From Markov Chains to Non-Equilibrium Particle Systems, Second Edition

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NEW JERSEY

LONDON

SINGAPORE * S ~ A N e ~ *A HONG l KONG

TAIPEI

9

CHENNAI

Published by

World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: Suite 202, 1060 Main Street, River Edge, NJ 07661

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British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

FROM MARKOV CHAINS TO NON-EQUILIBRIUM PARTICLE SYSTEMS (2nd Edition)

Copyright 0 2004 by World Scientific Publishing Co. Pte. Ltd AN rights reserved. This book, or parts there06 may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.

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ISBN 981-238-81 1-7

Printed in Singapore.

Contents Preface to the First Edition . . . . . . . . . . . . . . . . . . ix Preface to the Second Edition . . . . . . . . . . . . . . . . . xi Chapter 0 . An Overview of the Book: Starting from Markov Chains . 0.1. Three Classical Problems for Markov Chains 0.2. Probability Metrics and Coupling Methods . 0.3. Reversible Markov Chains . . . . . . . . 0.4. Large Deviations and Spectral Gap . . . . 0.5. Equilibrium Particle Systems . . . . . . . 0.6. Non-equilibrium Particle Systems . . . . .

. . . . . . .

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. . . . . . . 1 . . . . . . . 1 . . . . . . . 6 . . . .

. . . .

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13 15 17 19

Part I . General Jump Processes . . . . . . .

21

Chapter 1. Transition Function and its Laplace Transform . 23 1.1. Basic Properties of Transition Function . . . . . . . . . . 23 27 1.2. The q-Pair . . . . . . . . . . . . . . . . . . . . . . . 1.3. Differentiability . . . . . . . . . . . . . . . . . . . . . 38 1.4. Laplace Transforms . . . . . . . . . . . . . . . . . . . 51 57 1.5. Appendix . . . . . . . . . . . . . . . . . . . . . . . 1.6. Notes . . . . . . . . . . . . . . . . . . . . . . . . . 61

.

Chapter 2 Existence and Simple Constructions of Jump Processes . . . . . . . . . . . . 2.1. Minimal Nonnegative Solutions . . . . . . . . . 2.2. Kolmogorov Equations and Minimal Jump Process 2.3. Some Sufficient Conditions for Uniqueness . . . . 2.4. Kolmogorov Equations and q-Condition . . . . . 2.5. Entrance Space and Exit Space . . . . . . . . . 2.6. Construction of q-Processes with Single-Exit q-Pair 2.7. Notes . . . . . . . . . . . . . . . . . . . .

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Chapter 3. Uniqueness Criteria . . . . . . . . . . . . . . 3.1. Uniqueness Criteria Based on Kolmogorov Equations . . . 3.2. Uniqueness Criterion and Applications . . . . . . . . . . 3.3. Some Lemmas . . . . . . . . . . . . . . . . . . . . . 3.4. Proof of Uniqueness Criterion . . . . . . . . . . . . . 3.5. Notes . . . . . . . . . . . . . . . . . . . . . . . . .

V

. . . . .

62 62 70 79 85

88 93 96

. 97 . 97 . 102 113

. 115 119

CONTENTS

vi

Chapter 4. Recurrence. Ergodicity and Invariant Measures . . . . 4.1. Weak Convergence . . . . . . . . 4.2. General Results . . . . . . . . . . . 4.3. Markov Chains: Time-discrete Case . 4.4. Markov Chains: Time-continuous Case 4.5. Single Birth Processes . . . . . . . 4.6. Invariant Measures . . . . . . . . 4.7. Notes . . . . . . . . . . . . . . .

. . . .

. . . . . . . . . . .

. . . . . . . .

.

Chapter 5 Probability Metrics and Coupling 5.1. Minimum Lp-Metric . . . . . . . . . . . 5.2. Marginality and Regularity . . . . . . . . 5.3. Successful Coupling and Ergodicity . . . . 5.4. Optimal Markovian Couplings . . . . . . 5.5. Monotonicity . . . . . . . . . . . . . . . 5.6. Examples . . . . . . . . . . . . . . . . 5.7. Notes . . . . . . . . . . . . . . . . . .

. . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 120 . 120 . . . .

Methods . . . 173 . . . . . . . . 173 . . . . . . . . 184 . . . . . . . . 195 . . . . . . . . 203 . . . . . . . 210 . . . . . . . 216 . . . . . . . 223

Part I1. Symmetrizable Jump Processes . Chapter 6 . Symmetrizable Jump Processes and Dirichlet Forms . . . . . . . . . . . . . . . . . 6.1. Reversible Markov Processes . . . . . . . . . . . . . . . 6.2. Existence . . . . . . . . . . . . . . . . . . . . . . . 6.3. Equivalence of Backward and Forward Kolmogorov Equations 6.4. General Representation of Jump Processes . . . . . . . . . 6.5. Existence of Honest Reversible Jump Processes . . . . . . . 6.6. Uniqueness Criteria . . . . . . . . . . . . . . . . . . . 6.7. Basic Dirichlet Form . . . . . . . . . . . . . . . . . . 6.8. Regularity, Extension and Uniqueness . . . . . . . . . . . 6.9. Notes . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 7. Field Theory . . . . . . . . . . 7.1. Field Theory . . . . . . . . . . . . . . 7.2. Lattice Field . . . . . . . . . . . . . . 7.3. Electric Field . . . . . . . . . . . . . . 7.4. Transience of Symmetrizable Markov Chains 7.5. Random Walk on Lattice Fractals . . . . . 7.6. A Comparison Theorem . . . . . . . . . 7.7. Notes . . . . . . . . . . . . . . . . .

124 130 139 151 166 171

225

227 227 229 233 233 243 249 255 265 270

. . . . . . . 272 . . . . . . . . 272 . . . . . . . . 276 . . . . . . . . 280

. . . . . . . . 284 . . . . . . . . 298 . . . . . . . . 300 . . . . . . . . 302

CONTENTS

.

Chapter 8 Large Deviations 8.1. 8.2. 8.3. 8.4.

vii

. . . . . . . . . . . . . . . . 303

Introduction to Large Deviations . . Rate Function . . . . . . . . . . . Upper Estimates . . . . . . . . . . Notes . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

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. . . . 303 . . . 311 . . . 320 . . . 329

.

Chapter 9 Spectral Gap . . . . . . . . . . . . . . . . . . 330 General Case: an Equivalence . . . . . . . . . . . . . . 330 Coupling and Distance Method . . . . . . . . . . . . . . 340 Birth-Death Processes . . . . . . . . . . . . . . . . . . 348 . . . . . . 359 Splitting Procedure and Existence Criterion Cheeger’s Approach and Isoperimetric Constants . . . . 368 9.6. Notes . . . . . . . . . . . . . . . . . . . . . . . . . 380

9.1. 9.2. 9.3. 9.4. 9.5.

Part I11. Equilibrium Particle Systems Chapter 10. Random Fields

. . . . . 10.1. Introduction . . . . . . . . . . . . 10.2. Existence . . . . . . . . . . . . . 10.3. Uniqueness . . . . . . . . . . . . 10.4. 10.5. 10.6. 10.7. 10.8.

. . . .

. . . .

. . . .

Phase Transition: Peierls Method . . . . Ising Model on Lattice Fractals . . . . . Reflection Positivity and Phase Transitions Proof of the Chess-Board Estimates . . . Notes . . . . . . . . . . . . . . . . . .

.

Chapter 11 Reversible Spin Processes and Exclusion Processes . . . . . 11.1. Potentiality for Some Speed Functions . 11.2. Constructions of Gibbs States . . . . . 11.3. Criteria for Reversibility . . . . . . . 11.4. Notes . . . . . . . . . . . . . . . . .

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. . . .

. . . .

. . . .

. . 381

. . . 383 . . 383 . . 387 . . 391

. . . . . . . . 397 . . . . . . . . 399

. . . . . . . . 406 . . . . . . . . 416 . . . . . . . 421

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422 422 425 432 446

Chapter 12. Yang-Mills Lattice Field . . . . . . . . . . . . 447 12.1. 12.2. 12.3. 12.4.

Background . . . . . . . . . . . . . . . . . . . . . Spin Processes from Yang-Mills Lattice Fields . . . . . Diffusion Processes from Yang-Mills Lattice Fields Notes . . . . . . . . . . . . . . . . . . . . . . . .

. 447 . . 448 . . 457 . 466

CONTENTS

viii

Part IV . Non-equilibrium Particle Systems . . . . . . . . . . . . . . . . . . Chapter 13. Constructions of the Processes . . . . . . . . 13.1. Existence Theorems for the Processes . . . . . . . . . . 13.2. Existence Theorem for Reaction-Diffusion Processes . . . 13.3. Uniqueness Theorems for the Processes . . . . . . . . . 13.4. Examples . . . . . . . . . . . . . . . . . . . . . . . 13.5. Appendix . . . . . . . . . . . . . . . . . . . . . . . 13.6. Notes . . . . . . . . . . . . . . . . . . . . . . . .

467

. 469 . 469 . 486 . 493 502 510 . 513

.

Chapter 14 Existence of Stationary Distributions and Ergodicity . . . . . . . . . . . . . . . . . . . 514 14.1. General Results . . . . . . . . . . . . . . . . . . . . 514 14.2. Ergodicity for Polynomial Model . . . . . . . . . . . . . 521 14.3. Reversible Reaction-Diffusion Processes . . . . . . . . . . 532 538 14.4. Notes . . . . . . . . . . . . . . . . . . . . . . . . .

.

Chapter 15 Phase Transitions . . . . . . . . . . . . . . . 539 539 15.1. Duality . . . . . . . . . . . . . . . . . . . . . . . . 15.2. Linear Growth Model . . . . . . . . . . . . . . . . . . 542 15.3. Reaction-Diffusion Processes with Absorbing State * . . 547 15.4. Mean Field Method . . . . . . . . . . . . . . . . . . 550 15.5. Notes . . . . . . . . . . . . . . . . . . . . . . . . . 554 Chapter . 16. Hydrodynamic Limits . . . 16.1. Introduction: Main Results . . . . . 16.2. Preliminaries . . . . . . . . . . . . 16.3. Proof of Theorem 16.1 . . . . . . . 16.4. Proof of Theorem 16.3 . . . . . . . 16.5. Notes . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . .

. . .

555 555 . . . 559 . . . . 564 . . . . 570 . . . 571

. . . . . . . . . . . . . . . . . . . . . .

572

Author Index . . . . . . . . . . . . . . . . . . . . . .

589

Bibliography

Subject Index . . . .

. . . . . . . . . . . . . . . . 593

Preface t o the First Edition The main purpose of the book is to introduce some progress on probability theory and its applications to physics, made by Chinese probabilists, especially by a group at Beijing Normal University in the past 15 years. Up t o now, most of the work is only available for the Chinese-speaking people. In order to make the book as self-contained as possible and suitable for a wider range of readers, a fundamental part of the subject, contributed by many mathematicians from different countries, is also included. The book starts with some new contributions to thc classical subject Markov chains, then goes t o the general jump processes and symmetrizable jump processes, equilibrium particle systems and non-equilibrium particle systems, Accordingly the kook is divided into four parts. An elementary overlook of the kook is presented in Chapter 0. Some notes on thc bibliographies and open problems arc collected in the last section of each chapter. It is hoped that the book could be useful for both experts and newcomers, not only for mathematicians but also for the researchers in related areas such as mathematical physics, chemistry and biology. The present book is based on the book “Jump Processes and Particle Systems” by the author, published five years ago by the Press of Beijing Normal University. About 1/3 of the material is newly added. Even for the materials in the Chinese edition, they are either reorganized or simplified. Some of them are removed. A part of the Chinese book was used several times €or graduate students, the materials in Chapter 0 was even used twice for undergraduate students in a course on Stochastic Processes. Moreover, the gitlley proof of the present book has bcen used for gradiintc students in their second and third semesters. The author would like to express his warmest gratitude to Professor Z. T. Hou, Professor D. W, Stroock and Professor S. 3 . Yan for their teachings and advices. Their influences are contained almost everywhere in the book. In the past 15 years, the author has been benefited from a large number of colleagues, friends and students, it is too many to list individually here. However, most of their names appear in the “Notes” sections, as well as in the Bibliography and in the Index of the book. Their contributions and cooperations are greatly appreciated. The author is indebted to Professor x. F. Liu, Y . 3. Li, B. M. Wang, X. L. Wmg, J. Wu, S. Y . Zhang and Y . H. Zhang for reading the galley proof, correcting errors and ixlproving the quality of the presentations. It is a nice chance l o acknowledge thc financial support during thr. past years by fi’ok Ying-Tung Educational Foundation, Foundation of Institution of Higher Education for Doctoral Program, Foundation of State Education Commission for Outstanding Young Teachers and the ix

X

PREFACE TO THE

FIRSTEDITION

National Natural Science Foundation of China. Thanks are also expressed to the World Scientific for their efforts on publishing the book. M. F. Chen Beijing November 18, 1991

Preface t o the Second Edition The main change of this second edition is Chapter 5 on “Probability Metrics and Coupling Methods“ and Chapter 9 on “Spectral Gap” (or equivalently, “the first non-trivial eigcnvalue”) Actually, these two cha.pters have been rewritten, within the original text. In the former chapter, the topic of “optimal Markovian couplings” is added and the “stochastic cornprtrability” for jump processes is cornplited. In tlis latter cliapt,er, t,wo general results on estimating spectral gap l y couplings and two dual variational formula for spectral gap of birth-death processes are added. Moreover, a generalized Cheeger’s approach is renewed for unbounded jiirnp processes. Next, Sectiorr 4.5 on “Single Birth Processes” and Section 14.2 on “Ergodicity of Reactiondiffusion Processes“ are updated. But the original technical Section 14.3 is removed. Besides, a large number of recent publications are included. Numerous modifications, improvements or correct’ionsare made in almost every page. It is hoped that, t,he serious effort could improve the quality of the book and bring the reader to enjoy some of the recent developments. Roughly speaking, this book deals with two subjects: Markov Jump Processes (Parts I and 1) and Interacting Particle Systems (Parts m and IV). If one is interested only in the second subject, it is not necessary to read all of t,he first niae chapters, but instead, may have a look at Chapters 4, 5, 7: 9 plus s2.3 or so. A quick way to read the book is glancing at the element,ary Chapter 0, to get some impression about what studied in the hook, to have some test of the results, arid to choose what for the further reading. Some t.irnes, 1 feel crazy to writ’e such a thick book, this is due to the wider range of topics. Even though it can be shorten easily by moving sonic details but the resulting book would be much less readable. Anyhow, I belicve that the reader can make the book thin and thin. A concrete model t.hroughout the whole book is Schlogl’s (second) rnodeI, which is introduced at the beginning (Example 0.3) to show the power of our first main result and discussed right after the last theorem (Theorem 16.3) of bhe book about its unsolved problems. This model, completely different from Ising model, is typical from non-equilibrium statistical physics. Its generalization is t.he polynomial model or more generally, the class of reaction-diffusion processes. Locally, these models are Markov chains. But even in t,his case, the uniqueness problem of the process was opened for several years, though everyone working in this field believes so. From physical point of view, the Markov chains should be ergadic and this is finally proved in Chapter 4, Thus, to study the phase tra.nsit.ions, we have to go to the infinite dimensional setting. The first hard stone is the construction of the corresponding Markov processes. For which, the matherna.tical tool .is preI

xi

xii

PREFACE T O T H E SECOND

EDITION

pared in Chapter 5 and the construction is done in Chapter 13. The model is essentially irreversible, it can be reversible (equilibrium) only in a special case. The proof of a criterion for the reversibility is prepared In Chapter 7 arid completed in Chapter 14. The topics studicd in almost, every chapter are either led by or related to Schlligl’s rnodel, even though sometimes it is not explicitJy mentioned. Actually, the last four chapters are all devoted to the reaction-diffusion processes. The Schlijgl model possesses thc main characters of the current mathematics: infinite dimensional, non-linear, complex systems and so on. It provides us a chance to re-examine the well developed finite dimcnsional mathematics, to create new mathematical tools or new research topics. It is not surprising that many ideas and results from different branches of mathematics, as well ti physics, are used in the book. However, it is surprising that the methods developed in this book turn out to have a dccp application to Rierriaxiniari geometry and spectral theory. ‘l’his is clearly a different story. Since there are so much progress made in the past ten years or more, a large part of the new materials are out of the scope of this book, the author has decided to write a separate book under the title “Eigenvalues, Inequalities and Ergodic Theory”. It is a pleasure t o recall the fruitful cooperation with my previous students and colleagues: Y. H. Mao, F. Y. Wang, Y. Z. Wang, S. Y. Zhang, Y . H. Zhang et al. Their contributions heighten remarkably the quality of the book. The author acknowledges the financial support during the past years by the Research Fund for Doctoral Program of Higher Education, the National Natural Science Foundation of China, the Qiu Shi Science and Technology Foundation and the 973 Project. Thanks are also expressed to the World Scientific for their efforts on publishing this new edition of the book.

M. F. Chen Beijing August 29, 2003

Chapter 0

An Overview of the Book: Starting from Markov Chains In this chapter, we introduce some background of the topics, as well as some results and ideas, studied in this book. We emphasize Markov chains, and discuss our problems by using the language as elementary and concrete c2s possible. Besides, in order to save the space of this section, we omit most of the references which will be pointed out in the related “Notes” sections. 0.1 Three Classical Problems for Markov Chains For a given transition rate (Le., a Q-matrix Q = ( q z j ) on a countable state: space), the uniqueness of the Q-semigroup P ( t ) = (Pt3(t)), the recurrence and the positive recurrence of the corresponding Markov chain are three fundamental and clmsical problems, treated in many textbooks. As an addition, this seclion inlroduceu some practical results motivated from the study of a type of interacting particle systems, reaction-diffusion processes.

Definition 0.1. Let E be a countable set. Suppose that ( P z J ( t )is) a subMarkov transition probability matrix having the following properties.

(1) Normal condition.

-p&) < 1, hj E E , t 2 0.

P&) 2 0 ,

3

(2) Chaprnan-Kolmogorov equation. P i 3 ( I + 6 )= C P i k ( t ) P k j ( S ) ,

i , j E E , t , s 3 0,

k

(3) Jump condition. limt-o Pij(t) = Sij for all i , j E E . It is well-known that for such a ( P z j ( t ) )we , have a Q-matrix Q = ( q i j ) deduced by (4) Q-condition. (Pij(t)- S i j ) tlim 40

/ t = qij

where

1

for all i , j E E ,

2

0 AN OVERVIEW OF

THE

BOOK

Because of the &-condition, we often call P ( t ) = (Pij(t)) a Q-process. Unless otherwise st,ated, t,hroughout this chapter, we suppose that the Q-matrix Q = ( q i j ) is totally stable and conservative. That is

C3.f z. q ”

4%< C X ] ,

i € E. (0.1) The first problem of our study is when there is only one Q-process P ( t ) = (Pij(t))for a given Q-matrix Q = ( q i j ) (Then, the matrix Q is often called regular). This problem was solved by Feller (1957) and Reuter (1957). = qi,

2J

Theorem 0.2 (Uniqueness criterion). For a given &-matrix Q = the Q-process (Pij(t))is unique if and only if (abbrev, iff) the equation

(A

+

q;)Ui

=CQijUj, 0

< ui < 1,

iEE

(qij),

(0.2)

j#l

has only the trivial solution uz = 0 for some (equivalently, for all) X

> 0.

Certainly, this criterion has rna.ny applicatjons. For instance, it gives us a cornpl& answer to the birth-death processes (cf; Corollary 0.8 below). However, it seems hard to apply the above criterion directly to the following examples. Example 0.3 (SchIiigl’s modcl). Let S be a finite set and IT = X:, where Z+ = { O , l , . . . }. The model is defined by the following Q-matrix Q -r ( q ( q y) : z,y E

E):

[

A1

(2“’)”

+

if y = z

A1

+ e,

ify=z-e,+e,

4 ( 4 = - d x >4 =

c

for other y

# z,

q(WA

Y#X

(1)

where 2 = ( ~ ( u: )u E S ) , is the usual combination, XI, ’ ! A 4 are positive constants, ( p ( u , v ): u,’u E S) is a transition probability matrix on S and eU is t h e element in E having value 1 a t u and 0 elsewhere. 1

The Schlogl model is a model of chemical reaction with diffusion in a container. Suppose that the container consists of small vessels. In each vessel u f S , there is a reaction described by a birth-death process. The birth and death rates are given, respectively, by the above first two lines in the definition of (q(x,y)). Moreover, suppose that between any two vessels u and w, there is a diffusion, with rate given by the third line of the definition. This model was introduced by F. Schlogl (1972) as a typical model of nonequilibrium systems. See Haken (1983) for related references.

0.1 THREE CLASSICAL PROBLEMS FOR MARKOV CHAINS

3

Example 0.4 (Dual chain of spin system). Let S be a countable set, and X be the set of all finite subsets of 5'. For A E X ,let IAl denote the number of elements in A . For various concrete models, their &-matrices (q(A,B) : A , B E X)usually satisfy the following condition:

for some constant C, c E R := (-m, m). A particular case is that

F : I'A(A\u)=B

uEA

where

4).

2 0,

supc(.u) < CO? U

and supuc(u) C A p ( u , A )JAl
0 for some (equivalently, for all) X 2 0. Conversely, these conditions plus p 2 0 are also necessary.


.) -'(t,i2,*)llvar

< 2*1'i2 [ T > t ] - 0

Furthermore, if the process has a stationary distribution

ast--tO. T,

then

and so the process is ergodic.

As another application of the coupling technique, we discuss the monotonicity for Markov chains.

0 . 3 REVERSIBLE MARKOVCHAINS

13

Definition 0.24. Take E = {0,1,2,...} and let ( X k ( t ) ) i a o(k = 1,2) be two copies of a Markov chain ( X ( t ) ) t a owith different starting points. If

then we say t h a t the chain is monotone. One way to prove the monotonicity is using the coupling method. For example, applying the basic coupling to a Markov chain with regular Qmatrix Q = (yij) on Z+,we find that the condition:

is sufficient for the monotonicity of the Markov chain. Of course, if we use a different coupling, we will find different sufficient condition for the rnonotonicity. From this point of view, it is believable that condition (0.10) is not necessary for the monotonicity. A complete solution for the monotonicity for general jump processes, as well as other topics discussed in this section, are treated in Chapter 5 . The above two sections are based on Chen (1989a, 1991d), respectively.

0.3 Reversible Markov Chains

Definition 0.25. Let (Xt)t>o be a Markov process defined on (a,9, IF') with countable state space E . The process is called reversible if for any n 2 1, 0 < i l < . . < it, with

and any il, . ' .

, in c E ,

[X,, = i l , . . .

,xi,= in] = P [X,, = i n , . . , Xt,, = ill.

(Q.11)

Clearly, a reversible Markov chain (Xi)should be stationary. That is, xi := P [X, = i ] is independent of t 2 0. Actually,

Due t o the Markov property, (0.11) is equivalent to (0.12)

14

0 AN OVERVIEW

OF THE

BOOK

This implies that Tiqij

= rjq..ji)

i , j E E , t 2 0.

(0.13)

Since it is easy to get. Q = ( q i j ) in practice, but not ( P i j ( t ) ) ,we should start our study from a given Q-matrix. Thus, we are now at; the position as at the beginning of Section 0.1. Fur a given Q-matrix Q (yij) which is reversible with respect; to a probability measure (ni)in the sense of (0.13), we would like to know when there is one, and when there is precise one, such Q-process (Pij(5)) so that (0.12) holds. To state our main results, let us relax the probability measure (ri)by an arbitrary but non-trivial measure (7riTi). Then, we call Q = ( q i j ) (resp. ( P i j ( t ) ) )symmetrizable with respect t o (ni)if (0.13) (resp. (0.12)) holds. Finally, in this section, the only assumption for the Q-matrix ( q i j ) is the total stability: y i < 30 €or aIl i E E . Theorem 0.26. The minimal Q-process (Piyin(t)) is reversible (resp. symmetrizable) with respect to (T*)iff so is its Q-matrix.

Theorem 0.27. With respect to a probability measure ( x i ) , the reversible Q-process is unique iff the following conditions hold. (1) ( q i j ) is reversible with respect t o ( ~ i ) ~ (2) Ti(Yi - Cj+ 4ij) < 0 , (3) Equation (0.2) has only the trivial solution For general (ni): we have the following result.

Theorem 0.28. With respect t o a measure (q), there exits precisely one symmetrizable Q-process if the following conditions hold. (1) Q = ( y i j ) is symmetrizable with respect t o (ni). (2) C(T,(qi - Cj+iq i j ) < go or infi C j P;;~*’(A) > 0. (3) The only sumrnable solution t o Equation (0.2) is zero. We guess that condition (2) is still stronger than to be necessary. Thus, a complete criterion for the uniqueness of symmetrizable Q-process remains open. Besides, even though we have known a great deal about the general Q-process (cf. Section 3 4 , we only have a partial solution to the following problem.

Open Problem 0.29. What is the uniqueness criterion for honest reversible (resp. syrnmetrizable) @process? Here, “honest” means that C jPij(t) = 1 for all i E h‘ and t 2 0.

0.4 LARGEDEVIATIONS AND SPECTRAL GAP

15

In the study of symmetrizable Q-process, a new question arises. How can we justify whether a given Q-matrix is symrnetriz.able with respect to some measure (ri)? As a nice exercise, one may try to answer the question by himself for the Schliigl model. In general, this question is answered by us& ari analogue of the classical field theory in. analysis. It is interesting that the same idea can also be used ta study t,hc recurrence for symmetrizable Markov chnhs (see Chapter 7 for details).

0,4Large Dcviations and Spectral Gap Markov chains consist of a nice class of stochastic processes, not, only for their a lot of applications hut also for t,he!j, concrete beh.avior and simplicity. Tn the regular case, the paths of a hllarkov h a i n are simply step functions almost surely. We can even see thc jump law: starting from a state .i, the chain stays in i for a while according to the exponential distribution with parameter qi. Then, t.he chain jumps to j ( # i ) according to the distribution qij/yi (provided 4%> 0). Because of this reason, a large part, of the theory of stochastic processes was begun from hdarkov chains. Cowessely, Markov chains can be used to justify the power of a general theory for stochastic processes. Let us discuss the two topics expressed by the title of this section. In the Donsker-Varadhan's large deviation theory (an introduction to the theory is presented in Section 8.1), we are interested in the entropy (=rate function) :

(0.14) and -1 upper estimate: lim

t+w

lower estimate:


o be a Markov chain with transition probability P(t>= (Pij(t)) a.nd let Pi be the probability that the chain starts from i E E . Next, let 9 ( E ) be bhe set of probabilities on E , endowed with the weak topology. Set

and Q t , i = Pi o L t ' . Considering P ( t ) as an operator on b&: the set of bounded functions with uniform norm? it induces an infinitesimal generator

16

0 -4N OVERVIEW

O F T HE

BOOK

L with domain g ( L ) . Let g + ( L )be the set of strictly positive functions in

m)* In view of Markov chains, the entropy given by (0.14) is not satisfactory

since 9 ( L ) is quite poor, even the indicator Iti)(i E E) is usually not in g ( L ) . However, we have

Thesrcm 0.30. Given a regular Q-matrix Q = ( q i j ) , (1) if p E

9(&) satisfies C ,piqi < 00,

where %? = F

then

( L ) or anyone of the following sets:

> O for some E > O } , 0 < f < w},

&+ = (f : f 2 €O = b@+

(f

:

= &8n &+,

E

b@

=b

8

n go9

(2) If (gij) is reversible with respect to some 7r E P ( E ) , then p E 9 ( E ) ,we have an explicit expression as follows.

for every

This theorem is proved in Chapter 8. Some upper estimates are also studied there. Roughly speaking, the large deviations say that the exponential I ( p ) . For convergence rate of &t,z(C)is described by the entropy - iiifPEc reversible Markov processes, we have a different way 1,o look at the exponential corivergeiicc rate: IIP(l)f - ~ ( j ) l l - ~ ( f ) l l exp[-et], where 11 11 is the norm in L2(7r) and ~ ( fis)the mcan of f with respect to 7r. Let 0 denote the largest value of E > 0. The curlstant u is the rate we are looking for. As usual, the cortvergcnce rate is related to some spectral gap. Let L denote the generator with domain g ( L ) induced by P ( t ) on L 2 ( r ) and let gap(L) denote the infimum of the spectra of -L restricted to the orthogonal complement of the constant function 1. Then, we have the following result.

< 11s

Theorem 0.31.

+

0.5 EQUILIBRIUM PARTICLE SYSTEMS

17

For finite Markov chains 1vit.hQ-matrix &, gap(Q) is nothing but the first non-trivial eigenvalue of -&. Estimating gap(Q) is a traditional hard topic in mathematics. '1'0 compute gap(&) explicitly, one has to stop when the order of Q is higher than five. Surprisingly, we do have in a particular case a complete solution to the problem eve11 for some infinite matrices. Consider the birth-death &-matrix: q i , i + l = bi > 0 ( i 2 0), qi,i-1 = ai > 0 (i 3 1) arid q,j == 0 for all other i # j . Suppose that the process is ergodic and so we have a stationary distribution

nefine

{ {w~}+o wi is strictly increasing in i and C ixiwi 3 0}, "w = { there exists k 1 < k < so that wi = w is :

=

:

{wi.}i>o :

m

wiAk,

strictly increasing in [0,k] and

&(w) =

1

c

xi ~ i w =i 0},

00

bZ/Ji(Wi+l - Wi) j .= z +, l

&Wj?

i 2 0;

6 = sup

c

1 -c p j ,

i31 j G i - l ~ j b j>i j

Note that g i s simply a modification of W . Hence, only two notations W and l(w)are essential here.

Theorem 0.32. For the ergodic birth-death process as above, the following conclusions hold. (1) Variational: forrnula for the lower bound gap(D) = sup inf l i ( w ) - I , W G W 220

( 2 ) Variational formula for the upper bound: gap(D) = inf-sup &(w)-' W E Wi&J

(3) Explicit bounds and explicit criterion: 26-' particular, gap(D) > 0 iff S < 00.

3 gap(D) 2 (46)-1.In

The study on spectral gap is the aim of Chapter 9. 0.5 Equilibrium Particle Systems

Let us start from the simplest case. Consider a Q-process (P(t)) with Q-matrix

and state space E = {–1,1}. Assume that ab = 0,

otherwise, the model is trivial. Then

18

0 AN OVERVIEW OF

THE

BOOK

As the limit of Pij(t)( t -+ oo),we obtain the stabionary distribution n-1

+b),

a/(.

T+I

T

b/(a

+b).

In other words, there is only one stationary distribution, denoted by 19) = 1. ‘l’hc above process is a rnodel with single particle having two states f l . If we consider finite number of particles, say it’ E N,N 2 2, Then the state space becomes {-1, + l } N .Thc system can be nlso described by a Q-process (its operator is given by (0.15) below replacing Zd with N ) and we still have 1 4 1 = 1. What will happen if we replace N with a countable set? For instance, consider a particle system on the regular lattice Z d . At each site u E Zd, there is a particle with two states f l . Then the whole configurations consist of our state space {-l,+l}Zd, which is no longer countable. Hence, the system can notj be described by a Q-process. Now, we use c ( u , x ) ,instead of qzg,to describe the Osarisition rate of a particle changing its state. Given x E E , let if u = ‘11 (U.)(.> = if u.# v --q,

{

and define a formal generator as follows:

Rf(4=

c

c ( u , 4 [ f ( u . - .,

f(41

(0.15)

*

UEZ*

c

A particular choice of c(u, x) is c ( u , x) = exp rl‘hcIi we obtain the famous Ising model in astatistical physics. Now, corriplete different phenoinenon happens. For d = 2,we bave

1 9 1= 1,

if ,l3

1

< -log (1 + A) =: fl:2) x 0.44 2

,Bid)

For d 3, the picture is similar for a critical point > 0. But for d = I, we have 1 . 9 1 = 1. It should be clear now that the king model exhibits phase transitions which depend on the dimension d. Actually, t,his model has attracted a lot of attention in statistical physics, even in the 2-dimensional case (see for instance, McCoy and Wu (1973)). The Ising model, as well as a fundcamental part of the theory of random fields, including the typical methods-the Peierls method and the reflection positivity method for studying the phase transitions, are presented in Part JX. Based on the field theory, we introduce some simple criteria for the reversibility of spin processes and exclusion processes. Besides, two new developments

0.6 NON-EQUILIBRIUM PARTICLE SYSTEMS

19

in the field are included. The first one is to use the lattice fractals instead the regular lattice. Then, we do obtain some interesting results. For example, the Ising model on lattice Sierpinski gasket h m no phase transitions in any dimension bul the model on lattice Sierpinski carpet does exhibit the phase transitions in any dimension. The other one is to use some groups as the spin space instead of {-1, I l}, the latter one seems mainly suitable for the mctallic phasc transitions at low temperature. Howcvm, new progress on the superconductivity has been made recently by using ccramics instead of ferromagnetics. This explains why we have to consider more general spin space instead of {-1, +1}.

0.6 Non-equilibrium Particle Systems The Ising model discussed in the last section belongs to the equilibrium statistical physics. Having the knowledge about the equilibrium systems in mind, it is natural to ask what we can do for the non-equilibrium systems. A typical example is Schlogl's model (Example 0.3), replacing the finite set S with infinite one S = Z d . Thc formal generator can be written ~LS follows:

where X I , . . . , A4 and (p(u,u)) are the same as before, e, is the unit vector in E = Z s having value 1 at u E Zd and 0 elsewhere. This model is a special reaction-diffusion process studied in the last part of the book. I t may be helpful for our readers to compare the Schlogl model with the Ising model. (1) Clearly, the state space E = (-1, +l}"dfor Ising model is compact and so is L@(E).Thus, the process has at least one stationary distribution. But for Scl-ilogl model, the state space E = Z"+" is neither cornpact nor locally compact. (2) 'l'he king model is reversible, ils local Gibbs distributions are explicit. But the Schlijgl model hns no such advantage, except in a special case. (3) The generator ol the king model is locally bounded but it is not so for the Schlogl model.

These facts show that thc non-equilibrium particlc systems are more difficult to handle than the equilibrium systems.

20

0 AN OVERVIEW

OF T H E

BOOK

To construct an infinite dimensional Schlogl model, take a sequence {A,} of finite subsets of Zd so that A, t Z d . Then, we have a sequence of Markov -) : n 2 l} chains {P,(t) : n 2 l}. The next step is to prove that {Pn(t,x, is a Cauchy sequence. Thus, we have to use a probability metric, say W , for instance: as m 2 n --+ 00. w(P,(~ x,,.), ~ , ( t ,x,.)) + 0 From this line of the construction, we see a relation between the Markov chains and the interacting particle systems. Locally, particle systems are Markov chains. At this point, it explains why the title of the book is chosen. The constructions, the uniqueness of the processes as well as 15 concrete models are presented in Chapter 13. It will be proved in Chapter 14 that the reaction-diffusion processes often have at least one stationary distribution and sometimes they are ergodic. The reversible reaction-diffusion processes are always ergodic. For some special models, we will prove that there more than one stationary distributions. That is, the processes exhibit phase transitions (Chapter 15). Finally, we turn to discuss the relation between the processes and partial differential equations. It is known that the generator of &dimensional a2/ax?. Moreover, for Brownian motion {Bt}t>o is the Laplacian suitable g, f ( t ,x) := IE,g(Bt) satisfies the linear equation:

1

-=-C-f af 1 d2 at

2

,

2=1

ax?

If(0,x) = g ( 4 -

However, if we consider the reaction-diffusion equation (non-linear):

(0.16) where V is a polynomial, there is no hope to find a Markov process valued in Rd with such a generator since for a Markov process, its generator must be linear. Nevertheless, under some hypotheses on the initial distribution of the process and on the initial function p, it will be proved in the last chapter of the book that a limit of some mean of a scaled reaction-diffusion process provides a solution to Eq. (0.16). In other words, a reaction-diffusion process describes the microscopic behavior, and Eq. (0.16) describes the macroscopic behavior of a non-equilibrium system. In the last chapter, we will also prove that some solution to Eq. (0.16) are asymptotically stable but some of them are not. This result represents the critical phenomena of the systems, which corresponds more or less to the phase transitions for the microscopic processes.

PART I GENERAL JUMP PROCESSES

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Chapter 1

Transition Function and its Laplace Transform In this chapter, we first study some basic properties of sub-Markovian transition function of a jump process: continuity and differentiability. From which, we deduce the transition intensity, q-pair. Next, we study the one-toone correspondence between the transition function of a jump process and its Laplace transform. This enables us to use the fundamental tool, Laplace transform, instead of the transition function itself in the subsequent study. As well-known, the advantage of using the Laplace transform is reducing the integral equations to the linear algebraic ones.

1.1 Basic Properties of Transition Function Throughout the book, we use the following notations. Let ( E , 8 ) and ( X ,93) be two measurable spaces. Denote by f E €,'A3 the measurable mapping from ( E ,&) to (X, 3).However, if X = with Borel a-algebra 93 = B ( R ) ,we simply use the same notation € to denote the set of all measurable functions from ( E , € ) to (R,B(R)). Similarly, let r€ (resp. r€+, b&, b€+, 8+)denote the set of all measurable real-valued (resp. non-negative realvalued, bounded real-valued, bounded non-negative and non-negative but p+)denote the set of all may be +m) functions. Finally, let Y (resp. Y+, a-additive set functions (resp. finite measures, a-finite measures). Unless otherwise stated, the state space ( E , € ) considered in the book is a Pofash space with Borel a-algebra 8. Recall that a Polish space is a separable topological space that can be metrized by means of a complete metric.

Definition 1.1. We call P ( t , z , A )(t 2 0, z E E , A E €) a (sub-Markovian) transition function of a jump process if the following conditions hold. (1) For each (2) For each

t 2 0 and A E €, P ( t ,. , A ) E .8+. t 2 0 and z E E , P(t,z, .>E 9+and P ( t ,z, E ) < 1.

(3) Chapman-Kolmogorov equation (abbrev. CK-equation). For each t , s 2 0, z E E and A E €, P(t

+ s,

2,

A) =

J

P(t,z, d y ) P ( s ,y, A ) .

(4) For each z E E and A E 8, limt,o P ( t ,z, A ) = P(O,z,A ) = d(z, A ) , where 6(., A ) is the indicator of A , also denoted by 1,.

23

24

1 TRANSITION FUNCTION

A N D ITS

LAPLACETILANSFO

In this definition, the crucial point distinguishing to the transition function of a gcrieral Markov process is the last condition (4),which means the continuity at the origin and hence often quoted as continuous condition. However, because of this condition, the sample of the process are step functions, at least before the explosion time. Since this reason, we also call (4) the j u m p condition. In many cmcs, we do not want to distinguish different jump processes with the same transition function. Hence we olten call the transition function itself a jump process. In particiilar, we call it a Markav chain in the case of E being a countable set., denot.ed by matrix (Pij(t): i , j E E ) . A jump process P ( t , z , A ) is called honest, if for each t 2 0 arid z E f3, P ( t ,5 , h’)= 1. Otherwise, it is called non-honest. Theorem 1.2. For each 2 and g E b&+, Y(1,z,dy)f(y) i s uniformly continuous in t uniformly in f with I f 1 6 9. In particular, P ( t ,5 , A ) is uniformly continuous in

t

uniformly in A.

Proof: By conditions (3) and (2) of Definition 1.1, it follows that

In bhe last step we have used the fact that la - bl Thus, we have

< c for ail a , b E [O:c]

Now: the first assertion follows from this and Definition 1.1( 4 ) . I The next result shows a nice property of jump processes. Even though we need the result only in a few of cases, it is still included for completeness. Theorem 1.3. Let P ( t , z , A ) be a j u m p process on ( E , 8 ) . Then for each z E E and A E 8 , either P ( . , z , A )= 0 or P(a,.,A) > 0. Proof: If P ( t ,x 7A) is not honest, we may introduce a fictitious state A $! E , such that EA := E U {A} is again a Polish space and A is an isolated state. Moreover, € c &A := cr(& u {A}). Let

1.1 B A S I C PROPERTIES O F TRANSITION FUNCTION

25

Then we obt,ain an honest jump process P(t,x,A ) (t 3 0, 2 E EA, A E En). Clearly, if P(t,2,A ) possesses the properties described in the theorem, then so does P(t,x,A ) . Hence, we need only consider the honest jump processes. By CK-equation, we have

+

P(t s, x,A) 3 P( s , s , (x}) P ( t ,2, A ) .

> 0, then for all s > 1, P ( s , x , A )> 0. Froin this and the continuit.y of Y ( - , x ,A ) , it follows t b t P ( t , x , A ) > 0 for all t whenever z E A . Furthermore, for x $ A, there exists u(x,A ) E [O, 031 such that Hence, if P ( t ,.,A)

,P(t,x,A ) 2 0,

P ( t ,x,A ) > 0,


u(5,A ) .

Thus, what we need to prove is that for each either u(z, A ) = 0 or u(z,A ) = 03. Suppose that 0 < u ( 5 , A ) < 00

2

(1.2)

C E arid A E 6 , 5 .$ A,

(1.3)

for some x and A . Fix z and A. Set uo = u(x,A), u(y) = u(y,A) and Obviously, u and LJ are measurable. Since ( E , G ) is v(y) = u ( y ) A a Polish space, we can construct a Markov process X ( t ) on a probability P)with t,ransition function P ( t ,x,.) and initial state X ( 0 ) = 2 space ( Q , 9: (cf. Neveu (1965), p.83, Corollary), Let

Yo(t)= V ( X ( t ) ) . Then Yo(0)= uo:0 < Yo(t)< uo. Moreover

By the dominated convergence theorem, the right-hand side tends to 1 as h + 0. So Yo is contiiiuous in probability. On the other hand, since E can be embedded into a compact space X so that the completion (E,p) of E in X is again compact with metric p. There exists a measurable and separable version Y of Yo such that 0 Y ( t ) uo = Y(0). Now, it suffices to show that there exists A E 9such that P (A) = 0 and


A )

P(hl

y7 d z ) p(v(?/) - h7

'7

A)'

(1.6)

On the other hand, by (1.2), '(2) < '(9) - h implies that P(v(y) - h, z , A ) 0. Hence, (1.5) follows from (1.6). Next, by (1.5), we have

1

+

P [ Y ( t k ) < Y ( t )- h ] = This shows that

>

P ( t ,2,d y ) P ( h ,y, ( 2 : v ( z ) < v(y) - h}) = 0.

Y ( t + h) 3 Y ( t )- h,

P-a.s.

(1.7)

By the separability, we may choose an exceptional set so that (1.7) holds for all t , h 2 0. This proves a). In what follows, we will ignore the exceptional set. b) Prove that d

-Z(t) dt

=:

1,

a.e. t.

Because Z ( t ) is non-decreasing, the derivative Z' ( t ) exist almost everywhere. To compute Z'(t), let A(t,h) = IF [ Y ( t h ) # Y ( t ) ]We . have seen that limhtoA(t,h) = 0 for all t 2 0, so by the dominated convergence theorem,

+

1

oc,

lini

h+O

0

A(t,/ ~ ) e - ~ d=t 0.

Thus, we can choose a sequence {h,}, h,

3

0, such that

By the Fubini theorem, there is ZI set N with Lebesgue measure zero so that 00 for all t $ N . Then, by the Borel-Cantelli lemma, we have

c,A(t, h,)
/ 0 and A E 8. Applying the monotone class theorem, we obtain

1 . 3 DIFFERENTIABILITY for all f E b 6 . In particular, taking equation, we get

P ( t + t’, z,A) =

It

f

45

= P(t’, .,A) and applying the CK-

q ( z ) e d ” )( t - s )

.I

rl(s,z,dy)P(t’,y, A)ds

Starting from this and using (1.31) twice, we obtain

= P(t

+ t’,

2,A)

Hence

Thus, for each t’, there is a null set Ntt = Nt/(A),such that

b) We now start from (1.32) to construct the required version R ( t , z , A ) of r l ( t , z , A ) . First, we prove that the both sides of (1.32) are joint measurable in (8, t’). Actually, for fixed t‘, the right-hand side is measurable in t ; and for fixed t , it is continuous in t’. Hence it is joint measurable. The same conclusion holds for the left-hand side. Thus the set S = ( t ,t’) : t , t’ > 0, r1( t

{

+ t’, 5,A ) # f

r l ( t ,2,d y ) P(t’,y, A ) }

is joint measurable. By the Fubini theorem and (1.32), it follows that

11

dtdt‘ = 0.

1 TRANSITION FUNCTION AND ITS LAPLACE TRANSFORM

46

Let u = t , v = t

+ t‘, then JJM

dudv = 0, where

By using the Fubini theorem again, there is a riull set H , and then there is a null set H , for every 7~ 4 H , such that

~ $ 1 1 u, < v $ IIu.

r I (v,z, A) = ]~~(u, IC, dy) P ( v - IL, y, A ) ,

Next, we construct R ( v , x , A ) as follows. T& u‘ such that 0 < u’< v and u’$ 11, and set

R(v,z, A ) =

s

T ~ ( u5’ , d y ) P (v

w

(1.33)

> 0. Choose an arbitrary

- u’, y, A ) .

(1.34)

To justify this definition, we need to show the independence of the choice of u’.To do so, let wo > 0, 0 < u’,u” < vo, u’,u“ $! H be given. Then, for v $ HL U H;(I and v > u’ v u”,by (1.33), both of

1

rl(u’, z, d y ) P(v - u’, y, A ) and

s

rl(u“,z, dy) P(v - u”,Y,A )

equal to rl(v,z, A ) . In particular, this conclusion holds for v = zfo by the continuity of P ( . , y , A ) . Therefore, choosing u’or u”,we define the same

Rho, 2 , A ) . c) Finally, we prove that the kernel R(t,2 , A ) (t > 0, A E 8)constructed above satisfied the desired conditions. By (1.34), R ( t , z ,.)EL?+for each 1. Note that P ( ~ , xE,) is non-increasing. Thus, (2) follows from Lemma 1.18, Corollary 1.21 and (1.34). Next,, we: provc the continuity. Civcn t o > 0, choosc an arbitrary uo $ H so that 0 < uo < to. Then whenever > t o , we have

R(v,X,A)

s

T~(uO,

dy) p(u - uoj Y ~ A ) ,

which shows that R(.,2 , A ) is continuous in ( l o , m). Since to is arbitrary, R(.,z, A ) is continuous in (0,m). Now, we prove that for each A E 8,

R ( v , z , A )= r l ( v , x , A ) ,

a.e. v.

(1.35)

Actually, for the above vo and uo, by (1.33) and (1.34), we sce that (1.35) holds for all v > t o r v $! Hue. In other words, (1.35) holds for almost all

1 . 3 DIFFERENTIABILITY

47

> to. Letting to tend to zero along the rational numbers, we see that (1.35) holds for almost all w > 0. Finally, (1) follows from (1.35) and (1.31). Furthermore, in parallel to the proof of (1.32) by using (1.31), from Proposition 1.22 (l),it follows that

ZI.

+

R(t t', 2,A ) =

J

R(t,5 , d y ) P(t', y, A ) ,

t $ Np.

(1.36)

Both-hand sides are continuous in t , and so Eq.(1.36) indeed holds for all t. This proves (3). So far, we have completed the proof of the existence of $P(t, z, A ) ( t > 0).

Theorem 1.23. Let z satisfy 0 < q(x) < 00. Then for each A E 8, P ( t ,z, A ) (t > 0) has a continuous derivative P'(t, 2,A ) , which is a-additive on 8 , having total variation bounded from above by 2q(z). Moreover,

P ' ( t , z , A ) = q(z)( R ( t , z , A )- P ( t , z , A ) ) , P'(t + s , z , A ) =

J

P ' ( s , z ,dy) P ( t , y , A ) ,

Finally, if q ( z )= 0, then

s

t > 0, A E 8.

> 0, t 3 0, A

E

8.

(1.37) (1.38)

P'(t, 2,A ) = 0 for all t >, 0 and A E 8.

Proof: By Proposition 1.22 (1) and the continuity of R(t,z, A ) , it follows that P ( t ,z, A ) is differentiable and (1.37) holds. Then the properties of P'(t, z, A ) can be read out from (1.37). Finally, (1.38) follows from Proposition 1.22 (3), (1.37) and the CK-equation. The case that q ( z ) = 0 is trivial. H

Corollary 1.24. We have (1) limt,o R(t,2,{z}) = 0. (2) limt-0 P'(t,z, {x}) = -q(x).

+ t , x,{x}) 3 R(s,2,{z}) x P ( t ,x,{z}). So R(t,z, {z}) 2 P ( t ,x,{ z } ) z s - + oR(s,z, {z}).Hence

Proof: By Proposition 1.22 (3), we have R(s

lim R(t,z, {z}) 2 lim R(s,x,{z}). t 4 O

s-4

This shows that limt-0 R(t,2,{z}) exists. Setting A = {z} in Proposition 1.22 (l),we get

Then, part (1) follows by letting t --+ 0. From this and (1.37), we obtain part (2). H The next result is a complement to the previous one.

48

1 TRANSIT~ON FUNCTION

A N D ITS

LAPLACE'FRANSFORM

Corollary 1.25. Let n: $ A E 9, Then

(1) 1imt-o R ( t ,5 , A ) = q(a,A ) / q ( x ) . (2) lirnt,o P ( t ,55, A ) = q ( x , A ) .

Proof: Given

E

> 0, there is a S > 0 such that

ho r n Proposition 1.22 (3), it follows that

W I ' t , ~ , A 2>

s,"(.-. 9

1

dy ) P ( t , y , A ) 2 ( ~ - E ) R ( s , x , A ) .

NOW,the remainder of the proof is similar to the p w i 0 u . s one. Corollary 1.26. If c :=

SUP,~AQ(X)

are functions of bounded variation in


0, s

>, 0.

(1.39)

Proof: a) The proof of this part is similar t o those of Proposition 1.22. Set c := 8iIP,EA q(s). We have seen from the last proof that e r t P ( t , x , A ) is non-decreasing in t. Hence, by Theorern I .23 and Corollary 1.25 (2),we h a w

=:

U ( t ,5 , A ) ,

t 2 0.

On the other hand, by Theorem 1.23, we have IP'(t, x,A ) \ < 2q(z),and so U ( t ?x , A)

< 2q(a) + c.

(1.40)

1.3 DIFFERENTIABILITY

49

Thus, by the definition of U ( t ,2 , A ) , it follows that

Applying (1.41) once again, we obtain

Therefore

j” ecsU(s+ t’, rt

=

Z,

A)ds.

This shows that for each t’ 3 0, there is a null set Ntj such that

U ( t + t ’ , ~A,) =

.i

t $ Ntt.

P(t’,5 , d y ) U ( t ,y , A ) ,

(1.42)

In particular

U(t

+ t’,Z,A) 3

t $ Nv,

P(t’, 2 , d y ) U ( t ,Y, A ) ,

(1.43)

/En


0, and then there is a null set Ht for each t @ H , such that

+

+

/

U ( t t’,z,A)= P ( t ’ , z , d y ) U ( t ,y, A ) ,

O < t $ H , t’$Ht.

(1.45)

Now, we prove that Ht = 8 for each 0 < t $ H . Suppose that there are a to: 0 < t o $ H and tb $ H,,so that (1.45) does not hold. Then, by (1.44), we should have

Hence, whenever t’

> tb, we have

Since P(t’ - tb, z, {z}) 2 give us

=/

e-q(x)(t‘-th)

+

> 0, the above two facts and (1.44)

+

P(t’ - tb, 2 , d z ) U(t0 tb,X , A ) < U(to t’,2 , A ) .

This shows that for all t’

> tb,

1.4 LAPLACETRANSFORMS

51

It is clearly in contradiction with (1.45). So for all t: 0 < t f H , Ht = 8. In other words, we have proved that

s

U ( t + t ’ , x , A ) = P(t’,x,dy)U(t,y,A), Finally, for given t , t’ we obtain

O 0, choosing tl f N: 0 < tl < t , by (1.46) and (1.44),

U ( t + t‘, X,A ) = U(t1 + (t + t‘ - t l ) ,X ,A )

=s

P(t

+ t‘ - ti,

X,

d y ) u(1.1, y, A )

/

< W’,2,d z ) w, z,A), which is thc required converse inequality of (1.44). I 1.4 Laplace Transforms

In this section, we prove the one-to-one correspondence between a jump process and its Laplace transform. Recall that for a jump process P( t,x,A) (t >, 0, IC E E , A E. &‘), its Laplace transform P(X,x,A) (A > 0, x E E , A E 8)is defined by

P(A,2,A ) =

irn

e F x t P ( tIC, , A)&.

Lemma 1.28. Let P ( X , z , A ) be the Laplace transform of a jump process P ( t ,%,A).Then the following properties hold. (1) For each X > 0 and A E 8, P(X,., A ) E 8+.For each X > 0 and x E E , P(A,2,-) E Y+. (2) Normal condition. For each X > 0, z E E and A E 8,

0 < AP(A,z,A) < 1. (3) Resolvent equation. For each A, > 0, 2 E E and A E 6 ,

P(A,2,A ) - P(P,x,A)

+

- P)

1w,

x,d Y ) P(P,Y,A) = 0.

(4) Continuous condition or jump condition. For each x E Y, and A E 8 , limx+4MXY(A,x,A)= S(x,A). Let ( q( x) ,q( z, A)) (x E E , A E 0, x E E , A E 8) is the Laplace transform of a jump process P ( t , x ,A) (t 2 0, z E E , A E 8) iff conditions (1)-(4) of Lemma 1.28 hold. A jump process P ( t , x , A ) is honest iff for all X > 0 and z E E , XP(X,x,E)= 1. Proof: Clearly, we need only to prove that under conditions (1)-(4), P(X, z,A) is the Laplace transform of a jump process P ( t , z , A ) . Once this is done, the last assertion follows immediately. The idea of the proof goes as follows. Choose an appreciate Banach space. Construct a family of resolvent operators {PA X > 0) on the Banach space corresponding to P(X,Z,A). Then apply the Hille-Yosida theorem to determine a strongly continuous semigroup. Finally, construct a required jump process in terms of the semigroup. a) Choose a Banach space. In contrast to the usual choice b 8 with the the set of signed measures with finite total uniform norm 11 . )Iu, we use 2, variation. Define the linear operation (ClCpl+ C2P2)

(4= C l c p l ( 4 + c,(P2(A), CI,C2

E

R,

cplIcp2

E

A E 8,

2

and the norm n

n

Ip(Ai)l: {Ai}? ~8mutually disjoint and

llpll= sup{ i=l

Ai a= 1

+

1

=E, n 2 1

which is the total variation of 9:llyll = p+(E) p-(E). Then (2,) . 11) is a Banach space. b) Construct a family of linear operators {PA: X > 0} on 2. Given P(X,z,A)(X > 0, 2 E E , A E 8 ) satisfying conditions (1)-(4) of Lemma 1.28. Let

(PPd (A) =

/ cp(W

P(X,z , A ) ,

A

E

8.

The domain PA) of PA is chosen to be 3.Then PA is a linear operator from 9to itself and is bounded: IlP~ll< 1/X. Next, we use two steps to show that PA is a resolvent operator. c) PA is a one-to-one mapping. Take p E 2 and let A: be the JordanHahn decomposition of p - XpP,. Then II'P - XPPAII = (cp

-

XCPPX) (A:) -

('p - X(PPA)

(A!)

1 TRANSITION FUNCTION A N D ITS LAPLACE TRANSFO

54

where JcpJ= cpt

+ cp-.

Thus, for z E AX,we have

0 6 1 - XP(X,s,AX) = S(x,AX) - XP(X,z,AX) < 1- XP(X,x, {x}). And for z $ AX, we have

0 2 -XP(X, Z,AX) = S(Z, Ax) - XP(X, Z, A x ) 2 -XP(X,x,E \ {Z}) 2 XP(X,Z, {x}) - 1. Hence 16(~, AX) - XP(X,5, Ax)I 6 1 - XP(X,

5,

{z}).

By condition (4) and the dominated convergence theorem, it follows that

(1.47) Now, to prove that PA, is one-to-one, it suffices to show that cpPx, = 0 implies that cp = 0. But by ( 3 ) , cpP~,,= 0 implies that cpPx = 0 for all X > 0. In particular, from (1.47), we see that cp = 0. This finishes the proof of assertion c). Let &?(PA)c 9denote the range of PA. Define an operator Rx on &?(PA) as follows: XI - n A = PT'. We now prove d) R A is independent of X > 0, denoted by R; its domain g ( R ) is dense in 9. We first prove that ~ ( R x is) independent of X > 0. Given cp E 9, by (3), we have cppp = (P + (A - P ) c p w 3 . This shows that 9 ( P p )c %'(PA).Exchanging X and p, it follows that

Next, we prove that for each cp E ~ ( P x=) 9 ( P p ) ,cps2x = 'PO,. Actually, for cp1, 9 2 E 2 so that cp = cplP~= cp~P,, from ( 3 ) , it follows that

+

But Pp is one-to-one, so cp2 - cp1+ (A - p)cplP~= 0. Hence (pP;l- cpP;' cp(X - p ) = 0. This gives us cpRx = pR,. Finally, we prove that g(n) := g(fI2x)is dense in 9. Note that for each cp E 9, XcpP, E g ( R ) . On the

1.4 LAPLACETRANSFORMS

55

other hand, from c), we have wen that limx-,, IIy-XpPxII = 0. This yrovcv the denseness. So far, we have proved that {PA : X > O} satisfies the hypotheses of the Hille-Yosida theorem. Hence, there exists uniquely a strorlgly continuous, contraction semigroup {Ti : t 3 0) with resolvent apcrators {PA : X > 0) such that PA = J r e - x t 7 j d t , where the integral is in the Bochner sense. Take S, = 6(x:.) E 9and set P ( t , x , A )= 6,Tt(A). We prove that e) P ( t ,x,A ) is a jump process. By thc representation theorem (cf. Yosida (1978), p.248), we have

6,Tt(A) = lim n-02

M 1 [6,(~nb'2P,)'~](A) m. m=O

Since b . ( A )and (6.Px)(A)= P(X,.,A) E &+., we hwe [S.(tnW,)](A) E &+By induction, for each 7 n 2 1, [6.(tnRPn)m](A) E &,.. Therefore P ( t ,-,A ) = (6.Tt)(A)E &+. Obviously, P(0,x,A ) = 6 ( x ,A ) . Next, since exp(-tnJ) and exp(tn2P,) are d l bounded operators from 2+into itself, b, E LZk,so S,Tt E 9+, md hence P ( t ,5,.) 6 2+. By contractivity,

P ( t ,2,I";)

=

(6,Tt) ( E )

< 1.

By strong continuity, P ( t ,2 , A ) = (S,TL)(A)is continuous in t. Now: the remainder i s l o check the CK-equation. Note that pS. = p and so

.By semigroup property,

WE

have

1 TRANSITION FUNCTION AND ITS LAPLACE TRANSFO

56

Since the one-to-one correspondence between P ( t ,x,A ) and its Laplace transform P ( X , z , A ) ,f r o m now on, we also call P(X,.,A) a jump process (q-process). If P(X,x,A) satisfies conditions (1)-(4) of Lemma 1.28, then it determines uniquely a jump process P(t,x,A), and so one can define the correThus, Lemma 1.28 shows that spondent 9. Corollary 1.30. P(X,z, A ) is a jump process with q-pair ( q ( z ) ,q(x,A ) )(x E E 9) deduced by Theorems 1.4 and 1.5 iff conditions (1)-(5)5) of Lemma 1.28 hold.

E, A

As we have seen, this corollary is essentially not new since the q-condition lays on P ( t , x , A ) but not on P(X,x,A). The next result is much more meaningful. Corollary 1.31. Use the hypotheses and notations in Remark 1.8. Then P(X,z,A)is a q-process iff conditions (1)-(4)4) of Lemma 1.28 and the following q-condition all hold. lim X[XP(X,

x+co

2,A

n En) - S(Z, A n En)]= q(z,A n En) - q(x)6(x,A n En), A€€,

n31.

Proof: By Lemma 1.28, the q-condition here is clearly necessary. Now, suppose that conditions (1)-(4) are all satisfied. By Theorem 1.29, there exists a jump process P ( t ,x,A ) , and hence EL q-pair (Q(x),Q(x,A ) )(x E E , A E 9) Then from the q-condition here and the proof of Lemma 1.28, it follows that q(x) = Q(x),x E E. Moreover, Q(Z,

Put

A n En) = q(x,A n En),

x E E, A

E

8, n 3 1.

-

En = {x E E : n - 1 < ~ ( zXi+j,

(2.9)

f E X

a controlling equation of Eq. (2.6).

Theorern 2.6 (Comparison Theorem). Let f* be the minimal solution t o f q . (2.6). Then for any solution f t o Eq. (2.9), we have f f*.

>

Proof: Simply use induction and the first successive approximation scheme to show that f 2 4'") and then let n -+ 00. H By Theorem 2.2, for each A E d , we may define a map mA from 2 into itself as follows: m A ( d = f*. Then we have the following interesting result. Theorem 2.7. mA is a cone mapping. For { A n } c d ,A,, %, gn t g , we have A E d , g E A? and mA,,gn t m,g.

TA

and (9,)

C

Proof: By definit.ion, g E 53'. Similarly, it is easy to check that A E d.On and the other hand, by the comparison theorem, we have f: := mA,gn

f: Set

fl* = lirnTL-.oof;.

Anf;

+gn.

Then

2 f*by the minimum properly. is a solution to Eq. (2.6) and so Hence Note that by the order-preserving property, f* = m,g mA,gn = f: for all n , so f* 2 f*. Therefore f* = = limn4m mAngn.

>

Corollary 2.8. Let G be a countable set, {a, : s E G} C E+, then

Proof: When G is finite, the assertion follows from the first assertion of Theorem 2.7 and induction. Then the assertion for general case follows by using the second assertion of Theorem 2.7. I The next result, is the second successive approximation scheme for the minimal solution, its proof is similar t o the previous one and hence omitted.

65

2.1 MINIMALNONNEGATIVE SOLUTIONS

Theorem 2.9. Let {g"}?

c 3.Define

In particular, if we set

then mAg = f* = CrzIf ( n ) .

7

Tlieurcr:m 2.10, Let be a non-negative solution t o equation (2.6) so t h a t f" < p i * for some constant p 2 1. Then ,f = f". Moreover, for any initial f ( " ) 0 f(') G pf*, setting = Af"(n) g (n2 0), we have flcn) f * as

I("+')


0, .x E E , A

t 8"

R,cpeating the above proof in the opposite may, it follows that

It is easy to see that the integrands are all bounded and continuous in t . Hence (B) follows by the uniqueness theorem of Laplace transform. The proof of (P) ( F A ) is similar t o that ( B ) (Bx) and hence is omitted. To show that ( F A ) ( F ) , we need more work except the trivial case that q(x) = 0. Now, assume that q(a) > 0. At the beginning of Sectiori 1.3, we proved that

*

+

+


0,

2 1,

(2.22)

where

To see that the resolvent equation follows from (2.22), note that by the second successive approximation scheme,

Thus, by summing the both sides of (2.22) from 1 to 30, we obtain the required equation. We now prove (2.22) by using induction. When n = 1, at each point (x:A), the bath sides of (2.22) have the same value as

76

2 EXISTENCE AND SIMPLE

CONSTRUCTIONS OF

JUMP PROCESSES

Suppose that (2.22) holds for n - 1. Then, from

c n

( p - A)

jm(A)++l-k)

(A

k=l

n- 1

+

= ( p - A)P(1)(A)F(n)(p) ( p - A)

c~(X)P("(A)~("-"(( k=l

= ( p - X)[(A

+ q ) - l ] P ( q p )+ n ( A ) ( P - l ) ( A )

-P-l)(p))

we see that (2.22) holds also for n and hence we are done. b) Prove that Pmin(A, z, .) is the minimal solution to ( F A ) . We use again the notations given by (2.23). Moreover, for fixed set

5

E

E,

Once we prove that

i;Cn)(x) = P ( n ) ( ~ ) ,

A

> 0, n 2 1,

(2.24)

then the assertion follows immediately since the minimal solution to ( F A )is P(n)(A, 2 , .). To prove (2.24), we adopt induction again. When n = 1, (2.24) is trivial. When n = 2, we have

c,"==,

So (2.24) holds. Suppose that it holds for n- 1 and n, then, by the monotone class theorem, we have

P("+')(A)= P(")(A)Q[(A + q)-'1] = F(")(A)Q[(X + ~)-l1] = n(A)P(n-l)(A)Q[(X + q)-lI] = rI(A)P(n-l)(A)Q[(X = rI(A)p(n)(A) = rI(A)F(")(A) -

,(n+"(X).

+ q)-'I]

2.2 KOLMOGOROV EQUATIONS AND MINIMAL JUMPPROCESS

77

Therefore, (2.24) holds for all n. c) Prove the minimum property. Because every q-process P(X,2,A ) satisfies the backward Kolmogorov inequality (2.19), which is a controlling equation of (Bx),so the property follows by the comparison theorem. d) Prove that Pmin(X, z, A ) is the Laplace transform of the minimal solution Pmin(t, 2 , A ) to (B). First, we have

1

00

edxtP(')(t,x,A)dt= 0 = P(O)(X,z,A),

X

> 0, x E E , A E 8.

Next, suppose that



But condition (1) implies ip 3 -n(4ip +x-c x-c

whenever X have

> c. Thus, by the cornparisori theorem, whenever X > c V 0, we P"'"(X)(P

b) (2)

cp

x+y

+ uniqueness.

6'p 0, which certainly implies the uniqueness since Pmin(A) is the minimal one. For this, we use the resolvent equation:

P(X)l- P ( p )14- (A - p ) P ( X ) P ( p ) l= 0. Thus, if pP(pu)l .- 1 for some p

> 0, then it follows that

1 x P(X)1- - , i . -?(X)1 CL El

= P(X)I.

And so XP(A)l = 1 for all X > 0. Next, we prove that (1) j (3) + uniqueness. c) (1) + (3). This st,ep is not needed for the proof but included €or completeness. Let (1) hold. Then

2 . 3 SOME

SUFFICIEN'I' CONDITIONS FOR UNIQUENESS

This implies that Q y ( x )= 0 whenever c

1

81

+ q(2) = 0. We have

t

(c+q)t

or

1

- 11 >,

e(C+q)SQg&,

E

ipeCi

2

,

e-Y(L--s)Q)ecsds +

t20.

On the other hand, by Theorern 2.21 arid Theorem 2.12 (l),it follows that for each t >, 0, P"'"(t)ip is minirnal solution to the equation

Combining these two facts and using the comparison theorem, it follows that

pmin(t)y 6 yect < 00,

t 2 0,

which is just (3). d) (3) => uniqueness. Use the forward equation

(2.29)

(F):

Since the q-pair is conservative, by Theorem 1.15 and condition (3): it follows that

The null set depends on 2 , By using the forward equation again, in virtue. of condition (3) and the dominated convergence theorem, we obtain

d a.e. t. - p m y t , 2 , E ) = 0, dt However, the left-hand side is indeed continuous in t by Theorem 1.15, so this equat,ion holds for all t . Therefore

Pmin(t, 2,E ) = constant =Pmin(O, 2 , E ) = I,

x E E.

Now, the uniqueness follows from the minimum property of Pmin(t). W

2 EXISTENCE AND SIMPLE CONSTRUCTIONS OF JUMPPROCESS

82

Definition 2.23. T h e q-pair (q(x),q(x,A ) ) is called bounded if supzGEq(z)

< 03. Corollary 2.24. If a y-pair is conservative and bounded, then the q-process is unique. T h a t is P ( t )= etR = (tQ)n/n!.

c;=,

Proof: The conditions for the case (1) of Theorem 2.22 are satisfied wilh c = 0 and ‘p =: A4 = supxEEq ( s ) .

{&}? c 8 and ~1 E T€+. Suppose that (1) En t E , supzEE, q(x) < 00, limndm inf,,js, p(x) = 00,

Theorem 2.25. Let 00

where inf

0=

by the usual convention.

(2) There exists a constant c E IR such that flp

< CV.

(2.30)

Then the q-process is unique.

Proof: The idea of the proof is using an approximation by a sequence of jump processes with bounded q-pairs. For this, let

Then for each n, the q-pair (9, (x))qn (z, A ) ) is bounded and conservative. It x,A ) . Clearly, replacing determines uniquely a q-process, denoted by Pn(X, c and ( q ( x ) , q ( x , A ) with ) c+ = c V 0 and (qn(x),qn(z,A)), respectively, condition (2.30) remains true. Thus, according to the first step of the previous proof, we have

On the other hand? when

But when x $

5

En,we have

E

En,we have

2.3 S O M E

SUFFICIENT CONDITIONS FOR UNIQUENESS

83

Thus, in any cases, we have

X > O , X E E ,A C E , , n 2 1 .

Hence, by Corollary 2.24, Theorem 2.21 and the comparison theorem, we obtain X

Pmin(X, z,A ) 2 P,(X, z,A ) ,

> 0, z E El A c En, n 2 1.

Combining this with what mentioned at the beginning of the proof, we arrive at: 2,En) 2

XP"'"(X,

XP,(X,

2 , En) =

1 - XP,(X,X,

E;)

So we have XP"'*(X,

z,E ) = lim X P ~ ' ~ ( Xz, , E,)

= 1,

A>

C+.

It follows that the minimal q-process is honest and hence the q-process is unique. Next, we compare the above conditions. The purpose we introduce condition (1) of Theorem 2.22 is mainly for conditions (2) and (3) there. It is clearly more practical and provides the useful estimates (2.28) and (2.29). However, if we pay attention only to the uniqueness, then the first case of Theorem 2.22 is indeed a simple consequence of Theorem 2.25. To see this, taking En = {z E E : q ( z ) < n } and using the same cp and c given by Theorem 2.22 (l), it is easy to check that the hypotheses of Theorem 2.25 are satisfied. The next example shows that the conditions for case (1) of Theorem 2.22 are really stronger than the hypotheses of Theorem 2.25.

Example 2.26. Take E = { 1 , 2 , . . - } . Let { q l , q 2 , . . . ) be the prime numbers in the natural order. Define

Then the hypotheses of Theorem 2.25 are satisfied but not conditions for case (1) of Theorem 2.22.

84

2 EXISTENCE AND SIMPLE

CONSTRUCTIONS OF

JUMPPROCESSES

Proof: Take

Since

= 00. Because 'pi is are convergence-equivalent, it follows that lirni,,cp, increasing, it is now easy to see that the hypotheses of Theorem2.25 are satisfied with En = {1,2, - * - , n}. Next, we prove that the conditions for case (1) of Theorem2.22 do not hold. Actually, we can even prove that the conditions for case (2) of Theorem 2.22 do not hold. Note that in the present case, the solution to ( Bx ) is as follows:

PZj(A) =

(A

+

4i . . .Qj-l 4i).* * (A 4j)

+

j

> 2,

So we have 00

M

00

Since limj(2-1) j--tm

aj+l

=Xlim 3-00

j = o , 4j+l

it follows that the above series diverges for all A done. I

A>O,

> 0 and hence we are

We now introduce a criterion (its special case is Theorem 0.6) which shows that Theorem 2.25 is sharp. For further discussions about Theorem 2.25, see Section 3.2.

Theorem 2.27. The minimal q-process is not honest (equivalently, the 4processes are not unique) if the inequality Rcp 2 ccp has a solution cp E b6' with supzEEcp(z) > 0 for some (equivalently, for any) c > 0. Conversely, these conditions plus cp 2 0 are also necessary.

2.4 KOLMOGOROV EQUATIONS AND 9-CONDITION Proof: Sufficiency. Choose

A0

85

> 0 and zo so that


0 and A E 8 n G,, we have (2.32) IP(X)QIAI 6 1 + SUP(X+ d y ) ) / X . YEA

Next, by the resolvent equation,

2.4 KOLMOGOROV EQUATIONS A N D CONDITION

87

On the other hand, for fixed z E E and A E €, P ( X , s , A ) is decreasing in A. Hence for X 2 XO > 0, we have

P(X)QIA< P ( X ) [ I G , Q ] I A+ P ( X O ) [ I G ~ Q ] I A A , E gnG,. Letting X --t 00 and then n + 00, by (2.32) and the dominated convergence theorem, it follows that limx+cc P(X)QIA= 0 for all A E € n G,. Furthermore, by (2.32), we can prove that lim~.+mpLP(p)P(X)QIA = 0. Thus, we obtain

On the one hand, we have proved the continuity: limxdm XP(X) = I . So by Theorem 1.14 (4) we have

lim XP(X)QIA2 QIA 2 0 ,

x-+m

A E 8 n G,, n 2 1.

(2.34)

On the other hand, by (2.33), (2.34) and using Theorem 1.14(4) again, we have

Since P ( p ,z, {z}) > 0, this shows that

lim XP(X)QIA< QIA,

x-+m

A E € n G,, n 2 1.

(2.35)

n 2 1.

(2.36)

Combining (2.34) with (2.35), we obtain lim XP(X)QIA= QIA,

x+cc

A

E

Therefore, by (F,) and Theorem 1.14, we have

which is just the q-condition.

€nG,,

88

2 EXISTENCE AND SIMPLE CONSTRUCTIONS OF JUMP PROCESSES

2.5 Entrance Space and Exit Space

Suppose that P(X,2,A ) is a q-process satisfying the equation (Bx).Then .,A) is a non-negative bounded for fixed A > 0 and A , P(X, . , A ) - Pmin(X, solution to the homogeneous equation

f = rr(A)f,

f Eb8.

Thus, if the q-processes are not unique, then the above equation should have non-trivial solution. The solutions to the equation consist of the exit space. This explains intuitively why we introduce this exit space and how it concerns with the uniqueness problem. Similarly, if we consider the jump processes satisfying the forward equation, we will need the entrance space. The main purpose of this section is to prove that the dimensions of these spaces are independent of X > 0. For this, we need a lot of preparations. The related notations will become clear in the proof of Lemma 2.43. Introduce a kernel R(X,p)(X,p > 0), not necessarily non-negative, on ( E ,8') as follows: R(X,p ; 2,A ) = 6(z, A )

+ (A - p)Pmin(p,z,A ) ,

Equivalently R(X,p ) = I

2

E E , A E 8'.

+ (A - p)P"'"(p).

(2.37)

(2.38)

By resolvent equation, it is easy to prove the following result. Lemma 2.31. We have

ww,

w,

WP, 4= m p t ,A> = R(P,4 , P"'"(p)R(p, A) = R ( p ,X)Pmin(p)= Prni"(X>.

In particular,

w, P>R(P,4

= 1.

(2.39) (2.40)

(2.41)

The following notion is again due to the resolvent equation. Definition 2.32.

(1) A family of functions {fx E b 8 + : X > 0) is called consistent, if for all A, p > 0, we have fx = R ( p , X)f,. (2) A family of measures {px E 2'+: X > 0) is called consistent, if for all A, p > 0, we have px = cp,R(p, A).

2.5 ENTRANCE SPACE AND EXITSPACE

89

Next, define

{f E b€+ : ( X I - n)f = O}, YA= {'P E 2+: 'P(A)= 'P&((X + ~ ) - ' I A ) ,A E &}, W,l = {'p E 2+: c p ( X 1 - Q ) I A= 0, for all A E d n En and n 2 l}, X > 0. 52-A =

By using the method of the proof of Proposition 2.29, we obtain

> 0, we

Lemma 2.33. For every X Lemma 2.34. Fix X

have ?'A'

= Y,l.

> 0.

(1) If f , g E T&+ and

( X I - Q)f

z9,

(2.42)

then

f 2 P"'"(X)g. Moreover, if the sign of the equality in (2.42) holds and so does (2.43). (2) If p , U E p(E,) < 00, U ( & ) < co for all n and

(2.43)

52-~= {0},

then

2+,

p(XI

- Q ) I A2

U(A)

for all

A E E n En and n 2 1,

(2.44)

then

p 2 UP"'"( A).

(2.45)

Furthermore, if p ( E ) < 00,the sign of the equality in (2.44) holds and "yx = {0}, then so does (2.45).

Proof: Since the proofs for (1) and (2) are similar, here we prove (2) only. Note that

By (2.44), we have P[(X

+ q)II 2 PQ + U.

+ ~ ) - ' I A we , see that p is a solution to the equation cp 2 (pQ[(X + q)-l.1] + U[(A + q ) - ' I ] . (2.46)

Postmultiplying by (A

On the other hand, since P(A,2,.) is the minimal solution to ( F A ) by , Theorem 2.12 (2), it follows that UPm'"(X) is the minimal solution to (2.46), replacing the inequality by equality. Thus, the first assertion of (2) follows by applying the comparison theorem. Moreover, if the equality in (2.44) holds and p ( E ) < 00, then we have p - UP"'"(A) E YA,and so the last assertion is obvious. W

90

2 EXISTENCE AND

SIMPLE CONSTRUCTIONS O F

JUMP PROCESSES

Definition 2.35. (1) A family of functions

c b&+

{fk};,l

is called linear independent, if

*

for any { c ~ } ; = c ~ R, C;=,c k f k = 0 ck = 0, IC = 1,2, ,TI. (2) A family of measures { c p k } ; = , c A?+ is called linear independent, if for any { c k } ; = , c R, c k p k = 0 =+ ck = 0, k = 1 , 2 , - . . ,n.

xi=,

As usual, the number

c

sup{n : {fk};=l is called the dimension of %A, dim Wx.

%A

is linear independent}

denoted by dim%x.

Similarly, we can define

Lemma 2.36. (1) Given XO

>

fx,

Set f p = R(Xo,p)fxo, p is consistent. (2) Given XO > 0 and 'px, E "yx,,. Set 'pp = 'pxoR(Xo,p),p cp, E W, and {cp, : p > 0) is consistent.

f,

E

9,and

0 and

E %A,.

{f,: p > 0)

Proof: a) Prove that

f,>, 0 for all p > 0.

f, = fx, + 0 But for p >, Xo, since fx,

Actually, for p

>

0. Then

>

0. Then

< Xo, we have

0 - P)~"'"(p)fx0.

E %xo, we have

Thus, Lemma 2.34 (1) implies that

Hence we also have

f,

= fx,

+

(A0

- P)pmi"(P)fxo2

0.

b) Prove the measurability. Because fx,, R ( X o , p ) 1 ~E b&, so does f,= R(Xo,p)fx, by the monotone class theorem. c) Prove that f, E % ,, p > 0. Since Pmin(X)I~ satisfies (Bx)and fx, E %A, it follows that Qfp

= QR(XO, ~

) f ~ o

+ (Xo - p)QP"'"(p)fx,, = 0 0 + dfx, + ( A 0 - P ) ( P + Y)pmin(p)fAo - (A0 - p)fxo = ( P + Y)[fxo + (A0 - P)P"'"(P)fx,l = Qfx,

= (P

+ 4)fW

CL

> 0.

2.5 ENTRANCE SPACE

91

ANT) E X I T SPACE

d) The consistency of {fp : p > 0) follows from (2.39). Combining this with a), b) and c), we obtain assertion (1). The proof of (2) is quite similar. Clearly, q, E 2 ' ( p > 0). Next, for P 6 Xo, we certainly have pp 2 0. Conversely, for p 2 Xo, since 'pAoE WA,, it follows that

HAVEtHUS, BY lEMMA 2.34(2), WE

ENCEh

The consisTherefore we always have (pp 2 0 and furthermore y , E 2+. tency of {p,} is obvious. Now, by using (F,) and the fact that 'px, E Wj,,, we obtain

Theorem 2.37. Both dim92A and dim WA are independent of X Proof: Let

{ j r ) } L = ,c %j,

> 0.

be linear independent. Set

By Lemrrias 2.36 (1) and 2.31, we have

k= 1

k=l

k=l

{fr'}

{fik)}.

Thus, the independence of implies the independence of In other words, dim 9?Lp2 dim 92~. But X and p are symmetric, hence dim %-?,A = dim

ep.

Similarly, we can prove that dim"'= dimWp. In view of the above theorern, we can use dim% and dim?' instead of dim 7 1 ~ and dim 'Y' respectively.

92

2 EXISTENCE AND SIMPLE CONSTRUCTIONS OF ,JUMP PR.0CESSES

Definition 2.38. We call and WA the exit space and the entrancc space respectively. The elements of are called the exit solutions. If dimo& < 00, it is said t o be finite exit. In particular, corresponding t o dim kP = 0 and dim W = I , we have zero-exit and single-exit. Similarly, we can define the entrance solutions, finitc cntrance, zero-entrance and single-entrance.

For each x E E , set d ( x ) = q ( x ) - q ( x , E ) , which is called the nonconservative quantity at 2. Lemma 2.39. For each X > 0, {zA(x) := 1 - XP"'"(X,z,E) : x C E } is the maximal solution t o t h e equation u = rI(A)u id/(X

+ q),

0

< u < 1.

(2.47)

Moreover, it can be obtained by the following procedure. Let

Zip' = 1, Then z r ) for each X

1 xA as n

n 3 1.

(2.48)

particular, if the given g-pair is conservative, then > 0, zA is the maximal solution t o the equation -3

a3. In

u = n(x)u,

0 0) is consistent: z p = zA -t- (A - p)P"'"(X)z,. (2) Let {uA : X > 0) be a consistent family of functions, then uA # 0 or TL, = 0 is independent of X > 0. Moreover, u x 1 0 as X T ,30. (3) If PIX) defined by (2.50) is a jump process, then XpA(E> 1 for all

X


0.

Proof: The consistency of {xA : X > 0} is due t o the fact that P"li"(X satisfies the resolvent equation. Assertion (2) is easy t o check. Assertion (3) is obvious because of the normal condition of jump processes. Because of Lemma 2.42 (2), we may define u .-= 1imx ,O uA.Tn particular, z

L:

lirnA..+o xA - 1.

In order to guarantee P(X) defined by (2.50) being a jump process, the key point is the resolvent equation, which is solved by the next lemina.


0) be a consistent family of functions, 0 # uA 1 and yX E Y+,XyA(E) 1 for all X > 0. Then P(X) = Pmi"(X) uAp satisfies the resolvent equation iff there exists a consistent family {vx : X > 0) of measures such that pA= r n ~ q where ~ , r n A > 0 and m i 1 - Xqxu =: c 2 0 is independent of X > 0.


0 by (2.51), and so P(A) = Pmin(X). Hencc we have nothing to do in t,his trivial case. Assume that ywo# 0 for some pa, Then 'pi, # 0 for all p > 0. Let

Then { q x : X > 0) is consistent. In particular, t,nking X = p0 in (2.511, we

see that rlp(4 =

Note that

[...I

" +(P

-

POhq&IY,Lb41.

does not depend on A . When p,oup = 0, then 1

+ (p -

pO)'pp,up= 1 > 0. Othcrwise,


0 , p > 0. Thus, we may define

+

(px = rnxqx,

mx > 0, X > 0.

(2.52)

c) Substituting (2.52) into (2.51), it follows that {mx}must satisfy

sET

tHAT IS

tHEN WE HAVE

m,-l

- pL"(X,p) = mx-1

- XU(X,Pu).

On t,he other hand, by the consistency of {u,},we have 21,

+

= up ( p - X ) P m i n ( X ) u , ,

(2.53)

2.6 CONSTRUCTION OF q-PROCESSES WITH SINGLE-EXIT q-PAIR

95

and so ux = u - XPrnL"(A)u.

(2.54)

Next, Xqxu = XqpR(p,X)u = qJXu

+ (p -

= vp(Xu

+ ( p - X)XP"'n(X)u)

- .A,>

= cL77pU

- b - Q7,ux.

(2.55)

That is ( p - A)+,

p ) = ( p - A ) q A 7 L p = p/?jp7A - X q x l L .

Substituting this into (2.53) we obtain m i 1 - pq g u = m i - Xqhu. Hence, rn,' - X?jxu=: c is independent of X > 0. d) So far, we have proved the hypalheses are necessary. The suficieiicy is easier, simply substituting pAgiven by (2.52) into (2.51) and using the consistency of {qX}. I

Lemma 2.44. Given a consistent family {qx} of measures. Set 'px = q A / [ c + Xqx(E)]with c 0. Then P(A) defined by (2.50) is a jump process and it is

>

indeed a q-process satisfying

(Bx)if the

q-pair is conservative.

Proof: In general, since Pmin(X) is a jump process, the continuous condition for P(A) follows from Lemma 2.42 (2). In the conservative case, since zX E %A, P(A) satisfies ( B x ) . Hence the q-condition and so the continuous condition are automatic. Combining the above lemmas, we obtain a complete construction theorem.

Theorem 2.45. Given a conservative and single-exit q-pair, every q-process can be obtained by the following procedure. Choose an arbitrarily consistent family {qx} of measures and a constant c 2 0 and then set

where 0/0 = 0 by convention. The q-process is honest iff c = 0.

Corollary 2.46. Let the q-pair be conservative. Given 9 E p+with

,Prr"rl(X)(E) < 00, then for every c 2 0, P(A) := Prrlin(X) a q-process.

x > 0,

(2.56)

+ zxpY'T'i"(X)/[c + A9PT11i11(A)(is

Proof; We need only to check that {pPmi"(X): X this follows from (2.40). I

> 0)

is coiisistent. But

Combining Corollary 2.46 with Theorern 2,40, we obtain at last the following imiqueness criterion.

96

2 EXISTENCE AND SIMPLE CONSTKUCTIONS O F J U M P PROCESSES

Theorem 2.47. Given a conservative q-pair, the g-process is unique iff dim 9 2

0. Equivalently, the minimal q-process is honest.

2.7 ‘Notes For Markov chains, the theory of non-negative solutions to a. system of equations with non,nega.ti,vecoefficients is diie to Hou and Guo (1979), which goes back to Kantorovich and Krylov (1962). The general presentation here is talcen from Chen (1979). Theorem 2.12 is taken from Chen (19801, Chen and Zheng (1983). The ‘backward and forward Kolmogorov equations were first introdnced by Kolmogorov (1931). In a special case, Theorem 2.21 was obtained by Feller (l940), sometimes called the Feller’s construction in the literature, a generalization was obtained by Hu (1966). Here, we study the general case in details. The equation ( F A )was introduced by Chen (1980)) it is locally equivalent to the; ordinary ( F ) . The equivalent result Proposition 2.29 is taken from Chen arid Zheng (19133). Theorem 2.22 and Theorem 2.25 were first appeared in Chen (1986a,b). The later one is an analogue of Stroock and Vartzdhan (1979). Basis (1976, 1980) used a stronger version of the conditions for the cases (1) a.nd (3) of Theorem 2.22. For Theorem 2.25, the author was benefit.ed by a conversa.tion wi’Lh S. Z. Tang. The Example 2.26 is due to J. L. Zheng (oral cornmunication). It should point out that a slight modification of Theorern 2.25 is available for tirne-inhomogeneous jump processes. See Zheng and Zbeng (1986), Zheng (1993). For more related results, refer also to Chebotarev (1988), Hamza arid Klebaner (1995)) Kerstirig and Klebaner (1995). For the applications to quantum mechanics, refer t o the survey article by Konstantinov, Maslov and Chebotarev (1990), Thc materials of Section 2.4 are mainly taken from Chen and Zheng (1983). For Evlarkov chains, the part of Theorem 2.30 related to ( B n )is due to Feller (1957) and R.euter (1,957))and thc gen,erd form is due t o Hu (1966). For Markov chains, the other part of Theorem 2.30 related to (F,) is due to Hou (1982). Here is th,e natural generalization of the previous results. The author learnt thc name “consistent family” from Yang (1981). For Markov chains, corresponding to Corollary 2.46, there is a probabilistic construction, due to Doob (1945), and hence called noob’s construction. Tn the same situation, Theorem 2.47 is due t o Feller (1‘357}and Neuter (1957), the general form is due to Hu (1966). Essentially, Lemma 2,43 is due to Reiiter (1959) I

Chapter 3

Uniqueness Criteria This chapler begins with the study of uniqueness problem for the jump proccsses satisfying the backward or forward Kolmogorov equations. Then we study the uniqueness problcm for general jump processes. Besides, we introduce some applications of the uniqueness critcria. 3.1 U n i q u e n e s s Criteria Based on Kolrnogorov Equations

As we will see later, the Kolmogorov equations play a special role in the study of the uniqueness problem. The problem becomes easier if we restrict ourselves to those jump proccsscs satisfying one of the equations. This is the reason why we first deal with these cases. Definition 3 . 1 . A q-process satisfying (BA)(resp. ( F A ) )is called a Bq-process (resp. Fq-process). Theorem 3 . 2 . The Bq-process is unique iff dim GY = 0

Proof: Non-uniqueness ===+dim% # 0. This is clear as we explained at the beginning of Section 2.5. We now prove that “dim% # 0 ==+ nonuniqueness” by reducing the non-conservative case to the conservative one. For a different proof, see Proposition 6.28. a) Note that a given q-pair ( q ( x ) ,q ( x , d y ) ) induces naturally a conservative q-pair ( q a ( x ) , qn(J:, d y ) ) on the enlarged state space EA = E U {A}. &A = a ( € u {A}) as follows:

x E E , A E 85 ’ qa(J:, A ) = q(x, A \ {A}) -k I A ( A ) d ( Z ) , 4ab) = 4 ( 4 , J: E E , s a ( 4 = 0. By the last condition, for any q-process PA(X> with q-pair ( ~ ( x ~) ,( 2d y,) ) , we have Pa(&A , {A}) = l / X and so PA(&A, E ) = 0 for all X > 0. This plus Theorem 1.15(1) implics that the restrzctzon P(A) of PA(X)to ( E ,8 ) is not only a jump process but also satisfies the backward equation ( B x ) corresponding t o the q-pair (q(z),q(z,dy)), and hence is a Bq-process by Theorem 2.30. b) It is easy to check that the q-pair (qA(x),qA(x,dy)) is zero-exit iff so is the q-pair ( q ( x ) ,q ( x ,d y ) ) . Now, by the assumption and Corollary 2.46, there are infinitely many q-processes with the same q-pair ( ~ ( z ~) , ( xdy)) , but having different restrictions. H To study the problem for the Fq-processes, we need some preparation. The next result is about the decomposition of a consistent family of measures. 97

3 UNIQUENESS CRITERIA

98

Lemma 3.3. Let { q X } be a family of measures. Then it is consistent iff there exist a K. E y+and a consistent family of measures { f j ~E 'YX : X > 0) such that

Moreover, K and {qX : X uniquely by {qX : A > 0).

> O}

in the decomposition (3.2) are determined

Proof: a) The sufficiency is obvious. We now prove the necessity. Let {qx : X > 0) be a consistent family of measures. Then 7"

+ (v - X)qvPrni"(X)= rlx 2 0.

ENCEh fIX V AND LET

Define

KV(A)= vqy(A) - ~ ~ ( O I A ) ,A E 8 0 En,rt 3 1.

(3.4)

Then O 0) is a consistent family of measures. Thus, we assume that (q(x)>q ( x , A ) ) is non-conservative. Then from Lemma 2.44, it follows that (3.12) still defines a jump process. This proves the first assertion. b) Because Pmin(X) is a q-process, by Corollary 2.28, P(A)given by (3.12) is a q-process iff

+

A E 8 n E,,, n. 2 1.

lim X2zAqA(A)/[c XqA(E)J= 0, x+m c) Note that Lemma 2.42 (1) and (2) give us

AP"'"(A)l

t

1

as

x T 00.

(3.13).

haveWe

On t,he other hand, by Lemma 3.4, we have Xqx(E)

Hence

as x

+ x+oo lim

lim Xq,(E) = K ( E )

A+

+ 03.

00

X.il,(E).

+

(3.15)

(3,16)

Xqx(E) K ( E )= 0, then from (3.15) and (3.16), it follows that K = 7, = 0, so qX 0 and lience P(A) = Prnin(X) for all X > 0. Thus, this situation is trivial. In what follows, we assume that

Now, if limx,,

=3

Iim X7jA(E) ' + a 3

+ K ( E )# 0.

3.1 UNIQUENESS CRITERIA BASEDON KOLMOGOROV EQUATIONS101

d) Observe lim x - lim [I - X P ~ ' ~ ( =X0,) ~ ]

A 4 0 0 x-x-.cO

and note that zA satisfies Eq. (2.47). We obtain lim X r x ( x ) = d ( s ) ,

zE

A-00

e) Fix vo > 0. For each N(v,, E , A) so that

Thus, by (3.9))whenever

11

E

> 0 and A

E & I-

F.

En! there exists an AT

=

> vo, we have

From this and (3.9)) we see that lim ~,,QIA= 0,

v+w

A E € n En,n 2 1.

Therefore, by Lemma 3.3, (3.4) and (3.9)) we obtain

Moreover,

f ) Finally, combining d), e) with c), we get

l?rorom this and a), b) and c>, it follows that P(A>is a g-process iff one of (1)-(4) holds. Now, we can state the main result of this section.

Thcorerri 3.6. The Fq-process is unique iff either the minimal q-process is honest or dim "t/ = 0.

102

3 UNIQUENESS CRITERIA

Proof: a) Sufficiency. The assertion is trivial if the minimal q-process is honest. Otherwise, for any Fq-process P(A), we have

P(A,2,.) - PrnyX,z, ,)

E

w,.

Hence dim Y' = 0 also implies the uniqueness. b) Necessity. Assume that the minimal q-process is non-honest and dim '9' # 0. Fix A0 > 0, take qA, E W', \ (0) and set QA = qAoR(Xo,X).By Lemma 2.36 (21, { f j ~E Wx : X > 0) is consistent. Since zx # 0, setting K, = 0 and c = 0 in Proposition 3.5, we obtain a Fq-process which is different to Pmin(X). This completes the proof.

3.2 Uniqueness Criterion and Applications

Lemma 3.7. For each A X > 0.

E

&, either irifXEnPmin(A: z, E ) > 0 or = 0 for all

Proof: Suppose that infXEAPmin(Xo, x,E ) > 0 for some XO. Then by the resolvent equation,

P y A j = P"'"(A0)

+ (A,

- X)Pmin(A)Pmi"(Xo),

we have

On the other hand, as we have seen in (3.13) t#hatXPmin(X, x,E ) is increasing in A, we also have

Denote by

E0 = {x E E : d ( z ) := q ( z )- q ( 2 , E ) > 0) the set of all non-conservative points. We can now state our main criterion.

Theorem 3.8 (Uniqucness Criterion). Given a q-pair (q(z),q(z,A ) ) ,the q-process is unique iff the following conditions hold simultaneously.

(1) For some X > 0 (hence for all X > 0 ) , infxEEoPmin(X,z,E) > 0. (2) dim% = 0. (3) Either the q-pair is conservative or even it is not but still dim W = 0.

3.2 UNIQUENESS CRITERION AND APPLICATIONS

103

The theorem will be proved in Section 3.4. In this section, we introduce some of its applications. Before doing so, let us make a remark about the probability meanings of the conditions. Consider the minimal Markov chain (Xt)t>o on a probability space (R, 9, P). Its successive jump times are T, is called the denoted by TO = 0 < T~ < T~ < . . . 00. Then 7 := first infinity of the Markov chain. Next, let


0, there are infinitely many isolated jumps during ( T ( w ) - E , ~ ( w ) ) } .

Remark 3.9. Using the above notations, we have (1) Condition (1) of Theorem 3.8 holds iff there exists a to > 0 such that infiEEoPi(7 > t o ) > 0. (2) Condition (2) of Theorem 3.8 is equivalent t o say that Pi(fi)=O for all i.

The proof of Remark 3.9 was presented in Hou and Guo (1978), Section 12.10. Corollary 3.10. If q ( x , E ) = 0 for all z E E , then the q-process is unique iff SUPzEEq ( 2 ) < OO. Proof: Clearly, dim 92 = dim 'Y = 0. Since P m i n ( A ) = (A+q)-lI, it follows that

To study bounded q-pair, we need the following

Lemma 3.11. Let r), E W, \ ( 0 ) . Then J r),(dz)q(z,E)= 00. In particular, if supzEEq ( z ,E)< 00,then dim 'V = 0. Proof: We prove here only the first assertion. From r), E "yx = Wi, it follows that r),(X q)I = r),Q. In particular, XqA(E) q X ( q ) = r),Q(E). Thus, if the assertion is not true, then we could have

+

+

which is a contradiction. W The next result improves Corollary 2.24. Corollary 3.12. For bounded q-pair, the q-process is unique. Proof: a) Set c = supxEEq(z) < 00. For f, E

@A,

we have

3 UNIQUENESSCRITERIA

104 Hence

and so fx = 0. This provies that dimU&= 0. b) If the q-pair is conservative, then Eo = S and so the conditions of Thee rem 3.8 me all satisfied. In particular, we have proved again Corollary 2.24. c ) By rCheorem 2.21 arid Theorem 2.12 (I). Pmi"(X)l is the minimal solution to the equation u = r I ( X ) u + (A + 4 ) - 1 .

We thcn have

This implies condition (1)of the criterion. Finally: By a), c ) and Lernma3.11! the conditions of Theorern 3.8 are dl satisfied. I

As a special case of the above theorem, we have Corollary 3.13. Let E" be finite. Then q-process is unique iff both &-processss and Fq-process are unique.

Now, we want to sh,ow that Theorem 2.25 can be deduced from our main criterion. Indeed, since the q-pair discussed there is conservative, the uniqueKXSS criterion becorries dirr&%= 0. O n t,he other hand, dim% = 0 is equivalent t)o the minimal q-process being honest, which is the point we used to arrive at the uniqueness. Here, we want to show that EL dual approach enables 11s to arrive at dim% = 0. To do so: 'we introduce a comparison lemma, which is a dual of the comparison theorem for the minimal solution.

L L W L ~3.14 L ~ (comparison Lcmrnu). Let ni be a non-negative kernel, gi E 8+satisfying nil gi 1,,i = 1, 2. Denote by f i the maximal solution t o the equation

+



0 for all k 2 i } < 00.

The corresponding Q-process is called a single birth Q-process.

3 UNIQUENESSCRITERIA

106

In the case that yi,i+l > 0 for all i 2 0, we have N = 0. When N 3 1, to be distinguished, we also call the matrix (resp. Q-process) a single birth Q-matrix (resp. Q-process) with absorbing boundary.

Theorem 3-16" Given a single birth Q-matrix Q : (qij), (1) when N = 0, the Q-process is unique iff R := CrZN rn,

and q f ) = that

c i = o q k j for

xa= 0.

=. m,

where

z < k and k 2 1, Here we use the convention

(2) When N >, 1, choose arbitrarily a positive ijol, define in (3.20) and (3.19) with N = 0, replacing

$)

and En as

and qn,n+l, respectively,

with i#) and tjn,n+l:

Then the (qij)– process is unique iff

Proof: First, we consider the general case that N 3 0. a) By Theorems 2.47 and 2.40, it suffices to show that the maximal solution (ur)to the equation

equals zero identically for some fixed X > 0. When AT 2 1, the set {0,1, . . , N - 1) consists a closed subclass of the chain and so u: = 0 for all i 6 N - 1. b) Define

(3.21)

By induction and ( N O ) , it is easy to check that

3.2

UNIQUENESS CRITERION AND

APPLICATIONS

107

c) Let (ui)be a solution to the equation ( l + q i ) u i = x q i j u j , i 2 0 with

uk=O

for all k < N - 1 and u N = l . (3.23)

j#i

We now prove that (ui)is unbounded iff z,"==,Gn = 00. From (3.23), it follows that

[c n- 1

uN

= 1, un+1- un = q;,;+i

qik)(uk+l-

uk)

+ un

k=O

and hence ui as i

Gk

1

2N

(3.24)

1'. The key of the proof is to show that

< uk+l - u k < ( u N + 1 - uN

+ukGk,

k 2 N.

(3.25)

To check (3.25), we use induction. Noting that

haveand by ()3.24 we

Suppose that (3.25) holds for all k: N < k < n – 1 and we now consider the case that k = n. Then

and

=

+ un5iin,

( u N + l - uN)pLN)

n2N+l.

d) Having (3.22) and (3.25) in mind, it is now easy to complete the proof of the conclusion mentioned in c). If u, := limn+,un < 00, then, by (3.25), we have

108

3

Conversely, let

it follows that

UNIQUENESS

CRITERIA

.k < 00. Because

nkuk+l/ukand

log (uk+l/uk), and then

converge or diverge simultaneously whenever and (3.22), we get

U ~ + ~ /--fU 1. ~

Next, by (3.25)

Combining these facts together, we obtain the main conclusion. e) We now prove (3.19). Actually, by (3.20) and (3.21) with N = 0 and induction, if 6 k = mk for all k 6 n - 1, then we have

n-1

j=O

= m,

We have thus proved the first assertion of the theorem. f ) Finally, consider the case that N 2 1. From the expression of ( f i n ) , it is clear that one can regard {0,1, . . , N - 1) as a single absorbing state 0. If necessary, relabelling N , N 1,.. . as 1,2,. . . , respectively, we get a new Q-matrix and then the quantities &) (0 6 i < n ) and gn,n+l. For which, the

+

3.2 UNIQUENESS CRITERION AND APPLICATIONS

109

state 0 is the only absorbing one. However, for simplicity, we assume N = 1 and use the original ( q i j ) instead of the new Q-matrix. Then

Note that even though qol = 0 but (FLZ) : k 2 i 2 0) is still well defined by (3.20) with N = 0. Repeating the proof e), it follows that

,

n-1

n

\

On the other hand, define co = 1 and

Comparing this with n-1

k=O

and FAo) = 1, it is clear that cn = F?) for all n 2 0. Collecting these facts together, we obtain n

k=l

Therefore, we have returned to the case (1) with qol = 1. Now, part (2) of the theorem follows, because the sequence { F F ' : n 2 k 2 0} does not depend on gol and so the condition " R = 00" is independent of qol > 0. H Alternative Proof: We can reduce the case of N 2 1 to N = 0 by using a probabilistic approach, which goes back to Pakes and Tavark (1981). Let (qij) be a local modification of ( q i j ) , up to N-1 for instance. Denote by ( X t ) and the minimal processes determined by ( q i j ) and (t&) respectively. We want to show that ( X t ) has at most finite number of jumps in every Let (2,)denote the minimal process finite time-interval iff so does determined by the Q-matrix (&): & = 0 for all i 6 N - 1 and @ij = q i j for all i 2 N . Note that for each i ,< N - 1, stays at i with the exponential law having parameter qi and then jumps to other states. This is the only way yields more jumps than Due to the conditional independence and the fact N < 00, such jumps can happen at most finite times in a finite time-interval. Thus, (57,) has at most finite number of jumps more than in every finite time-interval. The same comparison holds for ( X , ) and ( X t ) . We have thus proved the assertion. H

(xi)

(xt). (x,)

(xi)

(zt)

(zt).

3 UNIQUENESS CRITERIA

110

Definition 3.17. A conservative Q-matrix Q = ( q i j ) is called a birth-death Q-matrix if qi,i+l=:

bi

> 0, i 2 0;

q i , i - 1 =: ai

> 0, i 3 1 and qij = 0 for

all li - j l

> 2.

The corresponding Q-process is called a birth-death process

Since the birth-death Q-matrix is a special type of single birth Q-mat,rix, as a consequence of the above result, we have Corollary 3.18. Given a birth-death Q-matrix, the Q-process is unique iff

where

po = 1,

pn

bob1 1

-~

bn-l

*

--

UlU2..

-

. an

,

n>l.

When bo i s replaced by 0, we obtain a birth-death Q-matrix with absorbing boundary. This necds only a little modification since the change of a finite number of thc transition rates qzJ docs riot iritcrfere the uniqueness property as mentioned above. Clearly, Theorem 3.16 is a very nice result since it is explicit and completely computable. Unfortunately, this is uncommon. Even in the twodimensional case, it is usually hard to justify whether a Q-matrix is zeroexit or not. The remainder of this section is devoted to study some sufficient conditions for the uniqueness of multidimensional Q-processes by comparing them with a single birth Q-process. The typical example in mind is Schlijgl’s model. Theorem 3.19. Let E be a countable set, Q = ( q ( z , y ) : ‘c,y c E ) be a conservative Q-matrix. Suppose that there exists a partition of E such t h a t Cr-oEk = JY and the following conditions hold.

{&}r

E3 for all k 2 0. (1) If q(x,y) > 0 and J: E Ek, then y E (2) AT := inf {i : q ( z , y ) > 0 for all z E Ek and all k 2 i} (3) CI, := sup{q(z) : z E Ek} < 00 for all k 2 0.

xYEEk+I

Define a conservative Q-matrix Q = ( y z j : i , j E

< 00.

Z+)as follows: (3.26) other cases of j

tf the (qij)-process is unique (i.e., R or then so is the (q(z,y))-process.

# i.

defined by Theorem 3.16 equals

oo),

3.2 UNIQUENESS CRITERION AND APPLICATIONS

111

Proof: By the uniqueness criterion, it suffices to show that the equation (A

+ q ( z ) ) u ( z=) 1q ( 2 ,

0 6 ).(.

Y>U(Y),

6 1,

x

E

E

(3.27)

Y#X

has only the trivial solution. Suppose that Eq. (3.27) has a non-trivial solution ( u ( x ): 2 E E ) . Let Uk

= SUp{U(X) : X E Ek},

k 2 0.

(3.28)

<
0 arid d k ) E Ek so that Fk(X c k ) < x/2, u ( x ( k ) ) (1 - E k ) U k (3.29)

cri:

+

IZ: with (3.29), we obtain

Replacing

d k )in

(3.27), using conditions (1) and (3), (3.28) and

k-1

That is

?(

xuk 2 + j=o

UGEj

p(2(*);y))(uk-Uj)


0 iff there exists a K E T+,such t h a t

&(En)< 00, n 3 1,

K ( E )= 00

(3.32)

x > 0.

(3.33)

and

0 < nPm'"(A)l

< 00,

Proof: We first prove that the property (3.33) is independent of A. Actually, if ~P"'"(p)l= 0 for some p, then by the resolvent equation, we have KP'"(p)l

= KP" '" ( A) l

+ (A - p ) K P m ' " ( p ) P m i n ( A ) l .

This implies that ~ l ' ~ ' " ( X ) l = 0, which is a contradiction. On the other hand, since ~(p,P"'"(p)l)is decreasing in p, so r;Prn'"(X)l< 00 implies that ~,P"'"(p)l< 00 for all p 6 A. As for p > A, the resolvent equation gives us KPrn'"(/L)l

< KP'"(A)l

< 00.

3 UNIQUENESS CRITERIA

114

The sufficiency comes from 3;,

> KEpmin(X)l 2 C ( X ) K ( E ) .

To prove necessity, assume that c(X) = 0. Choose an infinite subset N of N := {l,2,. I . } so that CrLEN inf,CE, P"'"[A)l(z) < 00 and choose a sequence ( 5 , F En : n E N } such that 0 < z n E N P m i n ( A ) l ( x n 0, then dim@ = 0. Proof: This is an easier consequence of Theorems 2.21 and 2.11. Lemma 3.23. (1) If f E bG) and Pmin(X)f = 0 , then f = 0. (2) If p E 9 and cpPmin(A)(A) = 0 for A E 8 n En,n 2 1,then p = 0.

Proof: a) From

(BA),

it follows that

f = ( X I -- I 2 ) P " ( X ) f = 0. Hence assertion { 1) follows. b) Let cp* be the John-Hahn decomposition of q, By the hypotheses, we have

cp+P"'"(X)(A) = pP-Pmin(X)(A),

A E c" n En, n >/ 1.

Now, the monotone class theorem implies that (p+Prni"(X)f = p-Pm'"(X)f

for all f E b8. Furthermore, for any f E 8,whenever one of the two sides exists, then the other side should exist and moreover! the both sides are equal. Applying (Fn),we obtain q+ ( A )=

C yipmi"(A) ( A 1 n

-f l ) I A E , =

C q-Pmin(A) ( X I n

for all A E 8,and so cp = 0 since cp E 9.

- fl2)I~~, = cp-

(A)

3.4 PROOF OF UNIQUENESS CRITERION Lemma 3.24. Let G be a kernel on ( E ) & )0, by f * the minimal solution t o the equation

< G < 1 and g E b&+.

115 Denote

f = Gf f g . Set

D = {x

E

E : g(x) > O}. Then supxEEf*(x:)= supzEDf*(x).

Proof: As usual, let f(1)

= g,

f(lz+')

+

= G f ( n ) 9,

n

> 1.

Then


and Proposition 2.18, we have

Thus

V(A)

:= Q ( A ) ( A I

- sl) 2 0,

A

E

8 n En, n 2 1

Clearly, for ewh X > 0 and z E , U ( A ,5,.) can be extended to 8 uniquely, denoted by U(A, 5 , ,) again. Using condition (2), Tlieorerri3.6 and Lemma 2.34(2), we obtain @[A) --- U(A)Pmin(A).From this and (3.34), it follows that P(A) = P"'"(A) U ( A ) P m y A ) . (3.35)

+

Hence

6 X P ( X ) l 6 1.

XT/(X)P"1'"(X)l

Besides, by condition (1) and Lemma3.7, we have 1 Combining these two facts together, we obtain

XU(X)l


U(p). (3.38)

3.4 PROOF OF UNIQUENESS CRITERION The right-hand side is non-negative whenever X

2 p.

117

Hence

and so p U ( p ) 3 (1 - p A - l ) X [ XP"'"(X)

-I]U(p),

X 2 p.

(3.39)

X(XPTi"(X) - I ) = R. Letting Note that Prnin(X) is a Bq-process, limx,, X --+ 00 on the right-hand side of (3.39) and using (3.36) and Theorem 1.14, we obtain pU(p) 3 RU(p). Thus

(3.40)

V(p) := (p1 - R)U(p) 2 0.

On the other hand, by condition (1)and Lemma 3.22, we know that dim 92 = 0. So Lemma 2.34 (1) implies that

Substituting this into (3.35),

P(X) = P"i"(X)

+ Pmin(X)v(X)Pmin(X).

(3.42)

Substituting (3.41) into (3.37), P"'"(X)[V(X)

- V(p)

+ (A - p)v(X)Prnin(X)Prnin(p)V(p)] = 0.

By Lemma 3.23(1), we have V(X) - V(p)

+ (A - p)V(X)Prni"(X)Prni"(p)V(p)= 0.

(3.43)

Hence, the resolvent equation gives us V(X) - V(p)

+ V(X)[Prni"(p)- P"'"(X)]V(p)

= 0.

(3.44)

Before moving further, we prove two facts: v(x)Pmin(X)V(p)l< 00,

(3.45)

V(X)Prni"(p)V(p)l< 00.

(3.46)

By (3.36) and (3.40), we have V(X)1 < 00 for X

> 0. But

3 UNIQUENESS CRITERIA

118

so

V(X)Prni"(p)V(p)l< p-lc(p)-lv(X)l

< 00.

(3.47)

This is just (3.46). Next, by the resolvent equation and (3.43), we have

P m i n ( X , s , A )1, V(X,x,A) J,

as X

t.

(3.48)

Thus, for X 6 p, from (3.46), it follows that


0, which completes the proof. I

3.5 NOTES

119

Proof of Theorem 3.8: Combining Theorems 3.26, 3.2 and 3.6 with Lemmas 3.22 and 3.25, we obtain the desired assertion. W 3.5 Notes In the above three chapters, we have chosen a short way to present our main uniqueness criteria for general q-processes. Some more refined results are delayed to Section 6.4, which should be considered as an essential part of the theory. Next, we mention a further topic related to this chapter. In practice, the most important jump processes are honest ones. But in physics, one pays more attention on the Fq-processes, i.e., the processes satisfying the Fokker-Planck equation. Corresponding to the equilibrium physics, there are so-called reversible (or more general, symmetrizable) jump processes, which are the main subject in Chapters 6 and 7. Thus, it is natural to study the existence and uniqueness problem for each type of jump processes. Moreover, one may even ask how many jump processes do we have in each case. To show that the answer is quite interesting, we mention that even for conservative q-pair, it can happen that the minimal jump process is nonhonest, but the honest Fq-process is still unique. Note that for such a jump process, its samples are not necessarily step functions, even have no left or right limits. Hence, to get a complete theory, it is not enough to consider only the samples which are right continuous having the left limits unless enlarging the state space. However, a quite complete picture for the above problem was obtained by Hou and Guo (1976) for Markov chains, and by Zheng (1982) for general jump processes. For Markov chains, Theorem 3.6 is due to Reuter (1962), Theorems 3.8 and 3.26 are due to Hou (1974). The proof adopted here is closed to Reuter (1976). For single birth matrix without absorbing boundary, Theorem 3.16 was first proved by Zhang (1984) by using a probabilistic approach. Then an analytic proof was presented in Yan and Chen (1986), in which an incorrect assertion was made for the Q-matrix with absorbing boundary. A correction was given by Li(1990b). A unified treatment was given in Chen(1999d). The present proof is a further simplification. The uniqueness criteria in the general context and the most of the materials discussed here are taken from Chen and Zheng (1983). The discussion about Theorem 2.25 is taken from Chen (1986b). Some more applications of the uniqueness criteria, especially for nonconservative Markov chains, were present,ed in Hou (1982), Chapter 8. A more complete theory of the uniqueness problem for Markov chains is explored in anderson (1991), Hou et a1 (1994, 2000).

Chapter 4

Recurrence, Ergodicity and Invariant Measures In this chapter, we first introduce some results about the weak convergence which are very useful but not popular yet in the literature. Next, we study the recurrence and the existence of stationary distributions for general jump processes, and then for Markov chains. Moreover, three types of ergodicity are also studied. Finally, we discuss the invariant measures for jump processes.

4.1 Weak Convergence

To begin this section, we recall iiwell-known result, its proof can be found from Billingsley (1968) or Stroock and Vsraclhan (1979). Let ( E , p ) be a metric space with Bore1 0-algebra f i . Denote by @ ( E ) the set of probability rricasiires on (I?,&'). TJet C&(E) (resp. U,(E)) dcnote the set of all bounded continuous (resp. bounded uniformly continuoils) functions on E .

Theorem 4.1. Given p , prL E , Y ( E ) (n 3 l ) , the following assertions are equivalent

(1) For each f E Cb(E),limndo;,pn(f) = p ( f ) . (2) For each f E U,(E), limndo;, p n ( f ) = ~ ( f ) . (3) For each closed subset C of E , p,(C) 6 p ( C ) . (4) For each open subset G of E , limn,,pn(G) p(G). (5) For each B E & with p ( d B ) = 0, limn-,o;,pn(B) = p(€?),where d B denotes the boundary of B.

>

Now, we return to our main setup, assume that E is a Polish space and choose a metric p so that ( E , p ) becomes a complete separable space with Bore1 u-algebra 6. Theorem 4.2. A subset A?c 9 ( k ! )is relatively compact in the weak topology iff for each E > 0, there exists a compact K, c E such t h a t

inf p(K,) 2 I

I.1E.I

- E,

where the weak topology is generated by the open sets

s

E > 0, (2 E qE), E c m . { PE *qE) : I P ( f ) - Q(S)I E } , This theorem is the set's form for the compactness. That is, using compact sets to describe the relative compactness. As the monotone class theorems having both the set's form and the functional form, the above criterion also has a functional form.

120

4.1WEAKCONVERGENCE

121

Definition 4.3, A function h E 8+ is called compact if for every d E [0,m) the set (x E E : k ( z ) 6 d } is compact. Note that a compact function is not necessarily continuous, for instance,

However, cvcry compact function should be closed I=lowcr scrni-continuom). T1ia.t is: for every d E LO, oo),the set [f d ] Is closed. Equivalently, lim, r'X) f(a,) 2 f (x ) far every sequence { x n } Y I K ~ hba,t; limn, ,.mxn - x. A function 'p E r S ; 011~d (resp. %') is compact iff limlz~n.,+mp(x) = 00 (resp. and p being closed), It, may be remarked that for the sufficiency in Theorems 4.2 and 4.4. below we med only msurne that, E is a separable metric space.




121

P E M nrl

Corresporiding to Theorem 4.1 (4):we also have a functional form.

122

4 RECURRENCE, ERGODICITY AND

INVARIANT

MEASURES

Theorem 4.5. p n converges weakly to p iff for every closed function f E &+, -n+m lim pn(f) 2 ~ ( f ) . Proof: Note that I B is lower semi-continuous for open set B. The condition is sufficient. Ta prove the necessity, we need only to consider bounded Actually, for a closed f E 8+,f A N is also closed and f a

This plus the monotone convergence theorcm gives us limn rm,un (f)2 p ( f ) . Thus, we havc rcdiiced the proof to the bounded case. Next, replacing f with f/Nif necessary, we may assume that 0 f < 1. Finally, set


0, A E S,

Q =(

E Cb(EA),

4 :~A E

S}.

Xu

pU(xzL, g u ) L / [ l + ~ u ( x yuU, ) ] , where (L is a positive summable sequence on S . Then ( E , p ) is a complete separable metric space. Given p, p n E .Yd(S),n 2 1, then pn p iff p n converges to p weakly in finite-dimensional distributions.

Remark 4.6. Define P ( Z , 3) =

=+

Proof: Clearly, if p n ===+ p , then p n converges to p weakly in finite-dimensional distributions. We now assume that pn converges to p weakly in finitedimensional distributions. Without loss of generality, assume that C , k, < 1. For each E > 0 and u E S , choose n compact PC,,, C E, such that

Set

then

Hcnce {pn : n 2 1) is relatively compact in the weak topology. Suppose that p n does not converge to p in the weak topology. Since the topology can be metrized by LBvy-Prohorov metric wf,there exist an E > 0 and a subsequence { n k } such that w(pn,,p) 2 E , k = 1,2,... . On the other hand, by the relative compactness proved above, we may choose a subsequence { p m k } & so that pmk p . This certainly implies that pr,, converges to p weakly in h e finite-dimentiional distributions. So by Kolniogorov extension theorcm, we have 0 = p . And hence, E:


. By assumption,

Applying Theorem 4.4 to the compact function hno, it follows that {pAh0 : P E A} is relatively compact and so the condition is also sufficient by a). I 4.2 General Results

For thc! remainder of this chapt.er, we often assume that the q-process is unique and study further the recurrence, the existence of stationary distributiori or invariant xlieasure and the crgodicity of the q-process.

Definition 4.8. We call a q-pair regular if it is totally stable, conservative and it determines uniquely a g-process.

Our first result reduces the recurrence of a q-process to the one of embedding process (cf. Section 4.3). Define

4.2 GENERAL RESULTS

125

Clearly, H(x, d y ) is a probabilily kernel, which is called an embedding jump process. Set

no = 1,

nnfl

= II II".

Theorem 4.9. Let (q(x),q(x,d y ) ) be conservative. Then we have

Here we use the convention O / O = 0, 0

8

oc = 00.0 = 0 and 1/0 = 00.

Proof: Since (Pmitl(A,x,A): x E E ) is the minimal solution to Eq. @A):

f = rI(A)f

+ (A + q)-%,

f E &+.

By the second successive approximation scheme, we have 00

P"'"(X,.,A)

=

CII(A)"((X+B n=O

Then, by the monotone convergence ~heorcm,we ohlain

Finally, the proof will be done once we show t,hat

This can be deduced by using induction on n. Acbually, when n = 0, both sides are the same I A / ~Assume . that the assertion holds for n - 1 and we now check the case of n ( 2 1). By inductive assumptim, we have

It suffices to show that

This is trivial if q(z) > 0. Let q(z) = 0. Then both sides are eqiial to ca if x E A . Otherwise, if x A , then both sides are 0 because of the convention 0/0 = 0 and 0 . 0 0 - 0. I Next, we study the exist,ence of stationary distribution. WE first consider the time-discrete case.

4

126

4

RECURRENCE, ERGODICITY AND

INVARIANT

MEASURES

Lemma 4.10. Let ( E , & ) be an arbitrary measurable space and P ( z , d y ) be a transition probability function on ( E ,8). Suppose that there exist h E d?+ and constants C E [O, 00), c E [0, 1) such that Ph 6 C

+ ch.

Then for each stationary distribution have

T(h)

(44

of P ( z , d y ) (i.e., TP =

T

T),

we

< C/(1 - c).

Proof: As usual, let P(O)= I , P(n+l)= P P(n).By induction, we have

C P ( " ) h 6 C ( l + c + . . . + c " - ' ) + c n h < -+cnh, 1-c

Set hN = h A N ,

N 3 1. Since T

( ~ N= )

n

> 1.

(4.2)

~ P ( " ) h r vit, follows that

Now, the assertion follows by letting N -+ 00. Denote by ZZp(E) the set of all Lipschitz continuous functions on the metric space ( E ,p ) .

Theorem 4.11. Let P ( z , d y ) be a transition probability function having the property: Pf E C6(E) for each f E ,+!?zp(E). Suppose that for some xo E E , there exist a compact h E d?+on E and a constant C E [O, 00) such that . , n 1

P(")h(xo) 6 C, n. .~ m= 1 Then, P has a stationary distribution particular, (4.1) implies (4.3).

n > 1.

7rx. having the

(4.3)

property ~ , ~ ( 6 h )C. In

Proof: Define

By (4.3) and Theorem 4.4, there is a subsequence {pnk}p=lsuch that pnk Some p as k m. Let f E 622p(E). Then p n k ( f )-+ p ( f ) , p n k P f pPf and --+

---f

4.2 GENERAL RESULTS

127

This proves the main assertion of the theorem. The other assertions are obvious. To study the time-continuous case, we need two lemmas. For the later use, we allow an operator acting on a column of functions. For instance, s1 acting on (fk : 1 k n) equals ( R f k : 1 k n).

<
i0. , , If x,'~< 1, then

This is impossible. So we have xfl = 1. Successively, we have

By the irreducibility, we indeed have x: = 1 for all i E E. b) Let xz < 0;)and ciilciliz . cin-lin > 0. If xzl = 00, then 0O

> x; 2 ciilz;l

= 00.

This is impossible. In the same way, we can prove that

< 00,.. . ,x;n < 00. Now, the irreducibility implies that x; < 0;)for all j x;z

E E.

c) By the localization theorem, if (x; : i E E ) is the solution given in (2), then (x; : i # j o ) is the minimal solution to the equation

xi

=

C cijxj + bi,

i # jo.

3#30

From this, the last assertion follows immediately.

H

132

4 RECURRENCE, ERGODICITY AND

INVARIANT

MEASURES

Lemma 4.20. Let (xik)*: i E E ) be the minimal solution t o the equation

xi = C c i j x j

+ bi

(k)

,

i E El k

= 1,2.

j

Suppose that

Clbil) < bi2) < C2b!’),

iEE

for some C1, C, E [O,m].Then

c1xj1)< *xj2)*6

c,xi (I)* ,

i

E

E.

Proof: Consider the case that C2bi” 2 bj2), i E E. By Corollary2.8, (C2xi1)*:iE E) is the minimal solution to the equation

Then, as an application of the comparison theorem, we have

czxy*2 $)*

,

iEE.

Similarly, we have

xi2)* 2 C l x i(I)* , Denote by and define

i E E. W

Pi the probabilityof the Markov chain (Xn) starting from i

In particular, M

n= I

Proposition 4.21. Let P = (Pij) be an irreducible Markov chain and H

#8

be a finite subset of E .

(1) The chain is recurrent iff f&

= 1 for all i $! H . Equivalently, for all i E E . (2) Let (9: : i E E ) be the minimal solution t o the equation

yi

PikYk

+ PiH,

E

fTH

=1

E,

Ic

where pi^ = CkEH P i k . Then the chain is recurrent iff yr = 00 for a l l i E E.

4.3 MARKOVCHAINS:TIME-DISCRETE CASE

133

Proof: a) Note that

f/A) = PiH,

f&+”

=

C

n 2 I.

Pzkfs,

WH

xr=lf!O,

Since f:H = it is easy to check, as we did in the proof of Proposition 2.14, that (f:H : i E E ) is actually the minimal solution to the equation

xi =

C Pijxj + P ~ H ,

i E E:

(4.6)

j@H

On the other hand, by Corollary 2.8, it is easy to check that

j € H n=1

Hence, by the localization theorem, (y: : i E E ) is the minimal solution to the equation

Take

C1 = ming;, JEN

Cz = maxy; j€H

By Eq. (4.6), Eq. (4.7) and Lemma 4.20, we obtain

+

Pick j , E H so that C1 = yT0. Then (1 yj*,)fjoH < y;o. Thus, if f j o H = 1, we must have y;o = 00 and hence y; = 00 for all i E E by Lemma 4.19 (1). b) Consider the special case that H = { j o } . Then, the first assertion follows from Proposition 2.14 and Lemma 4.19 (2). To prove the second assertion, note that by a) we have (1 y;o)f;o = y;” and so f;o = 1 y3t0 = 00. Conversely, let f;o < 1. Denote by f & ( r ) and y;(r), respectively, the minimal solutions to (4.6) and the equations given in the proposition when (Pij) is replaced by ( ~ P i j 0 ) ,< T < 1. It is clear that

+

>Y 3 )t

$7

The above argument gives us (1

fG0W 1 fG0

as ?- T 1.

+ ~;~(r))f;~(r)= y;”(~).

Hence

134

4 RECURRENCE, ERGODICITY AND

INVARIANT

MEASURES

This proves that fj*, < 1 + "y; < 00. c) We now return to general H . By the localization theorem, if for all i E E , then the minimal solution to the equation

f&

=1

equals 1 for all i $ H . This is equivalent t o say that f c H = 1 for all i E E . d) The second assertion for general W now follows from the expression of ya, b) and Lemma 4.19 (1). H For a given transition probability matrix P = (Pij),define a new transition probability matrix ?; = ( F i j : i , j E S) as follows:

-

,Paj =

{

&j

Pij

if i = 0 if i # 0.

Then 0 is an absorbing state. The following result is well known. Lemma 4.22. For each i E E , the limit ?ria = limndm Piu -3n) exists and

ii00

= 1.

Next, by the identitry

k-I

it follows that

Lcrrima 4.23. For each i Theorem 4.24. Let P the equation

=:

1, j z l = iiio.

( P i j )be irreducible. Then, the chain is recurrent iff

has a (finite) solution ( y i ) so that limi4m yi = 00 (i.e., (yi) is compact) for some finite H

# 0.

Proof: Trr vie'w of Eq. (4.6),(f& : i $ ,FT) is i,tideed the minimal solution to the equation xa = Papj - f ' P i f f , 1: 6 H .

c

j$H

Hence, we may regard H as a singleton {O}. a) Sufficiency. Let ( y i ) be a solution having the desired property. Then

4.3 MARKOV CHAINS:TIME-DISCRETE CASE

135

so and hence

it follows that iii~ = 1 for all i # 0. Letting n + 00, and then N + a, From this and Lemma 4.23, we see that f& = 1, i # 0. Finally, the assertion follows from

and the irreducibility of P. b) Necessity. Denote by (2,)the Markov chain with transition probability (&) and set SUCH THAT

Then f o ( n )= 0 for all n > 0 and f i ( n ) = 1 for all i 2 n. Because the originaI chain is recurrent, so we have f& = 1 for all i 2 0. Hence

Choose n k

1' such that

and define yi = c E , f i ( n k ) < possesses the desired properties.

00.

Then it is easy to check that (yi)

Theorem 4.25. Let P = (Pij) be irreducible. Then the chain is transient iff the equation JEE

has a non-constant bounded solution.

136

4 RECURRENCE, ERGODICITY AND INVARIANT MEASURES

Proof: Again, regard H as a singleton (0). Suppose that the chain is transient. Then there exists an i # 0 so that o:f = iii0 < 1. Note that i7io is the probability of Markov chain finally returning to 0 starting from i. We have

F

jEE

From this and i i o o = 1, we see that (yi = jiio, i E E ) is a desired solution. Conversely, suppose that the equation has a non-trivial bounded solution (yi). Without loss of generality, assume that yo = 1,

0 < yz

< 2,

2

E E.

Then

Furthermore

Letting n + 03, by Lemma 4.22, we get iiio 6 yi for all i E E . Because (yi) is not a constant, we have either yio < 1 or yio > 1 for some io. In the former case, we have i7iOo < 1. In the latter case, replacing (yi) with (2 - yi), we obtain the same conclusion. Now, the same argument given at the end of proof a) of Theorem 4.24 implies the required assertion. H Now, we turn to study the positive recurrence of the chain P = (Pij). We begin the study with a simple .result. Proposition 4.26. Let P = (Pij) be irreducible. If the equation

has a non-trivial solution (xi)so that Ci 1zil < 03, then the chain is positive recurrent. Conversely, only if the chain is positive recurrent, the equation

&+iPaj 6 Xj) xj 2 0, j E E ,

j EE CiEEZi < 0;)

has a non-trivial solution.

Proof: For simplicity, assume that the chain is aperiodic. Then, the assertions are easy to check by using the fact: the limit

exists and is independent of i, either

7rj 5 0

or

7rj

> 0 for all j

E

E. I

4.3 MARKOVCHAINS:TIME-DISCRETE CASE

137

Let P = (Pij) be an irreducible aperiodic Markov chain and (r2)bc a probability measure.

Definition 4.27.

(1) The chain is called ergodic (equivalently, positive recurrent) if P:?' + T j as n ---f oo for all i , j E E. (2) The chain is called geometrically ergodic if there is some j3 < 1such that for each i and j , IP)?' - 7rjl = O ( P n ) as n t 00. (3) The chain is called strongly ergodic or uniformly ergodic if sup, I P ~ I - T--3~ ~as n -+ oo.

o

Actually, these types of ergodicity have stronger properties. For instance, the geometrical ergodicity has the property: tpliere exist j3 < 3. and Ci, depending on i only, such that IPiT) ( - 7rjl Cip" for all i, j and n. Furthermore, we have


0. This is enough to guarantee the positive recurrence of the chain by the irreducihility. H

+

4.4 Markov Chains: Time-continuous Case This section deals with the recurrence and the ergodicity for Q-processes. define Given a conservative Q-matrix Q = ( q i j ) on E = Z+, nij

= l [ q i # O ~ (1- Sij) qij/qi

+ l [ q i = O ~ &j,

i , j E E.

Then @ I i j ) is a transition probability matrix and is often called embedding chain of the Q-process.

140

4 RECURRENCE, ERGODICITY AND

INV.4RTANT

MEASURES

Definition 4.32. We call the Markov chain P ( t ) = (Pi,(t)) recurrent if for each h > 0, P ( h ) is recurrent. Equivalently, S,”Pt,(t)dt = 00 for all i E E. Similarly, we call the Markov chain P(t) = (P%,(t)} positive recurrent or ergodic if so is P ( k ) for every h > 0 (cf Lemma 4.42 below). Equivalently, limt400 Pii(t)= T~ > 0 for all i E E .

By using the first; successive approximation scheme and Theorem 1.3, it is not difficult to prove the following simple fact. Lemma 4.33. For a given Q-matrix, t h e minimal q-process is irreducible iff so i s i t s Q-matrix.

As we mentioned in Section 4.2, thc recurrence of the q-process can be reduced to the one of the embedding chain. Let us copy Theorem4,9 as follows.

Theorem 4.34. Let Q = (q2,) be a conservative Q-matrix. Then

In particular, if Q is irreducible and regular, then (P,j(t))is recurrent iff so is its embedding chain.

Combining Theorem 4.34 with Theorem 4.24, we obtain

Theorem 4,35. An irrcducible conservative Q-matrix is regular with recurrent P ( t ) iff the equation 2 $ Jl CITijYj Yi,


0, (Pzj(h))is irreducible and aperiodic Markov chain. Hence t,he desired assertion follows from

4.4 MAR,KOV CHAINS:TIME-CONTINUOUS CASE

141

Theorem 4.37. Let Q = (qij)be a regular irreducible Q-matrix. Then P ( t ) is positive recurrent iff the equation

(4.10) has a summable, non-negative and non-trivial solution uniquely up t o a constant.

Proof: a) Let P ( t ) be positive recurrent. By Lemma 4.36, we have AND

Corresponding to the time-discrete case, we get rj = Ci rZPt,j(t).Next, from Theorem 4.17, it ibllows that

So (ri)is a solution to Eq. (4.10). b) Let (xi)be a summable, non-negative and non-trivial solution to Eq. (4.10). Without loss of generality, assume that

xi > 0,

c x i = 1;

2

E E.

a

Applying Theorem 4.17, we have

t 2 0, j E E .

- C.ZPij(t),

23. -

i

Letting t + 00, we obtain 0 < xj = x j . This proves not only the positive recurrcnce of P ( t ) but also the uniqueness of the solution to Eq. (4.10). The next two examples show that the positive recl.irrence, unlike the recurrence, of &-process cam not be rediiced 1,o the one of its embedding chain. Example 4.38. Take

Pa

0, i=l

n0,i = Pi, rIzj

= 0,

and qi = pa/2, 423. . = qiIIij, i not the Q-process.

221;

ITio=1,

i31;

otherwise

# j.

Then

IT

=

(Qj)

is positive recurrent but

142

4 REClJRRENCE, ERGODICITY AND INVARIANT MEASURES

Pro06 Clearly, ll = (Qj) has uniquely a stationary distribution as follows: 7rO : 1/2, T% = p , / 2 , i 2 1. On the other hand, since 0 = ( q i j ) is bounded, the Q-proccss is unique. By Theorem 4.37, if P ( t ) is positive recurrent, then xjqj = C i f jxiqij. Hence for every j 2 1, zjnj =

c

xi7rzrI, = xo7ropj.

i#j

xi

That is xj = xo. This implies that xi = 00, which is in contradiction with Theorem 4.37. Therefore P ( t ) is not positive recurrent. We have seen a,n example for which the embedding chain is positive recurrent but not the Q-process, The next example goes in the opposite way. Example 4.39. Consider a conservative birth-death Q-matrix: 0;=

hi, i 3 1,

C'glT/bi

bi + M such that recurrent but not II = {l&j).

Choose

bz


0,

2

2 0.

Then the Q-process is positive

00.

Proof: By Corollary 3.18, the Q-process is mique. The birt'h-death process is syniinetrizabie with respcct to ( p i ) (cf. Sections 6.1 and 6.2):

/GiP&) = jLj,Pji(t)> i , j where Pi =

Po = 1,

Because

cc'

bob1 bz-1 aIa2. ai + .

OC:1

c p i =l+box0

E ,F:,

i= 1

bi

,

t 2 0, 221

can not be determined completely by its embedding chain 11. So we need to study this problem more carefully, As .what we did for the tirncdiscrete case, we now study the three types of crgodicit.y. Definition 4.41.Let Q = (qi,j) be a regular irreducible Q-matrix and a probability measure.

( ~ i be )

(1) T h e chain P ( t ) is called exponentially ergod.icif there is some p > 0 such t h a t for each i a n d j , lPij(t)- ~jrjl= 0 ( e d t ) a5 t CXI. 4

( 2 ) The chain is called strongly ergodic or uniformly ergodic if supi IPij(t)-~JI4 0 as t + 00,

144

4

RECURRENCE, SRGODICITY AND INVARTANT MEASURES

For general Markov processes, Definition 4.41is ineaninghl once the pointwise convergence is replaced by the convergwr:e in total variation (cf. T h e orem 4.28 and Theorem 4.43below). The next result enables us to transfer the tirue-discrete case into the timecontinuous one. Lemma 4.42. Let ( X t ) t 2 0 be a Markov process. If for any h > 0, the skeleton (X(nh)),>o i s (resp. geometrically, strongly ) ergodic, then (X(t)) is (resp. exponentially, strongly ) ergodic.

Proof: Recall that for m y finite (signed) measure p on ( E , & ) ,its total variation norm can be represented as follows (by Hahn-Jordan decomposition):

For simplicity, in this proof, w e omit the subscript “Var”. a) The key fact in the proof is that. the lunction t - > IlvP(t)ll is nonincreasing on [0, m) for evcry bounded signed measure v on ( E ,-8).Actually,

Next, we show that for these types of ergodicity, the initial distribution p can be replaced by initial point mass. Actually, if

for all x c E , then for any p E

9((E),


0 and ( k - l)h, 6 !i < Ich (k: E Pi). By a), we have --)

and so n P ( t ) = T.Furthermore

4.4 MARKOVCHAINS: TIME-CONTINUOUS CASE

145

c) Suppose that ( X ( n h ) )is geometrically ergodic, i.e.,

/'T(dr)ll~(nhl+, -

= O(e-n,'(h)))

as n

--f

oo

for some P ( h ) > 0. Again, l e t t > 0 and (k - l ) h ,< t < k h ( k E we have

s

~ ( d z ) I l P (5t,,.)

-

TII

N), Then,

= O(e- t P ( h ) / h )*

d) Similarly, we can prove the assert,ion about the strong ergodicity.

H

Theorem 4.43. is ergodic iff[lP,,(t)= C j IPij(t) - 7tjl -+ 0 as all i E E . is exponentially ergodic iff IlPi.(t) - 7r]TIIvar = 0 ( e - P t ) as some ,L? > 0 and for all i E E . Equivalently, CinillPi.( n-l[var- O ( e - P t ) as t -+ cc for some p > 0. (3) The chain is strongly ergodic iff supi llPi . ( t ) - ~ [ = l vO(e-Pt) ~~ as t -+ m for some p > 0.

(1) The chain t --+ 00 for (2) The chain t -+ 00 for

Next, for a giver1 regular irreducible 6,)-matrix Q - (ye), let (X(t)),,, be the corresponding Markov chain defined on a probability space (12,F,IF). Its successive jumps are given by TO

= 0,

7,

= inf{t : t

> 7;2-1,X(t) # X ( T ~ - ~ ) } , n 2

Due to the regularity, we have T := finite subset of E and define aH = inf{t 2 following probabilistic criteria.

T~ T~

1.

= 00. Let H be a non-empty

:Xt E

H } . Then, we have the

Theorem 4.44. (1) The chain is ergodic iff R,a, < 00 for all 1: E H . (2) The chain is exponentially ergodic iff lE,eXo~~ < oc for all i E H , where 0 < X < qi for all i E F . (3) The chain is strongly ergodic iff supzEiaH < 00.

The analytic criteria for the t'hree types of ergodicity are the following. Theorem 4.45. Let H

# 8 be a

finite subset of E .

(1) The chain is ergodic iff the equation

(4.11)

146

4 R.ECURRENCE,ERGODICITY

AND INVARIANT

has a finite non-negative solution. (2) The chain is exponentially ergodic iff for some X i E E , the equation

MEASURES

> 0 with X < qi

for all

(4.12) has a finite non-negative solution. (3) The chain is strongly ergodic iff Eq. (4.11) has a bounded non-negative solution.

asReplacing (yi) in (4.12) with yi = yi + 1), one can rewrite ()4.12

To explain how to deduce these results, we need some preparritions. Define

n= 1

Lemma 4.46. We have f i(1) H - nZff~

=

f&+l)

cnil,f2,

n 2 1.

V H

Furthermore,

(fiH : i

E E ) is the minimal solution t o the equation

and the Markov chain is recurrent iff

f

i ..I ~

1 far all i E

Proof: Note that (1) f2H =

p,[qf= 71 1 = 1 - h

and

The r m a i n assertions are then obvious.

I

l?

4.4 MARKOV CHAINS:TIME-CONTINUOUS CASE

Lemma 4.47. We have

Proof: Obviously,

Next,

But by the strong Markov property, we have

and

By induction, this proves the required assertion. Lemma 4.48. For X E R with X

Then we have

I

< qi for all i E E , define

147

4 RECURRENCE, ERGODICITY

148

A N D INVARIANT MEASURES

In particular, ( e i H ( X ): i f E ) is the minimal solution t o the equation

Proof: The first assertion follows from Lemma 4.47 immediately. To prove the last assertion, apply Theorem 2.9 to

k$H

TZ-

1

and note that gi = fiH (i E E ) by Lemma 4.46. H Equivalence of Theorcm 4.45 and Theorem 4.44: a) Assume that the chain is recurrent. Then fiH = 1 for all i by Lemma 4.46. Hence by Lemma 4.48, ( e t H ( X ) : i E E ) is the minimal solution to the equation

On tmheother hand, since f i ~ ( t = ) lPi[uH> t]: by the Fubini theorem, we have

By the assumptions of Theorem 4.44, e i H ( X )< 00 for all i E H . From this, we show that eiH(X) < 00 for all i 4 H. Given i $ H , choose i o , i l , ,in with in = i so that io E H and -

-

a


-w. Thus, by compa,rison theorem, we

2 CP,,(X)(Cj - Xc),

i G E,

j

NOW,by the regularity, X Cj Pij(X) = 1, hence 1;2

XPaj(X)cj:

i E E,

x > 0.

j

Furthermore,

That is xzi

1

+

c

XPij(X)(Cj

- l),

i E E,

x > 0.

jEH

Since lim XP'j(X) = lim Pi,i(t)= rij = rj, x +o t4m

+

it follows that 0 2 1 C j E H nj(c, - 1).This s h w s that rj the irreducibility. I We now return to a condit'ion given in Theorem 4.14.

> 0 for all j by

150

4 RECURRENCE, ER.GODICITY AND

INVARIANT

MEASURES

Corollary 4.49. Let Q = (qi3)be a conservative irreducible Q-matrix. Suppose that there exist a compact h E T& and constants I: 0 and c > 0 such that IZh < C - ch. (4.15)

>

Then the Q-matrix is regular and the &-process is exponentially ergodic. Proof: The uniqueness of the process is a straightforward consequence of Theorem 2.25. Since h is compact, we can choose a finite H so that C ( c / 2 ) h -1 on H". Now, the asscrtion follows from Theorem 4.45. Applying Corollary 4.49 with h(x) = C , xu to Schlogl's model, it is easy to check that the model is exponentially ergodic. Actually, it, is strongly ergodic, as can be shown by Theorem 4.59.


1) is a non-negative solution to the equation

j=I

But the solution to this equation is also unique: i-1

3=1

for all i 3 1. Combining the

By induction, it is easy to check that zi = F,"' above facts, we obtain = 1 = Fo( 0 ) ,

xo

xis1 -xi

= Fi (0),

i 2 1.

This certainly implies the required assertion. d) To prove the second assertion, we use Theorem 4.45 (1). Let (ui)(uo= 0) be a non-negative solution to Eq. (4.11) with H = (0). Then k-1

j=O

j

k- 1

k-1

j=O

j =O

From this and induction, it follows that

where vk = u k S 1- u k , k 2 0. Hence L

.L

ic

, L

ic

4.5 SINGLE BIRTHPROCESSES This gives us d 6 u1 < 00. Conversely, a s u m e d

Obviously, we have C j f O q O j u= j qolul have

s=o

s=o

< 00.

155

Set

< 00. Moreover, for each k > 0, we

s=o

s=o

j=O

+

That is Cj qkjuj 1 = 0, k > 0. e) In view of Theorem 4.45 (2), the condition infi qi > 0 is indeed necessary. We need to construct a solution ( g i ) to Eq. (4.13) with H = (0) for a fixed A: 0 < X < infi qi. First, define an operator

This operator essentially comes from the study on spectral gap, and will be discussed in more details in Chapter 9. Next, define

Then p is increasing in i and cp1 = q&'. Let f = c q l 0 m for some c > 1. Then f is increasing and f l = cql0. Finally, define g = f I I ( f ) . Then g is increasing and

We now need a technical result, will be proved later, taken from Chen (2000b).

156

4 RECURRENCE, ERGODICITY AND

MEASURES

INVARIANT

Lemma 4.53. Let (mi)and (ni)be non-negative sequences, ni $ 0, satisfying

Define cpk =

Cizi nj. Then for every y E (0, l),we have

Proof: Let Mn = &2nmj. Fix N > i . Then by summation by parts formula and the assumption Mn ccp;', we get


0 is necessary by Theorems 4.44 (2) or 4.45 (2) [or Chapter 9, Lemma 9.7 with K = { k } ] . By Theorem 4.44(2), there exists a X with 0 < X < qi for all i such that EoeXoo < 00. Define eio(X) = e X t P i [ o o> t l d t , i 2 0. lim

C L o d k = lim dn F(0) n-w F p k=O k

n-mzn

I "

g1

I" +

Then IEieX"o = Xeio(X) 1. From the proof a) of Theorems 4.45 and 4.44, it follows that eio(X) < 03 for all i 2 1. Furthermore, EieXoo < 03 for all i 2 1. Note that if the starting point is not 0, then oo is equal to the first hitting time: T~ = inf{t > 0 : X ( t ) = O}. Hence lEieATo < 00 for all i

2 1. Define min)= E~T,". The Taylor's expansion (4.30)

leads us to estimate the moments mZ(").By a result due to Z. K. Wang [cf. Wang (1980, Chapter 3), or Hou and Guo (1981, Chapter 9), or Wang and Yang (1992)], we have

x c i-1

mi( l ) =

j=o

nx i-I

mjn)=

1

"

P A k=j+l

Pk,

1 " pkmf-'), j=o P A k = j + 1

C

n

2 2.

(4.31)

From these quoted books, one can also find the probabilistic interpretation while of S and R: S is the mean of the hitting time of 0 starting from "00)', R is the mean of the hitting time of "03') starting from 0. Actually, we have a general result as follows.

162

4 RECURRENCE, k h C O D I C I T Y A N D INVARIANT MEASUR

P r o p o s i t i o n 4.56. Far an regular and irreducible Q-process, (mjz := Eia;4 : i @ H ) is the minimal solution to the equation

Proof: Similar t o the proof of Lemma 4.48. W In the present case, obviously, m p ) 3 mLn) if k 3 i. By (4.31), it follows that

and

Hence, by induction, one gets

Combining this with (4.30), we obtain

which implies that

< 1.


0. Then the Q-process is

with rates b, = an always unique.

+

(1) The process is recurrent iff a < 6 or a = 6 2 p. (2) The process is ergodic (exponentially ergodic) iff a (3) The process is never strongly ergodic.

< 6.

1ti3

4.5 SINGLE BIRTH I"R0CESSES

Proof: a) The uniqueness assertion is easy since

Assertion (3) is also easy since

b) To prove the other assertions, we use

Kummer Test: Let (un) and (wn) be two sequences of positive numbers. l/v, = 03 and t,he limit K := limn+m ten exists, where Suppose that

xF

Then, the series respectively.

Cunconvmges or

diverges according to

K

> 0 or

Now, to prove recurrence, we consider the series Gnu,: un = Take v, = n. Then

[6

t&=n Q: -

So R = -too if a > 6, K

We have

K

-os if a

I I :

lf6(n+1)

< 6.

> 0 if 3 > 6, R < 0 if p < 6.

K

U(Y>

c

4 ( 4 Y)Ul 6

Q O l 9 .

PEE1

YfQ

When k 2 1, by using the argument which deduced (3.30), we obtain

j=1 \ y E E j

This shows that

uk

/

and moreover,

k-1 qkOuk

+

qkj(uk

- uj)

qk,k+l(Uk+l

- uk),

k 2 1.

j=1

Hence, we always have qkuk

6

qkjuji

>, 0.

j#O,k

Thus, by Lemma 4.51, Lemma 3.14 and Theorem 4.52, it follows that ui = 0. This contradicts with the assumption that u(x)$ 0. b) To prove positive recurrence, by Theorem 4.45, we need only to show that the equation

Y

Yf0

166

4 RECURRENCE, ERGODEITY AND

INVARIAN'I' h!fEASWRK;S

has a non-negative solukiori. For this, let ( u k )be a solution constructed in Lemma 4.54 and take u(x) = u k if z E EI,,

Now, for z # 6,there exists some k so that

k 2 0.

5 E Ek.

Hence

k-1

Y

On the other hand,

We have thus constructed a desired solution. c) The proofs for the remainder assertions are similar arid hence are omitted. I 4.6 Invariant Measures

An extension to stationary measure of a process P(d) = (Pij(t))or P ( P i j )is invariant measure, i.e., a non-trivial cr-finite measure 7r so that

n=;.rP(t)

or

=

T=TP

respectively. In this section, we study the existence and uniqueness of invariant measures. Of course, the uniqueness is in the sense of up to a constant factor. As usual, we restrict ourselves to the irreducible case. We begin our study with time-discrete case. the answers depend on recurrence or transience of the process. Let ( E ,X,, Pij)be an irreducible Markov chain with discrete times. Becausc of the irreducibility, if (n,) is an invariant measure, then 7rz > 0 for all i. For the positive recurrent case, the answer is quite sirriplc arid well-known.

4.6 INVARIANT MEASURES

167

Theorem 4.60. Let ( E ,X,,Pij) be an irreducible and positive recurrent Markov chain. Then there is precisely one invariant measure. More precisely, if the chain has period d ( 3 l), then the state E can be decomposed as a union of disjointed subclasses C O , .. Cd-1 so that Pk is an aperiodic chain on each CF Furthermore, the invariant measure is given by

Corollary 4.61. Let ( E ,X,, Pij) be an irreducible and non-positive recurrent Markov chain. Then i t s invariant measure, whenever exists, should be nonsumma ble.

Proof: See Proposition 4.26. For the recurrent case, the answer can be found in many books (cf. Chung (1967) Part 11, Theorem 9.7, for example). Here we state only the result.

Theorem 4.62. Let ( E ,X,,Pij) be an irreducible and null recurrent Markov chain. Then there is only one invariant measure, which is given by 00

n=O

where 8 is an arbitrary element in E and sP$) denotes the taboo probability: for a given B c E ,

B P23( o ) = b i j , B P23! n ) = P i [ x , = j , X I , . ' . ,Xn-l $ B ] . The transient situation is much more complicated. Let us look at some examples.

Example 4.63. Take Pi,i+l = pi and Pi,o = 1-pi for i 2 0, where pi E ( 0 , l ) such t h a t limn+m lJi=,p, > 0. Then the chain is irreducible and transient. It has no invariant measure.

Proof: Clearly, if (ri)with

7ro

=

1 is an invariant measure, then

Thus, there exists an invariant measure iff limn-,m n;=,p, = 0. On the other hand, n;=,p, describes the probability that the chain does not return pk to 0 in the first n + l steps starting from 0. Hence, condition = 0 means that the probability of the chain returns to 0 in finite steps starting from 0 equals 1. That is equivalently to the recurrence of the chain.

nLz0

168

4 RECURRENCE, ERGODICITY AND

INVARIANT MEASURES

Example 4.64. Take P,,,+l= p

ri'

1/2,

P,,%-1=

1 - p =: 4 )

Then the chain is irreducible and transient. measures.

Proof: Siippose that Vk+l

-

( T ~is )

Xk

2

= 0, 51, &2). . .

.

It has infinitely many invariant

an invariant measure of the chitin. Then

+ T L l P = 0,

k = 0, f l ,f 2 , .

2

'

.

To this equation, the solutions are as follows: 7rk = cl(p/q)k

+ c2,

k

= 0, fl: 42,.

..

Hence, for any c l , c2 0 , c1 + c2 = 1, (ni)is an invariant measure. This proves our assertion. H

Example 4.65. Take

Then the chain is irreducible and transient. It has precisely one invariant: measure.

Proof: Notice that the probability of the chain goes to infinity starting from 0 is 00 1/2rJ2i+l

- 1)/2i+1 > 0.

i=l

Hence the probability of the chain without return to 0 in finit'e steps starting from 0 is positive. So the chain is transient. Next, let (ni)be an invariant measure. Then

Set (7i-i)

1, then nL - 4. It is easy to sce that is positive, simply use induction. I

7r0 =

(T?)

is unique. To show that

Proposition 4.66. Let ( E ,X,, PiJ)be an irreducible, transient Markov chain. Without loss of generality, assume that E = 25,. Suppose t h a t the chain has an invariant measure, then there exists a simple path coming from infinity. T h a t is, there exist mutually distinct states i l , 22, such t h a t Pipil > 0, Pi3,i2 > 0, '

...

-

8

169

4.6 INVARIANT MEASURES

Proof: Define a, dual chain as follows: PzJ r 7 P J i , f T a , i , j Then

p::)

E

E.

= 7rjP:r)/7ri,i , j E E . Note that an irreducible chain is transient

iff

00

(4.34)

EP"'o0. n-0

We have

oc

M

n=O

n=U

x,,

So the dual chain ( E , Pt3)is irreducible and transicnt. Without loss of generality, we assume that the chain is aperiodic. Then, for almost all LJ, -

-

,..

PXl ( W ) , F O ( W ) , PJTZ ( W ) , T ? 1 ( w ) , are all positive. On the other hand, since the chain is transient,, for almost all w , in thc sequencc ( y o ( w ) , y l ( u ) ,. >,no clement can be appeared in infinitely many times. In other words, for almost all w , there exist infinitely many distinct XO( w ), XI ( w ), * . * so that ~ x l ( w ~ >, 0,~ o%(w),xl(w) (w) > 0 , ... a

This completes our proof. W We are now ready to present an existence criterion {br invariant measures. For simplicity, we take E = Z+ again. Moreover, we allow the chain to be sub-M arkovian : (4.35) CPij61, i € E j

As we did before, for H f f P2.7( 0 ) 6,j,

c E , define the taboo probability as follows: PiklPklkz * ''Pkn-]j, i,j E

ffp(n) z.7

E , n 2 1.

k 1 , . . ., k n - l $ ! f f

Theorem 4.67. Let ( E ,X,,Pij) be an irreducible and transient Markov chain, maybe sub-Markovian (i.e., (4.35) holds). Then, the chain has an invariant measure iff there exists an infinite subset K c E such t h a t

where

0 0 0 0

(4.36) r=j ,=I

This criterion is quite deep in the theory of hlarkov chains. Unfortunately, its proof is lengthy and so is omitted here. Refer to Harris (1957) and Veech (1963) for details. As a straightforward consequence of Theorem 4.67, we have

170

4 RECURRENCE, ERGODICITY AND INVARIANT MEASURES

Corollary 4.68. Under the hypotheses of Theorem 4.67, if for each i E E , there are only finite k E E so that Pki > 0, then the chain has an invariant measure. Now, we turn to study the time-continuous Markov chains. For the recurrent case, there exists uniquely an invariant measure, which can be seen from the time-discrete case at once. Thus, we consider only the transient case. Again, assume that the chain is irreducible. Hence, we have

0 < J, PZj(t)dt < w.

(4.37)

It is similar to the time-discrete case, in the present situation, we may have no invariant measure. But excessive measures do exist. Lemma 4.69. Let Q = ( q i j ) be an irreducible, regular Q-matrix and P ( t ) be transient. Then for each probability measure cr on E with finite support,

is a finite positive excessive measure of and t 2 0.

P ( t ) : pj 2

xi

p i P ; j ( t ) for

all j E E

Proof: By (4.37) and the assumption, the assertion is obvious. Now, it is natural to ask when an excessive measure becomes an invariant measure. More general, for a given ( p i ) , when we have pept = p

for some p

< O?

t2o

~(t),

(4.38)

If so, by Lemma 4.15, we should have (4.39) i#j

Define a dual chain as follows. &(t) = e - P t p j P j i ( t ) / p i . Clearly, Pij(t) is a Markov chain and (4.38) implies that

t

P ( t ) l = 1,

> 0.

(4.40)

F'urthermore, its Q-matrix is as follows:

4.. = p j.q j i / Pa, a3

4i = P +

Qi.

Thus, in order to having a pinvariant measure, it is necessary that pE

[

- infqi,

01. a

(4.41)

Moreover, (4.39) implies that the Q-matrix = (ijij) is indeed conservative. Next, assume (4.39) and (4.41), then the construction of the minimal process gives us Pij(t) = pein(t).Combining this with (4.40), we see that Q = (Qij) is regular. We have thus proved the following result.

4.7 NOTES

171

Theorem 4.70. Let Q = (qi,) be an irreducible and regular Q-matrix. Then ( p , ) is a p-invariant measure iff (4.39), (4.41) hold and = (qt2,)is regular. The dual chain used above is introduced in terms of an excessive measure, but we can also introduce a dual chain by means of a finite, positive excessive function, in view ol (4.37). The last tedinique even works for more general state space. Refer to Chcn and Stroock (1983).

4.7 Notcs

For a more complet,e theory of Markov chains, refer to Aldous and Fill (1994-), Anderson (1991)) Chung (1967)) Hou (19821, Hou et a1 (1994, ZOOO), Hu (1983, 1985), Wang (1980)) Wang and Yang (1992), Yang (1981). In particular, the complete proofs of the results in Sections 4.3 and 4.4 are included in Anderson’s book. The ergodicity has been studied for much more gcneral state space in the time-discrete case. Refer t o Nummelin (1984), Meyn and Tweedie (1993b) and references within for more details. For closely related results in the timc-continuous cmc for the general state space, refer to Down. Meyn and ‘rweedie (1995)) Meyn and Tweedie (1993a). Section 4.1 is mainly due to Dobrushin (1970). Remark 4.6 was pointed to the author by L. P. Huang. Lemma4.10 is due to Basis (1980). The particular case “c < 0” of Theorem 4.14 was appeared in Basis (1980) and Chen (1986b, 1989b)) based on a time-discrete analogue obtained by Dobriishin(l970). For Markov chains, the special case of Theorem 4.14 (i.e., Theorem 0.11) was obtained by Tweedie (1975, 1981). The present form of Theorem 4.14 seems to be new but quite natural. The proof given here simplifies greatly the original o i m . Theorem 4.9 was presented in Chen (1986b), for which the author was benefited from a conversation with S. W. He. The proof of Theorem 4.9 adopted here was actually contained in Feller (1957). The sufficiency of Theorem 4.24 is due to Foster (1953) and Kendall(l951). The necessity is due to Mertens, Samuel-Cahn and Zarnit (1978). Kendall (1959) introduced the term geometrically ergodic for irreducible Markov CzJ,f3G,where &, < 1. Vere-Jones (1962) chains for which P$’ - 7rJ showed that /3 can be taken to be independent of z and j. Then, Nummclin arid Tweedie (1978) showed that the coefficient C,, can be chosen independent of j . If we want to have a universal coefficient C , then thc gcomctrical ergodicity turns to be the uniform ergodicity as proved by Isamson and Luecke (1978). Theorem 4.28 (2) is due to Numrnelin and ’l’weedie (1978) and Numrnelin and Tuominen (1982). Theorem 4.28 (3) is due to Isaacson and Luecke (1978). For the special case that H = (0). Theorem 4.30 (2) goes back to Kingman (1964) and Theorem 4.30 (3) is due t o Huang and Isaacson (1976). Theorem 4.31 (l), (2) and (3) are due to Foster (1953)) Popov (1977)

I

I
+(U)= (P1-P2)-(Uc). Since

we have V(Pl,P2)3 ?jllP1 - P211Var.Conversely, for given PI and P2, we define a coupling is as follows:

where A is the diagonal in E : A = {(z,x): z E E } . Note that one may ignore lac in the above formula since (Pl-P2)+ and (PI-P2)- have different supports. Then

= (PI - P2)-(Uc)= 1 - (PI A P2)(E).

This gives us V(P1,P2)

< a IJP1- P2

as required.

I

To conclude this section, we study a dual expression of the W-metric. Note that the definition of W ( P l ,P2) is meaningful for any finite measures PI and P2, not necessarily probability measures. In what follows, unless otherwise stated, we will work in this general setup. For simplicity, rewrite Pi = (PI - P2)+ and Pi = (PI - P2)-. Let A1 be the support of Pi and A2 = A"1 which is the support of Pi.

Lemma 5.8. Using the above notations, we have

5.1 MINIMUM LP-METRIC

Proof: Let projcction:

181

P' be a coupling of Pi and Pl. Denote by 7r1 : E x E -r E Ike 7rL(z, y)

7

IC.

Define

E'(C) = P

l A Pd(7dC n D ) ) ,

where D = { (z, x ) : x E E } . We now prove that of PI and Pz. To do so, let B1 E E , then

R(B1 x E ) = Pi(&)

+

= ?;I t2 is a coupling

(Pl A P2)(B1) =

Pl(B1).

Similarly] wc have E(l3 x S,) = Pz(B2). From this, it, follows that

since the support of R' is D.

We now consider the simple case of E being finim. Writ,e E= {xl, . xn}, pij = p ( x c i , z j )and pi') = Pi({ai}), pj2) = P;({zi}).Then pl1)pj2)= 0, i = 1 , 2 , ' - . ,n. Moreover, 1

w ( PY ~),= i n f (

~ ~ , ~ p i j r "cij c i j3:

4

0, z j x : i j = p i(1), z i x ; j =yj( 2 ), 1 Q i , . j 611.).

Recause of the primal-dual relation for the linear programming problem, we have

=sup{ / f d ( r ; - P ; ) : f(Yd-f(Yz)

1

< P ( Y I , Y d , yk E A k , k = 1: 2 .

We have thus obtained the following resiilt. Lemma 5.9. For finite E , we have

wq, Pi)= sup

{Ifw; --

: f(Yl)

-

f(l/z)

< P(Y11Y2)'

Yk

1

fEAkr k = L 2

Theorem 5.10. Let zo E G , ,%(E) = { P E L @ ( E:)/ p ( q x , ) P ( d l c ) Then for P1,Pz E P o ( E ) , W(Pl,f'2) = sup = Sup

}

< ca}

"

I

fd(P1 - Pz) : f E %p(E), L,(f) < 1 ,

(5.5)

1

(5.6)

(s {1

fd(P1

I

1'2)

: f E b s p ( E ) )L(f)

61 ,

where Yip(E)denotes the set of Lipschitz continuous functions on E and L(f) denotes the Lipschitz constant of f E Y i p ( E ) ~

282

5

PROI3AI3ILITY

METKICSAND

COUPLING

METHODS

Proof: Dcxiote by L(P1,P2) the riglit-hand side of ( 5 . 5 ) . Obviously, L satisfies the trianglc incqualit,y. a) By Lemma 5.2, we can choose a coupling of PI and P2 so that W(P1,P2) is achieved at P . Then


as a subset of the separable space C ( Q ) with uniform topology and hence is separable. However, the proof does not mean the separability of ( b 2 z p ( E ) ,j j 1Iu). This is not surprising since the Lipschitx continuity is not a topological concept. We remark that the space bLFip(E) may riot be separuble with respect to the Tipschitz norm. To see this, consider E = R. Given E E R)define fc(z) =: 0 if J: 6 , 1. P,"'"(X, x:! A n En)< Pmin(A, 2,A n ET1),

5.2 MARGINALITY A N D REGULARITY Noting that

189

PFin(X, 2 , A n En) = 0 when IC @ En, we have

P,"'"(X, z,A

n E,)

t ,"lim cc

P,"'"(X, z,A n En)

< Pmin(X, x,A).

On the other hand, by using the previous proof a), replacing A with AnE,, we obtain

lim P,"'"(X, z , A n En) 3 Pmin(X, 2,A),

X

> 0,z E E , A E 8.

,-+a3

We have thus proved the first assertion and then the second one follows.

Theorem 5.19. If both of the marginals are regular j u m p processes, then so is every coupling Markov process. Conversely, if a Markovian coupling is a regular jump process, then so are i t s marginals.

-

Proof: a) Jump condition. Let Pk(t,xk,dyk) and P ( t ;zl, z2;dy,, d y 2 ) be

the marginal and coupled Markov processes respectively. By the marginality for processes, we have &;

3

2 1 ,z2; (21

1 x @2>)

&;z1,x2; ( 2 1 )

x

E2) -

F(t;x,,2,;E1 x (E2\

3 P(t;z1,22;{z,} x E2) - 1 + P(t;z1,22;E1x = z1, h)) .- 1 + P2(t,z2, {z2H.

w,

{.2)))

(4)

If both of the marginals are jump processes, then lim, ,o F(t;z,,x,; {zl)x {z,}) 3 1. This means the Markovian coupling P ( t ) must be a jump process. Conversely, since p@;~1,22;{ x~( 1 34 })

< F(t;z1,z2;{x,} x E2) = P 1 ( 4 ~ 1 , { ~ 1 } ) ,

F(t) is a jump process, then limt ,o Pl(t,zl, {zl}) 3 1 and so Pl(t) is also a jump process. Symmetrically, so is Pz(t). b) Equivalence of total stability. Assume that all the processes concerned are jump processes. Denote by ( q k ( z k ) ) q k ( z k , dyk)) the marginal q-pairs on g k ) , where

if

(a,

z2),Q(x,,x2;dyl, d y 2 ) ) a coupling q-pair on (El x Next, denote by (Q(z,, E2, g ) ,where

190

5

PROBABILITY METRICS AND COUPLING METHODS

We need to show that' @(Z) < 00 for all 53 E El x E2 iff q,(a,) V q 2 ( x 2 )< cm for all z1E El and x2 E E2. Clearly, it suffices to show that, Y l ( 4 vq z ( 4

s"(%.2)

q1h)

+Y

, W

(5.9)

Note that we can not use the conservativity nor iiniqiieness of the proccsses at this step. But Eq. (5.9) follows from a) and Theorem 1.4 immediately. c) Equivulence of corzservatzvity. From now on, we assume that all the q-pairs considered below are totally stable. The problem is that in general, we know that limt+o P ( t ,2 , A)/t = q(x,A ) only for x $ A f 9 rather than A E &. The last assertion holds once the q-pair is conservative (Theorems 1.5 and 1.13). Let the marginal q-pairs are conservative. We first prove that

Moreover, the convergence is uniform in A". To do so, by Lemma 1.7, choose {EP)}? c 9 k such t,hat E p ) T EI, as n -+ 00. We h a x seen from the proof a) that 1-F(t;z1,a2;{q}x

{.2})

< 1 - ~ l ( t , ~ l , { z l +} )1 - P 2 ( t , x z , { ~ , } ) .

Hence {Ein)x Ep))nbl c & and Ein) x E p ) t El x E2 as n + 00. Note that

=: I

+ II +m.

Besides, by the marginality for processes, we have

5.2 MARGINALITY AND REGULARITY

191

Fix n large enough so that zk E E f ) . Since n ( E p ) x E p ) ) E @, we have limt,oI = 0. Next, since the marginal q-pairs are conservative, we have limt,olI ql(zl, (E!"))') q2(z2,( E P ) ) ' ) . Collecting these facts together, we obtain

2


0,

Xk E Ek,

- 1, 2.

Ak E 8,,,k

This is again cieduced by using the comparison theorem. By assumption, the marginal q-pairs are regular and so is the coupling g-pair by Theorem 5.19. Herice

-

independent of r 2 .

Y(X;21:2 2 ; A , x Ez) = Pi (A, xl: A , )

Now thc inverse implication (5.8)*(5,7) 7(5.) follows from t h e uniqueness tbeorem of Laplace transform. H

5.3 Successful Coupling and Ergodicity Based on the results obtained in the last section, we assume for the remainder of this chapter, that all the q-pairs considered are regular. I n this section, we discuss a typical application of the W-metric and coupling methods. Let (X,', X,") ( t 2 0) be the path of a coupling jump process and set,

T = inf{t 3

o : X: = x,"}.

Definition 5.21. A coupling is called successful if I -

IFD"L*"qT< m] = 1, and

XL

f

,--. P " ~ + z [ x= , ~X; for a l l t 2 T ] = 1,

Suppose that

@'z11z:2

(5.12)

Tc2

(5.13)

x1 # x 2 .

is a successful coupling, then

[ I P ( t , X I.), - P(t,X,, . ) [IVar

< 2F-[T

> t ] ---t 0

Furthermore, if the process has a stationary distribution initial distribution p , we have

as t T,

-4

00.

then for any

5 PROBABILITY METRICSAND COUPLING METHODS

196

where P ( t ) is the transition probability function of the original process, By using this way, we prove the ergotlicity of the process. Now, it should be clear that the study of successful couplings is related to the distance of total variation. In general, the succcss of couplings is weaker than the recurrence of the process arid hence weaker than the crgodicity (See Example 5.50 below for instance). For the opposite direction, the next result gives us a reasonable solution.

Theorem 5.22, Let P be a probability kernel on a Polish space ( E , p , $ ) satisfying the following conditions. (1) P ( z , .) E 9'0for all z E l3, where gowas defined in Theorem 5.10. (2) There is a constant c E [O, 1) such that


0 and write t = [t/h]h+ ht,

b) Next, for each t > 0, by Theorem 5.22, there exists uniquely a 7rt E 9 such that 7rt = 7rtPt. The proof will be done once we show that 7rt = 7rs, since then we would have 7r = 7rPt and furthermore

c) The proof of 7rt = 7rs is based on the semigroup property plus the Noting uniqueness of the fixed point of the mapping cpt(p) = pPt, p E 90. that Pt 0 Ps(.t) = Pt+s(%) = Ps 0 'Pt(7rt) = Ps(Tt)r by the uniqueness of 7 r t , we have c p s ( 7 r t ) = 7rt and then 7rt = 7rs by the uniqueness of 7rs. 1 Before moving further, we mention that if we are interested only in whether the two marginals will meet or not, then we can ignore condition (5.13). In this case, we can even allow the two marginals to be different processes. On the other hand, if the two marginals are copies of a single process, it is often easy to modify the coupling process so that (5.13) holds. In this sense, condition (5.13) is not essential. Next, in the study of successful couplings for jump processes, we may and will fix a coupling q-pair (ij(z,,x 2 ) ,Q(z,,z 2 ;dy,, dy,)) and then justify whether the corresponding process is successful or not. Thus, our main task is to find some conditions, depending on the q-pair only, to ensure the success. In this section, we restrict ourselves t o the case that

( E l ,€1) = (E2,€2) = ( E ,8) and

01 = 0 2 .

Denote by ?(t;xl,z,; dy,, d y 2 ) the jump process determined by the coupling q-pair. Then, condition (5.13) becomes

P ( t ;z, z; A ) = 1,

t 2 0,

2

E E,

(5.14)

198

5

PRORARILITY hh3'l'RICS A N D COUPLING

METHODS

where A = { ( x , x ) : x E E } . Equivalently, $(x,x;Ac) = 0 for all z E E . Under (5.14), we have

and so condition (5.12) is now reduced to for all

Definition 5.24. A coupling q-pair is called successful if (5.14) and (5.15) hold.

s(A)

To state our criterion for success, let be the operator corresponding to lhe kcrriel @(x1,x2;dy,, &,)/(A n(xlLz,)) and let afi(A) denote the restriction of IT(x) to E' \ A. Set aii = ~ I - I ( o > .

+

Theorem 5.25. A coupling q-pair (ij(z,, z2),ij(al,x2;dy,, d y 2 ) ) is successful iff the following conditions hold.

(1) @(a, z; A') = 0 for all x E E . (2) q(xlrx2) > o for all ( x l , z2) E (3) T h e equation

has only the trivial solution

IL

E2 \ A.

= 0.

Proof: As we mentioned above: the first two conditions of the lheorern are necessary. Hence, we need only to show that condition (3) is cquivalent to (5.15) irndor the assumptions (1) and (2), Now, mxurne (1) and (2). By using the Laplxcc transform, (5.14) becomes

AF(A;x,x;A)= 1,

x E E, X > 0

(5.17)

and we can rewrite (5.15) as

Thus, we need only to show - that (3) and (5.18) arc equivalcnt. Fix X > 0. Sirice (AP(A;:cl, x2; A) : :cl, :c2 E E ) is the rriiriirrral sulution to the equatiori

5 . 3 SUCCESSFUL

by (5.17) and the localization theorem, we see that ( X s ( X ; x2) is the minimal solution to the equation

f(A)(s,Jz)

199

COUPLING AND ERGODICITY

a) :

51,~ 2 ;

= aii(x)f(X)(s,,s,)+~(zl,~,;A)/(X+~(al,22)),5 1

#

# 52.

Noting that

and using condition (2), we obtain

xF(x;~,, x 2

; ~ ) some f(xl, x2) as

x 1 0,

x1 # z2

and (f(x,, x2) : ( x l , x2) E: E2 \ A) is the minimal solution to

f(%.2)

= nfifl(z,, .2)

+

q

Y

h

3

52:

~ ) / d ( w4, 21 # zz.

Thus, ( h ( z l x2) , := 1- f(rc,. a,>: (xl, x 2 ) C E Z\ A ) is the maximal solution to Eq. (5.16). I Even though we have a general procedure to approximate the maximal solution to Eq. (5.16) (cf. Lemma 2.39). €€owever,such a procedure is somet i m e s not very practical, so we would like to propose some more ef1ective sufficieril conditions for success. Take 0 $ E2 arid set EQ = ( E 2\ A > U { H } , & = ' T { & ~ ~ E ~ \ (0)). A, Define a transition probability on (&I, &) as follnws: I"

-

V~(0,O) = I,

P0(z1,Z2;A)

[ 6 ( ~ ~ , ~\2(0)) ; - ~d-g"(z,;x2;A)IA(@)]/~(X~IZ~),

x1 # x,,

AE

&.

(5.19)

Intuitively, this transition probability is nothing but considering the set A as a single state 0.

Corollary 5.26. Assume t h a t (1) and (2) of Theorem 5.25 hold. Let h E &, h 3 0 and h ( 0 ) = 0. Suppose that there exist constants C > 0 and 0 c < 1 such that Poh C c h on -& (5.20)


C/[(l - c)(l - k ) ] such t h a t

P ~ ( J : ~ , zX ~I ; ~k ) for all (xLT1,z2) E Eo satisfying

h(x,, x2) 6 K .

(5.21)

Then the coupling q-pair is successful. In particular, if (5.21) is satisfied for all z1 # z 2 ,then the same conclusion holds w i t h o u t using condition (5.20).

5 P'KORARILITY METRICSANT) COUPLING METHODS

200

Proof: Consider the process (Z(n,) := (X'(n,),X2(n))),20 defined on a ,..

probalility space ( f 2 , 3 + ,IF) valued in (Eo,&) with transition probability Po. What st" need is to show that I E i r [ ~ ( ~ ) . + o---t] 0 as n t 00. Put

4, = .Ip(n)pe],

2 0. Since 6' is an absorbing state arid In-l = 0 ITL= 0, by (5.20), wc gct lEJ, = IE [I,-,lE(J, I Z ( n - l ) ) ] Jn

= In q q n ) ) ,

*

6 E [I,-lFoh(Z(n - I ) ) ] < IE [ In-1 (C c h ( Z ( n- 1))] = C E I n _ 1 + clEJn-l.

+

(5.22)

On the other hand, by (5.21), we have

EIn

= E [ irn-lFo(Z(n - 1);

E2\ A ) ]

6 IE [ In-lFo(Z(n - 1);E 2 \ A ) ; h ( Z ( n- 1)) < K ] + E [ In-lpo(Z(n - 1); E2 \ A); h ( Z ( n- 1))> K ]

< lclEIn!,_l + K-liEJ,-I.

(5.23)

Combining (5.22) with (5.23), we obtain

Since thc eigenvalucs of the matrix on the right-hand side are smaller than 1, we see that the left-hand side goes to zero as n + 00. 'l'he same proof, even sisnpler, will give us the lad assertion. W Corollary 5.27. Let E be endowed with metric p. Assume that conditions (I) and (2) of Theorem 5.25 hold. Suppose that q ( x ) is locally bounded. (1) If for every such that

T

> 0 there

exists a bounded function p : [O,T]

GO,

~('POP)(2,,~,)$.rl(~1,~,)

O, ~N )' }~,

-

c(F:+lQ

N

1.

n-1

Ph- 9=

- @9),

9 E 80,

k-0

by conditioris (1) and (2) of Theorem 5.25 and condition (5.24), if wc takc g = aT:= p o p , lhen

That is Let n …… and then N …… to get

This proves (5.27). Next, fix r2 > 0 and set F = f o p _ F'rom (1) and (2) of Theorem 5.25 and condition (5.26), it follows that P2F 6 F for all n 2 1. Hence, for p(x1,z2) := T E ( o : T ~ )E, ~ ~ J ~ FA (~ z0 ,( ~ ~) ) ~ ( x , , x ~ Letting ) . n -+m, by (5.2?), we see that


sr,]

(To,, STJ.

5

202

PROBABILITY

METRICSAND

COUPLING

METHODS


z2(T = 00) = 0. This shows that our coupling is successful. b) We now consider the converse case. Let x1 # x2. If @ z 1 ~ z 2 = GO] > 0, then @1,z2 [ T = m] > 0 and so there is nothing to do. Thus, we may and will assume that F ~ . " ~ [ TO,

k=O,l,*..,M.

(3)' There exists a sequence {A,}T of finite subsets of E2\ A so that

A,

1' E2\ A and a

positive function cp such that

(5.29)

ficp60

(5.30)

onE2\n.

Proof: Condition (2)' is clearly necessary. Now, assume that (1) and (2)' hold. Again, consider the subset A as a single absorbing state 8. Setting cp(8) = 0 and using the notation defined by (5.19), we have cp on Ee and


0. By (5.32) and

k = 1, 2.

(d+ W)'P'L(%)

, 0, we obtain a p-optimal measurable coupling Ft(xl, z2;dy,, dy,):

-

-t

x2) 6 P y ( x l , x,)

P

(5.38)

u

for all measurable coupling Pt of Pl(t) and Pz(t). Define

1 -6, G ( y 2 , A ) = --P tn

(?,A \ { z } ) ,

2 E E'1 x Ez,

A E 8 1 x 8 2 , 72 3 1,

By the marginality, we have

6 4 1 ( 4 + 42(Z,),

2 := ( Z 1 , 2 2 ) .

-(n)

(2,El x E2) < w for all X: E El x Ez. Next, since G c ) ( ~ k , A k=) q k ( z k , A k ) , we have G P ) ( x k , -% q k ( x k , Thus, for fixed 5, by Le Cam's Theorem, { G r ) ( x k ,-)}, is uniformly tight and so Hence supn21 G

a)

a).

-(n) is {G ( 5 ,.)}1L31 for fixed 5. Therefore, by Selection Theorem, there exists a transition rncasure y(2, -) such that for each ?, there is ( n ( 2 ) ) c { n } SO that &n(S))(Z, .) -% y(Z, .) as n ( 5 ) 4 00. Hence, we obtain a coiiservative y-pair (q(Z), G(Z, d j j ) ) which corresponds to an operator as follows.

fIf(2) =

1

q(?,dj)f(G) - q(z)f(2),

2E

x E2,

f

b(81

x

82).

From lhe construction, it is readily to see that is a coupling of 01 and 0 2 . Thus, it remains to check the p-optimality of G. To do so, we consider first bounded p. In this case, the proof is easier than Lemma 5.30. Because cp is bounded lower semi-continuous, by Theorem 4.5 and (5.38),replacing .$by Markovian coupling &t) with operator 6: we obtain flp(x1, x2)

5 PROBABILITY METRICSAND COUPLING METHODS

210

(2,,22) E

= fi(p(X1,X2),

El x &.

Finally, we consider general p. Let ( p n = cp A n. Then we have proved that there exists a (p,-optimal coupling operator of fl1 and slz such that for all coupling operator 6 of R1 and 0 2 :

a(n)

Following Lhe argument above, we know that { i j ( n ) ( Z ,d$)}n21 is uniformly tight, and so we can appIy Selection Theorem to find a transition measure Q(?, d y ) , as rz weak li,mj,t;of q(n(5))(ii,djj) as n(2)--t no. Then deGrie ari operator The cp-optimality now follows from Theorem 4.5 and the rnonotone convergence theorem. The construction oE an explicit optirrial coupling is usually not easy. Here we mention two results, taken from Chen (1994a), their proofs are omitted here.

n.

Theorem 5.37. Let p be a translation-invariant metric on Z+ and set uk:= p(k 1) - p ( k ) , k 2 0, where p(k> = p(0, k ) . Then, for birth-death process,

+

(1)

-

n, is p-optimal whenever iz

- il

=: k

7i.k

is decreasing in k. Moreover, we have for

2 1,

f L p ( i 1 , i 2 ) = (ailA M u , + , +(ail v bi&k

-

-

( b i , v a i z ) u b I .- (hi, A a t 2 ) U , - 2 ,

where u-1 = uo. (2) If uk is increasing in k , then fl, is p-optimal. Moreover, for k 2 1,

fl,p(i1,i2) = [(Ui,-%)

f

+ (bi, - bi, ,']w-,i."

- a21

>+

+(hi,

22 - i l

-biz)+]

=:

Uk-1.

Theorem 5.38. Let ( u k )be a positive sequence on Z., Define a distance p(m,n) = uj - Ejsnu j Then, every coupling mentioned in 50.2,

I Ej 0,

x u j / m j , j=O

where M = rn(r) 2 1 is a constant which will be determined later. Then, for ~ ~ ~ ( c p o p ) ( i l+q(il,i2) ,iz) < 0, by (5.55) replacing zhk by cvuk/?-nk, it suffices that

That is

Since u k

1 and m 2

1, it suffices that

Equivalently, ai

+

bi+k

Since the last term is negative for large enough a , by (5.52), this is equivalent

to or

Thus, one may first choose rn = m ( r ) large enough so that the right-hand side becomes posilive for all k T , and then the inequality holds for all k r whenever N is large enough. 1


, d tq) ( d 2 ) , A ) .

q(x(’),AC) = i=l

0

Our conclusion follows from Theorem 5.47. An alternative way, even simpler, to prove the monotonicity of the process is using the following coupling

(5.58) which gives us

Example 5.54. Let us consider a special case of the above example: d = 1, Pll = 1 and F(do is non-explosive. That is, P [ X t E El = 1 for all t 2 0. The following result explains the meaning of time-reversibility. Definition 6.1. A Markov process (Xt)t>o defined on (n,S,IP)with state space ( E , 8 ) is called reversible, if for any finite n 2 1, 0 t l < t 2 < ...


0. Equivalently, .for all f, g e be or e+ and all t > 0 Proof?: The last assertion follows from the monotone class theorem. By the , we haveassumption for all

and

Thus, if the process is reversible, then

So the condition is necesssary. Congives us versely, assume (6.4). This means that P(t) is self-adjoint on L2() , the space of all real square integrable functions with inner product: Let and Then, we habe

satisfy

and

Define

229

6.2 EXISTENCE

Now, the assertion follows by setting ft = I A ~ 1, 6 i Actually, we have also proved the following result.

< n.

I

Corollary 6.4, Every reversible measure of an honest P ( t ) is a stationary distribution.

The above theorem leads to the following Definition 6.5. Let P ( t ,2,d y ) be a sub-Markovian transition function. It is said t o be symmetrizable (resp. reversible) if there exists some a-finite (resp. probability) measure T such that (6.3) holds for all A , B E 8. Equivalently, (6.4) holds for all f , g E &. Similarly, by replacing P ( t , z , d y ) with q(z,d y ) , we can define a symmetrizable (reversible) q-pair.

It is worthy to mention that for f , g E E;, r ( f P ( t ) g )is meaningful but maybe infinite. In the L2-theory, which is the topic of the last two sections of this chapter, in order to avoid infinity, we usiinlly restrict f , g to the domain of P ( t ) . But in the first, six sections of the chapter, we do nal want to be involvcd in the L2-theor;y, so we allow (f,,q) = m. Lemma 6.6. If a jump process P ( t ) is symmetrizable with respect to a measure n,then so does its q-pair

Proof: Since

P ( t ,z,( 3 2 ) ) 2

e - q y

havewe Now, we split the proof into three steps. a)

b) By a), we obtain c0 In general, by a) and b), we have

6.2 Existence The first important result about symmetrizable q-processes is as follows:

230

6 SYMMETRIZABLE

JUMP PROCESSES

Theorem 6.7. The minimal q-process is symmetrizable with respect: to

T ifF so does its g-pair. In particular, for a given symmetrizable q-pair, there always exists a t least one symmetrizable q-process.

Proof: Recall that for f E S+,f1 denotes the kernel f ( x ) d ( s , d y ) . The necessity was proved in Lemma 6.6. 'lb prove the sufficiency, let

Then P r n i n ( A ) . -

C z ,P ( n ) ( X ) .

On the other hand, because

is symmetric in A and B , so P ( l ) ( X )is syrnmetrizable with respect to Suppose that F""j(A) is symmetrizable with respect to T T . That is

T.

By the monotone class theorem, this is equivalent to

Next, since I l ( X ) is symmetrizable with respect to

T ,we

have

Therefore, we obtain

here in the last step, we have used (2.24). Thus, P(")(X)is symmetrizable with respect to 7r for all n 1, and so is its sum Pmin(A).H Having the existence result in mind, we now discuss the uniqueness problem for symmetrizable q-processes. Clearly, we have

Corollary 6.8. If the q-pair (q(z),q(x,dy)) i s symmetrizable with respect t o x and the q-process is unique, then there is only one g-process which is symmetrizable with respect t o T .

6.2 EXISTENCE

23 1

In general, the uniqueness problem is quite hard, the remainder of this section is devoted to introduce a non-trivial sufficient condition. NoOe that

> 0,3A E €such that P(X,5 , A ) # Pmin(X, x,A ) } = {x E E : 3X > 0 such that P(A,z, E ) # Pmin(X, x,E ) ) = { x E E : VX > 0 such that P(X,2,E ) # Pmin(X, x,E ) }

{x E E

: 3X

is clearly &-measurable, hence we may introduce

Definition 6.9. We call y-processes P(X) and Pmin(X) are n-equivalent, if

~ { Ex E

:

3X

> 0, 3 A E &such

t h a t P ( A , x , A ) # Pmin(X, 2,A ) } = 0.

If P'"(X),k = 1 , 2 , are all n-equivalent t o Pmin(X), we call themselves n-equivalent.

Obviously, we have Lemma 6.10. Two n-equivalent q-processes are, or are not symmetrizable with respect t o

7r

simultaneously.

Lcmma 6.11. Let ( q ( z ) , q ( % , d y ) )be symrnetrizable with respect to n and { f x : X > 0) be a consistent family of functions. Then, the equality nfx = 0 holds or not simultaneously for a l l X > 0.

Proof: Let rfp= 0 for some p . Then, by consistency, we have

here in the second to the last step, we have used Theorem 6.7. W Lemma 6.12. Let the q-pair be symmetrizable with respect t o

7r.

( 3 ) If for some X > 0, dirn@A = 0 n-a.e., that is xu = 0 for all u E then dirri%x = 0 n-a.e. for all X > 0. In this case, we write dim@ (2) I f for some X > 0, there is u A E 9?!~ so t h a t

then, there is a consistent family of functions { u x E 0 < IIuxII1 < cc for all X > 0.

%A

:X

> 0)

@A,

0.

so t h a t

6 SYMMETRIZABLE JUMPPROCESSES

232

Proof: a) Let u x f %A for some A. Define up = R(X,p)uA for all p. Then, by Lemma 2.36, U~ E qpfor all p > 0 and { u p : p > 0} is consistent. Hence, the first assertion follows from Lemma 6.11. b) Let uA E q7dr, satisfy 0 < IIuxII1 < 00 for a fixed X > 0. Define the consistent family {uI1: p > 0 ) as. above. Then, tls we have just seen that nuc, > 0 for all p > 0. Moreover, by symmetry7 we obtain IIP1lyp)~xII1 =

(UA,

prnin(pII) 6 ru-yux, 1) = IIuxllI/F
0 such that P ( X , z , E )# Pmin(A,x,E)} c {x E E : 3X > 0 such that zA # 0}, which is a n-null set.

Proposition 6.14. Let 7rzA < 03. Then there exists n-almost honest syrnmetrizable Bg-process ifF the following conditions hold. (1) The q-pair is symmetrizable with respect t o n-. (2) The q-pair is n-almost conservative.

Proof: It will be proved later (Proposition 6.25) that for every Bq-process P(X>, x E E. d( s) = lim X [ 1 - XP(A, 5,E ) ] , A-00

This plus the n-almost honesty implies that n-d = 0. Hence the necessity of the conditions is clear. Next, when conditions (1) and (2) hold and dim% = 0, the existence is trivial. If dim % # 0 but dim % 0, then each Bq-process 7r

is x-equivalent t o ,"'"(A)

and x-almost honest. Finally, if dim%'#O, then

P(A) := P n i n ( X )

+ zApx/[Xpx(E)]

6.4 GENERALREPRESENTATION OF JUMP PROCESSES

233

provides one of the desired process, where p x = x [ z x l ] . What we are going to do for the remainder of this chapter is more or less related to the last two results. -4s we will see later, the uniqueness criterion for this special type of jump processes can not be deduced directly from the one for general jump processes. Actually, it is even not known completely. 6.3 Equivalence of Backward and Forward Kolmogorov Equations 111 this section, we prove that the two Kolrnogorov equations are almost equivalent.

Definition 6.15. W e say t h a t the backward (resp. forward) equation ( B A )(resp. (FA)) holds 7r-a.e. if there is a Ir-null set N so t h a t for all 2 $ N , a l l A E 8 and X > 0, ( B A )(resp. ( F A ) holds. Theorem 6.16. Let P(X) be a n-symmetrizable q-process. Then P(A) satisfies (BA),n-a.e. iff P(A) satisfies (FA),7r-a.e.

Proof: Let P(A) satisfy (Bx),r-a.e. Choose { E n } so that En n(E,) < 00 and supzEE, q(x) 6 n. Then, by Lemma 6.6, we have

E with

Tdking Radon-Nikodym clcrivative, we obtain

The exceptional set depends on X > 0, A and n. However, for fixed z and A, by the resolvent equation, P(A,x,A) is continuous in A, so we can choose an exceptional set depending on A and n only. Furthermore, since ( E , & ) is Polish space, 6 is countably generated, we can choose an exceptional set independent of A and n. Similarly, we can prove the other assertion. 6.4 General Representation of Jump Processes

To study further the uniqueness problem of symmetrizable q-processes, we need to know more details about the exit and entrmce hoiindaries of thc processes. In this section, we first introduce some basic results on Feller's boundary theory. Then, we present a general representation of jump processes. This section can bc regarded as an addition to Chapter 2. For convenience, set %A(C)

= { u E %A

:

u < c}.

6 SYMMETRIZABLE JUMPPROCESSES

234

Lemma 6.17. Let iiA = P"'"(X)d, X > 0 and let tix be the maximal element in %x(l) which can be obtained by the procedure: fix X > 0 and let E1

U(n+l)

n(x)u(4,

n

B 0,

then u(") 1 GA. Next, {fiA}and { f i x } are consistent families. Moreover,

Proof: The approximation scheme for {fix}is obvious. By Lemma 2.39, we know that for each fixed X > 0, zA is the maximal solution to the equation Ic = r I ( X ) X

+ d(X + q ) - l ,

0

< Ic < 1.

On t,he other hand, P"'"(X)cl is the rnininial solution to the same equation, so as their difference, { E x } must be the maximal element in Wx(1).The consistency of {GX} follows from (2.40). Now, as the difference of two consistent families, { U x } must be also consistent. I Recall that for the consistent families { f ~ and } {qA}, we have

Definition 6.18. We call f := 1imx-o fx and 71 := 1imx-o qAthe canonical images of (fx) and {qA} respectively.

In what follows, for the consistent families {fx}, { u x } , {qA}, {qx} and so on, we use f , u , q and f j to denote their canonical images, respectively. Lemma 6.19. For the canonical images, we have the following decompositions:

where LW

F(x, A )

. I

lim Pmin(X, 2,A ) =

A--0

Pmin(t, Z,A)& =

rIn(IA/q)(x). n=O

Proof: Ry consistency, we have

f x = fJL

+ ( P - X)Prnil'(X)fp,

fp

= fx

+ (A - p )

Pmin(X)fp.

Letting X 1 0 in the left equality, we obtain the first desired decomposition o f f . Let,ting p J. 0 in the right equality, we obtain the other decomposition of f. The proof for the decompositions of rl is similar. I

6.4 GENERALREPKESENTATION OF JUMP PROCESSES

235

Next, recall that Eo = {x : d(z) > 0) and set

Lemma 6.20. For the consistent families {G,}, {GA) and their canonical images ii,U we have

(6.7) (6.8)

(6.9) (6.10) (6.11) where uo is the maximal element in

having the property:

Proof: All assertions are clear except the ones about we have u, ii, = xx < 1.

14'.

By Lerrima 6.17,

+

Define

1

uo := lim (1 - a x X-+O

-

iiA) = 1 - u - fi 2 0.

Then, we have (6.11). Using Lemma 6.17, (6.8) and (6.9), we obtain (6.12). To show that uo E %, noting that by the definition of uo,we have XPrn'"(X)l

as x

J. uo

1 0.

If q(x) = 0, then we certainly have uo(x)= lfuO(s).Otherwise, the equation (Bx) also implies that u O ( z )= LIu"(x). Finally, for any solution f to the equation f = XPmin(X)f, 0 < f 6 1, we have

< xPmi"(x)l,

f = XPmin(X)f

arid so f

< lirri~-+OXP"'"(X)l

= uo.

Lemma 6.21.

(1) A family

{f,: X 3 0)

of functions is consistent iff it has the representa-

tion : f,

= prriin

(

W

I

+ A,

6 SYMMETRIZABLE JUMPPROCESSES

236

where w satisfies Pmin(X)w E b€+ for some (and hence for a l l ) A > 0 and (f, E %'A} is a consistent family. Moreover, w and hence fx are determined uniquely by = (XI -

UT

(2) A family {q, : X

n)f,, fx J, 0, > 0 ) of

-+

as X

w

T 00.

measures is consistent iff it has the representa-

tion:

77,

Xfx

2

#emin (A) + q x ,

where K E ?+ satisfies ~ E P ~ ~ E ~ 9 ~+ for ( Xsome ) (and hence for a l l ) X > 0 and { f j x E W,} is a consistent family. Moreover, K. and hence f j x are determined uniquely by

44 = % ( X I

- %4)7

rlx

-1 0, h ( 4 44 +

as

AEmE,,

T 00, n21.

Proof: Assertion (2) follows by combining Lemma 3.3 with Lemma 3.4. In a similar way, we can prove assertion (1). W Corollary 6.22. As A

t 00,

we have

-

P1x

-1 0:

6, 10,

X U , -+ 0; AfiA --t d.

> 0 } be a consistent family of measures. := AqAu0 < 03 independent of X > 0.

Lemma 6.23. Let {q, : X ' 7

Next, let

{fx

f E bd?. Then

> 0) X7,f T

:X

Then

be a consistent family of functions with canonical image as X 00. In particular, for a E Eo, if we set

."X(.)

= Pmin(X, 2,{ a } ) d ( a ) ,

then {u:) has canonical image ua = v fAl

.- Xq,ua ._

'LI"

= v;

vn,

a ) d ( a ) . Moreover,

T

+ q,({u})d(u)

0 0 7

independent of p

> 0.

Proof: Actually, it wus proved in (2.55) that

h f = PVpf +

- ChpfX7

so the second assertion follows. Similarly, by Lemma 6.20, we can prove the first one. Now? it is clear that, u: = 21;

+ ( A - p)q&,

a E Eo.

But Xu: 6 d(a), so the last assertion follows by the dominated convergence theorem. W

6.4 GENERAL REPRESENTATTON O F J U M P PROCESSES

237

Definition 6.24. Let P(X) be a q-process. We say that the backward equation at point II: holds if for all X > 0 and all A , P(X)pA(z)= P I ( X ) P ( X ) I A ( X ) f (A f q ( x ) ) - l I A ( x ) . The ncxt result improves Theorem 1.15.

Proposition 6.25. The backward equation a t z holds iff d ( z ) =: lim X [ 1 - XP(X,2 , E ) ]=: D ( z ) .

x

(6.13)

'M

In general, we have the inequality replacing "=" with " 3 " ,

Proof: a) As we did in Section 1.1, enlarging ( E ,8 ) by a fictitious state A, we obtairi ( E n ,&"a), Define

Clearly, PA(^) is a q-process on ( E n ,&A) with qa = 9 . 1 ~50 . the limit

lim X ( l

x+cQ

AP(X,x,E ) ) = lim A2P,(X, x-+m

x:(A})

exists for all x E E . On the other hand, by Lemma 6.17 and Corollary 6.22 we have

D ( z ) : = lirn X ( l - XP(X,z,E)) 6 lirn Axx(.) x-tm A400 = lim (XG, XG,) = d ( z ) . x-+m

+

This proves the last assertion. b) By a), we have q,(A) = 0 and

for all II: E E and for A in a ring and hence for all A E 8'. Hence, applying the backward Kolmogorov inequaiity to both of P(A)and Pn(X),we obtain

6 SYMMETRIZABLE JUMP PROCESSES

238

where

r

Thus, whenever D ( z ) = d ( z ) , the above every inequality should become equality. In other words, (Bx) at 2 holds. c) Conversely, if (Bx) at x holds, lhen

Lemma 6.26. Let P(X) be a q-process and set u x = 1 - XY(X)l. Then, we have u x >, l l ( X ) u x .

Proof: By using the resolvent equation with some computation, we obtain

Now, the assertion follows by letting We can now state the general representation theorem of q-processes. Theorem 6.27. Every g-process has th e following form

P(X) = Pi'1(X)

+ B(X) + X ( X ) F ( X ) ,

X

> 0,

where

X ( X ) := P"""(X)[dl], X

>0

(6.14)

and F(X) are kernels having the following properties. (1) For foxed x and A , B ( . , z , A ) and b;'(.,z,A)are continuous. (2) For fixed X and J:,

B ( X , z ,.), F(X,J:,*) E .A?+, XB(X)l(z)< 1, XF(X)l(J:) F ( X ) I ( X ) = o for J: 4 E'.

< 1,

(3) For fixed X and A , B(X,+ , A )E e ~ ( l / X )F(X, ? A ) E T&+. (4) If B(X)= 0, then F(X) satisfies the resolvent condition e ,

F ( p ) = F ( A)

+ (A -p )F ( A) pmin (T+ d F ( p ) )= F ( A) { I+ (A ,p( p)} . (,I)

-

(5) The q-condition holds:

lirri A2X(A)F(X)I~ = 0, x-+m

A E &nE,, n 2 1.

6.4 GENERAL REPRESENTATION OF J U M P PROCESSES

239

Furthermore, if P(A>satisfies the backward equation a t x,then F(X,x,*) 5 0. If P ( X ) satisfies the forward equation, then for every 2,B(A,x,.), F(X,x,.) E WA. Finally, P ( X ) is honest iff

XB(A)l = for all X

and

XP(A)l = 1

(6.15)

on E*

0.

Proof: a) Let P(X) be a g-process. Since every q-process satisfies the Kolmogosov inequality a,nd Pmin(A> satisfies ( B x ) ,wc have

Moreover, by Proposition 6.25, for the conservative point 5: d ( s ) = 0, the inequdity becomes equality for all X > 0 and all A . Hence, we may define

F ( X , s , A )=

the left-hand aide of (6.16)/d(x)

if x

(=_

Eo

if x $ E'.

(6.17)

Clca.rly, we have

and F(X,x:.) = 0 whenever P(X)satisfies the backward equation at 2. Moreover, for fixed 2 and A, E(X,5 , A) is continuous in X and so is X(A)F(A)(x,A). From this, msertkm ( I ) follows. b) Next, note that Lemma 2.34 (1) implies

P(A) - Prni"(X) >J P"'"(X)(dF(X)). So we may define B(X) = P(X) - Pmin(A) - Prni"(X)(dF(A)) 2 0.

01' course, we have

Now, P(X>can be written as follows:

P(A)= P"'"(Aj

Preniultiglying ( X I

-..

+ X ( X ) F ( X )+ B(X).

(6.18)

st), we obtain

( X I - ( 1 )P(A)= ( A 1 -fl)P*li" (A) + (A& S2)X(A) F ( A) = I+ ( X I - fl)X (A) F ( A).

6 SYMMETRIZABLE JUMPPROCESS

240

Noting that, by Eq. ( B A ) we , have (XI

-

fl)X(X)J = ( A T

-

st>Pmin(X)(df) = d f .

(6.19)

Conibining the above two facts:, we arrive at ( X I - a)P(A)l(rc)= 13. d(x)F(A)l(x),

X > 0: x E ED.

(6.20)

On the other hand, by Lemma 6.26, we have ( X I - R)P(X)l

< 1+ d / X ,

x > 0.

It follows that X F ( X ) l < 1 for all X > 0. c> It is emy to check that if B(X) = 0, then P(A) satisfies the resolvent equation iff

But

Hence, the above equality becomes

Now, applying (6.19) to this eqrrality, we obtain (4). d) Suppose that P(A) satisfies ( F A ) .Then

B(A)(M - n)IA+ X(X)F(X)(XI- n ) I A = 0,

A E 8 n E,,,

Premultiplying ( X I - a),we obtain d F ( A ) ( X I- ~ ) I = A 0,

A E 8 n En, n 2 1.

Moreover, we also have

B(A)(AI- 0 ) I A = 0,

A

E8

n E,,

2 1.

TI

31

6.4 GENERAL REPRESENTATION OF JUMP PROCESSES

241

This shows that for fixed X and z, B(X),F ( X ) E W,. e) We now prove the last assertion. The sufficiency is obvious. To prove the necessity, let P(X) be honest. By (6.20), we obtain X+Xd(z)F(X)1(2) = (XI-R)XP(X)l(z) = (AI-fl)l(x) = X+d(x), z E EO. Hence, X F ( X ) l = 1 on Eo. Substituting this into the expression of P(X),we obtain

XB(X)l+ X(X)XF(X)l

1 = X P ( X ) l = XP"'"(X)l+ = 1- z , =1

+ XB(X)l+ P"'"(X)d

+ XB(X)l- zx + f i x .

Thus, by Lemma 6.17, we have

XB(X)l = 2,

- 6,

= u,.

f ) Finally, we consider the q-condition. By Corollary 2.28, P(X) is a qprocess iff

But we have proved that B(X)l E

e~(l/X so ), by Lemma 6.17 we have

From this and Corollary 6.22, we obtain


0.

P ( X ) = P"'"(X)

Then

+ u,cpx,

X

>0

(6.21)

is a q-process iff either c p = ~ 0 or c p can ~ be obtained by the following procedure.

(1) Take

K.

E p+ so that K.P"'"(X) E 2f+.

(6.22)

6 SYMMETRIZABLE JUMPPROCESSES

242

(2) Take a consistent family { f j x E Wx} of measures so that

+

(6.23) (6.24)

q x := nPrni"(A) ?jA, 21A

r

:= X f j A i i

< 03, A

21

00.

Set

' 5 = X?jxuoC=

independent of A.

0;)

(6.25)

(3) Next, take constant c so that

ao+ n(u0 + ii)

+ 21 < c.

(6.26)

(4) Finally, take

cpx = q J [ c

+

K(U -

+

(6.27)

Ux) Xfj,U].

Moreover, the q-process P(X) constructed above must satisfy ( B x ) . It satisfies ( F A )iff n = 0. It is honest iff the q-pair is conservative and c = ' 5

+ KUO.

(6.28)

Furthermore, if d i m 9 = 1, then we have constructed all Bq-processes. If d i m 9 = 1 and the q-pair is conservative, then we have indeed constructed all q- processes.

Proof: Let (6.21) hold and c p $0. ~ By Lemma 2.43, we have rn;'

- xqxii =: c

independent of A.

Now, the normal condition gives us

XE,cp,(E)

< ii,

(6.29)

3- i i x .

E 9 ~ (and l) But iiA 00

iix = P"'"(X)d

= CIqX)"d/(X

+ 4) G zx G 1,

0

premultipIying II(A)" in (6.29) and then letting m + 03, we see that (6.29) is equivalent to Xcpx(E) 1. That is X q A ( l - U) 6 c. On the other hand,


1.

TEE,

Lemma 6.29. Let (q(x),q ( x ,dy)) be symmetric with respect t o be a consistent family. Then

7r

and { f ~ }

(1) XxfA is increasing in X. for all X > 0 and nf, < .x for some (equivalently, (2) If moreover f, E for a l l ) X > 0, then qx := 7 r I f ~ l E) WA for all X > 0 and { q x : X > 0) is consistent.

Proof: a) By Lemma 6.11 or is independent of > 0. Next, by consistency and symmetry,

.This proves the first assertion and b) Let

and hence c) Finally, by symmetry, we have

which proves the last assertion.

Then

6 SYMMETRIZABLE JUMPPROCESSES

244

Lemma 6.30. Let ( q ( z )q(z, , d y ) ) be symmetrizable with respect to

T.

Then

(6.30)

Proof: By Lemma 6.29, we need only to show that limx,m XxGA = n d . But XTGx = X ( P m i n ( X ) , d ) = (d, XP"'"(X)l)

== (d, 1 - zx).

So the assertion follows from the consistency of x , ~ Lemma6.21 , plus the monotone convergence theorem. I Lemma 6.31. Suppose that the q-condition given in Theorem 6.27 holds. Then, we have

lim X F ( X , z , A ) = 0,

x+co

J: E

Eo, A E € nE,, n 2 1 .

Proof:

Proposition 6.32. Let the given q-pair (q(x),q(s, d y ) ) be symmetrizable with respect t o 7r. Then, for the existence of an honest .rr-syrnmetrizableq-process, it is necessary that one of the following conditions holds.

(1) n d = 00. (2) limx,, XnU, = 00. (3) nd limx,, XnU,


, (JE,, uozx). Letting X + 0 and then n noting that zx 7 (1 - uo) as X L 0, we obtain (uol 1 - uo) = 0.

-+

03,

6 SYMMETRIZABLE JUMP PROCESSES

246

Lemma 6.34. Let (q(x),q ( x , d y ) ) be reversible with respect t o

T. Define

If U x # 0 and U x # 0, then U x and U x , and hence fjx and jjx are linear independent. The images of fj, and i j x are f j = n [ 4 and ij = 7r[UI],respectively. Moreover, lim Xijx(A)= ~ ( d l ~ ) , A E 8 n En, x+m

n 2 1.

Proof: Since { f i x } and {GA} are consistent, the consistency of f j x and i j x follow from Lemma 6.29. If clUx c2Ux = 0 holds for some X > 0, then c2(XI - R)G, = 0 since f i x E %A. But from Gx # 0 we know that d # 0. Hence (XI - R)U, = d # 0.

+

This implies that c2 = 0. Next, from fix # 0, it follows that el = 0. Finally, since the property whether f i x and fix are zero or not is independent of A, we have thus proved the linear independence. As for the last three equalities, simply apply Lemma 6.20 and Lemma 6.21. W

Lemma 6.35. Let fjx and i j x be the same as above. Set

Then, we have

(1) w:~

some wab as X

(2) wi2 = wil for all X

co,a, b = 1 , Z . Moreover,

> 0 and so w12 = w21.

Proof: The first assertion is a straightforward consequence of Lemma 6.34 and Lemma 6.23. By using these lemmas again, we get

and so wi2=

wil

for all X

> 0. W

6.5 EXISTENCE OF HONESTREVERSIBLE JUMP PROCESSES247

Write

W(X) = (w?

4 1 )

WZl

For a given matrix

R(X)=

:;(

rY)

>

r i2 we write

P(X) = P"'"(X)

p).

+ (fix, ux)R(X)

(6.31)

Lemma 6.36. Let (q(z),q(z,d y ) ) be reversible with respect t o 7r. Then P(X) is a reversible q-process if the following conditions hold.

(1) R(X) is non-negative and symmetric. (2) R(X)W(X)l < 1, x > 0. (3) (4) Moreover, the process is honest iff the equality in (2) holds.

Proof: Clearly, condition (1) implies that P(X) 2 0. Write

Then Xp",(E) = Xri1q,((E)

+ X.;%jx(E),

2

= 1,2.

On the other hand, by (6.11)) we have

here we have used Lemma 6.33: qxuo = (uO,u.x)

< (uO,q)6 (uO,1 - uO)= 0,

X

> 0,

(6.32)

and ljxuo = 0, X > 0. Thus, condition (2) implies the normal condition X P ( X ) l < 1 and furthermore the process is honest iff (2) becomes equality by (6.15).

6 SYMMETRIZABLE JUMP PROCESSES

248

To show that P(X) satisfies the resolvent equation, by consistency of U , and G, plus some computations, it suffices that

cpt - 9; + (A - p)cp:P(pu>= 0 ,

k = 1,2.

This can be deduced by using consistency of 5j,, ?jA and condition (3). Clearly, (4) follows from (3). Finally, since limx,oo XU, = 0 and limA+-ooXii, = d , we have lim X ( I A - X P ( X ) I A )

A-+m

= lim X ( I A - X P m i n ( X ) I ~) - (0, d ) x-+m

So the q-condition is also satisfied. 1

Lemma 6.37. The hypotheses and the notations are the same as in the previous lemma. Suppose in addition that 0 < .rrd < lim XnuA< co. A-+m

Then, we can choose a matrix R(X) so that P(X) defined by (6.31) is an honest reversible q-process.

Proof: Because nd

> 0, so Eo # 0 and GA # 0. Moreover, lim Xnii, 2 .ird > 0, A-+m

we also have ii,

# 0.

Now, take hl = lim Xnii,,

hz = x d ,

A-+W

r1 = (hl - h2)/h?,

r2

= l/h1.

Clearly, r1 2 0, r2 > 0. Next, take = [rl

+ ri(w22- w?~)]/A,

ri2 = ril = T~ [ 1 - r 2 ( d 2- w;')>I/A,

ri2 = r,"(wll - wil)/A, where

n = [ 1- r2(w12 - w,12 ) ]2 + T;(wll

- w:1)(w22

- w?) - r1(w1l

- w?)

and w $ ~and wab(a,b = 1,2) are defined by Lemma 6.35. It is easy to check that wab(a,b = 1 , 2 ) are finite and so are w:~. The computation of R(X) comes from Lemma 6.36 (3). Letting p i 00, we obtain R(X) = R ( I - (W - w ( X ) ) R ) l-.

From this, it is easy to show that R(X) constructed here satisfies all conditioris oC Lenrxna, 6.36 and moreover, the process is honest. We can now state our main criterion.

6.6 UNIQUENESS CRITERIA

249

Theorem 6.38. Let ( q ( x ) , q ( z , d y ) )be reversible with respect t o T . Then there exists a n-almost honest and .rr-reversible q-process iff one of conditions (1)-(3) of Proposition 6.32 holds. Proof: The necessity g v a proved in Proposition 6.32. To prove the sufficiency, we discuss the problem according to different cases. a) Let ( q ( z )q, ( 5 , dy)) be conservative. By Proposition 6.28,

> 0, i.e., fjA(3) > 0, then

is a q-process, it is clearly reversible. When niix it is actually honest. Ot'herwise, T [ 2E

E

:

XPrni"(Xj(;c)l < 11 6 T [ U X # 01 = 0,

and so Prnin(X) is a n-almost honest q-process. b) Let dim@ 0 and Eo 0. That is, the q-pair is n-almost zero-exit and n-almost conservative. In this case, we still have r [ z E E : XPrn'"(X,2,E)< 13 =7r[zx # O ]

c) Let Eo 0 and 0 < limx,,XnuA constructed in a) satisfies 7r

[. E 15

: AP(A, x,E )

< 7r[ux# O ]

< 00.

+ T [ i i A # 01

=o.

In this case, the q-process

< 11 6 T [ d # 01 4-7r [d = 0, A P ( X ) l < I ] -n[zA-uX #O,d-O] =

piA# 0, a = 01 = 0.

d) Let 0 < nd 6 limx+m X.rr?IA < m. In this case, an honest reversible q-process was constructed by Lemma6.37. e) Let n d = 00 or lirnAds X7riiA = 00. By Lemma 6.30, we have limxdm Xnzx = 00. Then, qX = ijx := 7r1tzx-l] satisfies Proposition 3.5 (4),so by Proposition 3.5, P ( x ) = Pmin(X) -k Z A ~ ~ A / ( A ~ ) A ( E ) ) is an honest q-process, which is clearly reversible.

I

6.6 Uniqueness Criteria

In this section, we fix a symmetrizable q-pair (q(z),q(z,dy)) with a symmetrizing measure T , which is also fixed. The uniqueness studied here is naturally in the sense of ;.r-equivalence.

6 SYMMETRIZABLE JUMPPROCESSES

250

Proposition 6.39. If %'AnL'(n) = (01, then there is only one n-syrnmetrizable Dq-process.

Proof: Suppose that the Bq-processes are not unique. Then, by Theorcrn 6.27, we should have a Q-process with the form P(X) = P"'"(X) 3. B(X) with B(X,.,A) E for all X > 0 and A E 8 and nB(Xo)l > 0 for some Xo > 0. Choose large enough n so that n ( B ( X o ) I ~ > , ) 0. But then for this n, we also have


0.

Proposition 6.40. If

(1) %A n L1(n) = {O} (2) n d

and

< oc,

hold, then there is only one n-symmetrizable q-process.

Proof: Let P(X) be an arbitrary .Ir-symmetrizable q-process. Then by Theorem 6.27, it has the representation:

+

+

P(X) = Prni"(X) B(X) X ( X ) F ( X ) . But by condition (1) and the proof of the last proposition, we see that B(X,+, l?) = 0, T-ax. On the other hand, by Lerrirria 6.31, we have lim XF(X).[A

A+cQ

= 0,

AE

&nnE,,

n 2 1.

Hence lim

X-CO

( x P ~ ~ ~ ( x ) I E~, X , F(X)I)

= lim X-+m

(I,yn, XPmin(X)dXF(X)l) (by symmetry of P"'"(X>)

= lim (1, XP"'"(X)dXF(X)I,gn) (by symmetry of P(X) and Pmi"(X X-+m

=

lim (XP"'"(X)l, ~ X I ; I ( X ) I E ~ ) (by symmetry of P"'"(X))

x-+m

= 0,

72

2 1.

6.6 UNIQUENESS CRITERIA

251

Here in the last step, we have used condition (2) and the dominated convergence theorem. Moreover, by Theorem 6.27, F ( p ) = F(X)( I + (A - p ) P ( p ) ) . Hence

Combining these facts together, we obtain

and so

Thus, there exists a n-null set N such that

Note that the jump condition gives us

By Theorem 1.14, it follows that limA,m the other hand, by symmetry,

XPmin(X)(I~cdXF(X)l) = 0. On

(1, P"'"(X)IN) = ( I N , P"'"(X)l) = 0.

So we can choose a n-null set fi so that Pmin(X, z,N ) = 0 for all z @ Therefore, we have actually obtained

N.

From this, it follows that

Now, by Proposition 6.25, we see that P(X,z, .) satisfies the backward equaTo see this, we can either use tion. Therefore, P(X) is equivalent to Prn1"(X), condition (1) and Proposition 6.39 to claim the uniqueness of Bq-processes or use Theorem 6.27 to claim that F ( X ) l = 0, n-a.e.

6 SYMMETRIZABLE JUMPPROCESSES

252

Theorem 6.41. There exists only one 7r-symmetrizable q-process if the following three conditions all hold.

(1) ( q ( x ) q(x, , dy)) is symmetrizable with respect t o 7 r . (2) %A n L ~ ( T=) {o}. (3) Either inf{Pmin(X)l(z): z E E o } > 0 or 7rd < 00. Proof: By Proposition 6.40, we need only consider the q-process having the form P(X) = Prni"(X) X ( X ) F ( X )

+

and consider only the case that inf{Pmin(X)l(x) : x E E o } > 0. By Thecrem 6.27 (4), we have

Hence

Letting p

--f

00,

we get

XF(X)IA 2 F(X)QIA,

X

> 0, A E € n En, n 3 1.

Thus on € n En, n >, 1.

U(X) := F(X)(XI- 0) 2 0

(6.33)

Of course, for every X > 0 and z, U(X,x,.) can be extended uniquely as an element in p+.We use the same notation U(X) to denote the extension. By Lemma 2.34 (a), it follows that

Set

V(X) = F ( X ) - U(X)Pmin(X)2 0,

X

> 0.

(6.34)

Then by (6.33) and (6.34) we obtain, for each xo E E , that

V(X)(XI- Q ) I A ( z ~=) 0,

X

> 0, A E € n En,n 2 1.

That is

v ( X ) I ~ ( z o=) V(X)Q((X

+ q)-'I~)(z~),

X

> 0, A E 8

(6.35)

6.6 UNIQUENESS CRITERIA

253

by the monotone class theorem. On the other hand, by symmetry, T ( A )= 0 implies that

2 2

1 1

E ( ~ ~ ) P " ' " z, ( X(,x } ) d ( z ) F ( Xz, , A) n(dx)Pmin(X, x,{z})d(x)V(X, z,A),

A

> 0.

This shows that for all zo out of a .rr-null subset of Eo, say E;, we have

V(X,z,;)

0.

Define

Then

and so by (6.35), we have

That is

Combining the above facts with condition

(a), we see that

Note that

(IAPmin (A) (dIq),V(X)1) < (Pmin (A) (dIE;), V(A) 1) 6 (Prni"(A)(dIq),F ( A ) l ) 6 (p"'"(X)(dIE;), =

1)/A

( I q ,dPmi"(X)l)/A = 0,

X

> 0, A

E 8.

6 SYMMETRIZABLE JUMPPROCESSES

254

We arrive at

P( A) = Prnitl (A) 3- Fin (A) [ d I ]u (A) Prrlin (A).

(6.36)

Now, we start to iise the condition

Since

12X F ( X ) l = XU(A)Pmin(X)l2 c(X)XU(X)l, we have XU(X)l

< c(X)-l

for X > 0 and so


0.

Besides, by Lemma 6.17, we have P ' " ( X ) d 6 1,

x > 0,

5

E E.

We have got everything we need; the next step is copy the proof of T h e e rem 3.26 starting from (3.42). 1 'Yhe Iast result of this section is a uniqiieness criterion for reversible qprocesses. Theorem 6.42. The reversible q-process is unique iff the following three conditions all hold.

(1) (q(x),~ ( zd ,y ) ) is reversible with respect: t o 7 ~ . (2) nd < 00. (3) @A = (0). Moreover, if (I) holds but (2) or (3) does not hold, then there are infinitely many reversible q-processes.

Proof: The sufficiency follows from Proposition 6.40. Clearly, condition (1) is necessary. We now assume (1). a) If .rrd = oc, or lirnA*m X.irii, = 30, then in the last step of the proof of Theorem 6.38, replacing P(A) with

wc obtain infinitely many reversible y-processes with parameter c 2 0. 17) If (3) does not hold and limh-oa AntiA < cx3. Then by Lemma 6.11, Lemma 6.29 arid Lemma 3.4, wc have xTT1LA> U for all X > 0 and

255

6.7 BASICDIRICHLET FORM Setting I E =0,

qx = ~ [ i i x I ] , X > 0, A E 8

in Propositiori 6.28, we obtain w = lim X(fiA,'lL) x4w

< x+w lim XnG,
0, we have actually obtained infinitely many reversible q-processes. W

6.7 Basic Dirichlet Form

In this and the next sections, we use the L2-theory, more precisely, the Dirichlet forms, to study the symmetrizable jump processes. Thus, d l processes considered in these sections are symmetrizable. Besides, we fix a a-finite measure T on ( E ,8')and denote by the space of all square integrable functions on ( E ,8, T ) with inner product and norm

respectively. Occa++mally, we also consider the P ( 7 r ) ( p E [l,031) space, in that case, the LP-norm is denoted by 11 . [ I p . The following result is our shrtirig point, which is even more than what we need at the moment.

Lemma 6,43. Let P (t,2 , dy) be a jump process with an excessive measure T (ie, T 2 n P ( t ) for all t 2 0). Then {P(t)}t>o:

can be extended uniquely t o P ( n ) ( p E [ l , ~ as ) )a strongly continuous contraction semigroup.

Proof: Note that

256

6 SYMMETRIZABLE JUMPPROCESSES

By Holder inequality and the excessive property, we have

=/7r(W(z)IP(1-

W , z ,{.I)).

Combining these facts with the jump condition and using the dominated convergence theorem, we obtain

This proves the strong continuity. As for the contraction, we have

From now on, we discuss only the case that p = 2. The above extension is again denoted by {P(t)}t20.According to the ordinary semigroup theory, { P ( t ) } t pdeduces an infinitesimal generator L as follows. If for some g E

L2(T ),

( P ( t ) f - f ) / t - g -+ o in ~ ~ ( 7 r ) as t j.0, then we define Lf = g . Such functions f consist of the domain of L, denoted by !3(L). It turns out that the generator L is self-adjoint (densely defined) on L2(7r). However, the domain g ( L ) is usually quite poor. For instance, even for countable E , the simple indicator I { i ) may not belong the domain of L. Thus, it is worthy to find a weaker version of the generator L. Actually, one such version is at hand, that is the weakly infinitesimal generator Instead of the strong convergence, we consider

E.

for all h E L2(7r). Such g E L2(7r)consist of the weak domain g ( L ) . But this version is not very useful since in the L2-context the weak and the strong topologies are quite closed each other.

6.7 BASICDIRICHLET FORM

257

Since the space I.’((n) is fixed, we also call the semigroup {P(t)}t>osymmetric on L2(7r).Recall that in thc finitc dimensional case, cvcry quadratic form can be represented as a sum of squares under an nppreciatcd orthonorma1 basis. The representation simplifies greatly the classification of the quadratic surfaces. As a gcneralization to the infinite dimensional case, a spectral family { E , : a E R} on L 2 ( r )is used instead of the orthonormal basis. More precisely, every self-adjoint non-negative definite operator -L has uniquely a spectral representation as follows:

-L = Furthermore, for every continuous

io lo adE,.

‘p :

P(-L) :=

[0,oo) + [0, oo),the operator

@I

P(.>dE*

is again non-negative definite. In particular, we have the strongly continuous contraction semigroup ( P ( t )= exp(tl)) and the resolvent (P(A) = ( X I L)-l : A > 0), respectively, as follows.

Note that in this general setup, the semigroup P ( t ) (resp. the resolvent P(A)) is not necessarily sub-Markovian: i.e., 0 6 P ( t ) f 1 for f E L2(7r with 0 6 f 6 1. But in our particular situation, P ( t ) is nothing but the semigroup induced by the transition function P ( t , x , d y ) and P(A) is its Laplace transform. Set


o.A remaining question is whether it is a q-process or not. The answer is affirmative. To show t.his, we need more work. Recall that the Eq. ( B A plays ) a crucial role in Ihe study of jump processes. In the present situation, an analogue is as follows:

Lemma 6.54. Let (q(s),q(x,dy)) be a symmetric q-pair and {P(A): X be a symmetric resolvent on L2(;7).Suppose that q(x, *) 0}

(6.42)

and Eq. (6.41) holds. Then ( B x ) holds, r-a.e.

Proof: Fix f = I A E L 2 ( r ) and let g E L2(q(z)r(dx))n L2(n). Then g E !3(D*)and

By (6.41), we have

6.8 REGULARITY, EXTENSION AND

265

UNIQUENESS

Clearly, this equality can be extended to for all g E Ly((.ir).RepIacing g with (A g)-lg, we get

+

(P(X).f,9 ) = (QP(X).f, (X+q)-lg> = ( W ~ ) ~ ( ~ ) f + ( ~ + 9Ld -g 1ffL7~ ( T Thus, for each A ,

P(X)IA2 0, XP(X)l < 1 and P(X)IA = f l ( X ) Y ( X ) I A (A q)-'JA

+

(6.43)

hold almost everywhere. The exceptional set depends on A and A. Since ( E ,8 ) is separable, we can choose an exceptional set depending on X only. On the other hand, the strongly continuous means that ( P ( X ) )determined by a dense set of X > 0. By using this arid (6.42), the exceptional set can be chosen so that it is also indeprndcnt of X > 0. Now, dmotc the r-null set by N . Then (6.43) holds for all 2 $ N, X > 0 and A E 8 , II

Theorem 6.55. Suppose that (E,S) is locally compact and (6,421 holds. Then ( D * ,9 ( D * ) )determines uniquely a symmetric q-process on L'(K). Proof: By the construction of resolvent (P*(X))from the Dirichlet form (cf. Fukusbirna (1980), Theorem 1.3.1), we have

P"L"4)

c 9(D*),

D*

D*(P*(X)f, 9) + W * ( X ) f9 ,) = ( f d , f E L2(7rr7),g E 9 ( D * ) .

So the proof of Lemma 6.54 tells us

P*( X) IA ( Z2) 0, XP*(X)l(s) < 1 and p * ( A ) I A ( z )= n ( A ) P * ( X ) I A ( x ) (A q ( T ) ) - ' I A ( x ) , 2 $2 N,X > 0 , A E 8,

+ +

(6.44)

where N is a .rr-null set independent of X and A. Letting X ---f 00 in the equation in (6.44), we see that the jump condition holds for all II: $ AT. Using the equation again, it is easy to check that q-condition also holds for all z $ N. Finally, the local compactness enables us to find a kernel P(X,Z, d y ) SO that P(X,. , A ) = P(X)IA. W 6.8 Regularity, Extension and Uniqueness

We have seen from the last section that the basic Dirichlet form corresponds to a symmetric g-process on L2(.rr). On the other hand, we have proved that there is thc minimal symmetric q-process, which then correIt is interesting sponds to a Dirichlet form, denoted by (Dmin,9(Dmin)). that the two Dirichlet forms constructed by different approaches are not the same in general.

266

6 SYMMETRIZABLE JUMP PROCESSES

Proposition 6.56. Let (g(x), g(x, d y ) ) be conservative and symmetric on L2(7r).Suppose that dim% < 1. 7T

(1) If dim% 0 or dim%#O but r z x = 00, then there is only one Dirichlet form: Dm'" = D". 7T (2) If dim %#0 and r z x < 00, then there are infinitely many Dirichlet forms having the same representation:

where c 2 0. Moreover, if c1 2 cz 2 0, then

Dmin(f, f) 2 DC1( f ,f ) 2 DCZ(f,f ) 2 D * ( f ,f ) for all f E g ( ~ c~a (~ D "'l )~ c g(D'2) ~ ) c g(O*). (3) In any cases, there is a t most one honest symmetric g-process.

Proof: The case of dim% 0 was treated in Proposition 6.13, We now as7r sume that dim W f O , equivalently, K Z > ~ 0. By Theorem 2.45, all g-processes

{7lx : X > 0} is a corisistent family of measures and c 2 0 is a constant. Now, by symmetry, we have

where

7 r ( z x I ~ ) q x (= B )K ( X ~ I B ) ~ ~ ( Afor ) all A,B E 8 In particular, we have qx I= 0 if 7rzx = 00 since qx is a finite measure and zx 6 1. Otherwise, we should have qx = c ( X ) n [ z x I ]where , c(X) is a constant depending on X only. Then, by using consistency of {vx : X > 0} and { n [ z x I ]: X > 0}, it follows that c(X) is actually independent of X > 0. Now, the Dirichlet form can be computed by using the formula

P ( f , f ) = lim X(f - XPC(X)f,f). x+cc

Finally, it is obvious that the symmetric q-process corresponding to Dc is honest iff c = 0. For birth-death processes, we have dim % 1. The Q-matrix is symmetrizable with respect to


, Then the symmetric q-process is unique iff the basic Dirichlet form is regular, i.e., 9(Dmin)= B ( D * ) . Equivalently, (6.49) and (6.50) hold. In this case, the unique Dirichlet form is just the basic one.

One may ask whether the basic Dirichlet form being the maximum in any cases, The answer is negative. Example 6.63. Take E = {0,1,2,.. . } and yiJ = 0 for all i 00. Then P$"(A) = & / ( A 4%). On the other hand,

+

PZj(X) :=

Pgin(x)+ (A + Y i Y) (i X + Y j )

x

# j , xi l/q, < (6.51)

is also a Q-process. The latter one is honest and syminetrizable with respect t o the measureri = l/qi. For this Q-matrix, we have 9(Dmin)= B ( D * ) c @ ( D )but 9(D4) # 9(D).

Proof: Obviously, we have

Since

270

6 SYMMETRIZABLE JUMPPROCESSES

we have Dmin= D* and 9 ( D m i n= ) 9 ( D * ) . On the other hand,

LeL f E g(D*), then

Thus B ( D * ) C g ( D ) . But 1 E g ( D ) ! 1 # g ( D * ) and so g ( D * ) # 9 ( D ) .

In contrast to the smallest extension; there is a Krein extension. It is known that the Krein extension is so large that it is sometimes even not sub-Markovian (cf, Fukushima (1980), Theorem 2.3.2). But we would like to point out here that the Krcin extension is so~netirriesnot large enough since t.here are some Dirichlet forms with larger domain than the Krein's. To show this, let us return to the above exitlnpk. In this c u e : it, is easy to check tha.t .N, = ( 0 ) (see Fukushima (1980), (2.3.5) for notat.ion), hence DrDin = D' = D K ( t h e Krein extension): and so 9 ( D K ) c 9 ( D ) but g ' ( D K )# 9(D) The main reason is that the symmetric q-process (Pij(X))given by (6.51) does not sat'isfy the equation ( B A ) . More precisely, the non-conservative part of the Martin boundary is not considered by the Krein extension. 6,9 Notes The study of symmetrizable Markov chains was begun by Kolmogorov (1936a). He considered the Markov chains with finite state space and discrete time-parameters. Now, there are several books on this subject, both for Markov chains and for general Markov processes. See E'ukushima (19801, Kelly (1979), Silverstein (1976) and Qian and Hou (1979), for instance. A lot of progress has been made in t.he past years, refer to Fukushima, Oshima and Takeda (1994)! Ma and Rijckner (1992).

6.9 NOTES

271

This chapter presents a general theory for symmetrizable jump processes. However, the results on this class of jump processes are still not as complete as those on the general jump processes. The uniqueness criterion for honest reversible q-processes remains open. This problem is certainly important in practice. The first three sections are taken from Chen (1980), where the reversible cases were studied, some generalization was due to Zheng (1981, 1983a). But the second sufficient condition of Proposition 6.13 was appeared in Qian (1978) . For Section 6.3, we refer to Yang (1981). Theorem 6.21 is due to Reuter (1957). The proof adopted here is taken from Zheng (1981). For Markov chains, the existence criterion for honest reversible Q-processes was obtained by Hou, Guo and Chen (1979) in the conservative case, and then in the non-conservative case by Chen and Zhang (1984). The generalization, including Lemma 6.33, was appeared in Chen (1986b). The uniqueness for reversible Q-processes in the conservative case was also appeared in Hou, Guo and Chen (1979). Some results for non-conservative case were obtained by Hou and Chen (1980). Theorem 6.42 was appeared in Chen (1986b), for which, the author benefited from Y. L. Dai, R. H. Ouyan and H. J. Zhang. Theorem 6.41 are taken from Chen (1991~). Theorem 6.50 is well known in the theory of Dirichlet forms (cf. Fukushima (1980)) under the local compactness hypothesis, which is removed by Bouleau and Hirsch (1986). For Markov chains, Proposition 6.59 was proved by Silverstein (1974), Proposition 6.56 was appeared in Hou and Chen (1980). The last result shows that even the general q-processes are not unique but the honest and symmetric one can still be unique. This indicates the difficulty to obtaining a criterion for honest and symmetric q-processes. The remainder of Section 6.7 and 6.8 are taken from Chen (1991~). One topic, not included in the book, in the study of jump processes is bo construct all q-processes for a given q-pair. For the symmetrizable case, refer to Chen (1982)) Chen (198613) and Zheng (1981, 1983a).

Chapter 7

Field Theory In the study of symmetrizahle Q-processes, a new question arises, i.e., for a given 9-matrix, how can wc justify it is symrnetrizsble or not. If so, haw can we find out a symmetrizing measure. Next, for a given regular syinnietriznkle Q-matrix, it is easy to know whether the corresponding Qprocess is positive recurrent or not, but it is riot so emy to understand the riull recurrence or transience of the process. These questions are discussed in this chapter. Our main tool is the field theory which will be also used in Chapters 11 and 14. 7.1 Field Theory Let E be an arbitrary set, T be an index set, and let a: T x E x E [-00,+00] satisfying

+ R :=

Hypotheses 7.1.

# 2 , 0 < a ( t ,x,2) < 00. (2) Co-zero property. a ( t , x ? Z )= 0 iff a(t,Z., x) = 0. (1) For each t E T and x, 2 E E , x

Definition 7.2. For given 5 , 2 E E , x # 5,5 is called reachable directly from x at time t , denoted by x 2, if a ( t , x,Z)> 0; 2 is called reachable , 2 ).,. . , dn)in E from z at time t , denoted by x A it, if there are d l ) d such t h a t Then, L ( t ) := (x,

z i1 x(1) i I.

*

J:(2)

4, . . . 4, .I4

t

f

5,

,dn),2 ) is called a path from x to i?

at time t.

Let d ( t ):. { a ( t ,x.2 ) : x,3: E E>(t E T).Denote by 2 ( t ) the collection of patlls of 0 iff so does for all t > 0. Hence 2 ( t ) , D ( t ) and E,(t) are all independent of t > 0. Definition 7.9. Let

Pij(t) = 0 u Pji(t)= 0

for all

t 2 0, i , j E E.

Then the field corresponding t o P ( t ) is called a chain field, also denoted by

P(t>.

7 FIELD THEORY

276

Proposition 7.10. A chain field P ( t ) is a potential field (equivalently, P ( t ) is symmetrizable) iff it is conservative field. Then, all potential functions are given by

v&) = -log A,(l)

+ ce(tf,

i E Ef,e

E

D,

where (Xi(1)) is defined by (7.4) and c e ( l E D ) is an arbitrary function on T .

Proof: Without loss of generality, assume that P ( t ) is irreducible. Choose a reference point A and reference path and define ?ai(t), ?ia(t), &(t) by (7.2) and (7.4) respectively. Since P ( t ) is a potential field, (&(t))defined by (7.4) is a symmetrizing function: Ai(t)Pij(t)= Xj(t)Pji(t)

for all t 2 0, i and j.

Then, by CK-equation and induction, we obtain

Xi(t)Pij(rnt)= Aj(t)Pji(mt)

for all m 3 1.

Noting that AA(t) f 1 and that P ( t ) is irreducible, we have

and so

xi(m/n>= A ~ ( I=>F A ~ ( I ) / & ( I )

for all rn, n.

Combining these two facts, it follows that

A&) Next, for any potential

= A&))

t E T.

V ( t )= { G ( t ) } of P ( t ) ,we have

7.2 Lattice Field

In the last section, the study on the symmetrizability is reduced to the potentiality of a field. Then, due to the path-independence, to justify whether a given field d ( t ) is a potential field or not, it suffices to check the work done by the field along the minimal closed paths. This idea should be clear intuitively and it is actually the key point why such an elementary tool being helpful. The main purpose of this section is to illustrate this idea, further applications will be presented in the next part (Chapters 11 and 14).

7.2 LATTICE FIELD

277

Lemma 7.11. Let S be a finite set and denote by E = { O , l } S product space of ( 0 , l ) . Define

the usual

Take

Then d = ( u ( x , 2 ) : x,2 E E ) is a potential field iff the following quadrilateral condition holds:

where

Proof: For each u S, define a transform (u): E E as follows:

x[(u) = ux, Then for every path L =

(2 =

x

E , u f s.

E

x(*),dr), a

4

, d")), we have

In this proof, we also use x c ( u l )...C(un> to denote the path L , where z denotes the starting point aiid ((u,) denotes the section di-') 3 ~ ( 2 ) . By the path-independence, we have

Using these notations, we may rewrite (7.5) as

Clearly w(z((u) .. ' ((?in>) =0

for any closed path L

= (x = do), x(I),

- . - da)= x).

(7.9)

7 FIELD THEORY

278

Since every closed path consists of an even number of edges, n = 2m for some positive integer m. We use induction on m. When m = 1, then u1 = u2 and hence (7.9) follows from (7.8). Suppose that (7.9) holds for n = 2(m - 1). Then for n = 2m, there is a k such that 2 k n,uk = u1 and ue # u1 for 2 L < k - 1. Applying (7.6) and (7.7), we obtain

<
and ( E ,P:3,x i ) which are equivalent in the sense that C-l aZJ/a&& C , then they are transient or recurrent simultaneously.


0. (7.19)

Cdi a

7.4 TRANSIENCE OF SYMMETRIZABLE MARKOVCHAINS Since

xi IdiJ
0 or = 0. i) a,i > 0. In this case, we claim that c,i = 0 and hence vi = 0. Otherwise, using k=a, j=iork=i, j = a Ekj

=

{ O7

Ckj,

otherwise

instead of (cij), we would obtain a different C E H ( F ) , anti-symmetric and having a smaller norm llCll < [[ cF 11. This is impossible.

7.4 TRANSIENCE OF SYMMETRIZABLE MARKOV CHAINS

295

ii) aai = 0. Define tiai

= hia = c

> 0,

6kj

= akj

in other cases,

and the correspondent l?, fi, g ( F ) and EF. It is easy to check that l l E ~ l l< J J c F and J J so 2;, = cF by the uniqueness of c F . Therefore, by i), we still have cai = C,i = 0 and furthermore vi = 0. e ) Let T = inf{n 2 0 : X , 4 F * } and set rO0

l

o

o

In this paragraph, we prove that (7.24) (7.25) First, we have

Next, by (7.22) and (7.23), we have

3

Hence

Furthermore,

jEF'

296

7 FIELD THEORY

This shows that (w - V ) ( X n A ~is )a Pi-martingale. But w = V = 0 on E \ F', so we have v = V. To prove (7.25)) fix k E F* and set d i j = a i j ( g i k - g j ~ ) Since .

it follows that

On the other hand, noting that

dij

+ d j i = 0, we obtain

which is just what we required. f ) Finally, we use (7.24) and (7.25) to prove (7.21). First, by (7.251, we have

Next, by (7.22) and (7.23)) we have

From this and (7.24), it follows that

which is what we required.

I

Example 7.24. The simple random walk in three-dimension is transient.

7.4 TRANSIENCE OF SYMMETRIZABLE MAR.KOV CHAINS

297

Proof: Take

and set

Ni = { j E z3: li - j l = 1). Clearly, ( u ; ~satisfies ) condition (1) of Theorem 7.23. Since for j E Ni,we have

Condition (3) holds. We now check condition (2). Let M be the unit cube centered at the origin and with length 2m of edges. Then

i€ M j € Ni

i € d M jENi\M

On the other hand, let (el, e2, e3) be the coordinate bases in R3 and set

z [ f+ ( ~+ 3

g(E) =

j(.

Eei)

- Eei) - :!f(x)],

i=l

where f is harmonic in a neighbor of z. Then, by the Taylor expansion formula, 1 g(E) = -(Af)E2 O(E4)= 0 ( E 4 ) . 2

+

Using these two estimates and the fact that A(l/lxl) = 0 (x # 0), it is easy to justify condition (2) of Theorem 7.23. Alternative proof: The proof consists of two steps. Firstly, construct a tree T3 with finite resistance. We begin the construction with drawing 3 branches from the root, denoted by t l l , t 1 2 and t 1 3 respectively. Each branch has 1 ohm. At the n-th step, each branch t n - l , k (1 6 k < 3n-1) constructed at the ( n - 1)-step splits into 3 branches and each of the new branch has 2n- 1 ohm. Suppose that there are no overlap. By the equivalence principle, it is easy to see that

n=O

Secondly, embed the tree T 3 into Z3. Put the root of the tree at the origin. At each step, let the rays go ahead along the tree coordinate directions.

3 FIELDTHEORY

298

+

Whenever a ray intersect the plane x+ y z = 2n - 1 for some n, it splits into three rays. If two rays pass though each other, we simply let them “bounce” (disconnect). Since this tree is a subgraph of Z3 with finite resistance, we have proved the transience of the simple random walk in Z3. 7.5 Random Walk on Lattice Fractals

To illustrate some applications of thc results obtained in the last two sections, we discuss in this section the simple random walk on two lattice fractals, Sierpinski gasket and Sierpinski carpet. As an example of the construction of general lattice fractals, we explain how to construct the lattice Sierpinski gasket. To do so, let us recall the usual construction of Sierpinski gasket in Rd(d 2 2). Starting from d 1 points ( 0 = q,,x l ?. - , xd} with length Ix,- xj I = 2. The d+ 1 vertices with d(d l ) / 2 (adjacent) edges make a d-dimensional polyhedron, denoted by H y ) . Next, let xzJbe the midpoint of the line jointing x2 and x3, write x,, = 2%:and let be the graph with vertices {xZ3: 0 6 i, j d + 1) and edges between x , and ~ xtk, x3e, 0 k, t! d, k # j , !# i. Clearly, H i d )consists of d 1 d-dimensional polyhedrons, with each pair sharing exactly onc vertex. , }nro. The Repeating this procedure, we obtain a decreasing sequence { H (4

+

+

+


0.


0 such that 1 for every E E [-6, oo),and so

As a function of

E,

F f has a minimum 0 at d -FFf(O) = dE

s

E

+ ~f E b&+

= 0: therefore

62fdp = 0.

This implies that p ( q f ) = p ( Q f ) for each f E b€‘+ and hence for each f E rd?+, which is then equivalent to the claim that p is a stationary distribution of ( P ( t ) }(Theorem 4.17).

8 LARGEDEVIATIONS

314

Conversely, assume that p is a stationary distribution of { P ( t ) } .F'rom the last part of the proof of Donsker and Varadhan (1975) [Lemma 2.51, it follows that

On the other hand, the proof of Donsker and Varadhan (1975) [Lemma 3.11 also works in our case, hence we have

Combining these two facts together, we obtain the required assertion. Now, we would like to give a clearer expression for I ( p ) . For this we need a hypothesis.

Hypothesis 8.10 ( H I ) . There exist a 0-finite measure X and an 6' x 8measurable function q(z,y) such that q ( z , d y ) = q(z,y)X(dy)

for all

z,y E E .

Theorem 8.11. Under ( H I ) ,for each p E 9 ( q ) satisfying p 0, X-a.e. on E+. Again, for

I/

Under

0" made in the last step, we prove that there is a probability measure a such that a 0 on E and aq < 00. For this, choose a sequence of disjoint sets {&}? C € such that 0 < A(&) < 00 and B, = E . Set

Cf"

Then f E d oand

+ f(~)-']-',

Now, if we take &(z)= [q(z)

a ( A )=

then the measure a defined by

A

E

8

< aq
O on E (i.e,, E = Ed-). (2) ( E , @ )is locally compact, ~ ( x ) is locally bounded and (3) ( H I ) and ( H z )hold. Then, for every p

0 and the assumption in (2): we have

Set

Because f is increasing, the assertion for case ( 2 ) follows from Theorem 8.28 and

c) Choose 0

< d < D < 00 and define

8.4 NOTES Then fo = d,

fi

tD

as i

329

co. Finally

8.4 Notes

For Section 8.1, refer to Stroock (1984). Theorem 8.17 is taken from Chen (1990a). The remainder of Sections 8.2 and 8.3 are taken from Chen and Lu (1990a,b) with some improvements. See Jain (1990) for recent progrcss on the lower estimates. Two recent books on large deviations are Demba and Zeitouni (1993), Deuschel and Stroock (1989).

Chapter 9

Spectral Gap In this chapter, we study the exponential L2-convergence. We prove in Section 9.1 that the exponential convergence rate for a Markov process can be described by the L2-spectral gap of its generator or Dirichlet form. In the reversible case, we prove an equivalence, or even coincidence, of this convergence and the exponentially ergodic convergence. Two main results for estimating the spectral gap by couplings and two approximating procedure for jump processes are introduced in Section 9.2. Section 9.3 is devoted to birth-death processes, for which we have a complete solution to the topic studied in this chapter. In the last two sections, another powerful tool-a —a generalized Cheeger’s method is studied. It works in a very general setup but the resulting estimates are usually less explicit. 9.1 G e n e r a l Case: a n Equivalence Let {P(t)}t20 be a positive, strongly continuous, Markovian contraction semigroup (i.e., P ( t ) l = 1 for all t 3 0) on L 2 ( n )with stationary distribution T,not necessarily symmetric. Again, denote by L and .9(L), respectively, the infinitesimal generator and its domain induced by {P(t)}t2o. We say that P ( t ) converges exponentially i n the L2(n)-norm [I 11 if there is a positive E so that for all f E L 2 ( r ) ,

On the other hand, since 1 E .9(L) and L1 = 0, the vector 1is an eigenvector of L with eigenvalue 0. One may seek for the next-to-largest eigenvalue of the self-adjoint (resp. the symmetrized part of a non-self-adjoint) generator L. That is, to seek for the infimum of the spectrum of -L restricted to the orthogonal complement space of 1: {f E L2(i.) : r ( f )= 0} n B ( L ) . This leads us to define the s p e c t r a l gap of L: gap(L) = inf {

-

( L f , f ) : f E g ( L ) , r(f)= 0 and llfll

= 1).

(9.2)

Our first step is to show that (9.1) and (9.2) are closely linked. To do so, let D ( f ) denote the limit

330

9.1 GENERAL CASE:

AN

EQUIVALENCE

331

provided the limit exists, here the equality is due to the fact that T is a stationary distribution of P ( t ) . Such functions f E L2(.rr)with D ( f ) < 00 consist of the domain a ( D ) of D. Clearly, 9 ( L ) C Q ( D ) . In the case of P ( t ) being symmetric on L'(T), D ( f ) coincides with D ( f ,f) introduced in Section 6.7. This explains why we choose the notations D(f) and , 9 ( D ) . Next, define gap(D) = inf { D ( f ): f E g ( D ) , ~ ( f=) 0 and

llfll

= l}.

By definition, it is clear that the condition f E a ( D ) in the last line becomes unnecessary in the symmetric case. Because we are working in the regular case, which is natural since the process is assumed to be ergodic. Thus, it would be much better if we could use gap(D) instead of gap(L). Finally, define u(t) = -sup { log IlP(t)fll : ~ ( f=) 0 and l l f l l = l}. Since

( I P (+~s)fll 6 e-'(t)\\P(s)fII 6

e-'(t)-u(s)

Ilf 11,

by the contractivity and semigroup properties, it follows that u(-)is superadditive and u(0) = 0. Hence, the limit

u := lim - = inf tJ0

t

t>O

4

-

(9.3)

t

is well defined.

Theorem 9.1. We have u = gap(D) = gap(L). Proof: Clearly, gap(D) ,< gap(L) since D ( f ) = (-Lf,f) on 9 ( L ) . To prove u >, gap(L), simply use the fact: d = 2 ( P ( t ) f , L P ( t ) f ) G -2gap(L)IIP(t)fl12, -IIP(t)fl12 dt

f E W )7 , 0 ) = 0 and llfll = 1, and the denseness of 9 ( L ) in L 2 ( r ) . Finally, let f E 9 ( D ) with and llfll = 1, then 1 D ( f ) = lim -(f tL0 t

-~

~ ( f=) 0

1 ( t ) ff) , 2 lim -(I - ePut) = u. tl0 t

Hence gap(D) 2 u. I At the moment, except the fact 9 ( L ) c 9(D), the knowledge of 9 ( D ) is quite limited. However, it will be clear later, whenever we have a little more information about the generator, the domain 9 ( D ) is actually manageable. The next obvious facts will be helpful for our further study.

9 SPECTRALGAP

332

Lemma 9.2.

-

(1) D ( f ) 3 0 for all f E .9(D). 9 ( D ) is dense i n L2(?r). (2) fEQ(D)*g:=cf + d ~ g ( Da)n d D ( g ) = c 2 D ( f )for all c, d c R . (3) fl 9 E .9(D) a n d f + 9 D ( f + 9 ) G W ( f+)&?>I.

.w)

Bcfore moving further, we would like to mention that the non-symmetric o given as case can be often reduced to the symmetric one. Let { P ( i ) } ~be above. Define a bilinear form D ( f ,9) on Q ( L ) by D ( f , 9) : - (Lf, 9 ) . Next, define a dual (or adjoint,) 6 of D as follows: 6(f,g) = D ( g , j ) , f , g E g ( L ) and set = (D+ 6) /2. Then -is is a syrnmctric form and so we have a norm 2 -f'j I( 1 0: Ilfli, - (f) llf1I2. Naturally, one can extend the domain of D to ~ ~ by g . Therefore, we the completion of 9 ( L ) with respect to 11. ( 1 denoted can define

+

gap(l;)) = inf

{o(f) : f E 3, ~ ( f=) 0 and l l f l l

= l}.

Noting that the last inequality D ( f ) 3 CT in the proof of Theorem 9.1 holds first for f E Q ( L ) and then for f E g , moreover, since

-

1

D ( f )=- Z(D(f)

+ m)) = Wf),

f E 9,

the proof of Theorem 9.1 shows lhat o = gap@) = gap(L). Hence we have proved the following result. Corollary 9.3.

(T

= gap(D) = gap())

= gap(L).

One often uses 9, rather than 9 ( D ) defined above, as the domain of D. This is because in general the bilinear form may not be regular. To use the semi-group corresponding to (0,Q), the theory of non-symmetric Dirichlet, forms should be helpful. Refer to Ma and Rockner (1992) (especially, Theorems 2.15 arid 2.18) for details. We will return to this problem more cnrcfully (see Theorem 9.12 below)+ mlPnext, coinparison result is useful t o compare the spectral gap of a complex process wit,h a simpler one. It shows, for hlarkov chains for instance, that a local pcrturbation does not interfere the L2-exponential convergence. The proof of thc thcorcm is straightforward and hence is omitted. I -

Theorem 9 . 4 (Comparison Theorem). Let ( D ,9 ( D ) ) a n d (6,9( be two forms with g ( D ) c defined on L2(,rr)a n d L2(?) respectively. Suppose t h a t there exist constants A , B and C such t h a t

9(D),

+ CD(f), ~ . P ( ~ ) / [ A B+ c ~ ~ P ( E ) ]

O ( f )6 A W ) , hen gap(D)

Varn(f>6 BVar;(f)

f

E .9t(D)*

9.1GENERAL CASE:A N EQUIVALENCE

333

For tensor product P ( t ) of { P k ( t ) }with generator Lk (it corresponds to a Markov process with independent components), the spectral gap takes a simple form. Denote by L the generator of P ( t ) on the product space with product measure

7r

=

nk

Tk.

Theorem 9.5 (Additive Theorem). gap(L) = infk gap(Lk). Proof: Choosing the functions to be depending only on the kth coordinate in the definition, it follows that gap(L) < gap(Lk). To prove the inverse assertion, it suffices to consider two components. Then, by induction the assertion holds for finitely many components and a limiting procedure gives it for infinitely marly components, since the functions depending on finitely many components are denso in Lz(x).~ e ft satisfy n(f)= o and l l f l l = 1. Exprcss f as f(x,y> = h ( z ,y) h,l (x) hz(y), whew J h,(.,y)dn, = 0 for a,e. y, J h,(s,.)&r2 = 0 for a.e. z, hldxr = 0 and / h2dn2 = 0. Then h, h,l and hz fire orthogonal in L 2 ( , r r ) an'd so are P ( t ) h ,Pl(t)h,l and F$(t)hz.The conclusion now follows from

+

+

IIW)fl12 = lIP(t)hl12+ IIP(tP1112 + IIP(t)h,2II2 = IlP(t)h.1l2+ l l ~ l ( ~112) + h Ilf3(t)hzIl2 and IlP(t)hll = IIPl(t)P2(t)h,ll6 e-l(gap(Ll)+gap(Lz))((hll. Now, we study the spectral gap for jump processes. staled, we consder only the regular j u m p processes.

Unless otherwise

As we have seen from Lemma 6.43, for a given q-process P ( t ,z, dy) with stationary distribution x,P ( t ) f := J P ( t ,-,dy)f(y), f E bd? can be extended to L2(T)uniquely as a non-negative, strongly continuous contraction semigroup. Thus, the above discussions are applicable to the present situation. , dy)), as we did befbre, define Next, for a given q-pair ( q ( x ) q(x, Xq(&

dy) = n(dz)q(z,dY),

For f E g ( D ) , by Fatou's lemma and Theorem 1.14 (4), it follows that 3o

> ~ ( f=)lim 1/ ' x ( c i z ) ( ~ ( t )-[ ff(z>lz)(z>~ * ( f ) . 110 2t

Therefore, we have 9 ( D ) C 9 ( D * ) . Note that here 7rq may not be symmetric. Choose and fix a sequence {En} c d? such that En 1 E and SUP,^^,^ q(x) = mpz< 00, n 2 1. Assume that mTL = n for simplicity. Define

X

=

md&={g:=cf+d:

{f

E

Lo3(x): {f # 0)

EX, c , d ~ R } .

c

some En}

9 SPECTRAL GAP

334 Lemma 9.6. XL c a ( D ) .

Proof: By the regularity of q-pair, we have

On the other hand, since

where

it follows that

Note that 7r is an invariant probability of { P ( t ) } t 2 0 ,7 r ( q f 2 ) = .rr(Qf2). Combining the above facts together, we arrive at

for f E X . Now, the conclusion follows from Lemma 9.2. I This simple result already enables us to get an upper bound for gap(D).

Lemma 9.7. We have 1 gap(D) 6 2 inf{k(K) :

o < 7r(K)< 1, I K

EX},

(9.5)

where

In particular, for Markov chains, gap(D) 6 infk q k / ( 1 - nk).

+

Proof: For 1, E A' with 0 < r ( K ) < 1, set f = CIK d. Choose c and d E R such that ~ ( f = ) 0 and llfll = 1. The first assertion follows from Lemma 9.6 by computing D * ( f ) . Then the second assertion follows by taking K = { k } plus the invariance of 7r.

9.1 GENERAL CASE: AN EQUIVALENCE

Lemma 9.8. Let

Choose and define

335

such that

Proof: It suffices to prove the first asserbion. Clearly, gn E g ( 0 ) and gn ---f g in L1(7r). Hence sup, .lr(gn) < 0s. Next, since

n ( s 3 z 7r(!121E,,) we have limn

+

,oo~ ( g , ) ~ / r ( g ; )=

..(.g2)

as 72

= 00

+ 00,

0. Therefore, by definition,

Definition 9.9. We call %? c 9 ( D * ) a core of D * , if '3 is dense in .9(D*) with respect t o the norm 11 lip: l l f l l ~ * = D * ( f ) llf1I2.

+

Lemma 9.10. If n(q) < 00, then Xj is a core of D*,

Proof: We need only to show that X is a core of D*. Let f E g ( D * ) , Replacing J with JrrL= (-m) V ( J A m) if necessary, we nay assume that f is bounded. Next, set fn = JIB,. Then

Theorem 9.11. If 3' is a core of D*,then gap(D) = inf{D*(f) : n(f)= 0 and = inf{D*(f) :

f E

IlfJl

= l}

XL, n(f)= 0 and l l f l l

= l}.

Proof; Since g ( 0 )- 9 ( D * ) ,D and D* coincide on 9 ( D ) ,by Theorem 9.1, it sufices to show that for each f E g(L)*) wilh n(1)= 0 and l l f l l = 1, there

9 SPECTRAL GAP

336

exists (fn} c Xi with 7i-(fm) = 0 and I I f n I I = 1 such that D * ( f n )--t D * ( f ) . Because X is a core of D*, by Lemma 9.2, we can find a sequence { f n } c ;U, with 7 r ( f n > = 0 and /lfnll = 1 such that fn -+ f in Ij . IIp-iiorm. Note that D * ( f n- f ) -+ 0 implies that D * ( f n )is bounded in n. On the other hand, by Schwarz inequality,

and so D * ( f r L-+ ) D * ( f ) as 71 oc. The assertion now foIlows. I We now specify the above result to Markov chains. Let Q = (qiJ : i, j E E) be an irreducible regular Q-matrix. Suppose that the Q-process ( P Z J ( t has )) a stationary distribution T . Define --j

It is easy to check t,hat, (&) is a conservative Q-matrix with stationary measure ( x i ) : so is (f&j). By Theorem 1.70, the Q-matrix ( & j ) is regular. Moreover, ( i j i j ) is reversible with respective to the same probability measure (7ri), and so is t,he corresponding minimal Q-process.

Theorem 9.12. Let Q = ( q z j : i , j E E ) be an irreducible regular Q-matrix. Suppose that the Q-process (Pij(t))has a stationary distribution x and the Q-matrix ( 4 ; j ) defined above is regular. Then

fi)'

:

f E XI,,~ ( f=) 0 and

1

=1 .

Proof: By Theorem 9.11, we need only to prove that X is a core of D*. From Corollary 6.62, we. know that (gtJ) is regular iff X is a core of D.But

We claim that the cores of 75 and D* are the same. The above result is a special case of Corollary 9.3, which says that gap(L) = gap((L z ) / 2 ) , where 2 is the adjoint operator of L. Before moving further, uFe now consider two irreversible Markov chains.

+

337

9.1 GENERAL CASE: AN EQUIVALENCE

Example 9.13. Take -1/2

1/2

0

Q=(! f f l i / 4 . Thus, in

Then gap(Q) = 1 but the eigenvalues of Q are 0, -5/4 general, Spec( (Q

+ 0)/2)

Example 9.14. Let

# Re. Spec(Q).

qk = qk,k-I

and qij = 0 for all other j

# i.

=

I, qok

Then gap(Q)


, (1 - f i ) - 2 &p}


o.

Definition 9.17. We say that g is an eigenfuction of L corresponding t o X in weak sense if g satisfies the eigen-equation Lg = -Xg pointwise. Note that the eigenfunction defined above may not belong to L2(7r).

Theorem 9.18. Let ( E , p ) be a metric space and let { X t } t > be ~ a reversible Markov process with operator L. Denote by g the eigenfunction corresponding t o XI := gap(L) in weak sense. Next, let ( X t , x ) be the coupled process, starting from ( I C , ~ ) ,with coupling operator and let y : E x E --+ [O,m) satisfy y(z,y) = 0 iff IC = y. Suppose that

z

(1) 9 E %J(L)l

(2) Y E %J(Z), (3) LY(Z,Y) 6 -QY(.,Y) for all 2 # Y, (4) g is Lipschitz with respect t o y in the sense that

Then, we have gap(L) = XI

2 a.

Proof: By conditions (2) and (3), we have

Next, by condition (1) and the definition of g , rt

rt

is a P"-martingale with respect to the natural flow of a-algebras { 9 t } t > 0 . In particular, giz) = E"[ g ( X t ) XI g ( X s ) d s ] .Because of the coupling property,

+

9 SPECTRAL GAP

342

Thus, we obtain

Define the coupling time T = inf {t>0: Xt = Yt}. Thenn

Noting that g is not a constant, we have cg,y = 0. Divifding both sides by Noting that g is not a constant, we have cg,y = 0. Divifding both sides by cg,y, we obtain

for all t. This implies that A 1 2 a as required. H Condition (3) in Theorem 9.18 is essential, for which one needs l o cliovse not. only a good coupling but; also a good distance. This leads to the study on optimal couplings discussed ia Chapter 5. The other conditions in Theorem 9.18 can often be relaxed or avoided by using a localizing procedure (cf. Theorems 9.20 or 9.22 below). The next weaker result is useful! it is actually relat,ed to the strong ergodicity of the process. Theorem 9.19. Let { X t } t 2 0 , L , X I and 9 be the same as in the last theorem. Suppose that (1) 9 E

%&)I

( 2 ) s'lP,+y

19(4 - 9(?)1/)1 < m.

-

Then for every coupling I W Y , we have gap(L) = XI 2 ( supzpy@,")-I,

Proof: Set $(x,y) = g(z) - g(y). By the martingale formulation as we did in the last proof, we have

9.2 COUPLING

AND DIS'rANCE

METHOD

343

Hence

-

sup,.,IE"~YT
=

c

2 I, we have

1

A - d o , A1 + 2[ d o ,0 ) - P ( 0 : j)l.

bk M k ,

k> 1

< 0.

Thus, it suffices to shcw that J := Ck21 b k [ p ( k , j )- p ( 0 , j ) l j = 1, J blgl f z k 2 2 b k [ p ( k ,1) - p(0, I)] = h g l -k z k 2 2 bkgk j 2 2, J =

bklbk k>l

$1

- (.Qj -

291)l

When 0. When

348

9 SPECTRALGAP

since j 2 2 and qo > b l . This example is also called star model since its graphic structure. There is a center at 0, from which there exists only one bond to each k. Thus, the geometric distance can be defined a s follows. First, we have p(0,k). Then, for i, j # 0, i # j , p ( i , j ) = p ( i , 0) p ( 0 , j). Based on this structure, a “path method” goes as follows. Replace q k o = 1/2 by a more general qkO = q k ( k 2 1). By Theorems 2.40 or 2.47, it is easy to check that the process is always unique. Clearly, the stationary distribution is ni = p i / Z , i 2 0 where PO = 1, pi = bi/qi for i 2 1 and 2 = c k 2 1 p k . For every f E L2(7r) with n(f)= 0 and l l f l l = 1, we have

+

Thus, gap(D) 2 2 - l i n f ~ l q i . When qi = 1/2, we return to the original model but the estimate of the last method is not sharp. Clearly, the problem does not come from the second inequality but the first one, which is based on the graphic distance. 9.3 Birt h-Deat h Processes

In this section, we study mainly the spectral gap for birth-death processes. This is crucial since the birth-death processes are often used as a tool to compare with some general (even infinite dimensional) process. For instance, the lower bounds obtained in this section is available for general reversible Markov chains on Z+ with qi,i+l > 0 and qi,i-1 > 0, provided the deduced birth-death Q-matrix is regular. Recall that for a positive recurrent birth-death process with birth rate bi > 0 (i 2 0) and death rate ai > 0 (i 2 l),the reversible measure (xi)is the following:

9.3 BIRTH-DEATH PROCESSES

349

Let Y be the set of all positive sequences (vi : i 3 0) and define

R&)

+ bi - aJvi-1 - bZ+lVi = Aa(i)- Ab(i)+ ai (1 + b i + l ( l - 4, = ai+1

a0

where A a ( i ) = ai+l

:=o,

v-1 := 1,

- ai,Ab(i)= bi+l

- bi.

i 3 0,

(9.12)

Next, let

W = ( { W i } i > o : wi is strictly increasing in i and ~ ( w2 )0},

-

W = { {wi}i>O : there exists k : 1 < k

< 00 so that wi = W i A k , w is

strictly increasing in [0, k] and n(w)= 0},

(9.13) Note that @is simply a modification of W . Hence, only two notations W and I ( w ) are essential here. The main results can be collected a s follows.

Theorem 9.25. Consider the ergodic birth-death process as above. We have the following conclusions.

(I) Dzflerence form of the variational formula f i r the lower bound (9.14)

(2) Summation f o r m of the variational formula for the lower bound: (9.15) (3) Summation f o r m of the variational formula for the upper bound gap(D) = inf sup I ~ ( w ) - ’ . w€?w i>o

(9.16)

Moreover, the supremum in (9.14) and (9.15), and the infimum in (9.16) can all be attained. 1 (4) Explicit bounds and explicit criterion: Define ‘po =0, ‘pi=

xjGip1

2 1, Q ~ ( Y ) = C p ~ c ~ ~ for? ~p-2 L j1,Q: = [ , i + k ] ~ ~ > ~ S(y) = supn2l Qn(r),6 = 6(1) and 6’= SUP,^^ Cyl: Q>vjn),where ~ ( is~ a probability 1 measure on {0,1,.. , k - l} with density vjk)= ( p $ ~ j > - ’ / Z ((and ~ ) Z(’”)is the normalizing constant). Then we have 6 < 6’< 26 and for2

+

In particular, gap(D) > 0 iff 6

< 00.

9 SPECTRAL GAP

350

In view of (9.12) and (9.13), one sees that the difference form (9.14) and the summation form (9.15) are quite different but there is indeed a correspondence between (iii) and (wi> (Idernma9.30). As we will see soon each of them has its own advantage. By exchanging “sup” and “inf”, we obtain (9.16) from (9.15), ignoring the difference of W and %? Thus, (9.15) and (9.16) are dual one to the other. All these formulas are completely different from the classical variational formula: gap(D) = inf {D(f) : ~ ( f=) 0 and

Ilfll

(9.17)

= l}.

Because of the uniqueness assumpt.ion of the process and Corollary 6.62, we do not need the condition ‘’f E B ( D ) ” in the last formula. Clearly, for each t,est sequence ( u i ) , from (9.14) we obtain a lower bound of gap(D). In particular, according to a classification of the test sequence, we obtain t.he following result.

Corollary 9.26. Define Au(i) = ai+l

+

(1) Let vi = ~ [ ll/(i

- ai,a0 := 0,

Ab(i) = 1

3 - bi. ~

+ c)], r 2 1 , c E 10,m]. Then

Aa.(i)- Ab(i)

+1 [ai - bi+i]

if r = 1.

Z+C

Then

(2)

A u ( i ) - Ab(i) - cI

ai

i - 1+ c2 -.” c1

2

-1- ca

In particular, w e have Examples 9.27. The exact gaps for nine examples are listed in Table 9.1 in the next page. Here, for the sixth example, we need a restriction: l / k < a/b k/(k - 1 ) 2 ( k 2 2). The test sequence used in the last example is ui = - 1)/4 for even i and ‘ui = (*+ 1)/4 for odd i .


i

gaP(D) 2 (.A - d q ) 2 / ( c ~ c 22) 1/(4C?c2). p ~ j6 clpiai and C3>i p j a j 6 c2piui for all i 2 1, then

(A

gap(D) 2 - dG)2/c1 3 1/(4cIc2). (4) If ai = bi and i Y C3.>% . l / a j < c(y) for some y 2 1 and all i 3 1, then gap(D) 3 max {(4c(l))-', b:-'c(y)-l(l - y-')}. Parts (1)-(3) of the corollary are all sharp, but not part (4)) for the constant rates b, = b and ai = a. Conversely, part (4) is sharp for ai = bi = i2(i 2 1) but parts (1)-(3) fail. The proof of the corollary is delayed for a while. The simplest example to show the power of Theorem 9.25 (4) is the following one. Example 9.29. Let ai = bi = i Y for all .i 3 1 and bo is ergodic i f f y > 1 and gap(D) > 0 i f f y 3 2.

> 0.

Then the process

Partial proof of Theorem 9.25: Step 1. First, we prove in the next lemma the equivalence of (9.14) and (9.15). Lemma 9.30. When vi = ui+l/ui for positive (ui), rewrite

Ri(v)as Ri(u).

(1) Given w E W with ~ ( w=) 0, set

Then we have &(u) = li(w)-lfor all i 2 0. (2) Given positive (ui : i 2 0) such that inf+oRi(u)

limn+m bnpnttn

>

0. Then c :=

< 00. Set

wi = aiui-1 - biui + c / ( Z- l), i >, 0, where u-1 = 1. Then we have wi+' > wi for all i 2 0, w E L ' ( n ) , ~ ( w= ) c / Z ( Z - 1) 3 0 , C i 2 1 p i w i > 0 and Ii(w)-'>, Ri(u) for all i 2 0.

9 . 3 BIRTH-DEATH PROCESSES

353

Proof: a) It follows from the definition of (ui) that we obtain Since

Since ~ ( w=)0, we have

Thus On the other hand, by (9.18), we have

We have thus proved part (1)of the lemma. b) For part (2), we first prove the existence of the limit limn-+mbnpnun. To do so, take wi = aiui-1 - biu; bouo (i 2 0 ) for a moment. Note that

+

(wi+l - w i ) / u i

( a i + l ~i bi+lUi+l - aiui-1 > 0, i > 0.

+ biui)/ui (9.19)

= Ri(U)

We have wi < wi+l for all i 2 0. On the other hand, since wo = 0, we see that w1 > 0 and so wi > 0 for all i 2 1. Thus n

n

n

kj-lpj-luj-1 0I + ~ A , A ~ ~ , A C . A ( ~ , ~ ) I ( ~ -- I (Af I) ~( Z) () y ) l . Then

362

9 SPECTRALGAP

When J i s symmetric, by assumption, we have

When J is not symmetric, we have


o. Then, by symmetry of .rr(dz)p(t,5 , d y ) (Theorem 6.7), we have

i(f2/g,

Here, we have used the fact that a + l / a 2 2 for all a > 0. Hence j.Ptj) ( f , f - P t f ) . From the spectral theory, it is standard that the righthand side increases to D ( f ) as t 0 (cf. Section 6.7). Thus, it remains to show that the left-hand side converges to ( f 2 / l j , -06) as t L O . This can be done by using the dominated convergence theorem and the following facts:


we have a quadratic form D(*): defined by (9.33). For which, Xi") and are well defincd. Next, define an isoperimetric (or Cheeger's) constants as follows

Ar)

Theorem 9.43. For t h e quadratic form given by (9.33), under (9.34) and (9.35), we have

9.5 CHEEGER'S APPROACH A N D ISOPERIMETRIC CONSTANTS

369

Proof: a) First, we express h(") by the following functional form

By setting f = I A / K ( A )one , returns to the original set form of h(*). For the reverse assertion, simply consider the set A , = {f > y} for y 2 0. The proof is also not difficult:

's

tf(Z)>f(Y))

J(%%

dY)[f(Z)

-

f(d1 + K ( " ) ( f )

(Co-area formula). Hence

The appearance of K makes the notations heavier. To avoid this, one can enlarge the state space to E* = E U (m}. Regarding K as a killing measure on E * , the form D ( f , g ) can be extended to the product space E* x E* but expressed by using a measure J* only. At the same time, one can extend f to a function f * on E*: f* = f 1 ~ More . precisely, define J*(") on E* x E* bY E 47 x 8 J'")(C),

c c =A x or +> x A, A E 8 c = {m} x {w}.

Then, we have

L.

J*(")(dz,E*)(f*(.)( =

L

J ( " ) ( d z ,E ) ( f ( z )f( K ( " ) ( ( f l )

9 SPECTRALGAP

370

b) Take f with 7 r ( f 2 ) = 1, by a), Cauchy-Schware inequality and conditions (9.34) and (9.35), we have

The right-hand side is bounded above by 1. Solving this quadratic inequality in D(')(f), one obtains D(')(f) 2 1c ) Repeating the above proof but by a more careful use of Cauchy-Schwarz inequality, we obtain

d-,

6 D ( f >[2 - D " ) ( f ) ] . From this and b), the required assertion follows. I We now turn to study XI. To do so, we need the following isoperimetric (or Cheeger's) constants.

1

/&a) = -

inf

2 dA)€(OJ)

J ( a ) ( Ax A") + $")(Ac x A ) I

(4 J(")( A x Ac)+ J(CY)(AC x A) 4 4 n

1 k(a)' == inf 2 n(A)~(0,1/2] 44 J ( O ) ( Ax A C ) J(.I(AC x A ) 1 -inf 2 n(A)~(0,1) r ( A )A x(AC)

+

The functional farm of these constants can be expressed as follows.

9.5 CHEEGER'S APPROACHANT) ISOPTSRTMETRIC CONSTANTS Lemma 9.44. For every

CL

371

3 0, we have

Proof: Because Q 3 0 is fixed, we can omit the superscript "(a)"everywhere in the proof. Denote by and ,@ the right-hand sides of the above and k' >, E l . We now prove quantities. By taking f = I A , we obtain k the reverse inequalities. a) For any f f L:(.rr) with J ~ ( d z ) ? r ( d y ) l f ( z) f ( y ) l = 1, by proof a) of Theorem 9.43, we have

This proves the first. equality of k("). Next, we show t'hat

whcrc Il.llp denotes the LP-norm. First, let n ( g ) = 0 with inf,,R IIy-,cllw Then, because r ( g ) = 0 and ~ ( -f ~ ( f ) = ) 0, we have

for all c E R. Hence, by Holder inequality, we have

for all c. This gives us

< 1.

372

9

SPECTRALGAP

On the other hand, for a given J E L1(n), set ATf = {f 2 n(f)} and A; = { J ,

K ; ~

Pf A!"

(9.38) the infimum is taken over all i.i.d. random variables X and Y with EX EX2 = 1.

r

0 and

Proof: a) First, we prove the first lower bound of (9.37). Let f 6 a ( D ) with .(I) = 0 arid r ( f 2= ) 1. Set y = f c, c E R. Similar to the proof c ) of Theorem 9.43, we have

+

for all P : 0 < /3 < Xi1) 6 2. By Lemma 9.44, we have

Thus, it remains to estimate

K ~ Set .

y = IElXl E (0, I]. Clearly,

Here in the second to the last step, we have used Jenseii's inequality,

We claim that when c = 0, I E I x 2 - Y*l >, 2 ( 1 -

IE(X1)= 2(1- 7).

(9.41)

To see this, note that IE(X2- Y21 21E(X2 - Y 2 ;Y 2< 1 < X 2 ) = 21E [ X 2 ;x 2>, 11P ( X 2 < 1) - 21E [ X 2 ;x 2 < 11P ( X 2 >, 1) = 2(P[X2 < 11 - I E [ X 2 ; x 2< 11).

9 SPECTRAL GAP

374 On the other hand,

Hence

Thus, we obtain

Letting p T X p ) , we obtain the first lower bound. b) For any B C E with n(B)> 0, define a local form as follows.

Obviously, D - ( ,a ) ( f ) = D -(a) B (91,). Moreover, xO(B>

:= inf{D(j> : f / p = 0,

llfll

= 1 ) = inf

( 5 B ( j ) : i 7 ( f 2 1 B ) = I}.

Let nB = T ( . n B)/.rr(B)and set

~ ~ s j a=)

inf

[ J ( a ) ( A(xB \ A )j + J ( " ) ( ( B\ A) x A ) ] / 2+ J(")( A x BC)

(4

ACB, rr(A)>O

- _1 inf 2 ACB,r(A)>O

$-)(A x A") + $")(A" x A )

44

Applying Theorem 9.43 to the local form on L 2 ( B ,6 n B , nB) generated by J B = T ( B ) - ~ J I and B ~K~B = d ( . ,Bc)IB,we obtain

c) Note that in

,by Theorem 9.33

9.5

375

CIIEEGER'S APPROACIi A N D ISOPERIMETRIC CONSTANTS

We obtain the second lower bound of (9.37). We now study a different way to renormalize the general quadratic forms. In contrast to the previous approach, we now keep ( J , K ) to be the same but change the L2-space. To do so, let p be a measurable function satisfying cyp

:= .Ir-essinfp > 0,

[ J ( d z ,E )

Pp := r ( p ) < 00 and

+ K(da)]/7r,(dz)< ,f&,

7rp-a.e.

where 7rp = pn/BpO,. In the non-symmetric case, assume also that (9.35) holds. hior jump processes, one may take p ( z ) = g ( s ) V r for some T 2 0. From this, one sees the main restriction of the present approach: J r ( d z ) q ( s )< 00, since we require that ~ ( p .=) 00. Except this point, the approach is not comparable with the previous one. Next, define

Theorem 9.46. Let P, ap,Pp, rp and X , i (i = 0 , l ) be given above. Then, we have xi

2 apAp,2/L&,

2

= 0, 1.

(9.43)

In particular,

(9.44) and when K

-f-

0,

(9.45) where

rp 2 1/2

is the unique solution t o the equation 2y'Lk; = 1 - 41 - 4(1 - ~ ) ~ t k ; ,

y E (0,l).

(9.46)

Proof: Here, we prove the assertions for i = 1 only since the proof €or i is similar and even simpler.

=. 0

9 SPECTRALGAP

376

To prove (9.43) for i = 1, by the assumption on lo, axid TJcmma 9.10, it is clear that L’(n) is dense in L2(7r,) in the 11 Ilo-norm. Noting that L2(n,j is just the domain of the: form D ( f ) on L‘(T,), by definition of’ A, and AP,, , i~ suffices to show that np(f2) - r,(f)’ a p [ n ( f 2-) ~ ( f ) ~ ] 3for / ~ every , f E J J M ( 7 i ) . The proof goes as follows.

-

>

To prove (9.44) and t,he second estimate of (9.45), simply apply Theorems 9.43 and 9.45 with 7~ replaced by 7rp and renormalizing constants r ( z , y ) 5 p,, s ( x ) = P,. Noting that for the modified k(”), we have k(“) = k/,’3,”, k, = k / P p and so on, where k = k(’). To prove the first estimate of (9.45),lct np(f)= 0 and 7 r p ( f 2 ) = 1. Similar to (9.39), we have

Solving t.his inequality in D ( f ) , we obtain D ( f >3 & ( I + c 2 ) -

Jm

and so &,1

+ c2)2 - P,”(l+ c2)M(x,Y,c)k;,

> &(1+ c2) - @(1

where M ( X ,Y ,c) =

+

-~.--

+

+ c)21)2

I

Thus

M ( X ,Y,c)k;

&J >

&

(y 1 c2 .-..

/-

1

+ J1 - M ( X ,Y,C ) k Z / ( l + c2)

*

Combining this with (9.40) and (9.41), we get

A1,

{

2 pp inf max 2y2k;, I YE [O?21

Jm}

= 24,(ypkp)2,

where y p is the unique solution to (9.46). To see that 7 , 2 1/2, note that JG 1 - 2/2 and solve the equation 2y2ki = 2(1 - ~ ) ~ k i H. The main advantage of the Cheeger’s approach is lhat it works in a very general setting. Here is an example.


1

Actually, it is known that for d-dimensional (d 2 2) Ising model, there ) that IF?l = 1 for 0 < Pc and 1 3 1 > 1 for ,O > &. In exists a PC E ( 0 , ~so this sense, we say that the model appears a phase transition. It is even known for this model that when d = 2, & = log(1 fi)E 0.4407. The tool in the proof is the contour method. Let Et denote the unit square parallel to the coordinate axes and centered at t E 2'. Denote by 8Et the boundary of Et. The union U t 8 E t is called the dual graph of E 2 . A closed path I' (without loop) on the dual graph with finite length is called a contour, its length is denoted by Irl. Next, let V c Z2 and set

4

+

t € V ,xr=-l

where dB denotes the boundary of B C R 2 . Note that the set a(a,) is not necessarily connected. Let V = VL be the square parallel to the coordinate axes, centered at the origin with length 2L of edges. Let P6il E 9 ( E ) be the conditional Gibbs distribution p v ( 3 , with boiindary (2t = fl,t E 1"). The key step for the proof is the following a)

Lemma 10.14 (Peierls inequality). Let l? be a fixad contour. Then P&[X

:

r c a(zv)]< exp[-2~1rI], v = v'.

Proof: A pair {s, 1 ) is called an edge, denoted by {:St), if 1s - 61 = 1. Set v == / { ( s t ): { s , d } n V # @}I. For each :c, let denole the number of the edges ( s t ) in the above set with x, = xt and as = -51 respectively. Then

x'

Because every edge of El which separates +1 and -1 belongs t o a(z,), we = 1) - Ifi(zv>l.By using these notations, the have C - = Id(zv)l. So Hsmiltonisn can be rewritten as follows:

xt

10 RANDQM FIELDS

398

where z equals 1 at the sites out of V . Thus

To estimate the quantity on the right-hand side, we make a simple transform. Let

Define a mapping Xr 3 xv

bv)t =

-+

-xt

z;

E

X F as follows:

il 1 is in the inside of I' if t is in the outside of r

Clearly, this mapping is one-to-one, it,s role is removing I' from 3(z,). have a ( x v >= ~ ( X G u) r , l a ( ~ ~=)ia(q)l i Irl.

We

+

Hence

Corollary 10.15. PG,3[xo= -I] 6

9(2e4@- 9) e4P(e4i3 - 9)2 '

1 2

p > - log3.

Proof: Let 0 E I n t r denote that the origin 0 is in the inside of zo = -1. Then there exists a contour l? so that 0 E I n t r c a(+) the boundary is 1). Thus, by the Peierls inequality, we have

I?. Let (since

10.5 ISING MODELON LATTICEFRACTALS

399

where c, = I{r : Irl = n and 0 E Intr}l. Now, we estimate c,. Fix the length n of contours. Starting from the origin, a contour with length n can at most pass through [ n / 2 ] squares along the positive direction of the first coordinate axis. In other words, there are at most [ n / 2 ]intersection points with the axis along the positive direction. Thus, if we set to(I’) = min{t : (t,O) E r},then

I{to(r): Irl = n, 0 E Int r}l < [ n / 2 ]< n / 2 . On the other hand, starting from the intersection point on the first coordinate axis, at the first step, we go along the positive direction of the second coordinate axis. Then, at each step, we have at most three directions to go. Moreover, since r is closed, at the last step, we have to go back to the starting point and so there is only one way to go. Combining these facts n . 3n-2/2. Finally, the length of a closed path together, we arrive at c, must be even, therefore


Po = 0.6587

Let L + 00, then VL T Z d . Denote by P$ a weak limit of P?P as L + 00. Then P$ E 92 and

P,”Xo = -13 < 1 / 2 ,

p 2 p,.

P,-[X, = +1] < l / 2 ,

p 3 Po.

By symmetry, This shows that F‘j!

> 1.

# P i and hence

10.5 Ising Model on Lattice Fractals In this section, we study the Ising model on irregular lattice, especially, on the lattice Sierpinski gasket G(d) and Sierpinski carpet F ( d )introduced in Section 7.5, The spin space is again {-1, +l}. The configuration x is now a mapping: G(d)(resp. F ( d ) ) + {-1, +l}. The Hamiltonian of the Ising model on G(d) is given by

c

H(x)= (st):

ZsXtr s,tEG(d)

10 RANDOM FIELDS

400

where ( s t ) denotes the bond of the nearest neighbor s, 1 E G(d) i a the usual sense. Replacing G(d)with F ( d ) ,we have a Hamiltonian H on F ( d )for the Ising model. Consider first the Ising model on G(’). Note that every point in G(2)has four neighbors. This structure is exact the same as the model on the regular lattice 2’. From this point of view, one may guess that the model on G ( 2 ) has a phase transition. However, the answer is the opposite!

Theorem 10.16. For any d

2, the king model on the lattice Sierpinski gasket (.$dl has no phase transition.

To prove this result., we need some preparations. Recall that Y(+)denotes the collection of Gibbs states with potential = ( @ A ) : 6,&)

=

cu

x,xt

if A = { s , t ) a.nd Is - tl = 1 otherwise.

By Definition 10.1 (3), for each p E Y(@)and V E S, a version of the condit.iona1probability p ( x V = ylz, = itu,,u # V ) is given by

Let pv denote the measure p v (. x 2 ) arid let g ( V )denote t,he closed convex hull of {pV,* : 5 E {-1: 1}’jV}. Proposition 10.17.

(1) Vl c VZimplies that Y(Vl> 3 Y(V2). (2) p E Y(@)iff for every u E S, a version of t h e conditional probability p{x, = yy.141x.1, =: yv, v # u} is given by

(3)

q a ) = n,qv).

(4) g(@) is non-empty, convex, and compact.

ProoE The first assertion follows from the consistericy condition since every p E Y’(Vz>can be expressed as a convex combination of the elements in

10.5 ISlNG MODELON

LA’I’TICE

FRACTALS

401

$$‘(Vl). Next, let p E Y(@).Then

-

-

,

L

exp [ / Y Z k u E A @ a ( 4+ CXP

1

-

1 + exP

[ - 2P CA:ucn

[oc,:,,, ,

U€V*

9A(Y)]

This proves the “only if” part of (2). To prove the ‘‘if” part, it suffices to show that yv,y is the only probability measure un {-1, l}v whose one point conditional probabilities are given by the above right-hand side It is equivalent to say that the potential of the corresponding field is unique up to 8 constant. This is obvious in point of view of the Geld theory, since V is fiuite and pv,, is positive everywhere in {-l,l)‘+ By definition, we have Y(Q) c g ( V )and so 9(@) c n,Y(V). Thc inverse inclusion Y(V) c $?(a)follows from the proof of (2). Finally, the last assertion follows from (l),(3) and the compactness of the state space. I Proposition 10.17 will be extended t o more general situations in the next chnpt er.

n,

Lemma 10.18.

/ bPp(f1f2 > b). By the assumptions, for every a E [O, A] and b E [I?,11, we have

P"(fl

A-u + f2 z a ) 2 1-a'

10 RANDOM FIELDS

412 b) Next, we prove that

for some e , 6 > 0. Actually, by the assumptions, A' and A2 and hence f' and f 2 are symmetric under 7 . It suffices to show that this estimate holds for at least one k , which is then implied by the following two conditions:

+

3~ > 0 so that P"f' f 2 2 a , f 1 f 2 < b) > e. ( 10.17) 36 > 0 so that {(x,y) : 0 < z < y < 1, x y 2 a , xy 6 b }

+

c {(x,Y) : Y > 1/2 + 6) holds for some a

< 1 and b > 0.

(10.18)

By (10.16),

PP(fl+ f 2 3 a ,

f1f2

< b) 3 PP(f'+ f 2 2 a ) + PP(f1f2< b) - 1 A-a b-B a-+-1. 1-a b

So (10.17) follows from

A-u 1-a

+-b -bB

+

> 1.

(10.19)

On the other hand, if we set x y = a' 2 a , xy = b' 6 b, then x and y (x y) are two roots of the equation z2 - a'z b' = 0. This gives us


l. 2

2

( 10.20)

But both (10.19) and (10.20) can be deduced by

a 1 l , which contradicts our assumption. b) If for some a t , Iluill = 0, we claim that G(a1,a2, * 7u2n)= 0. Otherwise, we may choose a sequence ( b l , - ' * , bz,) such that +

1

-

and ( b , , * . . , bzn) contains a chain (a1 T U I , * 1 having the maximal length. According to the argument in a>, i t follows that ( b l , . . . ,bzn) must be (a), r u l , . . ,u1, r u l ) . Therefore

0 # G ( b l ; * *,bzn) = G ( a l , r ~ l , . ,* u- ~ , T u=~ l) l

~ l = ( (0,~ ~

which is a contradiction. c) Finally, we prove that (10.22) implies the triangle inequality of Ilu

+ bll = IG(u + b, r ( a + b ) ,

*.

. ,a

I\ . 11.

+ b, r ( a + b ) ) 11/2n

To prove Theorem 10.28, we consider first a particular case. Corollary 10.32. Let N be even. Suppose that p E EN) is reflection T E 91/2 and is RP. Then for every family {ft E b 8 : t E ZN}

invariant for all we have

In particular, for every f E b 2 8 , we have

10 RANDOM FIELDS

420

Proof: Consider first the reflections along the first coordinate direction. Set hT= 2n and let 9 be the vector space spanned by

Define to be the common value

Applying 'I'heorem 10.31 to the function obtain

2n-1

I

n

0,j

=

nt(zl,.., ,t(d)

f(j,t(~)

we

2n- 1

Repeating the argument to the other ( d - 1) directions, we obtain (10.25). Finally, (10.26) follows by setting $0 = f arid f i = 1, t E EN \ (0) in (10.25).

Proof of Theorem 10.28: a> The proof can be reduced to the special case that d = 1 by considering the coordinate directions succcssively, as we did in the proof of Corollary 10.32. b) For d .- 1, the assertion Eollows from Lhe abst8ract8 chess-board estimate. Actually, we need orily regard each block A e as a brick, iis [mi1/2, rn 3/21 (m E Z)used in the proof of Corollary 10.32. I

+

+

10.8 NOTES

42 1

10.8 Notes In the past thirty years or more, the random field has been one of the most active subject in probability theory and in statistical physics. Several books are now available: Dobrushin, Koteckjr and Shlosman (1992), Ellis (1985), Georgii (1988), Malyshev and Minlos (1991), Riinlos (ZOOO), Preston (1976) and Sinai (1982). A plenty of references can be found from these books. Theorem 10.6 is due to Dobrushin (1970). Actually: for the existence theorem, the weak continuous condition can be weaken and the set of parameters can be more general. See Dobrushin (1970) and Georgii (1988). Another generalization, based on RP, was presented in Shlosman (1986). Theorem 10.9 is an extension to the well-known Dobrushin’s uniqueness theorem, in which the minimum L1-metric was evaluated at each single site. The present theorem, due to Dobrushin and Shlosman (1985), treats with the metric in a firiite volurne globally. See hlaes and Shlosrrian (1991) for recent progress on this topic. Nole that the measurability question in the original ststcment of Theorem 10.9 was solved by Zhang (1999), that leads to the use of Theorem 5.32. The king model on lattice fractals was studied by Gefen, Aharony and Mandelbrot (1983, 1984) and by Gefen, Ahxon?, Shapir and Mandelbrot (1984). Theorem 10.16 was proved by J. L. Zheng in 1989 but it is published here for the first time. The proof given here is an analogue of those in proving the absence of phase transition for the model having finite range. Refer to Liggett (1985, Chapter 4) for details and original references. The dual graph used in the proof of Theorem 10.20 is due to J . L. Zheng. The corrhinatorial Lemma 10.21 is due to J. Wu, Again, Theorem 10,20 appears here for the first time, A complete proof was presented in Zhcng (1993). The proof uses some results from algebraic topology since the complex of coritoiirs, It should rriention that the proofs of Theorem 10.16 and Tlieorerri 10.20 are suitable for much more general situations. For instance, the same conclusion of Theorem 10.16 should be held for any lattice nested fractals. See also Yoshida and Higuchi (1996) for a related result. Refer to Mandelbrot (1982) and Falconer (1989) for more materials on fractals. For diffusion processes on fractals, refer to Barlow and Perkins (1988), Kusuoka (1987, 1989) and Linstrbm (1990) and the references within. For the Peierls method and its development, called Pirogov-Sinai method, refer to Sinai (1982). Here, we mention Dobridiin and Zahradnik (1985), Park (1988a,b). The R.P was int,roduced by Osterwaldcr arid Srhrader (1973) and so is also called the Osterwalder-Schrader positivity. Section 10.6 is taken from Shlosman (1986). Finally, Section 10.7 is mainly taken from Frohlich, Israel, Lieb and Simon (1978). Finally, for large deviations of random fields, refer to Ellis (1985) and Olla (1988).

Chapter 11

Reversible Spin Processes and Exclusion Processes This chapter deals with the reversibility of two important classes of particle systems, spin processes and exclusion processes. There are two reasons why we study this problem. One is that comparing with an irreversible process, a reversible process has nicer property and is easier to handle, so we should justify the reversibility of a given process at the beginning of the study. This leads also to the study of potentiality. On the other hand, in the equilibrium statistical physics, we are given (local) reversible measures (i.e., conditional Gibbs states) and the processes are constructed for describing the systems. Thus, the processes should be reversible. Then, a question arises, what rates we should take so that the corresponding process actually describes an equilibrium system. This again leads to the study of the reversibility.

11.1 Potentiality for Some Speed Functions In this section, we introduce some simple criteria for the potentiality of the speed functions of spin processes and exclusion processes. As we will see in the subsequent sect#ions,the potentiality is essential for the reversibility of the processes. Besides, the idea given here will be also used in Section 14.3 to study the reversibility for some reaction-diffusion processes.

Definition 11.1. Let S be a countable set, E = (0,l)’. Suppose that c(., -) : S x E -+ R and c(., ., .) : S x S x E -+ R satisfy the positivity condition: (1) c ( u , z ) > 0, u E s,z E E ; (2) c(u,v , 2 ) > 0, u , v E u # v; z E E , x(u)# x(v).

s,

Set

4&,4

=

{

c(u, z)

if $ = , z for some u E S if 5 # z,, z for any u,

where

( U z ) ( v= )

{

-

XV

if = if u # v,

( ( u , v ) z ) ( w= )

i

(11.1)

z(w)

if w # u , v

4.)

if w = 11 if w = v.

x(u)

Then we call Qs := ( q s ( z , Z ) : x,Z E E ) and Qe := ( q e ( z , Z ) : z,Z E E ) a field of spin speed functions and a field of exclusion speed functions, respectively.

422

11.1POTENTIALITY

FOR S O M E SPEED

FUNCTIONS

423

Theorem 11.2. (1) The field Q s := ( q s ( x , 5 ) : x , Z E E ) is a potential field iff the following quadrilateral condition

4%M JU+(% ,

u v z > ) c ( v ,4)= C(V, +(%

d ) C ( V , U U ~ ) ) C ( U ux) ,

holds for any u , v E S and x E E . (2) The field Q, := (q,-(x,$) : x,Z E E ) is a potential field iff the following

triangle condition c h 'u, 44%w ,(u,lJ)W'Lv,Ul ( w , u ) 4 = 4%70, +(w

I f ?(U,W)++?

u?( V , U ) X )

holds for any u,'u,w E S and x E E .

Proof: The first assertion is actually a restatement of Lemma 7.11. As in the proof of Lemma 7.11, we define 4u,.>

=

(u,v)Z

for u # 'u and x(u)# x(v>. But here ( 4> 03 > 0.

So there exists a xV.k E E such that

For each y E & ( V ) , U , I I E V with y(u)

# y(v),

by Lemma 11.9(2)! we have

where zv,k = ( z ~ , ~ )Hence, ~ , ~ by . the positive condition of ( c ( u ?T I , x)}, we get YEGmPKYl) x E ( S \ 1 4 s \ W Z V , k ) > 0,

v>

Therefore the restriction of Qe on E k ( V ) x z V k is a potential field. From Theorem 7.6 and Theorem 11.2 (2) we have

Now the triangle condition follows from the continuous condition and the fact that UVEsU z o& ( V ) x { z ~ , is~ dense ) in E. M7e now discuss the uniqueness problem for Gibbs states of a spin process with nearest neighbor speed function.

Definition 11.16. The speed function { c ( u , ~ ) is } said t o have nearest neighbor if, for each u E S, there is a au E S such that c ( u , .) E S ( U

u au):= &o(u u au)x E ( S \ {uu au}).

From now on we make the "nearest neighbor" assumption. Take S such that = Vn-l uav,-,, 1s

v,

v,

{Vn}yc

11.3 CRITERIA FOR RF,VERSIRII,~TY

439

and c(u, .) E G(Vn)

for each u E Vn-l.

(11.40)

For each n 2 1, we take Bvn as a reference point in E(Vn). For each w E E(Vn),we choose {y(l),. - . ,~(‘1)) c E(Vn)such that ,Q

:=

p

3

$(’) + . . . + y(‘“) + y(‘++l)

=: w

and define M V , x “s\v,, 11) x “qv,) and q.s(w x x:s\v,, ,Q x XS\V,) as usual (cf. (11.19)). Because of (11.40), we may rewrite these qS’sa5 &(dv, x xav,, w x zavn)and &(w x xav,, Ovn x zaVn),respectively.

Theorem 11.17. Suppose that ( c ( u , z ) ) is positive, nearest neighbor and satisfies the quadrilateral condition. Then there is a bijective mapping between the set of all Gibbs states for { c ( u , x ) }and the set of all the equivalent classes of linear independent positive solutions to the following equation

(11.41) E(BVn-l), n 2 1, where V, = 0, avo = V, and &(y,$)/yB(&y) = 1 as a convention when y = f. In particular, there is only one Gibbs state i f f (11.41) has only one positive

z

E

solution up t o a constant.

Proof: Noting thatj without any confusion, we can use exp [ - ’p(Bv,,-, x x,y x z ) ] to denote (TR(f?vn-l x z , y x z)/q8(y x z , dv,,-l x z ) since the pathindependence, where cp(y, 2) is the work done by the field Qsfrom y to z. a) Suppose that (Zn,z} is a positive solution to (11.41). Define

hn(Y x z ) = 2;’ ~ X P [ - L ~ ( ~ Vx , X_,~Y x

~ n , t

(11.42)

for each y E E(Vn-l), z E E(dVn-1>and n 2 1, where

c

zn=

exp[-cp(bn-l x

*I

Y x z>IZn,z.

(11.43)

y€E(Vn-11 z E E( aV, - 1

Clearly, /inis a probability meamre on E(VOZ). We now prove that b) {/Jn}n>l i s a consistent family. By the path-independence, wc have cp(&n

x w, Y x

x w>

= P(Ovn x w,Qvn_lx = ‘p(Qvnx w, x

+ x w) + 9+v,-1

x w) ’p(Qv,-, x z x w,Y x z x w) x

*, Y x 2)

(11.44)

440

11 SPIN

PROCESSES AND

EXCLUSION PROCESSES

for each y E E(I&-l), z E E(aVn-l) and w E E(aVn). Hence for each n 2 1,

=

C

exp [ -

and so, by (11.44),we get

for each y E E(Vn-l), z E E(aVn-l) and n 2 1. Therefore) by the Kolmogorov consistency theorem) there is uniquely a probability measure p on 8 such that pn is the projection of p on E(Vn). Next, we prove that c) the measure p obtained in b j is a Gibbs state. By Theorem 11.1(l),it is enough to prove that p is a reversible measure. But this follows immediately from (11.42), the path-independence and the following result.

Lemma 11.18. Let p be a probability measure on (E,&'),then

/L

E .%'(St,) ifF

for each y E E(V,) and u E Vn-l, where p., is the projection of p on

E(Vn).

11.3 CRITERIA FOR REVERSIBILITY

44

Proof: If p E 9 ( f l s ) , then applying Lemma 11.8(1) to f = I { y } x ~ ( ~ \ ~ , , (y E E(V,)) gives (11.45). Conversely, it is trivial that (11.45) implies (11.46) with the above f in the cases that u # V, or u E V,-I.

If u E V,

\ Vn-l

=

aV,-,, from (11.45)) it follows that

for each z E E(V,+1

~ P X = x}

\ V,).

E(S\Vn+i1

That is

4%z)p(d4 = UYX

=Ix E(S\V,+l)

4%+(W.

Summing up z over E(V,+1 \ V,), we get (11.46), and then p E 9 ( f l s ) by Lemma 11.8(1). The lemma is now proved. 1 We now return to our main proof. d) Let p E 9 ( f l S ) and f i n be the projection of p on E(V,). Then, by (11.45), we have

Summing up w over E(dV,), we get

:= ,5n(6V,-1 x z ) : z E E(aVn-l), n 2 l} is a positive solution Hence {5& to (11.41). Next, from (11.45) and Theorem 7.6, we see that

P7dY x

4 = fin(6Vn-l

x 4 e x P [ - P(ev,-l x

2,

Yx

43

for each E E(V,-1) and z E E(aVn-l). Taking 2,,= instead of 3,,= in (11.42) and (11.43), we obtain 2, = 1 and p, = G,, for each n 2 1. So p is the same as the measure obtained in b ) . We have thus proved that the mapping defined in u ) and b) is an onto-mapping. We finally prove that it is also an one-to-one mapping. e ) Let {&$ : z E E(aV,-l),n 2 l}, i = 1,2 be two positive solutions to (11.41). If %,Z (1) = n,z> E E(aK-l), 21 (11.47)

442

11 S P I N

PROCESSES A N D

EXCLUSION PROCESSES

(~22)

for a constant cy > 0, then 22)= by (11.43)] and hence p?) = p?) by (11.42). We see that {xt;} and {x?;} determinate the same Gibbs state p by b). Conversely, if { x c ' z } and {xk:'z} determinate same p E 9 ( f l S ) ,then for each n 2 1, p?) and p?), as the projection of p on E(V,), must coincide. In particular, by (11.42), we have for each z E E(aVn-l)l

(.zc))-'zSk = p:)(evn-l

x

.)

= pLn (2) (6 vn-l x

z) = (

z~))-'

But 2;) is independent of n 2 1, so (11.47) holds. Example 11.19. Let @ : S + R satisfy CBCuEBES l@(B)1.\B\ < oa for every U E Set < cxp

[

-

It

is equivalent

C(s)ds] Q(t)*(t).

Noticing Z(0) = 0 and the absolute continuity of Z ( t ) ,we see that

Next, by definition of Z(t>and the exchangeability of Q ( s ) Q ( t ) , we have

T h u s , by definition of Y ( t )and condition (13.9), we get

This proves (13.10). Finally, if the integral form in (13.9) is an equality, then every inequality in the proof becomes equality. I

Lemma 13.7. Let ( q ( x ) ,q(s,d y ) ) be a q-pair on an arbitrary space ( E ,€'I and let T , k E &/B(R$). If s1 r' 6 h on E , then Pmin(t) T r+J: Pmin(s)hds.


0,

El Finally, if we use " A (= B" to denote ''[A- BI


G (c,( t )P. (21,.,))

t 2 0,

(4+

/

t

0

(Cn(t - 4cp.(8,z2;n, 4)( ' W ) d S ,

q , x 2 E Eo, m 2 n 3 1, where Cn(t)was defined in Theorem 13.1. Furthermore, wAn(Pn(t,


B ) E 80,

w,

P(t>p< C ( I + p ) e c t ,

t 3 0, z

E E~

(13.25)

and

However, the convergence in (13.24) is not necessarily uniform in IC E EON, which is just the point why we need a different approach to prove the semigroup property.

Lemma 13.11, Let Y c Yip(E0)satisfy sup{L(g) : g E 9?} < 00. Then for each t 2 0 and x E Eo, we have (P7,(t)- P ( t ) ) g ( z ) 0 uniformly in g E Y as n -+ 00. ---f

Proof: Set gn(x:>= g ( x C ” n ) ,5 E Eo, n 2 1. Then supn2,L(g,) ,< L ( g ) . The assertion now follows from the fact

plus (13.26) and (13.25). I One main character of P ( t ) constructed by Theorem 13.8 is as follows.

13.1 EXISTENCE TIIEORGMS

FOR TEIE PROCESSES

483


P(s)= P ( t s) on & i p ( E ~ ) .Note that P,(t)P,(s) = Pn(t + 9 ) . Given f f yip(&), we have

+

and

gn = 0. Hence, by the dominated convergence theorem,

Iim P ( t ) ( P ( s )- P,(sj)f(x) = lim P(t)g,(s) = 0. n-+m

n-03


0 and a constant c
. Thus, the reaction part of the formal generator of the process is as follows:

where e, is the element in E: having value one at u,and zero at other sites. Moreover, we use the following convention:

=o:

q&j)

2

E

;z+, j $ z.,,

TIE

s.

'The second part of the generator of the process consists of diffusions between the vessels, which arc described by a transition probability matrix P = (p(u,v) : u: 'u E S). For instancc, if there are k particlcs in u: then the rate fiinction of the diffiision from u to is c, ( k ) p ( u ?v), where

c, 2 0,

C,(O)

= 0,

uE

s,

(13.37)

Hence, the diffusion part of the formal generator becomes

Finally, the formal generator of the reaction-diffusion processes can be expressed as follows:

W(.) Choose {An}? corresponding

c S, A, On,d

= flrf(4

+ W(4.

S. Replacing S with A,, we can define the and

nn,respectively.

13.2 EXISTENCE THEOREM FOR REACTION-DIFFUSION PROCESSES

487

As explained in the last section, we need to use a smaller space Eo instead of E : Eo = { L T f E : 1121( := Cu5,ku< oo}l where {k,} is a positive sequence such that

(13.38)

From now on, we call ( p ( u ,v)) satisfying (13.38) an M-controlled matrix. Then, starting from 2 E Eo? after a linear immigration: c,(k) = k : the number of particles adds to the site v is equal to z,p(u, v>.Hence

c,

and so the process will still stay in ED. Note that for a given (p(u,v)), the required (ku) always exists. For instance, take A1 > 1 arid u bounded positive sequence (d,,L)land set

This explains the source of the sequence (k,) discussed in the last section. Besides, under the assumption

(13.39) we may choose a sumrnable (d,) so that (k,) is also summable. Now, it is the position to discuss the conditions made in the last section. qu(Ol k)k and condition (13.16) becomes In the present case, ,& :=

cF=,

(13.40) u

Next, the set EOOintroduced before Corollary 13.15 becomes

To guarantee the denseness of EOOin ED,we adopt the assumption q&, kfQ

i

+ k)lkl
o} COO,

+ Ic) - qu(j1,jl + k ) ) k ,

(13.43)

j,

> j , 2 0,

Set

Obviously, Finally, set

Remark 13.16. If for every

is a birth-death Q-matrix,

then

Proof: Deniote by c2 the quantity given by the right-hand side. Clearly, in the present situation, and hence On the other hand,

< (32 - j&,

j,

> j , 3 0. I

We arc now ready to state the main result.

13.2 EXISTENCE THEOREM FOR REACTION-DIFFUSION PROCESSES

489

Theorem 13.17. Under conditions (13.37), and ()13.40—(13.43), there exists a Markov process with state space (E,E). Moreover, for each we have and

1() (2) (3) 4() For each 5() For each (6)

is continuous in t. is continuous in t.

Proof: a) Since

and

we have

where pu(x) = xu, b(u,w) = c p ( u , w) and

To show the regularity of R,, note that

490

13 CONSTRUCTIONS

OF THE PROCESSES

by the above inequality. Hence, the required assertion follows from Theorern 2.25. We have thus checked the first condition of Theorem 13.8. b) To check the second condition of Theorem 13.8, we use the coupling of marching soldiers. For the diffusion part, the coupling is

For the reaction part, at each u E An, the coupling of marching soldiers is as follows:

When u E Am \ An, let the particles evolve independebtly. Then the whole coupling for the process is

In particular, for

we have

13.2 EXISTENCE THEOREM FOR REACTION-DIFFUSION PROCESSES 491

On the other hand, for the reaction part, we have

then

492

13 CONSTRUCTIONS OF THE PROCESSES

By symmetry, this estimate also holds for 2, 6 y.,, Combining the above two estimates, we arrive at

(13.44) Therefore, condition (13.17) holds with the choice:

We have proved that the Lipschitz condition of Theorem 13.8 is satisfied. To obtain the estimate in part (1)of the theorem, we need a little more careful. By (13.44)) we have

on,nP W ( 2 , y) 6 c; P w ( G 9)+

Hence

1- P ( W , 4

-

c

v$An

P

h4

>

-C

I

P h4

13.3 UNIQUENESS THEOREMS FOR

THE PROCESSES

493

and so

From this, assertion (1) follows easily. c) It is easy to see that the hypotheses of Corollary 13.15 are also satisfied. Then, the conclusions of the theorem follow from Theorem 13.8 and Corollary 13.15 by some computations. 13.3 Uniqueness Theorems for the Processes

In this section, we prove some uniqueness theorems for the processes constructed in the previous sections. Two different approaches are used here. The first one is the usual semigroup approach. The second one is the weak maximum principle. Theorem 13.18. Suppose that the hypotheses of Theorem 13.8 hold and additionally, condition (13.30)and the following conditions are all satisfied.

(1) Growing condition:

CuEAn

where piy)(x) = ~ ~ ( z ) ~m(> , k1 ~) ,is the minimal number so that the above control holds and K1 is a constant.

(2) Moment condition:

Then there exists uniquely a Markov process having the properties listed in Theorem 13.8. Moreover,

where Em = {x E ,230 : p ( " ) ( z ) := Finally

xupu(x)mk, < co} and

K2

is a constant.

(13.46)

494

13 CONSTRUCTIONS O F THE PROCESSES

Proof: a) Since Em C Eo, Theorem 13.8 and Corollary 13.15 are applicable to the present case. Next, by the moment Condition and Lemma 4.13, we have ~~(t)pi:)(x) 6 (1 +pi:)(z)) exp [ ~ ~ t ]t ,2 0,

2

n 2 1.

E

By using an approximating argument, we obtain

t 3 0 , x E Em.

P(t)p(")(x)6 (1 + P ( ~ ) ( Z )exp ) [Kzt],

(13.47)

This proves not only (13.45) but also that Em is a closed set of the process. b) Let f E YZp(E0). By the growing condition, pnf(z)I

Since R,f(z)

< KIL(f)(l+drn)(5)),

+ Rf(z),

Ifif(.)I

2

E Em, n

2 1.

we obtain

6 K & ( f ) ( l +P(")(Z)),

J:

E

Gn.

In particular,

and so

for t

< 1, f E Yiip(E0) and z E Em.

Thus, by a) and the dominated convergence theorem, we get

= P(t)Rf(z),

t 2 0,

2

€

Em, f E pip(&).

Combining this with Corollary 13.15, we obtain (13.46). c) Finally, let Pk(t),k = 1 , 2 be semigroups on Tip(&) perties: i) Pk(t) is Lipschitzian on

having the pro-

YZp(Eo),

L ( P l , ( t ) f ) 6 L ( f ) e x p [ c 2 t ]f ,~ B p ( E o )Ic=1,2 , for some c 2 > 0 ; ii) Em is dense in Eo with respect to p and Em is closed for both Pl(t) and PZ( t ); iii) (13.46) holds for P ( t ) = Ph(t), k = 1,2.

13.3 UNIQUENESS THEOREMS FOR

THE

PROCESSES

495

From these, we claim that Pl(t) = Pz(t),t 2 0. Since the denseness of Em and the Lipschitz property, it suffices to show that P~(t)f(z) = P~(t)f( for all f E b-%p(E0),z E Em and t 2 0. But Em is a closed set, we may replace Eo with Em and consider Q ( t ) as a semigroup on 9Zp(Em). Then the conclusion follows easily from the Hille-Yosida theorem. Actually, for f E b9b(Ern)t let

F ( z ) = Fx(z)= Then F E &?Zp(E,)

Jom

whenever X

> c2. On the other hand,

Hence

P ( ~ ) F ( x-)F ( z ) e X h- 1 h

z E Em.

e-"P(t)f(z)dt,

h

F ( z )-

$1

h

e-'"(t)f(z)dt.

By iii), we have

RF(2) = XF(Z) - f ( z ) ,

z E Em.

Or

( X I - S2)F(z)= f ( ~ ) , 1~ E Em, > ~ 2 . (13.48) From this, we show that F is determined uniquely by f . To do so, let FI and F2 satisfy (13.48) and set g = Fl - Fz. Then

z E Em, x > c2.

Rg(z) = Xg(z),

By iii) again, d --P(t)g(z) = RP(t)g(z)= P(t)Rg(z)= XP(t)g(z)

dt

and so by the continuity,

e-xtP(t)g(z)- g ( z ) =

1

t d

(eWxsP(s)g(z))ds = 0.

0

Therefore g ( 2 ) = e-"P(t)g(z),

This gives us SUP I g ( 4

zEEm

x E Em,

> cz.

< e-xt VEEm S U P IS(Y)I*

Hence g = 0. Finally, for given two semigroups Pk(t), we have two functions Fk(z)defined above for which (13.48) hold and hence Fl = F2 as we have just proved. Therefore, Pl(t) = &(t) on b2ip(Em)by the uniqueness theorem of Laplace transform. Now, we apply the above theorem to the reaction-diffusion processes.

496

13

CONSTRUCTIONS O F THE PROCESSES

Theorem 13.19. Suppose that the hypotheses of Theorem 13.17 hold and additionally the following conditions are all satisfied.

(1) Growing condition: For a fixed rn 3 1,

(2) Moment condition: sup

c

Q&,

i

+ k) [ ( i + k ) m - im] < Kz(1 + P ) ,

iE

z+

kfO

(3) Transition condition: supv

xup ( u , w) < C,

Without loss of generality, assume that k, clusions of Theorems 13.17 and 13.18 hold.

00.

< co (by

(3) ). Then the con-

Proof: What we need is to check that the conditions given here imply the corresponding ones given in the previous theorem. a) For rn > 1, by the growing condition, transition condition, the C,inequality and the Holder inequality, we have


, 2, c, = 1 if m 1. Note that this estimate holds even for rn = 1. Combining this with conditions (2) and (3), we have checked the second condition of Theorem 13.18.

13.3 UNIQUENESS THEOREMS FOR

497

THE PROCESSES

b) Similarly, one can check that the first condition of Theorem 13.18 follows from the first one here since the diffusion part is M-controlled by the distance I] . 11, which was defined before (13.38):

The idea of the above theorem is keeping the function f in L@ip(EO)but restricting EOto Em. The next result goes to the opposite direction.

Theorem 13.20. The assumptions are the same as the previous one except the last two conditions are replaced, respectively, by the foltowing ones. (2)' Moment condition:

(3)' Transition condition: There are a positive summable sequence (k,) and a constant M ( m ) such that

Then there exists uniquely a Markov process having the properties listed in Theorem 13.17. Moreover, for each f E 9ipm(Eo):

Zip,(Eo) = {f : f is Lipschitz continuous with respect t o the metric

lb - Y l l m = c,IX,

- YzLlJi~}

and t > 0,

(13.49)

Remark 13.21. Condition (3) plus MI imply (3)'.

:= sup { p ( u ,v)'-~ : ZL,

v E S and p(u, v)

> 0}
/ N'. Next, choose N2 so that [ b i - c m a i ] / i A for ) all i 2 N 2 . Put N = N 1 V N 2 . Then for each n 2 1, some A E ( 0 , ~ and we have


1, as a function on H,, g is compact. Next, let F : [O,T]x X below and set

---f

W

be a bivariate continuous function bounded

Finally, let R be a linear operator on @(&',g) with f21 = 0 and satisfy:

( M z ) Rg

X

< ag + b on X for some constant a,b 2 0.

500

13 CONSTRUCTIONS

OF T H E PROCESSES

(M3) Moreover, for each T > 0, if (P, E @(F,g) (n 3 some no), as a function on [0,TI x Hn, achieves its infimum at some point (s,, x,) E [O, TI x Hn,r:

H,,r := {Z E H , : g(x) 6 then

-

T},

Ry,(s,, .)(x,) 2 0.

Theorem 13.23 (Weak maximum principle). Let F and fl be as above and satisfy (MI)-(M3). Suppose t h a t for each x E X , F ( . , x ) is continuously

.>

differentiable and

get,).

2 R F ( 4 x)) (t, z E x. F ( 0 , x ) 3 0, Then F 2 0 on [O,T]x X .

{

E [O, T ] x

Proof: Without loss of generality, assume that a

x

> 0. Define

- t ,x) + (T - t ) +~&e-at[g(x)+ b/a]. Then f E ( t ,.) E @ ( F g) , for sufficient small E > 0. Clearly we have f E ( tx) , = F(T

on [O, TI zfE+ QfE < --E on X . f E ( T.), 2 0

(a

x X,

(13.52)

Thus, we need only to show that f E 3 0 on [O,T]x X for sufficient small & > 0. Suppose that f E < 0 at some point (s,x). Since H , X , there must be an N so that ( s , x ) E H , for all n 2 N. Take T

I

= inf{F(t,

x) : (t,x) E [o,T ] x

x>I

. eaT/&.

If T = 0 then there is nothing to do. Otherwise, from ( M I ) and (M3), it follows that there is a compact Hn,r so that

3 0 on [O,T]x (Hn\ Hn,r). Thus, (s,x) E [0,T ] x Hn,r for all n 3 N . But [0,T ] X Hn,r is compact, achieves its minimum at some (s,, x,), .fE

fE(sn,2), and

6 fc(s, x) < 0,

a -f&(Sn,%)

at

2 0,

fE

n2N

n 2 N.

Combining this with (Ads), we obtain

This contradicts with (13.52) and therefore f E < 0 is impossible. We now apply the maximum principle to the reaction-diffusion processes.

13.3 UNIQUENESS THEOREMS FOR

THE

PROCESSES

501

Theorem 13.24. The assumptions are the same as Theorem 13.19 except the growing condition is removed and the moment condition holds for some m > 1. Then there exists uniquely a Markov process having the properties listed in Theorem 13.17. Besides,

for some constant K2.

u,

H,. Proof: a) Take Hi, = {z E EO : z, = 0, u $ A,} and set X = Then X is countable. Endow X with the discrete metric p. Next, take g(z) = p(")(z),

2

E X.

It is easy to see that g E C ( X ,p ) and is locally compact. Actually,

H,,?.

:= {.

E H,

:

g(x) < 7=}

is a finite set. However, if we use the metric induced by p ( z ) , then g is clearly not continuous with respect to p unless m = 1. For this, we have to be careful. Finally, let Pk(t),k = 1 , 2 be two semigroups constructed by Theorem 13.8, F ( t ) = Pl(t)f- P2(t)fand define fi to be the restriction of R on @(F',g):

6cp = R F ( t , .) + q R g ,

cp E

@(F,g).

By assumptions, it is easy to check, as we did in the proof of Theorem 13.8, that the operator 6 defined on @ ( F , g )satisfies the hypotheses of the maximum principle except (A&), which is often the key point in the applications. b) Fix T > 0 and suppose that cp,(s,z)

:= F ( s , z ) + " y g ( z ) +a?) E @(F,g),

(s,z) E

achieves its infimum at some point ( s ( ~ ) , z ( ~ )E) [O,T]x x(,) ke, E H,. whenever u E A,. We have

+

and

[O,T]x H , Note that

13 CONSTRUCTIONS OF THE PROCESSES

502 Thus

:= I

--+

+ II.

0,

n

--+ 00.

(XF))~~,

Here in the last step we have used p(")(a(")) = C , 1) are bounded, we have the integration by parts formula:

Let

Sincew

we have

Therefore

We have at last obtained the estimate

These facts are enough to deduce the existence and the uniqueness of the process, as mentioned in Remark 13.14. To conclude this section, we introduce two related models. For which, the details are omitted.

13.4 EXAMPLES

509

Exarriplc 13.37 (Limiting Gaussian process). Take S = Zd9,E, = R and pu(zu,yu) = 15, - gal, 71, E S . T,et { T , ~ : u,v E S} be a ramily of constants satisfying the following conditions. (1)

T,,

= Tlu-vl

(2)

Tuu

> 0,

2 0.

CvEZ",vfurvu

< Tuu.

(3) Cv~vueYlu-"l6 C < 00 for some constants C and y > 0. (4) The matrix (ruv : u,v E A) (A c Z d ) is positive definite. For each n 3 1 and u E A,, let px(z,dy) be a probability of which onto (Eu,gu) is the Gaussian measure with variance 1/ruuand mean )&,n\{ul ; C ~ T ~ ~ / T , Finally, ,.

the family of the local characteristics of the process are

as follows: 4.12(z)d y )

Cu~h, px(z, d y ) ,

q71.(z) = q n ( x >E A n ) ,

-

??,

2 1*

Then, Theorem 13.1 is suitable with the choice k , = e- YI u l , u E Zd, 0 < ;j < y and p ( z ) = Cuzuku.Actually, the limiting process is ergodic. See Basis (1980) for details.

To see the main difference between Theorems 13.1 and 13.8, let us return to Example 13.35 and using the same coupling there. We have seen chat Theorem 13.8 is suitable for this model. As for Theorem 13.1, the natural , ) )would be choice of (cu,,) and ( c w ( n m cuw =

{

u#fu,

P(U, 4 ~ w / k u ,

+ 1,

P(U>U)

U=W

4% d = Cn 6

and there are constants I 7E

flnh,

hL(&J,

21,

E

Eu

(14.1)

and rl E ( 0 , ~ such ) that

[O,cxl)

< K - qh,

(resp.

< 0),

(14.2)

where h(s)- CUE*,, hA(su)hL. Then

(1) For each n 2 1, the process P,(t) has a t least one stationary distribution 7rn

satisfying

< K/q

(resp. 6 const.).

7rn(hn)

(14.3)

(2) The process P ( t )constructed in Theorem 13.1 has a t least one stationary distribution 7 r , which can be obtained as a weak limit of a subsequence of the s7;r and satisfies


given

xu

in (13.2) (resp. (13.17)) also satisfy

ccuw < -q

< 0,

u E S and

(14 -6)

W

~ ( c U w 6 (K W

< cx),

u f S.

(14.7)

51 8

14 EXISTENCE O F STATIONARY DISTRIBU'I'IONS

AND

ERGODICITY

Then

(1) The process P ( t ) constructed in Theorem 13.1 (resp. Theorem 13.8) ha5 exact one

stationary distribution

T

satisfying

for every f E h%jf!(E) f l z i p ( & ) ( & ) (resp. f f ~ i p ( & ~ ) ( ~ o ) ) ~ (2) For each n 2 1, if the coefficients c,,(n,n) given in (13.8) vanish, then Pn(t) has exact one stationary distribution rn satisfying

Proof: Wc prom the first assertion only since the second one can bc proved in the same way. To do so, we first justify the conditions of Theorem 5.23, The first moment, condition is covered by (13.11) (resp. (13.25)) with 9" = { p E 9(&) : &)I < ca). Ncxt, by the main estimate (13.8) (resp. (13.22)), we have

where Cn(f) = exp [tC;] Thus, if we set I


,P(t7 2 2 , 9)- W ( W ,2 3 , * ) ,P ( t ,2 4 , 4)I 6 I W ( P ( 4 2 , , * ) ,P ( k z 2 , * ) ) W ( P ( t , z , , . ) ,P ( t , a 3 , - ) ) ] + IW(P(4Z,,.), P ( 4 z 3 , 9)- W ( P ( t , Z 3 , . )P, ( t ,2 4 , G K ( t )(p(.I t 2 3 ) + P ( 2 2 , .4)).

I) .

Hence, W ( P ( t z, l , .), P(t,x2,.)) is indeed continuous in (xl, x2).

As a straightforward consequence of the above theorems, for the reactiondiffusion processes, we have

Corollary 14.5. Under the hypotheses of Theorem 13.17, if (14.2) holds, then there exists exact one stationary distribution T. If c,+cM < 0, then the process is ergodic and W ( P ( t , z -), , T ) ( p ( z ) 7r(p))e-qt for ail z E Eo and t 3 0.

+


0, C = sup{(M q ) k b ( k ) - u ( k ) : k 3 0) and take K = CC,,,k,. Then C E [ O , m ) and K E [0,m). On the other hand, for h,(z,) = p,(z,) = z,,we have

6 So

1[Mx,+ b(z,) - u(x,)]k,.

14.2 ERGODICITY FOR POLYNOMIAL MODEL

521

Assertion (1) now follows from Theorem 14.2. For the linear growth model, some restriction is needed for the existence of stationary distribution, even in one-dimensional case (cf. Example 4.57). The assertions for the ergodicity follow from Theorem 14.3. H

Remark 14.7. Sometimes, we can choose a transition probability ( p , ( u v)) , on A, for each n 3 1 to keep the M-controlled condition. (For instance, it is the case if (p(u,v)) is translation invariant in S = Zd with p(u,u)= 0. See Remark 14.9 and the proof c) of Theorem 14.10 below.) Then, the constant M used in the above examples can be replaced by M - 1. In general, it seems not natural to involve the constant M in the ergodic conditions for a real model. However, some example shows that the ergodicity does depend on the choice of M and its corresponding sequence ( k , ) . On the other hand, for a given model, the choices of M are usually not unique. So it is natural to take E m i n := {x E E : c,~,Ic,(M) 0 :there exists a positive sequence ( k , ( M ) ) so t h a t ~ , p ( u , v ) k , ( ~ 0, 6, >/ 0. From now on, we will often use the following natural hypothesis.

522

14 EXISTENCE OF STATIONARY DISTRIBUTIONS AND ERGODICI

Hypothesis 14.8 (H). The transition probability p ( u , v) is translation invariant and irreducible in S = Zd, p(u,u) 0 and mo 2 1, 00,61,Sm0+l > 0.

=

Remark 14.9. Because of the translation invariance, for each h4 > 1, the sequence (k,) defined by

c 00

k,

M-np(n)(u,o),

:=

u Es

(14.13)

T&=O


P 2

< 61

(14.14)

363 P; +4 +(2 2pZ)) 363

(14.15)

respectively. Clearly, the pure birth rate PO plays no role in these conditions. The reason is that the distance used to deduce the conditions is the ordinary (in particular, translation invariant) distance on Z+and the coupling used there is the one of marching soldiers. All these are certainIy not necessary. In view of Theorems 5.37 and 5.38, there are two ways to improve the above results. The first one is introducing a refined translation invariant distance and adopt the coupling by reflection, based on Theorem 5.37. The second one is to use the classical coupling and a non-translation invariant distance, as suggested by Theorem 5.38. Both ways are meaningful. Here we adopt the second one as an illustration. Thus, we consider the following distance

P(k4

=

1

k >e E

c u jj$,wk < 00. Set u* = 0 v - ui).Suppose that there exists an & > 0 such that bk+lUk+l-(bk+ak+l+k+l-&)ur,+(ak+k)uk-1+u+ku*

6 0 , k > 0, (14.16)

where a0 = 0 and 'LL-1= 1. Then under (H),the reaction-diffusion are exponentially ergodic, uniformly in initial points.

processes

1.4.2 ERGODEITY FOR POLYNOMIAL MODEL

523

We remark that (14.14) and (14.15) can bededuced from (14.16) by setting = 1. For instance, for the second Schtogle model, (14.16) with Uk z 1 becomes 2PZk - 61 - 363k(k - 1) < 0. This is trivial when k = 0. Hence: the condition becomes Uk

This holds iff A := (2p2 - 383)2 - 1263(61 - 202) < 0. That is (14.15). The next two corollaries are out of the range of (14.14) and (14.15).

Corollary 14.11. Under (H), the processes are exponentially ergodic, uniformly in initial points, provided (1) dmo+1 is large enough for fixed & and 6 k , k 6 mo, or (2) PO is large enough for fixed 0,and 6 k , k 2 1. Proof: Take

Uk =

( k + 1)-l, k 2 0. Then ii

-

1: u*

2:

0 and (14.16)

becomes

Since the degree of ak is higher than that of bb, this inequality holds toor large enough k . Next, for fixc?d k , in case (1) (resp. (2)), the second (resp. first,) term on the left, can be arbitrarily negative for large enough Srnoil (resp. PO). Now, the assertion follows from Theorem 14.10.

Of course, in order to get a more precise ergodic region, some restriction is necessary. T h e one-paramet.er coefficienbs used in the next corollary have a deep reason, which will be explained at the end of 316.1.

Corollary 14.12. Consider the second Schlogl model with = 2a, 02 = 6a, and S, = Q > 0. Then, the processes are exponentially ergodic, uniformly in initial points, for all o 2 0.7303.

61 = 9a

Proof: Take

E

6 lo-', uo = 1, u1 = u2 = 3/2

+

E

(trick!),

Define k.1 = inf{k 3 2 : U ~ + I 2 uk). When Q = 0.7303, a numerical computation gives us !q = 15 ( k l can be smaller if a is bigger). We now show a technical point so that the computation can be stopped in finite steps.

14 EX~STENCE OF STATIONARY DISTR.IRUTIONS AND EKGOUIC

524

First, replace ukl+l.by ukl =: IL, (14.16) still holds. Next, suppose that we have already had ux: = uc+1 = g for some k 2 2. Then, in order for uk4-2

2 g,by definition, it suffices that (%+2

+ bk+l + 1

-

E

-

Uk+l)%

- (k

bc+2 Equivalently, !L>

Au(k

+ 2)u, + k + 1 2 2.

( k + 2)u, - ( k + 1) + 1) - A b ( k + 1) +'l -

E

where A a ( k ) = a k + l - u k . Thus, by induction, once this holds, we can indeed use uk = g instead of the original uk for all k 2 k l . In other words, the computation of (uk) can be stopped at k l 1. Next, since the resulting sequence (uk) satisfies (14.16) with u = u1 and u*= ii - 1, the conclusion of the corollary follows from Theorem 14.10. The remainder of this section is devoted to the proof of Theorem 14.10. For this, bhe m x t estiniate plays a key role.

+

Lemma 14.13 (Esti.matc of Moments). Under (H), for every m 2 1, there exists a decreasing function pm: ( 0 , ~3) [0, ca) such that

E z ( X u ( t ) m L, 0,

7~

E Zd and

x E E,",

(14.17)

where (X(t)>,,, is the reaction-diffusion process and

E; = {x E Eo : 2 , - zo for all

PL E

Zd}.

Proof: a) Let En denote the expected value of the process starting from xu = n. Note that Et c Em for all m 2 1, where Em was defined in the last chapter:

E,, =

{

II: E

EO : p ' " ) ( ~ ) := C ~ r k tl := inf{t > 0 : g n ( t ) M } and so g n ( t ) < KM for all t f ( O , t z ] . Furthermore, g n ( t ) < M for all t > t 2 up to t 3 := inf{t > 0 : g n ( t ) 2 M } . However, by (14.21) again, we have gn(tA) = 111 and gk(t3) < 0. Hence, the function gn goes down again, provided t 3 < 00. Thm, g, can never exceed M for all 1 2 t z . Combining this with (14.18), we obtain


K M . Define T K M = inf{t

2 o : gn(t) < K M } .

(14.23)

14.2 ERGO 111crr Y

F'DR

P oLY N o M IAL M oLIEI,

Then by (14.20), (14.21) and the fact that gn(t> 2 K M for all t have

527

< T K Mwe ,

N.ow, applying (iv) to obtain d

-dgt n ( t )

Write

E~

=E

< -&Elgn(t)T(gn(t)),

t < TKM.

(14.24)

Note that, on the one hand, we have

E ~ .

Here we have used the fact that gn(t) E [ K M , nm] for all t < T K Mand (iii). On the other hand, by (12.24), we have

Therefore, T K M< 00 and so g, 2 K M on [0, T K Mby ] the continuity of gn. By using (14.24) again, we get

That is

Set O0

Then

du

0 .

528

14 EXISTENCE OF

STATIONARY

DISTRIBUTIONS AND ERGODICI

Because r is strictly decreasing in (0, m), so its inverse function exists but also decreases. From (14.25), it follows that loggn(t) 6 T * ( E Z t )

< 00,

t

E

I?* not only

(0, T K M ] ,

i.e., gn(t) < exp [ r * ( E z t , ] , t E (0, 5%M]. Combining this with (14.23), it follows that cp&)

:=

t >o

(exp [ r * ( ~ ~ v (t K) ]M)) ,

provides a desired function. Proof of Theorem 14.10: The proof is split into five steps. a) First, consider the finite dimensional case. Let S be a finite additive group. Suppose that (p(u,v) : u,v E S ) is a translation-invariant transition probability :

p(u

+ w, v + w) = p(u,v)

for all u,v,w E S.

By using S instead of Zd, one can define an operator as in (14.11). For the diffusion part, for each u,we adopt the coupling of marching soldiers. (2, Y)

+

+

-+

.( - eu + ew, Y - eu + ew) .( - eu ew, Y) (2, Y - eu + ew)

+

at rate (xuA Yu)Pb,4, at rate (xu - Yu)+P(u, 4, at rate (yu - x,)+p(u, v).

For the reaction part, at each u E Zd, due to Theorem 5.38, we adopt the classical coupling. If xu = yu, then the two marginal processes evolve at exact the same rates. If xu # yu, then they jump independently

(x,Y)

.( + eu, Y) -+ (x - eu,y) (2,Y eu) ( x , v - eu) -+

-+

+

+

at at at at

rate rate rate rate

b(xu),

a(xu), b(Yu), u(yu).

Clearly, this coupling is order-preserved. b) Next, we make some computations. Denote by fiC the coupling operator defined above. Fix x 6 y and u E S, write xu = i j = yu. We have

,.,

R,-p(i,j)=


~ ~

(14.26)

529

14.2 ERGODICITY FOR POLYNOMIAL MODEL

where I k 2 1 is the indicator of the set { (z,y) : y, - 2, 2 l}. The last term on the right-hand side appears since p is not translation invariant. Now, by (14.16), for the first term on the right of (14.26), we have

e=i j-1

< -.>ut

- ( j - 2)'lL -

( j - 2)iu'

t=i

< -ep(i, j ) - ( j - i)u - iu*.

(14.27)

On the other hand, we have monotonicity, x translation invariance E"?Z,(t)

= +?zo(t),

C p ( v ,u)= C p ( 0 ,u - v) = 1, 21

V

where Z ( t ) = ( X ( t ) , Y ( t ) )for , any initial x

E"

= {X E

< y + X ( t ) < Y ( t ) ,a.s., and

Z$ : X,

< y, z,y E ES:

= x0,u E

S}.

Corresponding to the last two terms on the right of (14.26), we have

Here is the main place we have to pay for the method. Because of the interaction, one can not replace u ~ ~and( ~~l ~ ), ( -~ u ) ~ , ( ~ with ) u ~ ~and( U Y V ( t )- U X u ( t ) ' respectively. Otherwise, the system would become independent one for which the process is exponentially ergodic. Combining (14.26) and (14.27) with (14.28), we obtain

Hence, we arrive at

c) Let AN = [-N + l , N l d c Zd and regard AN as the torus SN = Z d / ( 2 N Z d ) ,the factor group. On S N ,we can introduce a shifting operator

530

14 EXISTENCE OF STATIONARY DISTRIBUTIONS A N D ERGODICI

in a natural way and transition invariance is meaningful. Next, for a given translation-invariant transition probability p ( u , u) on Z d , we can introduce p N ( u ,v) on S N with the same property:

Here we have identified u E SN as an element in Zd. Clearly, this ( p N ( u ,v) : u,u E S,) possesses the M-controlling property mentioned in Remark 14.9. Applying a) and b) to the present case with an obvious change of notations, we get IE>Yp(Zo(t)) 2~"(Zo(l))e-"'"-'', t21 (14.29)


0 for all t 0, this proves that Yom(0o) Set zA n = (xuA n : u E S ) . Then

>

x0(t),< ~ " " " ( t I, F-a.s.

This proves the ergodicity of the process. Actually, the convergence is exy). ponential, uniform in initial points (z,

Remark 14.14. Because p is a metric on Z+, p(z,y) = ~ , p ( z U , y U ) k U defines a metric on EO ( (Eo,p ) is a Polish space), and so we have a minimum L'-distance W ( P ,&) = inf,- s p ( x ,y)F(dz, dy). In this notation, we have indeed proved that U

21

+o

as

t -+ 00.

To conclude this section, we consider the reaction-diffusion processes with an absorbing state 6 = (6, = 0 : u E S). Note that the above proof needs only a little modification. Since the process Xo((t)stays at 8 for all t 2 0, thus, in (14.26) for instance, we can simply set i = 0 and 2, = 0. After some suitable modifications, we can prove the following result.

532

14

EXIS'I'ENCE O F STATIONARY DISTRIBCTIONS A N D

ERGODEI

Theorem 14.15. Let (uk) be a positive sequence on Z + with uo = 1 and .li := ~ u p ~ > , < ~ u03.k Suppose t h a t there exists an E > U such t h a t

where a0 = 0 and u-1 = 1. Then under (H) but b~ = 0, the reaction-diffusion processes with absorbing state 0 are exponentially ergodic, uniformly in initial points.

Applying Theorem 14.15 to the sequence Uk = 1 again, th e sufficient condition becomes infk>I(nk - b k ) / k > 0. Thus, for t.he first; Srhlijgl model, we obtain the same ergodic condition as (14.14). For the second model, the condition becomes

which is weaker than (111.15). In principle, the original process is easier to be ergodic than the absorbing one, due to monotonicity. Clearly, we should have analogue of Corpllaries 14.11 ( I ) and 14.12.

14.3 Reversible Heaction-Diffusion Processes It is natural to ask when a reaction-diffusion process is rcversible. For simplicity, we discuss this problem only for polynomial model.

other cases of 5 #

u E Z*,

x,Z

E

E.

2,

(14.33)

Assume that P O , 61,Jmotl > 0 as in the last section, then b ( k ) > 0 €or all k 3 0 and u(k) > 0 for all k 1. In order for ( q ( q 2 ) : x,2 E E ) to be a field, we need assume that p(u,'L')> 0

-

p(v,u)> 0,

u,v E E d .

(14.34)

Under this assumption, we can introduce paths, works and so OR to make a field ( q ( x , 2 ) )as in Section 7.1. Now, we look for the conditions under which (q(x?5))becomes a potential field. It is clear that for this field there is only one type of minimal closed paths. That is the triangle

14.3 REVERSIBLE REACTION-DIFFUSION

+ +

533

PROCESSES

+

+

Here, the path from z e, to z ew is due to the diffusion (z e,) + (x e,) - e, e , = z e,. For the works done by the field along these paths to be zero, it is necessary and sufficient that the triangle condition

+

+

holds for all x E E and u,ZI E Zd,u # v. Fix u # v and let z satisfy z, = z,. From the last equality, it follows that

Furthermore, the same equality gives us

(’ u(k

l)b(Ic) =:X > 0, independent of k. Equivalently, ” =A. 1) 6k+l

+

(14.36)

Based on these discussions, it is not difficult to prove (as we did several times in Chapters 7 and 11) the following result. Lemma 14.16. Under (14.34), the above field (q(z,5): z,2 E E ) is a potential field iff (14.35) and (14.36) hold.

Having the potentiality of the field at hand, the next step is to construct all Gibbs states and then prove that the Gibbs states coincide with the reversible measures or the process. However, since the potential function of the field is exactly the same as that of the field induced by the independent product of birth-death processes, we guess that a Gibbs state (i.e., a reversible measure) may be obtained by the product of the reversible measures for the birthdeath processes. That is, the independent product of Poisson measures with parameter X since (Ic l)b(Ic) = Xu(k l), Ic >, 0. The main purpose of this section is to prove this conjecture and furthermore the uniqueness of the reversible measure.

+

+

Theorem 14.17. Consider the polynomial model for which the birth-death rates satisfy Po, 61, bmo+l > 0. Suppose that ( p ( u ,v)) is translation invariant and p(u,u) = 0. Then the process is reversible iff (q(z,iE))is a potential field, i.e., (14.34), (14.35) and (14.36) hold. Furthermore, the only reversible measure is the product of the Poisson measures with parameter A.

To prove this theorem, we need some preparations. Lemma 14.18. Under the hypotheses of Theorem 14.17, if the process is reversible with respect t o T , then

(14.37) J

J

534

14 EXISTENCE OF STATIONARY DISTRIBUTIONS AND ERGODICITY

Proof: a) Let ~ ( ~ ) (=x CUES ) x z k u , m E N. Denote by x A n the element in Eo with value xu A n at every u f S . Observe that

as n + 00. By Lemma 14.13 and the monotonicity of the process, for every stationary distribution T of P ( t ) ,we have

.(p‘”’)

= s 7 r ( d s ) l E , p ( m ) ( X ( t )= )

cJ k,

7r(dx)E,Xu(t)m

U

In particular, 7r(Em)= 1,

rn E N,

(14.38)

where Em = {x : p(”)(x) < m}. b) Given f , g E b29g.t ( E ) , by Theorem 13.19, (14.38) and the dominated convergence theorem, we have

The proof is completed.

Lemma 14.19. Condition (14.37) hol;ds iff 1() (2) hold for all

and

and for all

14.3 REVERSIBLE REACTION-DIFFUSION PROCESSES

535

Proof: a) Sufficiency. Note that

1

fR9dn - 1 9 R f d n

=:I+II+III. By (l),we have

1=

c/ c1

u(&J

[ g ( z ) f ( J-: eu) - f ( z ) s ( J-: eu>]7r(d4= -II*

U

On the other hand, by (2): we have ZuP(zL, 4

uL,'u

=

1s

9 ( z - eu

+ ev)f(47@.:)

~ v P ( Wu>9(.>f(. ,

+ eu - ev).rr(dz)

u,v

=TJ

w ( u , M z ) f ( s - eu + ev)7r(dz),

here in the last step, we have exchanged u and ZI. This completes the proof of sufficiency. b) To prove the necessity, we first consider condition (2). Take y E E*, A G Zd = S. f = I l y l x E ~ \ "and g = I { y + e , - e t ) x E S \ ~ ,

If { s , t } is not contained in A, s E A but t f A for instance. Then, using the expression

f

=

I{yXy'}XES\A'

Y'E%

fY',

=: Y'EE

where A' = A U t , we may replace A and f with A' and fY', respectively, so that {s, t } c A'. Now, assume that {s,t } C A. We have

0= =

J [fog

c u,u

-g

~ l d n

2uP(u, 4 [ 9 b - eu

+ ew)f(z) - f(.

- eu

+ eu)g(.)]

X(d4

536

14 EXISTENCE OF

STATIONARY

DISTRIBUTIONS AND ERGODICI

This gives (2). As for condition (I), the proof is similar but using g = I { g + e 8 1 x E instead ~\~ of the above g .

Proof of Theorem 14.17: a) Let the process be reversible and denote by 55'the set of all reversible measures of the process. As usual, set

Let n E .%'. Applying 1,e:Mma 14.19 (1) to the function f = b,n(s : 2, = n) = a n + l ~ (:xxu = n 1).This gives us

+

7+

: 5, = 0) = z-I,

n(2 :

2,

= n) = pn/Z.

we have

(14.39)

it ~follows , Next, applying Lemma 14.19 (1) to the function f = I [ l s , = n , Z v = m that b,.ir(z : 2 , = nl z, = rn) = u ~ + ~ T :( 2z, = n 1, x, = m). So

+

Suinrning over n 2 0 gives T(Z

: Z, = 0, Z, = m ) = T ( X : Z, = m ) / Z = T ( Z :

:I:,

= O)T(:C : :cW = m )

Substituting this into (14.40) and using (14.39) we get n(2 :

2, = 12

+ 1,2, = m ) = pn+17r(x : 5, = m ) / Z = n ( x : xu = n + 1)v(x : x2.= m).

By induction, it is now easy to check that

This proves that 1221' = 1 and the unique stationary distribution is the independent of ( p n / Z : n 2 0) which is just the stationary distribution of the marginal birth-death processes. Finally, applying Lemma 14.19 (2) to the function f=1[2u=n, r , = n a + l ~ , we obtain

+

(n I)p(u,v)n(z: zll = n -t 1, x, = m> = (,m l)p(v, u>n(z: 2, = n, 2 , = rn + 1).

+

(14.41)

14.3 REVERSIBLE REACTION-DIFFUSION PROCESSES

537

In particular, setting m = n, we get p ( u , u ) = p(u, u),and so p ( u , u)= 0 a p ( u , u ) = 0. This proves the necessity of (14.34) and (14.35). Now, assume that p ( u , u ) > 0. Then (14.41) implies that

Thus

+

( n 1)bn T ( 2 : 2 , = m) = (m+ 1)7r(2 : 2 , = m+ 1). %+I Summing over m 2 0, we obtain

c

(n + 1)bn = an+1 m=l 03

m T ( X :

x, = m ) =: A,

independent of n 2 0, which gives us (14.36). By Lemma 14.16, we have proved that ( q ( 2 , 2)) defined above is a potential field. b) Let (14.34), (14.35) and (14.36) hold. Then (14.37) holds. In the finite dimensional case, (14.37) implies the reversibility of the process with respect to the independent product of Poisson measures with mean A. To pass through from the finite dimensional case to the infinite dimensional one, let f, g E b%?yl(E). We need to show that

which follows from the proof d) in the proofs of Theorems 14.1 and 14.2 (regarding gdTn and g d r as dTn and d x respectively). A further problem concerned with the reversible reaction-diffusion processes is the ergodicity. By using the monotonicity and the estimate of the first moment, we see that the process X n ( t ) starting from 5 , = n will have a weak limit for some sequence t k ---f 00. In particular, we have a stationary distribution p as a limit of X o ( t ) starting from 0 = (0, = 0). Because of Lemma 14.1% it is even true that X W ( t ) := limn+m X n ( t ) ( t >, 1) has a weak limit ,G. Again, by monotonicity, for any stationary distribution p , we should have p < p. Thus, whenever p = p, the process must be ergodic. In the present case, since p and ji both are translation invariant, the last assertion is equivalent to say that the only translation invariant stationary distribution is the reversible measure p given in Theorem 14.17. Based on these observations plus some computations on free energy, Ding, Durrett and Liggett (1990) proved the following result. Theorem 14.20. Under the hypotheses of Theorem 14.17, the reversible reaction-diffusion processes are ergodic.

538

14. EXISTENCE OF STATIONARY DISTRIBUTIONS AND ERGODICI

Refer to the cited paper for a proof. Refer also to 515.3 for more results on this class of processes. It is a good chance to look at the spectral gap for this infinite dimensional process.

Theorem 14.21. For reversible polynomial model, we have gap(S1) 2 gap(Q) > 0, where Q is the birth-death Q-matrix appeared in the reaction part of the process.

Proof: The Dirichlet form of the process consists of the reaction part and diffusion part. Ignoring the diffusion part, we get a smaller one, which is the sum of the forms of independent birth-death processes. By Theorem 9.5, we have gap(R) 3 gap(&). To see that gap(Q) > 0, one may use Theorem 9.25 (4). An easier way to see this qualitative result goes as follows. Applying Corollary 4.49 to hi = 1 i shows that the birth-death process is exponentially ergodic and so the assertion follows by Theorem 9.15 (2).

+

14.4Notes Section 14.1 is taken from Chen (198613, 1989b), but the proofs are now simplified. Some analogies of Theorems 14.1 and 14.3 were appeared in Basis (1980). An analogue of Theorem 14.2 was obtained by Huang (1987). Under the hypotheses of Theorem 14.3, the existence of stationary distribution is due to Zhang (1999). Meanwhile, some progress about the ergodicity for reaction-diffusion processes were made by Ding, Durrett and Liggett (1990) and Neuhauser (1990). In the paper by Ding et al, Lemma 14.13 was appeared. The present proof is an extension to the original one and works also for non-polynomial case, a complete exploration is contained in Chen, Ding and Zhu (1994). The sufficient conditions (14.14) and (14.15) were proved in Chen (1986b). Then, it was proved by Neuhauser (1990) that the processes are ergodic when pis and 62s are all large enough. Part (2) of Corollary 14.11 is taken from Chen (1990). Sections 14.2 is taken from Chen (1990, 1995), some other criteria and their comparison are also appeared there. Finally, Theorem 14.17 is mainly based on Zhu (1990). The ergodicity for reaction-diffusion processes with absorbing state was studied by Li (1995) with more direct approach (without using coupling). A related result will be stated at the end of Section 15.3. Based on Chapter 8, the large deviation principle for reversible reactiondiffusion processes was proved by Chen (1996a,b) completely in the finite dimensional case and partially in the infinite dimensional case. Comparing the irreversible case with the reversible one, it seems that the ergodic theorems may be further improved.

Chapter 15

Phase Transitions This chapter is devoted to the study on phase transitions for the reactiondiffusion processes. Of course, we have to restrict ourselves to some more concrete rriodels. Our first model is the linear growth model for which we adopt the moments method. The second iiiodel is rioriliriear but having 8 = (@, = 0 : u E Ed) as an absorbing state, for which we usc thc graph rcprcscntation method. Finally, &s an approximation, we study a time-inhomogeneous Q-process, instead of the infinite dimensional reactiondiffusion processes, to exhibit the phase transition. The last approach is often called the mean field method in the statistical physics. As a preparation, in the next section, we introduce an important “dual” formula. 15.1 Duality

Unless othcrwise stated. throughout this chapter, assume that ( p ( u ,u)) is a genord random walk (Lc: a translat,ion-invariant transition probability on S = E d ) with p(u,u) 0. Moreover, the rates for the diffusion part of the reaction-diffusion processes are fixed: x:,p(u, v). That is

=

Qdf(4

= .&uP(U,

).

[f(. - eu + e l J - f ( 4 ] ,

(15.1)

u.v

Define

and the dual operator Q X Y ) = CY(.)P(./4[f(Y

- eu + e u > f ( w ) ] , -

Y

E(f)-

PL,V

Note that the rate ( p ( v , u ) )for diffusions used in the last line is the dual of the original ( p ( u ,v)). Next, define a dual function (Poisson polynomial) D : ~ ( f x) E 4 Z+ as follows

W

+

where n(’) = 1, n ( k )= ,n(n- 1) - - ( n - k l), k 2 1. The main rcason we adopt the notation n ( k )is that €or a Poisson random varia.ble E with mean A, we have E$k) = A k , lc 2 0. The next result exhibits the dual relation between !& and sZ;C, in terms of the function D. 539

15 PHASE TRANSITIONS

540

Lemma 15.1. Let S be finite. Then (fldD(Y7

9)(4= (flm, 4)(Y),

r, Y E E.

Furthermore

ED(y,X(t))= W Y ( t ) , z ) ,

z, y E El

where IE is the expectation of the independent product of the R&process starting from y and the Rd-process starting from 2.

Proof: a) By definition,

x

[ (u) (u)- 1) 2

(2

(.(V)

+1)(y(v))-

x (u) 2 (u) (Y(U))z(v) I:y(V))].

Note that

We have

Here, in the last step, we have used C, p ( 7 ~11) , z 1. b) Next, set d N ) ( y ,z) = D ( y )z A N),whcre 5 A By a), we have

N

= (xu A

N

:

E S).

54 I

15.1 DUALITY

Hence

d d.5

-ED(")(Y(t

- s ) , X ( s ) )= 0,

0 6 s 6 t.

Integrating over s , we obtain

W Now, t,he second assertion of the lemma follows by letting N 4 03. Actually, (15.2) is a consequence of the integration by part.s formula. Regard X ( t ) and Y ( t )as bivariate processes with state space E x E and denote by Pd(t) and P;(t) their semigroups, respectively. Then, by Theorem 13.40, we have

Now, (15.2) follows from

For the infinite dimensional c s e , it should pointed out that the d i d relation between f l d and 02 given by Lemma 15.1 is not meaningful since D(y, -) is not a Lipschitz function. To avoid this, one may replace D with D I N ) ,and then handle the infinite dimensional case directly. But here, we would like to continue our study by starting from the finite dimensional cases. Note that corresponding t o a:,the process on E ( f j is a Markov chain. Each E(")(n3 0) is a closed set for the chain. In particular, on E('), the Markov chain has &-matrix Q = P' - I , where P = ( p ( u ,u)). Denote by p * ( t , u!u) the corresponding Q-process. Since the particles evolve independently, the transition probability of the chain on E(f)is given by

whore 0 =

(QT8

= 0). In particular, for finite S , by (15.2), we have

On the other hand, since D ( N ) ( y , f bVg!(E) and D ( N ) ( y ' , x )< 00, by using a limiting procedure (a.pplying Theorem 13.8, Lemma 5.14 and the proof a>of Theorem 4.38 to the left- and the right-hand sides of (15.3), a )

542

15 PHASE TRANSITI

respectively), we see that (15.3) still holds for S = Zd. Again, letting N + 00, we get

We have thus proved that the zero range process (Rd-process) has a dual (Ri-process) which is also a zero range process (and so the dual is called self-dual). For the reaction-diffusion process with generator R = Rd+R,, even though the above dual formula does not hold, but the same argument leads to the following result.

Proposition 15.2. For the polynomial model or the linear growth process with diffusion part as above, we have

Y E E(f),

2 f

(15.4)

Emo+[yI,

where for the polynomial model, mo was given in Example 13.27 and for the linear growth process, mo is setting to be 0.

This formula is a starting point for the study in Chapter 16. In the next section, we will use this formula to compute the first two moments of the linear growth processes.

15.2 Linear Growth Model In this section, we study the linear growth model:

b ( k ) = Alk) u ( k ) = A&,

k 2 0,

A1,AZ

> 0;

(15.5)

with diffusion part given in the last section. As was mentioned before, since ( p ( u ,w)) is a random walk, we can choose a positive summable sequence (ku) and an M > 1 so that

Recall that the convergence in finite-dimensional distributions coincides with the weak convergence in ~ ( E owith ) respect to the induced topology (cf. ) Remark 4.6). As usual, we call P ( t ) ergodic if there is a p E ~ ( E osuch that vP(t) + p as t + 00 for all v E P ( E 0 ) . The phase transition of this model is described as follows.

15.2 LINEARGROWTHMODEL

543

Theorem 15.3. For the linear growth model, the following conclusions hold. (1) If XI

t

< A,,

then the process is ergodic. More precisely, v P ( t ) v E 9 ( E o ) ,where So is the point mass a t 8. Az, then the process is non-ergodic.

+ 60 as

-+ 00 for any

(2) If A1

>

The key to prove this theorem is based on the fact that the first moment for the model is known explicitly. Actually, by (15.4), for IyI = 1, y = eu(say!), we have

W q Y , X ( t ) )= & L ( t ) Y’

+

= CP*(t,U,u’)zu’ (A, -A,) u’

It

dsCp*(t,’

S,U,U’)lE,X,‘(S).

From this, we obtain the following result. Lemma 15.4. Let p u ( z ) = x,, z E Eo, u E S = Zd. Then

P(t)p,(x) = e(X1-Xz)t

CP*(t) u , v)p,(J;).

(15.6)

A direct way to prove this lemma goes as follows. By Theorems 13.19 or 13.20, we have $P(t)f = P(t)flf for all f E Yip(Eo)(E). Since p, E B p ( E o ) ( E )and f l p u ( z ) = ( h- h)p.ll(.)

+&

m ( P ( W

- SU,),

21

we get

This differential equation is linear, so its solution is unique. A simple computation shows that the solution is given by (15.6). Next, we consider the second moment which is the case that IyI =.2. Fix y = e, e,. Since f12,D(y,z)= 2(X1 - Xz)D(y, x) ~X~J;,S, by (15.4), we have

+

+

E A Y , X ( 4 ) = C P * ( t , Y ,Y’)D(Y’,4 Y’

544

15 PHASE TRANSITIONS

fROM THIS, WE CLAIM THAT THE SOLUTION OF THE SECOND MOMENT IS GIVEN BY (15.8) BELOW. lEMMA 15.5.

Proof: Denote by by gf)(t,y) the right-hand side of (15.8). Then

Combining this with (15.8), we see that f(t,y) is a solution to Eq. (15.7). .But the solution is unique, so we have proved the required conclusiuon

15.2 LINEARGROWTHMODEL

545

Before going further, let us point out that the upper bound of P ( t ) D ( y ,x)

(Ivl = 2), which is what we need only to prove for the main theorem, can be also obtained in a different way without using the "dual" formula. Consider first the case that I S1 < 00. Then, the estimate follows from Lemma 4.12. Since the estimate is independent of 151 ' , the assertion then follows by a truncation argument. Lemma 15.6. Let ( E , p , $ ) be a metric space. Given { p n } c 9 ( E ) and m, S > 0 , if pn + p and CJ := supn pn(pmt6) < 00, then pn(pm) p(p") as ---f

n

--f

00.

Proof: Obviously, lim:

,oop n ( p m )

p(p"). On the other hand,

Letting n + 00 and then N 00,we get Kn+oo pn(pm) < p(pm). Proof of Theorem 15.3: a) From Remark 14.9, we have seen that the process is ergodic whenever XI < X2. b) Now, let XI > X2. Denote by X1((t)the process starting from 1, which is the configuration having value 1 at every site u E Z d . Let pt be the distribution of X i (t)/lEXi ( t ) .Then pt E 9( [0,00)). Moreover, ---f

LNTPL(dT)

= EX,l(t)/IEX,l(t) = 1.

By Theorem 4.4, we can choose a sequence { t n } so that pt, 9( [ 0 ,00)). On the other hand,

* some p E

where yo = 2eo. But by Lcmma 15.4 and Lemma 15.5,we have

P(t)D(yO,1) = e2 ( x 1 - x z ) 1

(Cp*(t, 0, u))a U

15

546

PHASE 'I'RANSI TIONS

11;Iollows that sup t>O

/*

r2pt(d?-) < 00.

0

Thus, by Lemma 15.6, we can choose a constant c > 0 such that p ( ( c ,m)) 2 E > 0. Then pt,,((c,m)) 2 ~ / 2 > 0 for all n 2 no = no(&). Set A ( t ) = E X t ( t ) . Noting that either X & ( t )= 0 or X,'(t) 2 1, we obtain

P ( L ) ( p oA 1)(1)=E[xA(t,)A 11 2 lE[X,'(t,) A 1 : Xi(Ln) > cA(tn)] = P[XCx,'(tn)> cA(t,)] ((c, I.) 2 E/2. R u t p0A1 E bWyl!(E),it is impossible that 6,P(t) + 60 as t 4 co.Therefore, the process ( X ( t ) ) is non-ergodic. I In view of Theorern 15.3, the only remainder case for this model is that A, = A,. To discuss this situation, we need some notations. Let = pt,,,

9 = the set of stationary distributions of P ( t ) , 9 = the set of translation invariant measures, Ye= the set of the extreme points in 9, LPm(E)= { p E Y ( E ) : JxFp(dx) < oo}:

+

and set p(u,v ) = ( p ( u ,v ) p ( v , u ) ) / 2 . Theorem 15.7. Under the assumptions of Theorem 15.3, suppose additionally t h a t ( p ( u , v ) )is transient. Then

(1) For each p

/

> 0, there is uniquely an vP E 9 n 5@satisfying

/

xcovp(dx) = p,

W v V p ( d 2 ) = P(P

+ L) + 2x1

lm

F ( 4 Zt.7 v)&

where p ( t ) is the Q-process determined by the &-matrix 2(p- I ) (2) Let p E 9 n 9 n LP1(E),then there is a X E 9 ( ( [ 0m)) , such that

P =;J vPX(dP). (3) Let p E Yesatisfy J x o p ( d x ) = p Moreover, if p E 9n 9 2 ( E ) ,then


0.

k-*m t-+w

f

pP(t) = v p .

Zd.

15.3 REACTION-DIFFUSION PROCESSES

WITH

ABSORBINGSTATE

547

One of the key to prove this theorem is that the second moment for this model is also explicit. This is a common point in the study of such type of results (cf. Liggett and Spitzer (1981)). Since a complete proof of Theorem 15.7 is quite lengthy and this finer result does not interfere too much the picture of the phase transition, we omit the details here. Refer to Ding and Zheng (1989).

15.3 Reaction-Diffusion Processes with Absorbing State

It was proved in Section 14.2 that the reaction-diffusion processes are ergodic whenever the pure birth rate is large enough. In this section, we consider an opposite extreme case that the pure birth rate vanishes. For which, we show that the processes can be non-ergodic and hence there exist phase transitions. Theorem 15.8. Take S = Z1. Consider the reaction-diffusion process ( X ( t ) ) with birth-death rates b ( k ) = Xk and a ( k ) [a(O)= 01 being arbitrary and with diffusion coefficient x,p(u, w), where ( p ( u ,w)) is the simple random walk in Z1: p ( u , v ) = 1/2 iff Iu - wl = 1. Then A, := inf { A :

IP [ ~ ' ( ( t ) $ o for

where Xo((t)is the process starting from

all

t > 01 > 0} < 00,

xo : z: = 1 and zt = O(u # 0).

The proof of this theorem is based on a comparison of the process with an oriented percolation. Let A? = { (m,n ) E Z2 : m n is even, n 2 O}. For each z E 2, draw an oriented bond from z to z (-1,l) and to z (1,l). Suppose that each z E 2 is independently open (denoted by q ( z ) = 1) with probability p E (0,l) and closed (denoted by q(2) = 0) with probability 1 - p . Given z j E 2, j = 0, .. . , k , (zo, ' . ,z k ) is called an open path from zo t o zk, denoted by zo z k , if ( z j - 1 , z j ) is a bond and zj is open for all j . Next, let

+

+

+

+

y-t

The first problem in the study of the percolation is looking for p , := inf ( p : itD (st,)

Lemma 15.9. 1/2

> o}.

< p , < 80/81.

Proof: a) The lower bound is obtained by comparing the percolation with a branching process (0 = 1, P [El = 01 = (1 - P ) ~ ,IP[& = I] = 2p(1 p ) , IP [ 1 or E (1 6 1 respectiveIy. From this, we see that pc 2 1 / 2 . b) The upper bound is harder to estimate than the lower one. Here, we adopt the contour method. Let

For l C ~ l< co, let r be the boundary of the unbounded component of (R x (-1, co)) \ W N .Such a r is called a contour. Note that for a contour I? of length m to exist there must be at least m/4 sites outside of r which are closed (we will prove this below) and the shortest possible contour has length 2 N 4. As an analogue of the Peierls' inequality, we have

+

for p > 1 - 3-4. The right-hand side is less than 1 for sufficient large N . From this and IP[ICNI= 001 6 (N+l)IP[ICol = 001, we claim that p , 6 80/81.

(-2N,

- 1)

(0, -1)

Now, we return to prove the %/4" assertion as promised in the last paragraph. Let us orient I' in such a way so that the segment (0, -1) + (1,O) to be positive. Clearly, there are four types of oriented bonds: \ J \ / * . Label them by i = 1,.- . ,4,respectively. Set mi = the number of bonds of

15.3 REACTION-DIFFUSION PROCESSES

WITH

549

ABSORBINGSTATE

4

i-th type. Then, m = mi. Next, for a bond L of type i (z = 1,2), if we stand at the midpoint of 1 and face in the direction of the orientation of I?, then the site closest to our right hand, denoted by zg, must be closed. Note that howcvcr, two bonds .t and !’ may share the same site xg. Because 1’s and 2’s bonds decrease our z coordinate by 1 and 3’s and 4’s increase: it by 1, and on the other hand, the contour r starts at (0, -1) and ends at (-2N, - l ) , we have (ml m2) - (m3 r n 4 ) = 2N. Thus 2N+m m rnl -1 7743 = 2 2F

+

+

and hence (rnl f m2)/2 2 m/4 a s desired. For our purpose, we need to generalize the above result to a slight more general situation.

+

Definition 15.10. Let Il(rn,.)I[ = (Iml Inl)/2 and q : ( R , 2 ) + (0,l). We call 7 l-dependent if q(zl), . . . ,~ ( 2 , ) are independent for all zl,. . , z, E 3 with IIx, - zjll > 1 for i # j . +

Lemma 15.11. For l-dependent percolation q , we have p , < 1- 3-36. Proof: The previous proof works also for the l-dependent case except one estimate is changed as follows. Observe: that for each (m,n,)E 2, thcre are 9 sites in 2 with \i(m,n)II < 1, so for each I’of length rn there is a set of m/36 sites which are separated by more than 1 and which must be closed for the contour to exist. Therefore

P “Cl, for 1 - p < 3-36. 1

< w]

< corlst.(9(1

-

p)”/’y

Proof of Theorem 15.8: a) The idea is to make a comparing of the rcaction-diffusion process with a l-dependent oriented percolation. Clearly, for fixed site m E Zl,the reaction-diffusion process dominates a birth-death process (Bt)t>owith rates

An: 4n,n-1 = 44 + ?a. Let To = inf{t : Bt = 0). Take N and X so large that e-N/2 Qn.nS-1=

< 3-38

< 3-38, PI [ inf B~ < N] < 3-38. l 0 is a

constant depending on

+

+

/32,

61 and

+ 663) and ,OO E [0, c),

83.

Proof a> Let PO > 0. Then f ( 0 ) < 0 and f ( A ) > 0 for all sufficient large X > 0. Hence, there exists at least one positive root of f(A) and all positive roots belong to a finite interval (0, XI. This proves (913 1. b) P u t p ( i ) = i, ph, = p r\ N and T~ = inf{t : X ( t ) > R}+Since thr,

P,V(W

/Id) -

j

nsP,(X(s))ds

0

is a Pi-martingale, we have

(15.12)

Set C = supkao [ b ( k ) - u(C) + 2k] < w. From {15.12), it, follows (;hat,

Letting N + 03, we obtain

By Gronwall’s lemma, we have

~ p ( X (At T ~ ) )< i + Ct +

et-s(i

+ Cs)ds = -C + (i + C)et,

15.4 MEANFIELD hhTHOD




d

c - m(t).

-m(t) = lEin,p((x(t))6

dt

IIence

( 15.13) c) To prove the ergodicity, we adopt the coupling of marching soldiers. Given two stationary distribution r1 and 7r2 with the first moments m1 and m2, respectively. Denote by fit and the coupling operator and the expectation of the coupling process, respectively. Note that

We have

fit(X1(k)- X2(t)1 < cLIX1(t) - X2(t)l + IX'(t)

X2(t)I,

(15.14)

where ci is given by Remark 13.16. In virtue of (15.13) and (15.14), we get

% I X l ( t ) - X 2 ( t ) [= fiiltlXl(t) - X2((t)(< (Ci + l)EIX1(t) - X"(l)(. dt This plus the assumption c$

+ 1 < 0 gives us

d) To prove the last assertion of the corollary, consider first the case that

,& = 0 and rewrite f ( A ) as Ag(X):

+ ?(A

- k)

(A

+

U.J

k=4

Clearly, g(0) > 0, g(1)

bl) . .

.

(A *

+ bk-1)

ak

< 0 whenever 61 <s;

1 2

0) is the independent product of the Poisson measures for which P E ( 4= P ( 4 ,

zf

E Zd,

where p is a non-negative, bounded C2(Rd)-function with bounded first derivative. Denote by JE:c the expectation of the process with generator RE and initial distribution p'. The main result of this chapter is as follows. 555

16 HYDRODYNAMIC LIMITS

556

, r d )E Rd and t 2 0, the limit

Theorem 16.1. For all

T

where [TIE] = ( [ T ~ / E ] :. equation:

, [ r d / 4 E) E d , exists and satisfies the reaction-diffusion

I

=

(TI,.*

-

I f(0,4=P(4. -

Furthermore, for any n >, 1 :,rl, . . ,T , E Wd and t

> 0, we have

where

A particular ca,se is that p is a constant.. Then pE is translation invariant and hence f ( t ,r ) = f ( t , 0) satisfies the equation m+l

(16.4)

Of course, if f is a constant^ then it must be a non-negative root of the cyiiation

(16.5) j =O

j=1

Definition 16.2. A non-negative, spatially homogeneous solution fo(t) t o Eq. (16.2) is called an equilibrium if it satisfies Eq. (16.4). An equilibrium solution fo(t) t o Eq. (16.2) is called stable (resp. asymptotically stable) if for every E > 0, there exists a 6 > 0 such that for any solution f ( t , r ) t o Eq. (16.2), whenever If(0,r) - fo(0)I< 6, we have I f @ , r ) - fo(t)l < E for all t > 0 (resp. h 4 m If(tl4 - f"(t)l = 01. Theorem 16.3. Let p.A be the independent product of identical Poisson measures with parameter X > 0 and EzA be the expectation of the process with > Ah the generator flE and initial distribution p'. Denote by A1 > A2 > non-negative roots of (16.51, where X j has multiplicity mj. Then, the nonnegative equilibrium solution f ( t , r>5 X i is asymptotically stable iff mi is odd and

CjGi-l rnj is even.

16.1 INTRODUCTION: MAINRESULTS

557

This result describes the critical phenomenon for a non-equilibrium system in terms of the reaction-diffusion equation. We now compare Theorem 16.3 with the results obtained in the previous chapters. First, consider the reversible case. Because Pj = aSj+l for some Q: > 0, j = 0 , . . . , m, the equation (16.5) becomes m

m+ 1

m+l

j=O

j=1

j=1

which has only one non-negative solution 8 = a. By Theorem 16.3, the solution is asymptotically stable. This conclusion is consistent with Theorem 14.20, which says that there is no phase transition in the reversible case. Next, consider the first Schlogl model with PO = 0. Then, Eq. (16.5) has two roots: X I = (PI - &I)/& and A2 = 0. It is easy to see that X I is asymptotically stable but not X2. This conclusion is certainly reasonable since there is a phase transition whenever PI is large enough (Theorem 15.8). However, if PO > 0, then there is only one non-negative root and hence asymptotically stable. From this, one may conjecture that there would be no phase transition for the first Schlogl model and there would exist phase transition for the second Schlogl model since for which not every solution being asymptotically stable (we will come back to the second model soon). These conjectures remain unsolved. Certainly, in these two different contexts the objects are actually quite different. There is a scaling factor E - ~( E J, 0) in front of the diffusion rate z(u)p(u,v)in the study of hydrodynamics in order to obtain the Laplacian in the equation. Thus, to regard Eq. (16.2) as an approximation of the particle systems, as indicated by (16.1), the diffusion rate should be large. Alternatively, if we fix the diffusion rate to be 1, then the reaction rates u k and bk should be replaced by E ~ U ~ respectively. Thus, the above comparison makes sense for those u(k) and b ( k ) up to a sufficient small scaling factor. The above two theorems will be proved in Sections 16.3 and 16.4 respectively. Some preparations are presented in the next section. To conclude this section, we explain the reason why we choose the coefficients in Corollary 14.12. Note that for the second Schlogl model, the role played by each of the parameters ,& and & is not clear at all. It seems too hard and may not be necessary to consider the whole parameters. Based on the above observation and to keep the physical meaning (the details are given below), we fix p2 = 6 a ( a > 0), 61 = 9a and 63 = a. Then, when Po E (0,4a), there are three roots A1 > A2 > A3 2 0. By Theorem 16.3, XI and A3 are asymptotically stable but not A2. When PO = 401, we have A2 = 1 with ml = 2 and XI = 4, A1 is asymptotically stable but not X2. As for PO > 4a, there is only one non-negative root which is certainly asymptotically stable. Hence, we guess that the ergodic region of Po should be

558

16 HYDRODYNAMIC LIMITS

located in (4a,m) for sufficient small a. Of course, the assertion is true in the reversible case, for which, we have Po = 36a (Theorem 14.17). On the other hand, as mentioned in Durrett and Neuhauser (1994) that the reactiondiffusion equations are usually the end of the study of hydrodynamical limits of the reaction-diffusion processes. But we can also go to the opposite direction, i.e., using the reaction-diffusion equation to investigate the microscopic processes. The main point used in the above quoted paper to prove some kind of phase transitions for the reaction-diffusion processes with absorbing state xu = 0 is t o look for the critical value at which the speed of the traveling wave solution to (16.2) changes its sign. Let us mention, without details, that in our present situation, this critical value is Po = 2a. From this point of view, the phase transitions would be appeared when ,& f (0, 2a). Based on these considerations, we propose a typical non-trivial case, for which we have more precise picture as shown in Corollary 14.12. We now go to the details. The main point is that, to keep the essential meaning of the model, we should choose the parameters so that the equation

Po + p2x2 - S I X - 63X3 = 0 contains a non-asymptotically stable root. Of course, we can take a = 53 = 1. Let X = z ,&/3. Then, the equation is reduced to z3 p z 2 q = 0, where 1 2 1 2 4 = -Po 3@2&P = 61 -

+

+

,a,

+

zP;*

+

+

When y2/4 p 3 / 2 7 > 0, there is onIy one real root, which is necessarily positive and asymptotically stable. Hence, the only interesting case is that y2/4 + p3/27 < 0. Solving the equation y2/4 + p3/27 = 0 in variable Po, we obtain (1) -

Po

(2)

-

-/%(a&

- -P2(2&

Po -

- 961) - 2(@ - 3 6 1 ) ~ ’ ~

, 27 - 961) 2(@2”- 3 6 1 ) ~ ’ ~ 27

@g)<
0 iff PO > 4. If so, there is only one non-negative root. Next, q2/4 t-p3/27 < 0 iff ,& < 4. In that case, we have three non-negative roots given in the last formula. Finally, when PO = 4, we have XI = 2 with multiplicity 2 and a single root X I = 4. Thus, as mentioned at the second to the last paragraph, for every PO E (0,4] there is precise one non-asymptotically stable root but there is no such root for all Po E (4,oo). We have thus arrived at the desired position. In our particular situation (Po = 2)) the three roots are 2 2, 2 A.

+

a,

+

16.2 Preliminaries

Let co = suprp(r). Denote by po the independent product of identical Poisson measures with mean co. Lemma 16.4. For every

E

> 0, we

have

(1) pE )= CYl P"(t, 9 , d ) l - L E ( W9) , and (2) E&D(Y,X ( t > > = CYl P"(t, Y, Y/)P"(D(Y', + ds CYlP"(t - s, Y,Y ' ) q [ R A Y ' , where p'(t, y, y') = p ( ~ - ~y,t 3, ') .

9)

,

X(S))]

In view of part (2) of the proposition, to study the convergence of ELE,we need to estimate p " ( t , y, y'). For this, we have the following result.

Lemma 16.6. Let P = ( p ( u , u ) ) be a general random walk on Z d such that p(0,O) < 1 and let ( p ( t , u , u ) )be the Q-process with Q-matrix Q = P - I . Then there exists a constant C > 0 such that

p(t,u,v) < C / h ,

t > 0, u , v

E Zd.

Proof: Clearly, the characteristic function cpt(a)( a E Rd) of the distribution ( p ( t ,0, .)) is given by

16.2 PRELIMINARIES

56 1

As the convolution of the distribution p ( 0 , u) =: pu with itself n-times, the characterstic function of P(~)(O,u) should be as follows:

where U

Hence p t ( a ) = exp [t(cp(a)- l)].Next, by the inverse formula,

Note that

and that

{

exp t p , (cos

c,"=, uiai - I)} 6 1,

Besides, we may assume that

B

:= p(,',,.. +dl

uE

~

d

.

> 0 with nd # 0, Then

P ( t , o,u>

Here in the last step, we have changed the variables

pi = ai,

1

, 6/2 or t , t' 6. For the latter one, the proof is easy:


X1. Note that if B ( X , t o ) = A1 for some to > 0, then O(X, t ) -= XI for all t 2 t o since the uniqueness of the solution to Eq. (16.19). Thus

qx,t> x1, whenever X > XI. root and limk,, for X > A1

d -qx,

at

t 2 0,

(16.22)

On the ohher hand, since XI is the largest non-negative [C,"l-o&kj - CT2' djk3] = -00, by Eq. (16.4), we have

t ) = -6,+1(0

-X

p . .. (0- Xk)mkg(B)

< 0,

(16.23)

where 9 3 0 011 [0,ca}. This means that 8(X, - ) ( A > XI) is nan-increasing and so the limit limt,w B ( X , t ) 2 XI exists. By Eq. (16.4), we have

Letting t 7 00, since both the integral and its integrand on the right-hand side converge, we have m

m 1.1

This plus thc existence of lirnidW @(A, t ) certainly gives us thc desired assert ion.

16.5 NOTES b) Let X i

< X < Xi-1.

571

Then, as an analogue of (16.22), we have

mj is even (resp. odd), then f?(A,t) is nonMoreover, by (16.23), if increasing (resp. non-decreasing) in t. Therefore, the proof a) gives us

< <
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Author Index Deuschel, J.-D., 329, 576 Diaconis, P., 576 Ding, W. D., 302, 446, 513, 537, 538, 547, 554, 575, 576, 587, 588 Dittrich, P., 513, 571, 576 Dobrushin, R. L., 171, 223, 421, 576 Doeblin, W., 223, 576 Donsker, M. D., 314, 577 Doob, J. L., 96, 455, 577 Down, D., 171, 577 Dowson, D. C., 9, 577 Doyle, P. G., 302, 577 Dudley, R. M., 182, 223, 510, 577 Durrett, R., 302, 537, 538, 554, 558, 576, 577 Dynkin, E. B., 57, 59, 577

A Aharony, A,, 421, 578 Aizenman, M., 446, 572 Aldous, D. G., 171, 572 Anderson, W. J., 119, 171, 223, 572 Andjel, E. D., 513, 572 Arnold, B., 172, 581 Arnold, L., 513, 572 Austin, D. G., 572

B Barlow, M. T., 421, 572 Basis, V. Ya., 96, 171, 509, 513, 538, 572 Bebbington, M., 572 Billingsley, P., 120, 175, 572 Blumenthal, R. M., 572 Bobkov, S. G., 380, 572 Boldrighini, C., 554, 571, 572 Bouleau, N., 271, 572

E

C

Ellis, R. S., 421, 577

Carlen, E. A,, 572 Chebotarev, A. M., 96, 572, 581 Cheeger, J., 380, 572 Chen, A. Y . , 271, 572, 580, 587 Chen, D. Y., 572, 573 Chen, J. D., 208, 573 Chen, J. W., 172, 538, 573 Chen, M. F., 6, 61, 96, 119, 171, 172, 223, 271, 302, 329, 380, 446, 466, 513, 538, 571, 573, 574, 575, 576, 580, 587 Cheng, H. X., 61, 575 Chung, K. L., 5, 167, 171, 172, 575 Cohn, D. L., 57, 575

D Dai,-Y. L., 271, 446, 576 Daley, D. J., 211, 585 Dawson, D. A., 554, 576 DeMasi, A,, 554, 571, 572, 576 Dembo, A., 329, 576 Derman, C., 172, 576

589

F Falconer, K. J., 421, 577 Fei, Z. L., 580 Feller, W., 2, 61, 96, 171, 577, 578 Feng, J. F., 571, 572, 573, 578 Feng, S., 554, 578 Ferrari, P. A., 578 Fill, 3. A., 171, 572 Forbes, F., 578 Foster, F. G., 172, 578 Frohlich, J., 421, 578 Franqois, O., 578 Freedman, D., 578 Fritz, J., 571, 578 Fukushima, M., 270, 271, 578 Funaki, T., 571, 578

AUTHOR INDEX

590

G Gefen, Y . , 421, 578 Georgii, H. O., 421, 578 Getoor, R. K., 572 Givens, C. R., 9, 578 Gong, S., 451, 579 Gotze, F., 380, 572 Granovsky, B., 380, 579 Greven, A . , 513, 578 Griffeath, D., 223, 579 Gross, L., 380, 579 Guionnet, A., 446, 579 Guo, M. Z., 571, 579 Guo, Q. F., 96, 103, 119, 161, 172, 271, 580

K Kantorovich, L. V., 96, 581 Kelly, F. P., 172, 270, 581 Kendall, D. G., 61, 171, 172, 581 Kersting, G., 96, 581 Kingman, J. F. C., 61, 171, 581 Kipnis, C., 380, 571, 581 Klebaner, F. C., 96, 579, 581 Kolmogorov, A. N., 61, 96, 270, 581 Konstantinov, A. A., 96, 581 Koteckf, R., 421, 576 Krylov, V. I., 96, 581 Kusuoka, S., 421, 572, 576, 581 Kuznetsov, S. E., 60, 581

H 1

Haken, H., 2, 513, 579 Hamza, K., 96, 579 Han, D., 172, 530, 579 Harris, T. E., 169, 172, 579 He, S. W., 171 Herbst, I., 579 Higuchi, Y . , 421, 587 Hirsch, F., 271, 572 Holley, R., 446, 459, 513, 572, 579 Hou, Z.T., 6, 61, 96, 103, 119, 161, 171, 172, 270, 302, 579, 580, 584, 588 Hsu, P. L., 61, 580 Hu, D. H., 61, 96, 171, 554, 580 Hua, L. K., 451, 580 Huang, C. C., 171, 580 Huang, L. P., 171, 223, 538, 571, 575, 580 Hunt, G. A., 580 Hwang, C. R., 380, 581 Hwang-Ma, S. Y . , 380, 581

I Ikeda, N., 7, 179, 581 Isaacson, D., 171, 580, 581 Israel, R., 421, 578

J Jain, N. C., 329, 581

Lowe, M., 582 Lbpez, F. J., 583 Landau, B. V., 9, 577 Landim, C., 380, 571, 581, 582 Lawler, G. F., 380, 582 Levin, S., 554, 577 Li, J . P., 119, 530, 580, 582 Li, S. F., 223, 571, 575 Li, S. Q., 446, 582 Li, T. D., 582 Li, Y . , 446, 513, 554, 582 Li, Z. B., 513, 587 Li, Z. P., 554 Lieb, E. H., 421, 578 Liggett, T. M., 3, 223, 380, 421, 432, 446, 449, 450, 456, 513, 537, 538, 547, 576, 579, 582 Lin, X., 172, 588 Lindvall, T., 223, 571, 582 Linstr+m, T., 421, 582 Liu, X. I?., 57, 61, 586, 587, 588 Liu, X. J., 446, 576, 582 Liu, Z. M., 580 LoBve, M., 61, 70, 582 Lu, Y . G., 329, 575 Luecke, G. R., 171, 581 Lyons, T. J., 302, 583

591

AUTHORINDEX

M M a , Z. M., 60, 270, 583 blaes, C., 421, 583 Malyshev, V. A . , 421, 583 Mandelbrot, B. B., 421, 578, 583 Mao, Y . H., 172, 380, 583 Martinelli, F., 446, 583 Martinez, S, 583 Maslov, V. P., 96, 581 McCoy, B. M., 18, 583 Meise, C., 582, 583 Mertens, J. F., 171, 583 Meyn, S. P., 171, 455, 577, 583 Miclo, L., 380, 583 Minlos, R. A , , 421, 446, 583 Mountford, T. S., 550, 583

N Nash-Williams, C. St. 3. A . , 292, 299, 583 Nelssen, R. B., 223, 584 Neuhauser, C., 538, 554, 558, 577, 584 Neveu, J., 25, 584 Nicolis, G., 513, 584 Nummelin, E., 171, 584

0 Olkin, I., 9, 584 Olla, S., 421, 571, 581, 584 Oshima, Y . , 270, 578 Osterwalder, K., 421, 584 Ouyan, R. H., 271

P Pakes, A. G., 109, 584 Papanicolaou, G. C., 571, 579 Park, Y . M., 421, 584 Parthasarathy, K. R., 175, 584 Pellegrinotti, A . , 554, 571, 572 Percherski, E. A,, 223, 576 Perkins, E. A . , 421, 572 Perrut, A , 571, 584 Pirogov, S. A, 421 Pitt, L, 579 Pollett, P., 572 POPOV, N. S., 172, 584 .

Preston, C. J., 421, 457, 584 Presutti, E., 554, 571, 572, 576 Prigogine, I., 513, 584 Pukelsheim, R., 9, 584

Q Qian, M., 270, 584 Qian, M. P., 271, 572, 573, 584

R Rockner, M., 270, 380, 583, 585 Riischendorf, L., 9, 585 Rachev, S. T., 223, 584, 585 Redig, F., 583 Ren, K. L., 446, 584 Reuter, G. E. H., 2, 61, 96, 119, 172, 584 Roberts, G. O., 380, 585 Rogers, L. C. G., 223, 571, 582 Rosenthal, J. S., 380, 585 5 Saada, E., 583 Samuel-Cahn, E., 171, 583 Sanz, G., 583 Schlogl, F., 2, 585 Schonmann, R. H., 446, 585 Schrader, R., 421, 584 Sethuraman, S., 582 Shapir, Y., 421, 578 Sheu, S. J., 380, 581 Shiga, T., 513, 550, 585 Shlosman, S. B., 421, 457, 576, 583, 585 Shortt, R. M., 9, 578 Silverstein, M. L., 270, 585 Simon, B., 421, 578 Sinai, Ya. G., 421, 576, 585 Snell, J. L., 302, 577 Sokal, A . D., 60, 380, 582, 585 Spitzer, F., 513, 547, 563, 582, 585 Spohn, H., 571, 585 Stoyan, D., 211, 585 Strassen, V., 585 Stroock, D. W., 96, 120, 171, 305, 307, 324, 329, 446, 459, 510, 513, 572, 575, 576, 577, 578, 579, 585 Sullivan, W. G., 380, 585 Saulga, A., 182, 585

AUTHOR INDEX

592

T Takeda, M., 270, 578 Tang, S. Z., 96, 446, 456, 513, 585, 586 TavarB, S., 109, 584 Theodosopulu, M., 513, 572 Thomas, L. E., 380, 585, 586 Thorrison, €I., 223, 586 Trisch, A . , 446, 583 Tuominen, P., 171, 584, 586 Tweedie, R. L., 171, 172, 455, 577, 581, 583, 584, 586

V Vallender, S. S., 9, 586 van Doorn, E. A . , 380, 586 Varadhan, S. R. S., 96, 120, 314, 513, 571, 577, 579, 581, 582, 585, 586 Varopoulos, N . , 302, 586 Veech, W., 169, 172, 586 Vere-Jones, D., 171, 586 Villani, C., 223, 586

W Wang, F. Y . , 380, 466, 575, 585, 586 Wang, J. X., 57, 587 Wang, P. Z., 580 Wang, Z. K., 57, 61, 70, 1.61, 171, 586, 587 Wasserstein, L. N., 223, 587 Watanabe, S., 7, 179, 581 Williams, D., 587 Wu, R., 151, 172, 587 Wu, J., 421 Wu, L. D., 172, 587 Wu, T. T., 18, 583 Wu, X. Y., 446, 587

X Xiao, G. N., 580 Xu, X. J., 571, 575, 587

Y Yan, J. A., 510, 587 Yan, S. J., 57, 119, 172, 446, 513, 554, 575, 587 Yang, C. N., 582 Yang, X. Q., 96, 161, 171, 271, 587 Yoshida, N., 421, 587 Yosida, K., 35, 55, 587 Yuan, C. G., 580

Z Zahradnik, M., 421, 576 Zarnit, S., 171, 583 Zegarlinski, B., 446, 579, 585, 587 Zeifman, A. I., 380, 579, 587 Zeitouni, O., 329, 576 Zeng, W. Q., 446, 587 Zhang, H. J., 172, 271, 572, 580, 587, 588 Zhang, J. K., 119, 587 Zhang, S. Y., 538, 588 Zhang, Y . H., 151, 172, 223, 583, 587, 588 Zheng, J . L., 96, 223, 421, 588 Zheng, X. G., 6, 61, 96, 119, 271, 513, 547, 554, 572, 575, 576, 578, 582, 588 Zhou, J. Z., 580 Zhou, X. Y., 300, 302, 581, 588 Zhu, D. J . , 538, 575, 588 Zolotarev, V. M., 223, 588

Subject Index Special Symbols

r-specification, 384 R(X,P), 88 8, 122,383 a-isomorphic, 60 U,(E), 120

1-dependent, 549 11. l l z l l 184 C, 383 ( B ) , 70 ( B n ) , 86 ( B x ) , 70 Bq-process, 97 Cb(E), 120 %y.f(E), 411 ( D ,g ( D ) ) , 258, 331 d-system, 57 dim% 2 0, 231 dim%, 91 dm i ?,' 91 ~ W ( L 341 ), ( F ) , 70 (Fn), 86 ( F A ) , 70 Fq-process, 97 cp-optimal, 203, 205 cp-optimal Markovian coupling, 205 cp-optimal measurable coupling, 205 gap(D), 331 gap(L), 331 % p ( E ) , 126 2 - s y s t e m , 57 Xo, 363, 368 XO(A)I 359 X I , 359 Xi(B), 360 n-almost honest, 232 n-equivalent, 231 7r-system, 57 P ( E ) , 120 (P;in(t)), 5 Pm'"(X,x,A),74 Pmin(t,x,A),77 Q-condition, 1 q-condition, 56, 85 Q-matrix, 1 q-pair, 32 Q-process, 2, 32 r-boundary, 383

A absorbing, 32 abstract chess-board estimates, 417 additive theorem, 333 aperiodic, 130 asymptotically stable, 556 autocatalytic model, 503

B backward equation at point x, 237 backward Kolmogorov equation, 73 backward Kolmogorov inequality, 73, 74 basic coupling, 11, 13, 186 basic Dirichlet form, 264 birth-death Q-matrix, 17, 110, 142, 164, 325, 488, 538 birth-death process, 11, 110, 160, 162, 210, 217, 218, 220, 266, 346, 348, 349, 359, 377, 549, 551, 559 Boltzmann constant, 385 Brussel's model, 150, 509

C canonical Gibbs state, 429 canonical image, 234 chain field, 275 Cheeger's constant, 368, 370 chess-board estimate, 410 CK-equation (Chapman-Kolmogorov equation), l, 23 classical coupling, 10, 185, 347, 354, 528 closed function (lower semi-continuous function), 121, 122, 205, 209, 305 co-zero property, 272 coalescing process, 504 coefficient of anisotropy, 385 compact function, 5 , 121, 124, 174, 391 comparison lemma, 104

593

594

SUBJECT INDEX

comparison theorem, 64, 332 conditional Gibbs state with periodic boundary condition, 408 conditional Hamiltonian, 385 conductance, 281 cone mapping, 62 configuration, 383 conservative, 32, 33 consistency condition, 384 consistent family of functions, 88 consistent family of measures, 88 constrained system, 321 continuous condition, 24, 51 contour, 397, 403, 548 contraction, 256 controlling equation, 64 core, 267, 335, 433 coupled branching process, 504 coupled random walk process, 506 coupling, 6, 173, 324, 340, 341, 342, 347, 470, 471, 479, 490, 506, 507, 518, 519, 553, 562, 563 coupling by reflection, 11, 218 coupling of marching soldiers, 11, 186, 347, 490, 528, 553 coupling operator, 10, 185 current, 281 cyclicity, 417 cylindrical function, 411 cylindrical set, 411

E edge, 272 effective conductance, 281 effective resistance, 281 eigenfunction, 341 eigenfunction in weak sense, 341 electric potential, 281 embedding chain, 139 embedding jump process, 125 energy dissipation, 282 entrance solutions, 92 entrance space, 88, 92 entropy, 15, 304 equilibrium solution, 556 ergodic, 12, 17, 137, 140, 143, 145, 149, 150, 152, 163, 164, 165, 196, 308, 325, 338, 352, 411, 450, 455, 459, 518, 520, 522, 523, 532, 538, 542, 543, 550 estimate of moment, 524 estimate of the first moment, 472 excessive measure, 170 existence criterion, 361 exit solutions, 92 exit space, 88, 92 explicit bound, 17, 349 explicit criterion, 17, 349 exponential Lz-convergence, 331 exponential ergodicity, 143 exponentially growing, 391

F

D dimension of %A, 90 dimension of YA,90 Dirichlet eigenvalue, 359 Dirichlet form, 258 Dirichlet operator, 261 DLR-equation (Dobrushin-Lanford-Ruelle equation), 384 Donsker-Varadhan entropy, 311 Donsker-Varadhan theorem, 306 Doob’s construction, 96 dual graph, 397 duality, 539 Dynkin-class, 57

Feller’s construction, 96 field of exclusion speed functions, 422 field of spin speed functions, 422 finite dimensional generalized Potlach process, 221 finite entrance, 92 finite exit, 92 finite range, 384 finitely ramified fractal, 300 first infinity, 103 first moment condition, 470, 479 first successive approximation scheme, 63 flow, 284 Fokker-Planck equation, 70 forward Kolmogorov equation, 73

595

SUBJECT INDEX

forward Kolmogorov inequality, 74 Friedricks extension, 267

G generalized Potlatch process, 505 geometrically ergodic, 137 Gibbs distribution in V with boundary condition, 387 Gibbs distribution in V with boundary condition, 387 Gibbs random field, 387 Gibbs state, 387, 429 graph representation method, 539 Gronwall’s Lemma, 472 growing condition, 493, 496

H Hamiltonian, 384 Hamiltonian with periodic boundary condition, 408 homogeneous equation, 63 honest, 24

I I-function, 304 independent coupling, 10, 185 inner regular, 60, 205 instantaneous, 32 integration by parts formula, 484, 507, 508, 510, 541 interaction function, 385 invariant measure, 124, 128, 166 inverse temperature, 385 irreducible, 130, 134 Ising model, 18, 384 isoperimetric constant, 368, 370 1

J

L

L6vy-Prohorov metric, 7 Laplace transform, 37, 51 large deviation principle, 305 lattice field, 278 lattice Sierpinski carpet, 299, 300, 402 lattice Sierpinski gasket, 298, 299, 400 limiting Gaussian process, 509 linear growth process, 503 Lipschitz condition, 470, 479 localization theorem, 67 locally compact, 499 logarithmic Sobolev inequality, 358 Loth-Volterra model, 221 lower semi-continuous function (closed function), 121

M marginality, 6, 10, 174, 184 Markov chain, 24 maximal coupling, 208 maximal solution, 92 mean field method, 539 measurable coupling, 392 minimal non-negative solution, 63 minimal solution, 63 minimum LP-distance, 8, 173, 179, 391, 469, 531 minimum property, 63 moment condition, 493, 496, 497 moments method, 539 monotone, 211 monotone class theorem, 57

jump condition, 2, 24, 51 jump process, 23

N K Kantorovich-Rubinstein-Wassersteinmetric, 173 kernel, 40 Krein extension, 270

nearest neighbor, 438 non-conservative quantity at x, 92 non-honest , 24 normal condition, 1, 51

596

SUBJECT INDEX 0

Ohm’s law, 281 open path, 547 operator Q , 71 optimal Markovian coupling, 203 order-preserving coupling, 211, 220 Osterwalder-Schrader positivity, 406, 421

resolvent equation, 51 reversible, 13, 227, 339, 341, 344, 348, 433, 452, 455, 533, 537, 550 reversible q-pair, 229 reversible measures, 433

S P partition function, 385 partition of E , 85 path, 272 Peierls inequality, 397 periodic configuration, 408 phase transition, 18, 20, 397, 400, 402, 413, 455, 457, 542, 547, 550, 558 Pirogov-Sinai method, 421 Polish space, 23 polynomial model, 503, 520, 521, 532, 533, 538, 550, 555 positive measure, 438 positive recurrent, 137, 140 potential, 273 potential field, 273 Potlatch process, 505 Potts gauge model, 450 principal eigenvalue, 362 probability kernel, 40

Q quadratic form, 359, 368 quadrilateral condition, 277, 278, 423, 426, 429, 435, 437, 439

R random field, 384 rate function, 15, 304 reachable, 272 reachable directly, 272 realization, 404 recurrence, 124, 140 reflection positive, 406 regular, 2 regular g-pair, 124 resistance, 281

Schlogl’s first model, 503, 520, 522, 532, 557 Schlogl’s model, 2, 4, 6, 19, 79, 110, 150, 164, 221 Schlogl’s second model, 503, 520, 522, 532, 552, 557 Schwarz inequality, 417 second successive approximation scheme, 64 section, 272 self-dual, 542 shift, 407 Shlosman model, 452 single birth Q-matrix, 4, 105, 152, 324 single birth Q-process, 105, 112, 151, 160, 300 single-entrance, 92 single-exit, 92 smoothing process, 504 specification, 384 spectral gap, 16, 330, 340, 348, 359, 538 speed functions, 425 spin space, 383 stable, 32, 556 stationary distribution, 124, 126 statistical sum, 385 stochastic monotonicity, 21 1 strong continuity, 256 strongly ergodic, 137, 143 sub-Markovian, 257 successful, 12, 198, 202, 218, 220, 222 successful coupling, 195 symmetric, 257 symmetrizable, 14, 142 symmetrizable jump process, 229

SUBJECT INDEX

T taboo probability) 167 tight, 205 total energy dissipation) 282 total variation, 8 totally instantaneous, 32 totally stable) 33 transition condition) 496, 497 transition function, 23 transition measure) 40, 205 translation, 407 translation invariant, 408 triangle condition, 423, 426, 429, 435, 438, 443, 533

U uniformly ergodic, 137, 143 uniformly tight, 205 uniqueness criterion, 2, 3, 103, 115 uniqueness theorem of Laplace transform, 58 universal measurable set, 60 universal measurable space, 60

597

V Varadhan theorem) 305 variational formula, 17, 349 variational formula for Dirichlet form, 363 Volterra-Lotka model, 509

w Wasserstein metric, 8, 173, 179, 391, 469, 531 weak convergence, 7 weak convergence in finite-dimensional distributions) 123 weak domain) 341 weak maximum principle, 500 work, 273

Y Yang-Mills lattice field, 451 Z zero range, 384 zero range process, 503 zero-entrance) 92 zero-exit , 92