]. In Section 1.1 the extended Dirichlet space is formulated with no topological assumption on the underlying space following an idea of B. Schumland [134], while the Dirichlet space itself has been defined without topological assumption in the books by Bouleau-Hirsch [15] and by Ma-Röckner [119]. Analogous results to those in Section 1.5 hold for the non-symmetric sectorial Dirichlet form as well. In particular, the book [119] presents a direct construction of a Borel special standard process associated with a quasi-regular non-symmetric Dirichlet form as well as the necessity of the quasi-regularity. Further extensions to more general non-symmetric situation have been carried out by W. Stannat [141]. Theorem 2.1.11 on the irreducibility characterization is due to M. Takeda. All examples in Section 2.2 are finite dimensional. They are collected here mainly to identify explicitly the associated extended Dirichlet spaces and reflected Dirichlet spaces (later in Section 6.6). The theory of Dirichlet forms brings the analytic and probabilistic potential theories together in a natural and coherent way. Its use and application is wide ranging. It is especially better suited for the study involving non-smooth data

452

NOTES

or irregular or infinite dimensional state spaces. The Dirichlet form theory has played an important role in the investigation of various subjects in probability theory, analysis, and statistical physics, in finite as well as infinite dimensional spaces. See [5, 14, 17, 20, 29, 30, 36, 43, 59, 60, 100, 114, 118, 129, 130, 143, 144] and references therein for a sample. The concept of Dirichlet forms has also been a powerful tool in the analytic and probabilistic study of fractals. See the book by J. Kigami [105] and a recent paper by Barlow-Bass-KumagaiTeplyaev [6]. When (C, F) is a positivity preserving coercive form, the equivalence of local properties for (C, F) in analogy to that stated in Theorem 2.4.2 was established by B. Schmuland [133]. The formulation of Theorem 2.4.2 in this book is new; it is applicable to the strongly local part E c of a Dirichlet form (E, F). Theorem 2.4.3 on strongly local property is new. See Proposition I.5.1.3 of Bouleau-Hirsch [15] for other equivalent characterizations of strongly local property of a Dirichlet form. Many assertions made in Chapter 3 overlap with those in the book [73, Chapter 4] but some of them are proved quite differently. Especially the current proof of Theorem 3.3.9 on the regularity of the part of the Dirichlet form on an open set is new and probabilistic, while the proof in [73] is given by reinterpreting the celebrated spectral synthesis theorem in the analytic potential theory due to J. Deny [39, Chapter 4]. The methods of the proof employed in Chapter 4 are different from those in the book [73, §5.1, §5.2, §5.3] in various respects. Moreover, the probabilistic characterizations of the local and strongly local properties in Theorem 4.3.4 are obtained as direct consequences of the probabilistic descriptions of the Beurling-Deny formula contrarily to [73, §4.5]. Theorem 4.3.8 is an extension of Theorems I.5.2.3 and I.7.1.1 in Bouleau-Hirsch [15], where the same result is established for the local Dirichlet forms. The stochastic calculus in terms of Dirichlet forms has applications to infinite dimensional quasi-regular situations. For instance, the concept of BV functions over the abstract Wiener space has been formulated and studied in this way (cf. M. Fukushima [67, 68], Fukushima-Hino [72], M. Hino [87]). See L. Zambotti [155] for a similar approach to a SPDE with reflection. NOTES ON CHAPTER 5 The contents of Sections 5.1 and 5.2 are formulated for the family S of general smooth measures. Most of them have been presented in the book [73, §6.1, §6.2] but only for a subfamily of S consisting of positive Radon measures charging no E-polar sets. Among other things, the independence of the extended Dirichlet space of the trace Dirichlet form on the choice of measures µ ∈ S quasi supported by a common quasi closed set F is presented in Theorem 5.2.15 for the first time in this generality.

NOTES

453

The last part of Section 5.2 contains two important applications of the invariance properties under the time change by a fully supported PCAF. Especially construction of the resurrected Dirichlet form and the resurrected Hunt process is well carried out for a general regular Dirichlet form by removing its killing measure. This enables us to identify in the last example of Section 5.3 the Dirichlet form of a general one-dimensional minimal diffusion with possible killings inside. The complete characterization of the trace Dirichlet form in Sections 5.5 and 5.6 in terms of the Feller measures is based on Chen-FukushimaYing [26]. However, the proof of a main result (Theorem 2.6) in [26] (which corresponds to Theorem 5.5.8 of this monograph) contains a serious gap (the dual predictable projection was used incorrectly in the displays preceding [26, (2.29)]). This gap has been fixed through Lemma 5.5.4, Lemma 5.5.5, and Theorem 5.5.6 of this book. These three results as well as Exercise 5.5.7 are taken from [25]. Section 5.7 is based on Chen-Fukushima-Ying [27]. This paper also establishes the identification of the Lévy system of the time-changed process as is presented in Theorem 5.6.3 using a counterpart of Theorem 5.7.6 for a more general standard process X possessing a weak dual process under the condition that every semi-polar set is m-polar. Fitzsimmons-Getoor [57] has extended the contents of [27] to much more general right processes. However, the identification of the strongly local part of the trace Dirichlet form as is formulated in Theorem 5.6.2 is only achieved by the present approach. The first two examples of Section 5.8 are taken from FukushimaHe-Ying [71] and [26], respectively. Similar explicit expressions of the Feller kernel and related quantities along with their estimates for symmetric reflecting Brownian motions had been presented in P. Hsu [90]. See J. Kigami [106] for an explicit expression of the Douglas integral on the Cantor set induced by random walks on trees. The trace Dirichlet form on a quasi closed set F also arises naturally as a limit of a sequence of Dirichlet forms when their underlying measures are changing and convergent to a degenerate measure supported by F. See A. Kasue [101, Example 2.1] and Ogura-Tomisaki-Tsuchiya [125] for explicit expressions of trace Dirichlet forms obtained in this way, and Kuwae-Shioya [114] and Kim [109] for a general formulation of Mosco convergence of Dirichlet forms with varying state spaces. Section 5.4 is an adoption from R. K. Getoor [81] into the present symmetric setting. NOTES ON CHAPTER 6 Two variant notions of reflected Dirichlet forms were introduced by M. L. Silverstein in [138] and [139]. A fundamental tool used in both [138] and [139]

454

NOTES

is the “extended Markov process,” a continuous version of Hunt’s approximate Markov chain that has random birth time and death time. These two variant notions are proved to be equivalent in Z.-Q. Chen [16], using a direct martingale approach. Sections 6.1–6.3 are based on Chen [16] but the present formulation using the energy functional introduced in Section 5.4 is new. Section 6.4 is based on an unpublished paper by Z.-Q. Chen. Among the examples of reflected Dirichlet spaces presented in Section 6.5, (5◦ ) and (6◦ ) are taken from Chen-Fukushima [24] and Bogdan-Burdzy-Chen [14], respectively. All other examples appear here for the first time. Section 6.6 is essentially based on the book by M. L. Silverstein [138], but the presentation here is new. Section 6.7 is taken from Chen [18]. Theorem 6.7.2 was first established in Lyons and Zheng [118] for symmetric diffusions on Rn . The most general form of such a decomposition for symmetric Markov processes can be found in [19]. Röckner-Zhang [129, 130], Takeda [147], Kawabata-Takeda [102], and Eberle [46] have studied Markovian self-adjoint extensions of some local symmetric operators in relation to the Silverstein extensions. In [113], K. Kuwae gave a proof of the closedness of the active reflected Dirichlet form (E ref , Faref ) of a quasi-regular Dirichlet form (E, F) without decomposing F ref into the sum of Fe and HN. Along the way, he proved the results in Proposition 6.6.4 using a different method. NOTES ON CHAPTER 7 Section 7.1 is basically taken from Chen-Fukushima [23]. Theorem 7.2.1 is taken from Chen-Fukushima-Ying [26, §3]. The identification of the trace Dirichlet space with the space of finite Douglas integrals for the reflecting extension (Corollary 7.2.8) and for the reflected Dirichlet space (Theorem 7.2.9) appears here for the first time. Section 7.3 is based on Chen-Fukushima [23]. The current formulation of the lateral conditions goes back to W. Feller [50] and the second author [60]. When F is countable, the description of the resolvent in Theorem 7.4.4 has its counterpart in Chen-Fukushima [22] formulated for a more general non-symmetric Markov process possessing a weak dual. Constructions of onepoint and several-point extensions in Sections 7.5 and 7.7 have been dealt with by Chen-Fukushima-Ying [28] and Chen-Fukushima [22], respectively, under the weak duality setting. The uniqueness of the one-point extension stated in Theorem 7.5.4 is based on Fukushima-Tanaka [75, §6]. Its extension to the countably many-point case in Theorem 7.7.3 is new. Most examples in Section 7.6 are taken from Chen-Fukushima [21], while the last three examples in Section 7.8 are based on Chen-Fukushima [23, 24] and the second author [69]. There is a host of literature on boundary theory for Markov processes. For countable state Markov processes, see W. Feller [50], E. B. Dynkin [45],

NOTES

455

K. L. Chung [32], Hou-Guo [89], and X.-Q. Yang [152], and the references therein. The investigation of the boundary theory for the one-dimensional diffusions was accomplished by W. Feller [49], Itô-McKean [96, 97], K. Itô [95], and E. B. Dynkin [44] among others. Approaches to one-point extensions via excursion theory have been carried out by K. Itô [94], T. S. Salisbury [131], R. M. Blumenthal [12], FukushimaTanaka [75], Fitzsimmons-Getoor [56], and Chen-Fukushima-Ying [28], to name a few. As for higher dimensional diffusions with Wentzell’s boundary condition ([151]), see K. Taira [145] and the references therein for analytic approaches and S. Watanabe [150] and the references therein for probabilistic approaches.


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Catalogue of Some Useful Theorems

For readers’ convenience, we present an index of useful theorems, some of which may not be easily located by just using the table of contents of the book. Theorem 1.1.5: Extended Dirichlet space on a σ -finite measure space without topological assumption. (1.1.22) and (1.5.1): Useful approximating expressions of E( f , f ) for f ∈ Fe . Corollary 1.1.9: Fatou’s lemma for extended Dirichlet form (Fe , E). 2 (E; m) is α-excessive and v ∈ F with u ≤ v, then Lemma 1.2.3: If u ∈ L+ u ∈ F and Eα (u, u) ≤ Eα (v, v). Theorem 1.3.14: Relation between E-polar set and Cap1 -polar set; f is quasi continuous if and only if f ∈ C({Fk }) for some Cap1 -nest {Fk }. Lemma 1.3.15: Every f ∈ F admits a quasi continuous version in the restricted sense with respect to Cap1 -nest. Theorem 1.4.3: A Dirichlet form is quasi-regular if and only if it is quasihomeomorphic to a regular Dirichlet form. Theorem 1.5.2: Association of a nice Markov process to a quasi-regular Dirichlet form. Proposition 2.1.3: An irreducible symmetric Markovian semigroup is either transient or recurrent. Theorem 2.1.5: Dirichlet form characterization of transience. Theorem 2.1.8: Equivalent characterizations of recurrence. Theorem 2.1.9: Equivalent characterizations of transience. Proposition 2.1.10: Px (ζ < ∞) > 0 on E implies that the symmetric process X is transient in the Dirichlet form sense. Theorem 2.1.11 and Theorem 5.2.16: A criterion for a recurrent Dirichlet form to be irreducible. Theorem 2.3.2: For a transient Dirichlet form (E, F ), a set N is Cap(0) polar if and only if it is E-polar; every f ∈ Fe admits a Cap(0) -nest {Fk } so that f ∈ C∞ ({Fk }).

468

CATALOGUE OF SOME USEFUL THEOREMS

Theorem 2.3.4: Every f ∈ Fe admits a quasi continuous version. Here (E, F) may not be transient. Corollary 2.3.11: For a Borel set B, Cap1 (B) = 0 if and only if µ(B) = 0 for every µ ∈ S00 . Theorem 2.3.15: Structure of smooth measure. Theorem 2.4.2: Equivalent characterization of local property of a Dirichlet form that is invariant under quasi-homeomorphism. Theorem 2.4.3: Equivalent characterization of strongly local property of a Dirichlet form that is invariant under quasi-homeomorphism. Theorem 3.1.3: A set N is E-polar if and only if it is m-polar. Theorem 3.1.4: Cap1 -nest vs. strong nest; probabilistic characterization of E-nest under Px . Theorem 3.1.5: Any m-polar set is contained in a Borel properly exceptional set. Theorem 3.1.7: Finely continuous property for f ∈ F. Theorem 3.1.10: Any semipolar set is E-polar. Proposition 3.1.11, Corollary 3.1.14(ii): Equivalence of the absolute continuity of transition function and resolvent kernel. Theorem 3.1.12: Uniqueness of m-symmetric Hunt process associated with a regular Dirichlet form. Theorem 3.1.13: Uniqueness of m-symmetric right process properly associated with a quasi-regular Dirichlet form. Corollary 3.1.14(i): Any m-tight m-symmetric right process on a Radon space or any m-symmetric right process on a Lusin space is a special Borel standard process outside an m-inessential set. Lemma 3.1.17: Probabilistic characterization of E-nest under Pm . Theorem 3.2.2: Probabilistic representation HαB f of Eα -orthogonal projection PHBα f of f ∈ F for nearly Borel measurable set B. Theorem 3.3.3: Relation between quasi notion and fine notion such as quasi open vs. finely open and quasi continuity vs. fine continuity. Theorem 3.3.5: Equivalent characterization of quasi support of a measure. Corollary 3.3.6: If f > 0 m-a.e., then the measure f · m has full quasi support E. Theorem 3.3.8: For a nearly Borel quasi open set D, (E D , FD ) is a quasiregular Dirichlet form; relation between E D -polarity and E-polarity for a set N ⊂ D. Theorem 3.3.9: For a regular Dirichlet form (E, F) on E and an open set D ⊂ E, (E, FD ) is a regular Dirichlet form on L2 (D; m).

469


Theorem 3.4.8: For every u ∈ Fe ((E, F) not necessarily transient), HB u is a quasi continuous element in Fe that is E-orthogonal to Fe,E\B . Theorem 3.4.9: For a nearly Borel fine open set D, the extended Dirichlet space FD,e of (E, FD ) can be identified with Fe,D = {u ∈ Fe : u = 0 E-q.e. on Dc }. So we can use FeD to denote either of them. Lemma 3.5.1: For a transient process, a Cap(0) -nest is a strong nest. Theorem 3.5.2: Sample paths wander out to infinity on {ζ = ∞} for transient process. Corollary 3.5.3: For transient process X and f ∈ Fe , limt→ζ − f (Xt ) = 0 along the paths that admit no killings inside E. Theorem 3.5.4: If { fn } ⊂ Fe is both m-a.e. convergent and E-convergent to fn } that is uniform convergent some f ∈ Fe , then there is a subsequence of { to f on a strong nest. Here (E, F) may not be transient. Lemma 3.5.5: Consequences of recurrence for bounded excessive functions. Theorem 3.5.6: Probabilistic interpretation of a non-m-polar set and irreducible recurrence. Theorem 4.1.1: One-to-one correspondence between PCAFs and smooth measures. Theorem 4.2.1: Killing measure in terms of a Lévy system. ◦

Theorem 4.2.5: Completeness of (M, e) and the locally uniform conver◦ gence of Mn ∈ M. Theorem 4.2.6: Fukushima’s decomposition for u ∈ Fe . Proposition 4.3.1: Strongly local property of µcu for u ∈ Fe in terms of quasi open sets. Theorem 4.3.3: Beurling-Deny decomposition for extended Dirichlet space (Fe , E) of a regular Dirichlet form. Theorem 4.3.4: Probabilistic characterizations of the local property and the strongly local one for a regular Dirichlet form. Theorem 4.3.7: Derivative formula for energy measure µcu .

Theorem 4.3.8: For u ∈ bF , µcu ◦ u−1 is absolutely continuous with respect to the Lebesgue measure on R. Theorem 4.3.10: Normal contraction property in u for µcu . Theorem 5.1.3: Feynman-Kac transform and perturbed Dirichlet form (E µ , F µ ), where E µ = E + (·, ·)µ . Theorem 5.1.4: Equivalence of E µ -nest and E-nest. Theorem 5.1.5: Quasi-regularity of (E µ , F µ ) for smooth measure µ ∈ S.

470


Theorem 5.1.6: For regular Dirichlet form (E, F) and positive Radon measure µ charging no E-polar set, (E µ , F µ ) is a regular Dirichlet form on L2 (E; m). Proposition 5.1.9: ((F µ )e , E µ ) = ((Fe )µ , E µ ). ˇ F) ˇ Theorems 5.2.2 and 5.2.15: Characterization of the Dirichlet form (E, of a time-changed process and its extended Dirichlet space Fˇ e . Theorem 5.2.5: Transience/recurrence is invariant under time-change. Theorem 5.2.6: Restrictions of E-nest and E-quasi-continuity to the quasi ˇ ˇ support F of µ yield E-nest and E-quasi-continuity. ˇ F) ˇ and the proper association of Theorem 5.2.7: Quasi-regularity of (E, time-changed process. Theorem 5.2.8: Let F be the quasi support of a Radon measure µ. N ⊂ F ˇ is E-polar if and only if it is E-polar. Theorem 5.2.11: Equivalence of quasi notions for time-changed processes with full support. Theorem 5.2.13: For Radon measure µ, its time-changed process, properly restricted, is a Hunt process and its associated Dirichlet form is regular on ˇ the topological support F ∗ of µ. Moreover, F ∗ \ Fµ is E-polar. Proposition 5.2.14: Reduction of a right process properly associated with a regular Dirichlet form into a Hunt process. Theorem 5.2.16: If a regular Dirichlet form (E, F) is irreducible, then u ∈ Fe , E(u, u) = 0 implies u = constant. Theorem 5.2.17: Resurrected Dirichlet form and resurrected Hunt process obtained by the removal of killing measure. Theorem 5.3.1: Conformal invariance of two-dimensional absorbing Brownian motion. Example (2◦ ) of Section 5.3: Conformal invariance of two-dimensional reflecting Brownian motion. Theorem 5.3.4: Identification of the Dirichlet form and the extended Dirichlet space of a general one-dimensional minimal diffusion (possibly with killings inside). Theorem 5.5.6 and Exercise 5.5.7: For quasi open subset D ⊂ E and v ∈ Fe , with m0 := m|D , 1 lim Em0 (v(Xt ) − v(X0 ))2 ; t < τD t→0 t (v(x) − v(y))2 J(dx, dy). = µcv (D) + D×D


471

Theorems 5.6.2 and 5.6.3: Beurling-Deny decomposition of the extended Dirichlet space of a time-changed process. Lemma 5.7.2: σF = inf{t > 0 : Xt ∈ F or Xt− ∈ F} a.s. Theorem 5.7.5: Characterization of the entrance law induced by an exit system. Theorem 5.7.6: Feller measures as joint distributions of starting and end points of excursions. ◦

Theorem 6.1.2: Every M ∈ M is uniformly Px -square integrable for q.e. x. Theorem 6.1.8: Beurling-Deny type formula for e(M h ) for h ∈ HN. Theorem 6.2.13: A characterization of F ref in terms of finite E-energy. Theorem 6.2.14: (E ref , Faref ) is a Dirichlet form. Theorem 6.3.2: For a recurrent Dirichlet form (E, F ), (E ref , F ref ) = (E, Fe ). Theorem 6.4.2: A simpler characterization of F ref in terms of quasi open sets. Proposition 6.4.6: Reflected Dirichlet space (E ref , F ref ) is invariant under a full support time change. Theorem 6.4.12: A characterization of F ref for transient quasi-regular Dirichlet form (E, F). Theorem 6.6.5: Probabilistic meaning of a Silverstein extension. Corollary 6.6.6: If (E, F) is conservative, then (E ref , Faref ) = (E, F). Theorem 6.6.9: (E ref , Faref ) is the maximal Silverstein extension of (E, F). Theorem 6.6.10 and Remark 6.6.11(ii): Relation between (E ref , F ref ) and the extended Dirichlet space of (E ref , Faref ). Remark 6.6.11(i): Reflected Dirichlet space (E ref , F ref ) of (E, F) is also the reflected Dirichlet space of (E ref , Faref ). Theorem 6.7.2: Lyons-Zheng’s forward-backward martingale decomposition for u(Xt ) − u(X0 ). Theorem 6.7.4: For a smooth measure ν and Uν := Ex [Aνζ ], Uν ∈ Fe if and only if E Uν(x)ν(dx) < ∞. In this case, E(Uν, v) = E v(x)ν(dx) for v ∈ Fe . Here the Dirichlet form (E, F) is not assumed a priori to be transient. Theorem 6.7.13: Equivalence between probabilistic and analytic notions of harmonicity. Theorem 7.1.6: Relation between the extended Dirichlet space of X on E and the reflected Dirichlet space of the part process X 0 of X on a quasi open set E0 = E \ F.

472


Theorem 7.1.8: Relation between the Dirichlet space of X on E and the active reflected Dirichlet space of X 0 on E0 . Corollary 7.2.8: When X is a reflecting extension of X 0 , the trace on F of the Dirichlet space of X is identified with the space of functions on F of finite Douglas integrals. Theorem 7.3.5: Characterization of the L2 -infinitesimal generator of X in terms of a lateral condition on F involving the flux. Theorem 7.4.5: When F is countable and X admits no jumps from F to F or killings on F, the trace Dirichlet space of X is the closure of finitely supported functions with respect to the Douglas integral and X is uniquely determined by X 0 . The L2 -generator of X is characterized by the zero flux condition. Theorem 7.5.4: Uniqueness of the one-point extension and its equivalent characterizations without assuming the regularity of the associated Dirichlet form. Theorem 7.5.6: Construction of a one-point extension by a Poisson point process of excursions. Theorem 7.5.9: Construction of a one-point extension by darning a hole. Theorem 7.5.10: Expression of the entrance law for a darning of a hole in terms of the entrance law induced by an exit system. Theorem 7.5.12: Construction of a skew one-point extension. Theorem 7.6.3: Conformal invariance of the one-point extension of twodimensional Brownian motion by darning a compact hole. Theorem 7.7.3: Uniqueness of the countably many-point extension and its equivalent characterizations without assuming the regularity of the associated Dirichlet form. Theorem 7.7.4: Construction of a countably many-point extension by repeating one-point extensions. Theorem 7.7.5: Construction of a many-point extension by darning countably many holes. Theorem 7.8.1: Conformal invariance of the many-point extension of twodimensional Brownian motion by darning finitely many compact holes. Proposition 7.8.5: Two-point reflecting extension of a time-changed RBM on a closed domain with two branches of infinite cones. Proposition 7.8.7: A skew extension of a one-dimensional absorbing Brownian motion with countable boundary. Theorem A.4.1: Banach-Saks Theorem.

Index

absolute continuity condition, 99, 140, 422 absorbing Brownian motion, 127, 272, 319, 352 absorbing diffusion, 125, 358 additive functional, 423 m-equivalence, 423 finite càdlàg, 424 in the strict sense, 141, 423 martingale –, 147 positive continuous –, 131, 424 almost surely (a.s.), 418 approximating sequence of function in Fe , 5, 50 Banach-Saks Theorem, 442 Bessel process, 126 Beurling-Deny decomposition, 159 for quasi-regular Dirichlet form, 263 for time-changed process, 222 Beurling-Deny representation, 157 BL function, 69–74 BL(D), 69 Blumenthal’s 0-1 law, 417 boundary point approachable, 65 approachable in finite time, 125 exit –, 126 regular –, 66, 126 boundary theory, 300 Brownian motion, 127, 362, 377 absorbing –, 127 circular –, 353 on manifold, 75 reflecting, 128 reflecting –, 353 skew –, 354 standard –, 127 transience of –, 127 Walsh –, 362

Caph,α , 18, 21 Cap1 , 28, 77 càdlàg, 143, 424 canonical measure, 64, 125 canonical scale, 64, 125 canonical subprocess, 433 capacitable, 22 capacity, 77 (h, α)-, 18 0-order, 79 1-, 28, 77 Choquet K-, 21 tight, 27 carré du champ, 165 censored stable process, 190, 274, 323, 368 Cesàro mean, 11, 442 co-tangent space, 76 co-vector field, 76 collapsing a hole, 348 compound Poisson process, 59 conformal invariance of absorbing Brownian motion, 191 of Brownian motion with darning, 364 of Brownian motion with multiple darnings, 377 of reflecting Brownian motion, 192 conservative Dirichlet form, 46 Markovian kernel, 12 process, 399 semigroup, 46 continuous additive functional of zero energy, 149 continuous boundary, 73 contraction normal –, 4–12 unit –, 4, 5 convolution of measures, 56

474 convolution semigroup on Rn , 56 on unit circle, 62 core of Dirichlet form, 30 special standard –, 30 standard –, 30 darning a hole, 348 darning multiple holds for reflecting stable process, 379 darning multiple holes for Brownian motion, 377 defining set of additive functional, 423 derivation property, 159, 160 diffusion with jumps, 297 Dirichlet form, 4 conservative, 46 extended, 6 irreducible, 43, 46, 187 local, 30 of process, 33 of transition function, 15 One-dimensional strongly local –, 63 part of –, 108–111 quasi-regular, 26, 31, 33 recurrent, 45 regular, 27, 28, 33 strongly local, 30 trace –, 179 transient, 42 Dirichlet integral, 58 Dirichlet space, 4 extended, 5 Doob’s maximal inequality, 439 Douglas integral, 74, 195, 224, 311 dual predictable projection, 435 Dynkin class, 392 energy functional, 203, 245 energy measure µu , 164 energy of additive functional, 143 entrance law, 346 entrance time σ˙ B , 402 equilibrium potential, 77, 78, 105 0-order, 79 equivalent m-, 131 in the strict sense, 141

INDEX exceptional set of additive functional, 423 excessive function, 410, 413 α-, 16, 410 excessive measure, 418 excursion away from a set, 227 excursion interval, 227 excursion-reflected Brownian motion, 365 exit system, 227 extended Dirichlet space, 5, 36, 49, 80, 118 for one-dimensional diffusion, 68 for time-changed process, 221 for trace Dirichlet form, 186, 224 irreducible, 43 recurrent, 45 extended space of symmetric closed form, 5 extension (many-point) of a process, 369 existence, 373, 376 q.e. –, 376 uniqueness, 370, 376 Feller kernel, 233 Feller measure, 208 α-order –, 211 in terms of excursions, 230 supplementary –, 208 Feller process, 413 Feller’s canonical form, 64 Feynman-Kac semigroup, 168 filtration, 393 adapted, 393 admissible, 393 augmentation of, 397 minimum admissible, 396 minimum augmented admissible, 397 quasi-left-continuous, 412 right continuous, 393 fine topology, 418 finely continuous, 107, 418 q.e., 96 finely open, 417 flux at a point, 336, 338 flux functional, 327 forward and backward martingale decomposition, 284 Fourier transform, 57 Fukushima decomposition, 152 Gelfand transform, 32, 277 generator of resolvent, 440

475

INDEX of semigroup, 440 of symmetric closed form, 441 harmonic, 242, 287 analytic characterization, 290–295 under quasi-regular setting, 268 harmonic functions of finite energy, 246 hitting distribution, 416 α-order, 103, 416 hitting probability, 416 α-order, 416 hitting time σB , 402 Hopf’s maximal inequality, 38 Hunt process, 33, 406 equivalent, 100 ideal, 275 image Dirichlet form, 30 infinitesimal generator, 327 initial distribution, 397 intrinsic measure, 313, 314, 317 invariant set X-, 95, 96, 186, 407 {Tt }-, 38 for Borel right process, 407 for Borel standard process, 101 for Hunt process, 95, 407 for semigroup, 38, 42 inverse of non-decreasing function, 428 of PCAF, 429 irreducible Dirichlet form, 43, 46, 187 process, 118, 123 semigroup, 38, 39 irregular point for a set, 417 Itô’s formula, 439 jumping measure, 158 kernel, 12 m-symmetric, 13 0-order resolvent, 38 conservative, 12 Markovian, 12 probability, 12 resolvent, 13 killing, 433 killing measure, 144–158 Laplace-Beltrami operator, 75 lateral (boundary) condition, 327

lateral condition, 330 Lévy exponent, 56 Lévy measure, 56 Lévy system, 145, 436–438 Lévy-Khinchin formula, 56, 63 lifetime, 399 Lindelöf space, 26 Lindelöf theorem, 26 linear operator Markovian, 4 ◦ local Dirichlet space F loc , 163 local property, 30, 89, 158 strongly –, 30, 90, 158 Lusin space, 14, 398 Lyons-Zheng decomposition, 284 m-inessential set, 130, 419, 421 m-polar, 93 m-tight right process, 33 m-version, 98 MAF, 147, 434 of finite energy, 149 Markov process, 395 measurable, 400 progressive, 400 restricted to an invariant set, 407 strong, 401 Markov property, 396 strong, 401 Markovian kernel, 12 linear operator, 4 symmetric form, 4 martingale, 395 continuous part, 435 purely discontinuous part, 435 martingale additive functional, 147, 434 of finite energy, 149 measurable space, 391 measure of finite energy integral, 80 0-order, 87, 202 minimal diffusion, 125, 199, 271, 380, 386 monotone class theorem, 392 mutual energy, 143 nearly Borel measurable, 408 nest, 77 attached to smooth measure, 84 Cap(0) -, 79 Cap1 -, 29

476 nest Continued Caph,α -, 22 E-, 22, 103 m-regular, 26 strong, 94, 119 Newtonian convolution kernel, 62 no killing inside, 158, 340 null set, 391 one-dimensional Brownian motion with countable boundary, 386 one-dimensional diffusion, 271, 380 in interval, 63, 123, 199, 271, 380 one-point extension, 340 existence, 344, 348 q.e. –, 347 uniqueness, 341, 348 operator non-negative definite, 440 symmetric, 440 orthogonal decomposition E-, 114 E 0,ref -, 303, 305 Eα0,ref -, 305 Eα -, 84 martingale –, 153 Parseval formula, 57 part process, 109 PCAF, 131, 424 perturbation method for Dirichlet forms, 43 perturbed Dirichlet form (E µ , F µ ), 168 Poincaré inequality, 74 Poisson point process, 345 of excursions, 344, 346 polar, 417 E-, 22 Cap(0) -, 79 Cap1 -, 28, 83 Caph,α -, 22 m-, 93, 419, 421 potential of a measure, 285 α-, 80, 83, 136 0-order, 87, 202 potential operator, 38, 47 potential zero, 417 predictable quadratic variation of martingale, 148, 435 probability measure, 391

INDEX probability space, 391 projection operator, 440 properly associated with, 33, 35, 100 for time-changed process, 181, 185 properly exceptional set for Borel standard process, 101 for Hunt process, 96, 130 pure jump step process, 51 push forward measure, 30 for µcu under map u, 161 q.e., 73, 78, 94, 106, 419 q.e. equivalent, 106 q.e. excessive, 203 q.e. finely open, 106 q.e. purely excessive, 203 quadratic covariation of martingale, 436 of semimartingales, 438 quadratic variation of martingale, 436, 439 quasi closed, 106, 107 quasi continuous, 77, 97, 107 in the restricted sense, 29 quasi everywhere, 78, 94, 419 quasi-homeomorphic, 30 quasi-homeomorphism, 101 of Dirichlet forms, 30 quasi-left-continuous on (0, ∞), 405 on (0, ζ ), 405 quasi open, 106, 107 quasi-regular, 26, 31 quasi-separated, 334 quasi support, 107 full –, 108 of a measure, 175 Radon space, 14, 409 RBM, 192 recurrence, 118, 123 of time-changed process, 179 recurrent Dirichlet form, 45 extended Dirichlet space, 45 process, 118, 122 semigroup, 38, 39, 44–45 reduced function 0-order, 88 α-, 18, 84

477

INDEX reference function for transient Dirichlet form, 42 for transient semigroup, 42 reflected Dirichlet space, 246, 258, 280 active –, 247, 303 for part process, 302 for quasi-regular Dirichlet form, 265 invariance under time change, 266 of absorbing Brownian motion, 272 of censored stable process, 274 of symmetric Lévy process, 269 of symmetric step process, 270 recurrent case, 261 reflecting Brownian motion, 128, 273, 319, 380 conformal invariance, 192 time change of –, 196 trace of –, 194 reflecting diffusion, 124 reflecting extension, 316, 318, 379 reflecting stable process, 274, 323, 368 regular for Dirichlet form, 27 for trace Dirichlet form, 183 point for a set, 417, 422 regular step process, 51–55 resolvent kernel, 13 of closed symmetric form, 3 of semigroup, 3 strongly continuous contraction, 2 symmetric operator, 441 resolvent equation, 14, 82 restriction of a process to an invariant set, 407 of Dirichlet form, 43 of extended Dirichlet form, 43 resurrected Dirichlet form, 188 resurrected Dirichlet space extended –, 188 resurrected Hunt process, 188 Revuz measure, 426 Riemannian manifold, 75 Riesz convolution kernel, 60 Riesz potential, 60 right process, 410 m-tight –, 33, 35 Borel –, 401, 410, 413 under time change, 181 road map, 51, 54 rotation-invariant stable semigroup, 60

sample path, 393 sample space, 393 self-adjoint operator, 2, 440–441 semigroup conservative, 46 Feller –, 413 of closed symmetric form, 4 recurrent, 38 strongly continuous contraction, 2 symmetric operator, 441 transient, 38 semigroup property, 14 semipolar, 98, 417 shift operator, 399 Silverstein extension, 275, 277, 280 simple random walk, 59 skew Brownian motion, 354, 389 skew extension, 352 by identifying multi-points, 357 smooth measure, 84, 131, 202 in the strict sense, 141 Sobolev inequality, 61 Sobolev space of order 1, 58 spectral family, 440 spectral representation of self-adjoint operator, 441 speed function, 51 speed measure, 52 square field operator, 165 stable-like process, 295 standard process, 412 Borel –, 405 special –, 412 special Borel –, 33 stochastic integral w.r.t. semimartingale, 439 stochastic process, 393, 394, 434 indistinguishable, 394 left continuous, 394 measurable, 394 optional, 394 predictable, 394 right continuous, 394 Stone-Weierstrass theorem, 33 stopping time, 393 predictable, 435 totally inaccessible, 435 strong Feller, 129 submartingale, 395 supermartingale, 395

478 support of a measure, 16 of PCAF, 175, 427 sweeping out, 84 symmetric kernel, 13 operator, 440 process, 33 symmetric Cauchy process, 198 on unit circle, 63 symmetric form, 2, 440 approximating form of, 3 closed, 2 Markovian, 4 of semigroup, 3 symmetric Lévy process, 57, 269 symmetric stable process, 60 on unit circle, 63 symmetric stable semigroup on unit circle, 63 symmetric stable-like process, 295 symmetric step process, 270 symmetrizing measure, 52

tangent space, 76 terminal random variable, 242 under quasi-regular setting, 268 thin, 417 at a point, 417 time change, 190–202, 428–432 of RBM, 196, 273 time-changed process, 175–185, 431, 432 time-reversal, 284 topological support of a measure, 182 trace of Brownian motion, 237 of RBM, 232 trace Dirichlet form, 179, 184, 186 transience, 118 of time-changed process, 179 transient Dirichlet form, 42 process, 118, 119 semigroup, 38, 39, 42, 45–46 transition function, 13, 396 Feller –, 413 for Dirichlet form, 99 transition probability, 13 trap, 396

INDEX uniform domain, 382 universal Dirichlet space, 246 Walsh Brownian motion, 362 X 0 -fine limit function, 324 α-excessive regularization, 414 A+ c , 131 Acc1 , 141 (A.1), 340 (A◦ .1), 347 (A.2), 343 (A.2) , 343 (A.3), 343 (A◦ .3), 347 (A.4), 343 (A◦ .4), 347 (AC), 140, 422 (AC) , 422 B, 335 Br , 417 (B.1), 347 (B.2), 347 (B.3), 347 C, 224 C[α] , 313 (C.1), 375 (C.2), 375 (C.3), 375 eB , 78, 105 (0) eB , 114 e(A), 143 e(A, B), 143 E-supp[µ], 262 E-supp[u], 262 E-neighborhood, 262 E-nest, 22, 77, 103 E-q.e., 22 E-quasi-continuous, 22, 29, 77 E-quasi-everywhere, 22 E µ , 168 ◦ F loc , 163 Floc , 248 (F.1), 334 (F.2), 334 (F.3), 334 G, 311 G0 , 311 HB , 113, 117, 416 HαB , 103, 416

479

INDEX HBα , 84 HN, 269 H0 N, 303 (K.1), 375 (K.2), 375 (M.1), 370 (M◦ .1), 376 (M.2), 373 (M.3), 373 (M◦ .3), 376 (M.4), 373 (M◦ .4), 376 M [u] , 150 ◦ M, 148 µu , 153, 164 µcu , 153 j

µu , 153 µku , 153 N, 269, 303 N1 , 304 N [u] , 150 N , 307 N1 , 309 Nc , 149 N(f ), 327 O0 , 92 pB , 115, 416 pαB , 105, 416 PHB , 114 PHBα , 84, 104

π -system, 392 Q, 176 (RE.1), 316 (RE.2), 316 (RE.3), 316 (RE.4), 316 S, 84, 202 ◦ S, 181 S0 , 80 S00 , 83 (0) S0 , 87, 202 S1 , 141 SF , 176 S(E0 ), 207 S pur (E0 ), 207 σB , 402 σ˙ B , 402 σ -field, 391 Borel, 392 completion of, 391 universal completion of, 391 universally measurable, 392 (t.1), 13 (t.2), 13 (t.3), 13 (t.4), 13 (t.4) , 14 α Up,A , 172 , 280 k≥1 Gk , 243

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