Lecture Notes of the Unione Matematica Italiana
For further volumes: http://www.springer.com/series/7172
10
Editori...

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Lecture Notes of the Unione Matematica Italiana

For further volumes: http://www.springer.com/series/7172

10

Editorial Board

Franco Brezzi (Editor in Chief) IMATI-CNR Via Ferrata 5a 27100 Pavia, Italy e-mail: [email protected] John M. Ball Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB United Kingdom e-mail: [email protected] Alberto Bressan Department of Mathematics Penn State University University Park State College PA. 16802, USA e-mail: [email protected] Fabrizio Catanese Mathematisches Institut Universitatstraße 30 95447 Bayreuth, Germany e-mail: [email protected]

Persi Diaconis Department of Statistics Stanford University 450 Serra Mall Stanford, CA 94305-4065, USA e-mail: [email protected], [email protected] Nicola Fusco Dipartimento di Matematica e Applicazioni Universitá di Napoli “Federico II”, via Cintia Complesso Universitario di Monte S. Angelo 80126 Napoli, Italy e-mail: [email protected] Carlos E. Kenig Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637-1514 USA e-mail: [email protected] Fulvio Ricci Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7 56126 Pisa, Italy e-mail: [email protected]

Carlo Cercignani Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milano, Italy e-mail: [email protected]

Gerard Van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam, The Netherlands e-mail: [email protected]

Corrado De Concini Dipartimento di Matematica Universitá di Roma “La Sapienza” Piazzale Aldo Moro 2 00185 Roma, Italy e-mail: [email protected]

Cédric Villani Institut Henri Poincaré 11 rue Pierre et Marie Curie 75230 Paris Cedex 05 France e-mail: [email protected]

The Editorial Policy can be found at the back of the volume.

Sergio Albeverio Frederik Herzberg

•

Ruzong Fan

Hyperfinite Dirichlet Forms and Stochastic Processes

123

Sergio Albeverio University of Bonn Institute for Applied Mathematics and HCM Endenicher Allee 60 53115 Bonn Germany [email protected]

Frederik Herzberg Bielefeld University Institute of Mathematical Economics Universitätsstraße 25 33615 Bielefeld Germany [email protected]

Ruzong Fan Texas A and M University Department of Statistics College Station 77843, TX USA [email protected] Current address Biostatistics and Bioinformatics Branch Division of Epidemiology, Statistics & Prevention Eunice Kennedy Shriver National Institute of Child Health & Human Development 6100 Executive Blvd. MSC 7510, Bethesda, MD 20892 United States of America

ISSN 1862-9113 ISBN 978-3-642-19658-4 e-ISBN 978-3-642-19659-1 DOI 10.1007/978-3-642-19659-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011928508 Mathematics Subject Classification (2000): 03H05; 60J45 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

In memory of the dear father, Mr. Decai Li (1926–2003), of the second author.

Preface

The theory of stochastic processes has developed rapidly in the past decades. Martingale theory and the study of smooth diﬀusion processes as solutions of stochastic diﬀerential equations have been extended in several directions, such as the study of inﬁnite dimensional diﬀusion processes, the study of diﬀusion processes with non-smooth unbounded coeﬃcients, diﬀusion processes on manifolds and on singular spaces. The interplay between stochastic analysis and mathematical physics has been one of the most important and exciting research areas. One of the best techniques to deal with the problems of these areas is Dirichlet space theory. In the original framework of this theory, the state space is a locally compact separable metric space, e.g., Rd , or a d-dimensional manifold. This theory has given us a nice understanding about the property of diﬀusion processes with non-smooth unbounded coeﬃcients. Moreover, it has been fruitfully applied to mathematical physics. This framework has been generalized to state spaces which are more general topological spaces or some inﬁnite dimensional vector spaces or manifolds. Several key problems, such as the closability of quadratic forms and the construction of strong Markov processes associated with quasi-regular Dirichlet forms, have been solved. The study of inﬁnite dimensional stochastic analysis as well as the study of processes on singular structures (like fractals, trees, or general metric spaces) has enriched and extended the Dirichlet space theory. In the meantime, a new framework has been introduced into Dirichlet space theory by the development of nonstandard probabilistic analysis [25, 166]. As is well-known, nonstandard analysis is an alternative setting for analysis (and, indeed, all areas of mathematics), namely by enriching the set of real numbers by inﬁnitesimal and inﬁnite elements. It has its origin in seminal work by Schmieden, Laugwitz [325] and most notably Robinson [310]. By now, several textbooks and surveys exist on this theory and its applications (see, e.g. [25, 63, 125, 217]). Nonstandard analysis gives a novel approach to the theory of stochastic processes. In particular, it has led to hyperﬁnite symmetric Dirichlet space theory. Besides being interesting by itself, it has also many applications. In the ﬁrst part of the book, we extend the research to the

vii

viii

Preface

nonsymmetric case, and remove some restrictive conditions in the previous treatment of the subject (Chap. 5 of [25]). In addition, we shall apply the theory to present a new approach to inﬁnite dimensional stochastic analysis. In writing this book we have two main aims: (1) to give a presentation of research on nonsymmetric hyperﬁnite Dirichlet space theory and its applications in (standard) ﬁnite and inﬁnite dimensional stochastic analysis, Chaps. 1–4; (2) to ﬁnd nonstandard representations for a special class of (ﬁnite dimensional) Feller processes and their inﬁnitesimal generators, viz. stochastically continuous processes with stationary and independent increments (i.e., Lévy processes), Chap. 5. Chapter 6 is a complement to illustrate the usefulness of the hyperﬁnite probability spaces. The ﬁrst part (Chaps. 1–4) is based on Chap. 5 of Albeverio et al. [25] and the further in depth research of Sergio and Ruzong; the second part (Chaps. 5–6) is based on results obtained recently by Tom Lindstrøm and their extensions by Sergio and Frederik. As mentioned earlier, the interplay between stochastic analysis and mathematical physics has been one of the most important and exciting themes of research in the last decades. This is already a suﬃcient rationale for the research of the ﬁrst part of the present book. The motivation for including the second part, Chap. 5, into this book is that many of the issues discussed in the more general framework of the ﬁrst part, such as existence of standard parts of hyperﬁnite Markov chains, become much less technical to resolve for hyperﬁnite Lévy processes. Furthermore, the more restrictive setting of the second part also allows one to obtain ﬁner results on the relation between Lévy processes and their hyperﬁnite analogues, one example being a hyperﬁnite version of the Lévy–Khintchine formula. The contents of this book are arranged as follows: In Chap. 1, we introduce the framework of hyperﬁnite Dirichlet forms. We develop the potential theory of hyperﬁnite Dirichlet forms in Chap. 2. In Chap. 3, we consider standard representations of hyperﬁnite Markov chains under certain conditions, and translate the conditions on hyperﬁnite Markov chains into the language of hyperﬁnite Dirichlet forms. As an interesting and important application in classical stochastic analysis, we construct tight dual strong Markov processes associated with quasi-regular Dirichlet forms by using the language of hyperﬁnite Dirichlet forms in Chap. 4. The results show that hyperﬁnite Dirichlet space theory is a powerful tool to study classical problems. In the ﬁrst sections of Chap. 5, the notion of a hyperﬁnite Lévy process is introduced and its relation to hyperﬁnite random walks as well as to standard Lévy processes is investigated. These results can be used to show that the jump part of any Lévy process is essentially a hyperﬁnite convolution of Poisson processes. Finally, Chap. 6 is an epilogue, providing a rigorous motivation for the study of hyperﬁnite Loeb path spaces as generic probability spaces.

Preface

ix

The entire book is based on nonstandard analysis. For the reader’s convenience, we present some basic notions of nonstandard analysis, such as internal sets and saturation, linear spaces, Loeb measure spaces, structure of ∗ R and topology in the appendix. Because of its monographical character centered around the hyperﬁnite approach, the book has by no means the goal of including all aspects of recent developments in the theory of stochastic processes and its connections with Dirichlet forms theory or the theory of Lévy processes. For this, we rather refer to surveys and proceedings like Albeverio [2], Barndorﬀ-Nielsen et al. [73], and Ma et al. [275], respectively. The germ of this book goes back to the year 1989 when the second author, Ruzong Fan, worked on the construction of symmetric Markov processes associated with Dirichlet forms at Peking University, Beijing ([165] and Chap. 4). At that time, Ruzong was unaware that Sergio’s group was working on the same project using standard methods [41]. The second author, Dr. Zhiming Ma, of [41] did privately inquire Ruzong about the progress of Ruzong’s research in 1989 at the Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing. In response to Dr. Ma’s request of a private meeting, Ruzong presented his work to Dr. Ma in a classroom with Dr. Ma as the only audience. Dr. Ma, however, did not mention his ongoing work with Sergio in any way. Thus, Ruzong was totally unaware of Sergio’s research. In the spring of 1990, Ruzong ﬁrst realized this when he saw a manuscript of Albeverio and Ma [41] in Beijing with a surprise. These events notwithstanding, Ruzong continued to work on a “symmetric version” of Chaps. 1–4 using non-standard language when he was at Peking University till 1991 and when he visited the Humboldt-University, Berlin, between 1991 and 1992. Under Sergio’s supervision and encouragement, Ruzong extended the project to the current “nonsymmetric version” from 1992 to 1994 at Ruhr-University, Bochum. In 2006, Frederik kindly joined the project with a contribution on hyperﬁnite Lévy processes (Chap. 5) and the Epilogue (Chap. 6). In the summer of 2006, the three authors gathered at the University of Bonn to ﬁnalize this monograph. We gratefully acknowledge the manifold support of various institutions in the long process of work on this project. In the run-up to its completion, Sergio and Frederik were supported partially by the collaborative research center SFB 611 of the German Research Foundation (DFG), Germany; in addition, Ruzong’s visit to Bonn was partially funded through a research fellowship from the Alexander von Humboldt Foundation, Germany. Over the course of his career, Ruzong has received a lot of generous support from Sergio. As a Ph.D candidate in Beijing around 1987–1988, Ruzong was greatly fascinated by Sergio and Raphael Høegh-Krohn’s novel work on inﬁnite dimensional stochastic analysis, in which Ruzong ﬁnished his Ph.D thesis. Unfortunately, Ruzong got no chance to meet Raphael Høegh-Krohn; right before Ruzong went to Europe, he was shocked to learn that Raphael

x

Preface

Høegh-Krohn died of a heart attack. In a relatively isolated environment, Ruzong mostly worked on himself by reading numerous papers and books of Sergio and Raphael Høegh-Krohn; and many times, Ruzong had to spend a few days on a single equation or lemma to guess and to understand it. Whilst it seemed like a helpless or hopeless situation for Ruzong at that time, Ruzong eventually came to the forefront of research in areas of inﬁnite dimensional stochastic analysis: he studied the hard and central questions regarding Beurling–Deny formulae, representation of martingale additive functionals and absolute continuity of symmetric diﬀusion processes on Banach spaces, potential theory of symmetric hyperﬁnite Dirichlet forms, and construction of the symmetric strong Markov processes associated with quasi-regular Dirichlet forms by using the non-standard analysis language. This direction of research was initiated by Sergio, although Ruzong was unaware that Sergio’s group already worked on the construction of Markov processes using the language of standard stochastic analysis. In early 1989, Ruzong applied for a fellowship from the Alexander von Humboldt Foundation from Peking University, Beijing; soon after a rejection from the Foundation in the fall 1989, Ruzong received a warm letter from Sergio with encouragement and a kind oﬀer to nominate, as an academic host, Ruzong for the fellowship and by writing a strong letter of recommendation. This is just one anecdote to illustrate how Ruzong has constantly been able to count on Sergio’s help via communications by either mail or face-to-face conversations starting from 1989. Between 1992 and 1994, Sergio generously supported Ruzong at Ruhr-University Bochum to complete the main part of Chaps. 1–4 of this monograph, and helped Ruzong to pass the hard period of time in his career. The story of Ruzong is an example how Sergio has helped many young mathematicians to grow and to mature. Quite probably, Ruzong would have disappeared from academia a long time ago without the support of Sergio. In a true sense, Sergio has been an academic father ﬁgure for Ruzong when he desperately needed one. In recent years, after his departure from Sergio’s research group, Ruzong has been mainly working on statistical genetics guided by his beloved American mentor, Dr. Kenneth Lange, at the University of Michigan and UCLA. Nevertheless, Ruzong has fond memories and deep appreciation of numerous communications with his European academic father Sergio; and both Ruzong and Frederik are deeply grateful for Sergio’s mentoring. Thus, especially right after Sergio’s 70th birthday in 2009 – which also marks the 50th anniversary of his remarkable scientiﬁc career –, Ruzong and Frederik are sure that they will be joined by many other young mathematicians in thanking Sergio for his wonderful role in our professional and personal development and in wishing him all the best for the rest of his life: Not just continued productivity, but most of all good health, happiness, joy, and peace.

Preface

xi

We owe a huge debt of gratitude to our families: In the summer of 2006, Dr. Li Zhu (Ruzong’s wife) kindly took care of two young children when her husband was visiting Bonn. Their adorable daughter, Olivia Wenlu Fan, was with the second author in Germany for the “hot and interesting” summer of Bonn, where she liked everything except German milk. Frederik thanks his wife, Angélique Herzberg, for her love and manifold support with the words of Proverbs 31,10–12: “A wife of noble character [. . . ] is worth far more than rubies. Her husband [. . . ] lacks nothing of value. She brings him good [. . . ] all the days of her life.” We are all very grateful to our families for their love and understanding during the entire process of writing this book. Finally, we would like to thank Dr. Catriona Byrne as well as Susanne Denskus and Ute McCrory of Springer Verlag for their kind, unfainting editorial assistance in the long process of publishing this work.

Contents

1

Hyperfinite Dirichlet Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Hyperﬁnite Quadratic Forms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Domain of the Symmetric Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Resolvent of the Symmetric Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Weak Coercive Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Hyperﬁnite Dirichlet Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Hyperﬁnite Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Weak Coercive Quadratic Forms, Revisited . . . . . . . . . . . . . . . . . . . . .

1 2 6 17 26 36 52 61

2

Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Exceptional Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Exceptional Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Co-Exceptional Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Excessive Functions and Equilibrium Potentials . . . . . . . . . . . . . . . . 2.3 Capacity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Relation of Exceptionality and Capacity Theory . . . . . . . . . . . . . . . 2.5 Measures of Hyperﬁnite Energy Integrals . . . . . . . . . . . . . . . . . . . . . . . 2.6 Internal Additive Functionals and Associated Measures . . . . . . . 2.7 Fukushima’s Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Decomposition Under the Individual Probability Measures Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Decomposition Under the Whole Measure P . . . . . . . . . . . 2.8 Internal Multiplicative Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Internal multiplicative functionals . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Subordinate Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 Subprocesses .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Feynman-Kac Formulae .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Alternative Expression of Hyperﬁnite Dirichlet Forms . . . . . . . . . 2.10 Transformations of Symmetric Dirichlet Forms .. . . . . . . . . . . . . . . .

65 66 66 71 73 78 86 91 102 107 107 118 120 120 121 122 123 124 125

xiii

xiv

3

Contents

Standard Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Standard Parts of Hyperﬁnite Markov Chains . . . . . . . . . . . . . . . . . . 3.1.1 Inner Standard Part of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Strong Markov Processes and Modiﬁed Standard Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hyperﬁnite Dirichlet Forms and Markov Processes . . . . . . . . . . . . 3.2.1 Separation of Points .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Nearstandardly Concentrated Forms . . . . . . . . . . . . . . . . . . . . 3.2.3 Quasi-Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Construction of Strong Markov Processes .. . . . . . . . . . . . . .

129 130 131 136 145 146 151 155 160

4

Construction of Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Hyperﬁnite Lifts of Quasi-Regular Dirichlet Forms . . . . . . . . . . . . 4.3 Relation with Capacities.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Path Regularity of Hyperﬁnite Markov Chains .. . . . . . . . . . . . . . . . 4.5 Quasi-Continuity and Nearstandard Concentration . . . . . . . . . . . . 4.6 Construction of Strong Markov Processes .. . . . . . . . . . . . . . . . . . . . . . 4.7 Necessity for Existence of Dual Tight Markov Processes . . . . . .

165 166 170 177 180 181 192 197

5

Hyperfinite Lévy Processes and Applications . . . . . . . . . . . . . . . . . . 5.1 Standard Lévy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Characterizing Hyperﬁnite Lévy Processes .. . . . . . . . . . . . . . . . . . . . . 5.3 Hyperﬁnite Lévy Processes: Standard Parts . . . . . . . . . . . . . . . . . . . . 5.4 Hyperﬁnite Lévy-Khintchine Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Representation Theorem for Lévy Processes . . . . . . . . . . . . . . . . . . . . 5.6 Extensions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 200 203 217 224 232 239

6

Genericity of Loeb Path Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Adapted Probability Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Universality, Saturation, Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Hyperﬁnite Adapted Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Probability Logic and Markov Processes . . . . . . . . . . . . . . . . . . . . . . . .

243 244 245 246 247

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 General Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Structure of ∗ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Internal Sets and Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Loeb Measure.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 249 250 251 252 253

Historical Notes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Chapter 1

Hyperfinite Dirichlet Forms

The interplay between methods from functional analysis and the theory of stochastic processes is one of the most important and exciting aspects of mathematical physics today. It is a highly technical and sophisticated theory based on decades of research in both areas. Numerous papers have been written on the standard theory of Dirichlet forms. Apart from the articles and monographs cited below, other notable contributions to the area include: Albeverio and Bernabei [5], Albeverio, Kondratiev, and Röckner [32], Albeverio and Kondratiev [33], Albeverio and Ma [39], Albeverio, Rüdiger, and Wu [54], Bliedtner [94], Bouleau [98], Bouleau and Hirsch [99], Chen et al. [112], Chen, Ma, and Röckner [116], Eberle [149], Exner [154], Fabes, Fukushima, Gross, Kenig, Röckner, and Stroock [155], Fitzsimmons and Kuwae [172], Fukushima [177,179,180], Fukushima and Tanaka [185], Fukushima and Ying [188, 189], Gesztesy et al. [191, 192], Grothaus et al. [198], Hesse et al. [208], Jacob [218–220], Jacob and Moroz [221], Jacob and Schilling [222], Jost et al. [225], Kassmann [232], Kim et al. [240], Kumagai and Sturm [248], Le Jan [258], Liskevich and Röckner [265], Ma and Röckner [272, 273], Ma et al. [274], Mosco [283], Okura [292], Oshima [294, 295], D.W. Robinson [312], Röckner and Wang [317], Röckner and Zhang [319], Schmuland and Sun [329], Shiozawa and Takeda [331], da Silva et al. [332], Stannat [336, 338], Stroock [340], Sturm [343], Takeda [346, 347], Wu [363], and Yosida [364]. In this monograph, we present the theory of Dirichlet forms from a uniﬁed vantage point, using nonstandard analysis, thus viewing the continuum of the time line as a discrete lattice of inﬁnitesimal spacing. This approach is close in spirit to the discrete classical formulation of Dirichlet space theory in A. Beurling and J. Deny’s seminal article [87]. The discrete setup in this monograph permits to study the diﬀusion and the jump part by essentially the same methods. This setting being independent of special topological properties of the state space, it is also considerably less technical than other approaches. Thus, the theory has found its natural setting and no longer depends on choosing particular topological spaces; in particular, it is valid for both ﬁnite and inﬁnite dimensional spaces.

S. Albeverio et al., Hyperfinite Dirichlet Forms and Stochastic Processes, Lecture Notes of the Unione Matematica Italiana 10, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19659-1_1,

1

2

1 Hyperﬁnite Dirichlet Forms

Whilst Albeverio et al. [25], Chap. 5, only discussed symmetric hyperﬁnite Dirichlet forms and related Markov chains (refer to [165, 166] also), we shall extend the theory to the nonsymmetric case. We shall try to follow as much as possible the path suggested by the work on the symmetric case. An important sub-class of Markov process are Feller processes with stationary and independent increments (Lévy processes), and in recent years, these processes have attracted a lot of interest, including from nonstandard analysts. Initiated by T. Lindstrøm [263], a number of articles have been devoted to the investigation of hyperﬁnite Lévy processes. Chapter 5 of this monograph is a detailed exposition of Lindstrøm’s theory [263] and its subsequent continuation by Albeverio and Herzberg [14]. The book ends with an expository summary (without proofs) of the model theory of stochastic processes as developed by H.J. Keisler and his coauthors, who formulated and proved the “universality” of hyperﬁnite adapted probability spaces in a rigorous manner, and a short description of recent fundamental results about the deﬁnability of nonstandard universes. Meanwhile, our purpose in the ﬁrst chapter is to develop a general theory of hyperﬁnite quadratic forms. We shall set the scene in Sect. 1.1. Sections 1.2 and 1.3 will study the domains of symmetric parts, the standard parts and resolvents. We shall discuss the property of weak coercive quadratic forms in Sects. 1.4 and 1.7. In Sect. 1.5, we shall study Markov forms and begin the analysis of associated Markov chains and get the basic Beurling–Deny formula. We discuss the hyperﬁnite lifting theory of standard Dirichlet forms in Sect. 1.6.

1.1 Hyperfinite Quadratic Forms We shall develop a hyperﬁnite theory of nonnegative quadratic forms on inﬁnite dimensional spaces. It is well-known that in the Hilbert space case the theory of closed forms of this kind is equivalent to the theory of nonnegative operators. In fact, there is a natural correspondence between forms E(·, ·) and operators A given by E(u, u) = Au, u, where ·, · is the scalar product in the Hilbert space. We have chosen to present the theory in terms of forms and not operators for two reasons: partly because forms are real-valued, and this makes it simpler to take standard parts, but also because in most of our applications, the form is what is naturally given. Let H be an internal, hyperﬁnite dimensional linear space1 equipped with an inner product ·, · generating a norm || · ||. Let ∗ R be the nonstandard

1

The notions of hyperﬁnite dimensional linear space are given in Albeverio et al. [25].

1.1 Hyperﬁnite Quadratic Forms

3

real line2 . We call a map E : H × H −→ ∗ R nonnegative quadratic form if and only if for all α ∈ ∗ R, u, v, w ∈ H, E(u, u) ≥ 0, E(αu, v) = αE(u, v), E(u, αv) = αE(u, v), E(u + v, w) = E(u, w) + E(v, w), E(w, u + v) = E(w, u) + E(w, v). Since E(·, ·) is a nonnegative quadratic form on the hyperﬁnite dimensional space H, elementary linear algebra tells us that there is a unique nonnegative deﬁnite operator A : H −→ H such that E(u, v) = Au, v

for all

u, v ∈ H.

(1.1.1)

To see this, let ∗ N0 be the nonstandard integers3. Let {ei | 1 ≤ i ≤ N } be an N orthonormal basis of (H, ·, ·) for an N ∈ ∗ N. We put Aei = j=1 E(ei , ej )ej . Then (1.1.1) follows immediately. Hence, A is given by the matrix A = (E(ei , ej ))1≤i,j≤N , i.e., ⎛

⎞ E(e1 , e1 ) E(e1 , e2 ) . . . E(e1 , eN ) ⎜ E(e2 , e1 ) E(e2 , e2 ) . . . E(e2 , eN ) ⎟ ⎜ ⎟ A=⎜ ⎟. .. .. .. .. ⎝ ⎠ . . . . E(eN , e1 ) E(eN , e2 ) . . . E(eN , eN )

(1.1.2)

Moreover, Au, u ≥ 0 for all u ∈ H. This means that A is a hyperﬁnite dimensional matrix (not necessarily symmetric). Let Aˆ be the adjoint operator of A, that is, ˆ E(u, v) = u, Av

for all

u, v ∈ H.

ˆ are the By (1.1.2), we have that Aˆ is the transpose of A. If ||A|| and ||A|| ˆ ˆ operator norms of A and A, respectively, we have ||A|| = ||A||. We ﬁx an inﬁnitesimal4 Δt such that

2 ∗R

is the standard notation for the nonstandard real line, refer to Appendix, Albeverio et al. [25], Cutland [125], Davis [135], Hurd [216], Hurd and Loeb [217], Lindstrøm [262], Stroyan and Bayod [341], and Stroyan and Luxemburg [342]. 3 ∗ N is the standard notation for the nonstandard integers, refer to Appendix, Albeverio 0 et al. [25], Cutland [125], Davis [135], Hurd [216], Hurd and Loeb [217], Lindstrøm [262], Stroyan and Bayod [341], and Stroyan and Luxemburg [342]. 4

In the sense of nonstandard analysis, refer to Appendix, Albeverio et al. [25], Keisler [237, 238], Stroyan and Bayod [341], and Stroyan and Luxemburg [342].

4

1 Hyperﬁnite Dirichlet Forms

0 < Δt ≤

1 1 = . ˆ ||A|| ||A||

(1.1.3)

ˆ Δt by Let us deﬁne new operators QΔt and Q QΔt = I − ΔtA, ˆ Δt = I − ΔtA. ˆ Q ˆ Δt are nonnegative. The relation (1.1.3) implies that the operators QΔt and Q ˆ Δt are less than Because A is nonnegative, the operator norms of QΔt and Q ˆ ·) or equal to one. Similarly, we deﬁne the nonnegative quadratic co-form E(·, of E(·, ·) by ˆ v) = E(v, u) for all u, v ∈ H. E(u, Introduce a nonstandard time line T by T = {kΔt | k ∈ ∗ N0 }. ˆ t to be the operators For each element t = kΔt in T , deﬁne Qt and Q Qt = (QΔt )k , ˆ t = (Q ˆ Δt )k . Q ˆ t }t∈T are obviously semigroups. We shall call The families {Qt }t∈T and {Q t ˆ t }t∈T the co-semigroup associated with E(·, ·) {Q }t∈T the semigroup and {Q ˆ ·), A, A, ˆ T, Qt and Q ˆ t in and Δt, respectively. Whenever we refer to E(·, ·), E(·, the rest of this book, we shall assume that they are linked by above relations. In applications, the primary objects will often be the semigroup {Qt }t∈T ˆ t }t∈T . We can then deﬁne A and Aˆ (and hence E(·, ·)) by and co-semigroup {Q

1 I − QΔt , Δt 1 ˆ Δt . I −Q Aˆ = Δt

A=

The operator A is called the inﬁnitesimal generator of E(·, ·), and Aˆ is called the inﬁnitesimal co-generator of E(·, ·). For each t ∈ T, we may deﬁne approximations A(t) of A and Aˆ(t) of Aˆ by

1 I − Qt , t 1 ˆt . I −Q = t

A(t) = Aˆ(t)

(1.1.4)

1.1 Hyperﬁnite Quadratic Forms

5

From A(t) and Aˆ(t) , we get the forms E (t) (u, v) = A(t) u, v = u, Aˆ(t) v,

(1.1.5)

and Eˆ(t) (u, v) = E (t) (v, u) = Aˆ(t) u, v = A(t) v, u. ˚ ·) of We deﬁne the symmetric part E(·, ·) and anti-symmetric part E(·, E(·, ·) by 1 E(u, v) + E(v, u) , 2 1 ˚ E(u, v) = E(u, v) − E(v, u) . 2 E(u, v) =

For α ∈ ∗ R, α ≥ 0, we set E α (u, v) = E(u, v) + αu, v. Each of these forms generates a norm (possibly a semi-norm in the case α = 0):

E α (u, u) = Eα (u, u).

|u|α =

We recall that the original Hilbert space norm on H is denoted by || · ||. Similarly, we set for α ∈ ∗ R, α ≥ 0, Eα (u, v) = E(u, v) + αu, v, ˆ v) + αu, v. Eˆα (u, v) = E(u, t

Let A and {Q } be the generator and semigroup of E(·, ·), respectively. Then A=

1 A + Aˆ , 2

Q

Δt

=

1 Δt ˆ Δt and QkΔt = (QΔt )k , ∀k ∈ ∗ N. Q +Q 2

t

Since A and Q are nonnegative, self-adjoint operators, they have unique 1

t

nonnegative square roots, which we denote by A 2 and Q 2 , respectively.

6

1 Hyperﬁnite Dirichlet Forms

In the same manner as (1.1.4) and (1.1.5), we can deﬁne approximations (t)

A

of A and E (t)

A

(t)

=

(·, ·) of E(·, ·) by 1 t I −Q , t

E

(t)

(t)

(u, v) = A u, v,

t ∈ T.

If a nonnegative quadratic form E(·, ·) : H × H −→ ∗ R satisﬁes E(u, v) = E(v, u) for all u, v ∈ H, i.e., E(u, v) = E(u, v), we shall call it a nonnegative symmetric quadratic form. It is easy to see that a nonnegative quadratic form E(u, v) is symmetric if ˆ t , ∀t ∈ T. and only if A = Aˆ or Qt = Q In this book, we shall deal with nonnegative quadratic forms E(·, ·) and the related theory. For the framework, potential theory and applications of nonnegative symmetric quadratic form, we refer the reader to Albeverio et al. [25], Chap. 5, Sect. 5.1 and Fan [165, 166]. We shall utilize the known results of symmetric forms in our study, and extend them to the nonsymmetric case. In particular, we need the notion of the symmetric part E(·, ·) of E(·, ·), and the related notations. In Sect. 1.2, we shall deﬁne the domain D(E) of the t symmetric part E(·, ·) by using the semigroup {Q | t ∈ T }. We shall introduce the resolvent {Gα | α ∈ ∗ (−∞, 0)} of E(·, ·) in Sect. 1.3, and characterize the domain D(E) by this resolvent. In Sect. 1.4, we shall deﬁne the domain D(E) of E(·, ·) by its resolvent {Gα | α ∈ ∗ (−∞, 0)}; under the hyperﬁnite weak sector condition, we shall show that D(E) = D(E). In Sect. 1.5, we shall introduce hyperﬁnite Dirichlet forms and related Markov chains. For standard coercive forms, we shall construct their nonstandard representation in Sect. 1.6.

1.2 Domain of the Symmetric Part In this section, we shall deﬁne the domain D(E) of the symmetric part E(·, ·) for a hyperﬁnite nonnegative quadratic form E(·, ·). Before giving a strict deﬁnition (Deﬁnition 1.2.1), we shall mention an intuitive description. At ﬁrst, let Fin(H) be the set of all elements in H with ﬁnite norm. By deﬁning x ≈ y if ||x − y|| ≈ 0, we know from Proposition A.5.2 in the Appendix that the space5 ◦

H = Fin(H)/ ≈

5

≈ stands for diﬀering by an inﬁnitesimal, in the sense of nonstandard analysis, refer to Albeverio et al. [25], Cutland [125], Davis [135], Hurd [216], Hurd and Loeb [217], and Lindstrøm [262].

1.2 Domain of the Symmetric Part

7

is a Hilbert space with respect to the inner product (◦x, ◦y) = st(x, y), where ◦ x denotes the equivalence class of x and st : ∗ R −→ R is the mapping of standard part6 . We call (◦H, (·, ·)) the hull of (H, ·, ·). Consider the standard part E(·, ·) of the nonnegative symmetric quadratic form E(·, ·). If E(·, ·) is S-bounded, i.e., there exists a constant K ∈ R+ such that |E(u, v)| ≤ K||u||||v||

for all u, v ∈ H,

we can simply deﬁne E(·, ·) by E(◦u, ◦v) = ◦ E(u, v). If E(·, ·) is not S-bounded, we shall meet two diﬃculties. We no longer have that E(u, v) ≈ E(˜ u, v˜) whenever u ≈ u ˜ and v ≈ v˜, and there may be elements v ∈ Fin(H) such that E(˜ v , v˜) is inﬁnite for all v˜ ≈ v. The latter problem should not surprise us. It is an immediate consequence of the fact that unbounded forms on Hilbert spaces cannot be deﬁned everywhere. We v , v˜) is inﬁnite shall solve it by simply letting E(◦u, ◦v) be undeﬁned when E(˜ for all v˜ ∈ ◦v. The most natural solution to the ﬁrst problem may be to deﬁne E(◦u, ◦u) = inf{◦ E(v, v) | v ∈ ◦u},

(1.2.1)

and then extend E(·, ·) to be a bilinear form by the usual trick E(◦u, ◦v) =

1 ◦ E( u + ◦v, ◦u + ◦v) − E(◦u, ◦u) − E(◦v,◦v) . 2

The disadvantage of this approach is that it gives us little understanding of how the inﬁmum in (1.2.1) is obtained. For an easier access to the regularity properties of E(·, ·) and E(·, ·), we prefer a more indirect way of attack. Our plan is to deﬁne a subset D(E) of Fin(H) – we call it the domain of E(·, ·) – satisfying if ◦ E(u, u) < ∞, there is a v ∈ D(E) such that v ≈ u, if u, v ∈ D(E) and u ≈ v, then ◦ E(u, u) = ◦ E(v, v) < ∞.

(1.2.2) (1.2.3)

We then deﬁne E(·, ·) by E(◦u, ◦u) = ◦ E(v, v),

6

Refer to Albeverio et al. [25].

(1.2.4)

8

1 Hyperﬁnite Dirichlet Forms

when v ∈ D(E) ∩ ◦u. It turns out that the two deﬁnitions (1.2.1) and (1.2.4) agree (see Proposition 1.2.4). If we look at the standard nonsymmetric Dirichlet theory, see Albeverio et al. [9], Kim [241] and Ma and Röckner [270], the domain of a quadratic form is given from the very beginning. After that, the authors such as those of Ma and Röckner [270] introduced the symmetric and anti-symmetric parts (see page 15, [270]). This method makes the domains of the quadratic form and its symmetric part coincide. On the other hand, Albeverio et al. [25] has given us a very nice deﬁnition of domain for the symmetric hyperﬁnite quadratic forms by their semigroups. Therefore, we may deﬁne the domain t D(E) of E(·, ·) via the semigroup of {Q | t ∈ T }. In the next section, we shall discuss the property of the resolvent {Gα | α ∈ ∗ (−∞, 0)} of E(·, ·). We can deﬁne the domain of D(E) through {Gα | α ∈ ∗ (−∞, 0)}. Now it is very natural to ask: can we as well deﬁne the domain D(E) of E(·, ·) directly from {Qt | t ∈ T }? Here we would mention that it seems not easy to do the job. In Sect. 1.4, we shall deﬁne D(E) by means of the resolvent {Gα | α < 0} of E(·, ·). Under the hypothesis of weak sector condition, we shall prove D(E) = D(E) by showing that the two deﬁnitions satisfy (1.2.1). This is similar to the procedure in the standard nonsymmetric Dirichlet space theory, see, e.g., Albeverio et al. [9], Albeverio et al. [47], Albeverio and Ugolini [57], Kim [241], and Ma and Röckner [270]. (t)

Notice that even when E(·, ·) is not S-bounded, E (·, ·) is S-bounded for all non-inﬁnitesimal t. One of the motivations behind our deﬁnition of the domain D(E) is that we want to single out the elements where E(·, ·) is really approximated by the bounded forms E that ◦

(t)

E(u, u) = lim ◦ E t↓0 t≈0

(·, ·), t ≈ 0, i.e., those u ∈ H such (t)

(u, u).

(1.2.5)

We could have taken this to be our deﬁnition of D(E), but for technical and expository reasons we have chosen another one which we shall soon show to be equivalent to (1.2.5) (see Proposition 1.2.2). Definition 1.2.1. Let E(·, ·) be a nonnegative quadratic form on a hyperﬁnite dimensional linear space H. The domain D(E) of the symmetric part of E(·, ·) is the set of all u ∈ H satisfying (i) ◦ E1 (u, u) = ◦ E 1 (u, u) < ∞. t t (ii) For all t ≈ 0, E(Q u, Q u) ≈ E(u, u). Let us try to convey the intuition behind this deﬁnition. Thinking of A as a diﬀerential operator, the elements of D(E) are “smooth” functions and

1.2 Domain of the Symmetric Part

9

t

Q is a “smoothing” operator often given by an integral kernel. If an element t u is already smooth, then an inﬁnitesimal amount of smoothing Q , t ≈ 0, t t should not change it noticeably, and hence E(Q u, Q u) ≈ E(u, u). We shall give a partial justiﬁcation of this rather crude image later, when we show t that if ◦ E 1 (u, u) < ∞, then the “smoothed” elements Q u, t ≈ 0, are all in D(E) (Lemma 1.2.3, see also Corollary 1.2.3). Our ﬁrst task will be to establish a list of alternative deﬁnitions of D(E), among them (1.2.5). We begin with the following simple identity giving the relationship between E(·, ·) and E (t) (·, ·), and also the relationship between E(·, ·) and E

(t)

(·, ·) :

Lemma 1.2.1. For all u ∈ H and t ∈ T , we have (i) (ii) (iii)

(t)

E (t) (u, u) ≥ 0 and E (u, u) ≥ 0, Δt Δt ˆ s u), E (t) (u, u) = E(Qs u, u) = E(u, Q t t E

(t)

0≤s 0. For α ∈ ∗ (−∞, 0), we deﬁne for u, v ∈ ∗ K (α)

F (u, v) = −α(u + αRα u, v) ˆ α v, u) = −α(v + αR

and (α)

Fˆ (u, v) = −α(v + αRα v, u) ˆ α u, v). = −α(u + αR

Then ˆ α u, v) for all u ∈ F (u, v) = F (−αRα u, v) and (α) Fˆ (u, v) = F (−αR ∗ ∗ K, v ∈ (D(F )) and α ∈ (−∞, 0). ˆ α u, αR ˆ α u) ≤ (α) F (u, u) for all (ii) F (αRα u, αRα u) ≤ (α) F (u, u) and F (αR ∗ ∗ u ∈ K and α ∈ (−∞, 0). (iii) |(α) F 1 (u, v)| ≤ (C + 1) F1 (u, u) (α) F 1 (v, v) for all u ∈ ∗ (D(F )), v ∈ ∗ K and α ∈ ∗ (−∞, 0). (iv) F1 (αRα u, αRα u) ≤ (C + 1)2 F1 (u, u) for all u ∈ ∗ (D(F )) and α ∈ ∗ (−∞, 0). (i)

(α) ∗

Proof. The proof follows from Proposition 1.6.2.

When are the standard forms generated by two hyperﬁnite forms diﬀerent? The last result in this section we shall prove shows that to answer this question, it is enough to check whether the forms have the same resolvents. We recall that in Theorem 1.4.1 we found a way to construct a form from its resolvent. This representation will be helpful to solve our problem. Theorem 1.6.2. Let K be a Hilbert space and H be an S-dense, hyperﬁnite ˘ ·) be two hyperﬁnite weak dimensional subspace of ∗ K. Let E(·, ·) and E(·, ˘K (·, ·) on K, respeccoercive quadratic forms on H inducing EK (·, ·) and E ˘ α } be the resolvents of E(·, ·) and E(·, ˘ ·). Assume tively. Let {Gα } and {G ∗ that for some ﬁnite, non-inﬁnitesimal α ∈ (−∞, 0), there is a u ∈ H with ◦ ˘ α u are both nearstandard, but E1 (u, u) < ∞ such that v = Gα u, w = G ◦ ˘ ||v − w|| = 0. Then EK (·, ·) = EK (·, ·). Proof. Assume for contradiction that EK (·, ·) = E˘K (·, ·). Pick v˜ ≈ v, w ˜≈w ˘ w such that v˜ ∈ D(E), ˜ ∈ D(E). Notice that by Lemma 1.4.5, v ∈ D(E), w ∈ ˘ We have D(E).

1.7 Weak Coercive Quadratic Forms, Revisited

61

u, v − w ≈ u, v − w ˜ = E−α (v, v − w). ˜

(1.6.8)

Since v, w are nearstandard and EK (·, ·) = E˘K (·, ·), we have ◦

E −α (v, v − w) ˜ = EK (◦ v, ◦ v − ◦ w) − (◦ α)(◦ v, ◦ v − ◦ w) ˘K (◦ v, ◦ v − ◦ w) − (◦ α)(◦ v, ◦ v − ◦ w) =E v , v˜ − w). = ◦ E˘−α (˜

(1.6.9)

On the other hand, we have u, v − w ≈ u, v˜ − w = E˘−α (w, v˜ − w).

(1.6.10)

Combining the relations (1.6.8), (1.6.9), and (1.6.10), we see that v − w, v˜ − w) 0 = ◦ E˘−α (˜ ◦ ◦ ≥ |α| ||v − w||2 > 0.

The theorem is proved.

1.7 Weak Coercive Quadratic Forms, Revisited Let E(·, ·) be a hyperﬁnite weak coercive quadratic form on a hyperﬁnite dimensional space H. Let {Gα | α ∈ ∗ (−∞, 0)} be the resolvent of E(·, ·), and ˆ α | α ∈ ∗ (−∞, 0)} be the co-resolvent of E(·, ·), respectively. Let us still let {G denote by A the inﬁnitesimal generator of E(·, ·), and by Aˆ the inﬁnitesimal co-generator of E(·, ·). Such as in Sect. 1.1, we ﬁx an inﬁnitesimal Δt, and we ˆ Δt by deﬁne new operators QΔt and Q QΔt = I − ΔtA, ˆ Δt = I − ΔtA. ˆ Q Introduce a nonstandard time line T by T = {kΔt | k ∈ ∗ N0 }. For each ˆ t to element t = kΔt ∈ T , deﬁne the semigroup Qt and the co-semigroup Q be the families of operators Qt = (QΔt )k , ˆ t = (Q ˆ Δt )k , t ∈ T. Q

62

1 Hyperﬁnite Dirichlet Forms

For each t ∈ T, we may deﬁne approximations A(t) of A and Aˆ(t) of Aˆ by

1 I − Qt , t 1 ˆt . I −Q = t

A(t) = Aˆ(t)

From A(t) and Aˆ(t) , we get the forms E (t) (u, v) = A(t) u, v = u, Aˆ(t) v. Let E(·, ·) be the standard part of E(·, ·). Then E(·, ·) is closed. In addition, (E(·, ·), D(E)) satisﬁes the weak sector condition by Remark 1.4.1. Hence, (E(·, ·), D(E)) is a coercive closed form on ◦H. Let {Rβ | β ∈ (−∞, 0)} ˆ β | β ∈ (−∞, 0)} be the resolvent and co-resolvent of (E(·, ·), D(E)), and {R respectively. Similarly, let {Tt | t ∈ [0, ∞)} and {Tˆt | t ∈ [0, ∞)} be the semigroup and co-semigroup of (E(·, ·), D(E)), respectively. For t ∈ (0, ∞), we deﬁne E (t) (x, y) =

1 (x − Tt x, y), x, y ∈ ◦H. t

By Albeverio et al. [9], Theorem 3.4 (or referring to Proposition 1.6.4), we have Lemma 1.7.1. (i) Let x ∈ ◦H. Then x ∈ D(E) if and only if supt>0 E (t) (x, x) < ∞. (ii) For all x, y ∈ D(E), we have lim E (t) (x, y) = E(x, y). t↓0

(iii) For all x ∈ D(E), we have lim E1 (x − Tt x, x − Tt x) = 0. t↓0

By applying Lemma 1.7.1, we can see from the proof of Theorem 1.6.1 that there exists an inﬁnitesimal δ ∈ T such that (E(·, ·), D(E)) is the standard part of E (δ) (u, v) (referring to Remark 1.6.1, and replacing −γRγ by Qδ in the proof of Theorem 1.6.1). In addition, for any x ∈ ◦H, u ∈ x, and for all t ∈ [0, ∞), s ∈ {kδ | k ∈ ∗ N0 }, t = ◦ s, we have that Qs u ∈ Tt x. Since Qs1 +s2 ≈ Qs1 if s2 ≈ 0, we have that Qs u ∈ Tt x for all t ∈ [0, ∞), s ∈ T = {kΔt | k ∈ ∗ N0 }, t = ◦ s. Notice that (E(·, ·), D(E)) is the standard part of both E(u, v) and E (δ) (u, v), and so the resolvent of E(u, v) is almost the same as that of E (δ) (u, v) by Theorem 1.6.2. Summarizing above results, we have Theorem 1.7.1. Let E(·, ·) be a hyperﬁnite weak coercive quadratic form on a hyperﬁnite dimensional space H. Then, we have

1.7 Weak Coercive Quadratic Forms, Revisited

63

(i) Let u ∈ H. Then u ∈ D(E) if and only if sups ◦ E (s) (u, u) < ∞. (ii) For all u, v ∈ D(E), we have lim E (s) (u, v) = E(u, v).

◦ s↓0

(iii) For all u ∈ D(E), we have lim E1 (u − Qs u, u − Qs u) ≈ 0.

◦ s↓0

Proof. Let x = ◦ u. Then for all t ∈ [0, ∞), s ∈ T = {kΔt | k ∈ ∗ N0 }, t = ◦ s, we have that Qs u ∈ Tt x. By Lemma 1.7.1, the theorem follows easily.

Proposition 1.7.1. Let E(·, ·) be a hyperﬁnite weak coercive quadratic form on a hyperﬁnite dimensional linear space H. If ◦ E(u, u) < ∞, then for all ﬁnite s > 0, s ∈ T, and ◦ s = 0, we have ◦ [Qs u] ∈ D(E). Proof. Let x = ◦ u. For t = ◦ s, we have that Qs u ∈ Tt x. Therefore, we have [Qs u] = Tt x ∈ D(E).

◦

Chapter 2

Potential Theory of Hyperfinite Dirichlet Forms

Probabilistic potential theory has been a very important component in the study of hyperﬁnite Dirichlet space theory. It provides a probabilistic interpretation of potential theory; and, more generally, it establishes a beautiful bridge between functional analysis and the theory of Markov processes. There are many applications of this theory, especially in the area of inﬁnite dimensional stochastic analysis and mathematical physics. Our purpose in this chapter is to develop the probabilistic potential theory associated with hyperﬁnite Dirichlet forms and the related Markov chains. The motivation is twofold. On the one hand, we want to establish a relationship between the standard Dirichlet space theory and the hyperﬁnite counterpart. On the other hand, we want to provide new methods for the theory of hyperﬁnite Dirichlet forms itself. Inﬁnite dimensional stochastic analysis has been developed extensively in the last decades. We hope to convince the reader that nonstandard analysis can provide a new tool to deal with problems in this exciting area, see particularly Chap. 4, for example. The arrangement of the present chapter is as follows. In Sect. 2.1, we shall deﬁne exceptional sets for non-symmetric hyperﬁnite Markov chains. Sect. 2.2 will discuss excessive functions and equilibrium potentials. Moreover, we introduce a capacity theory for hyperﬁnite quadratic forms and show that it is a Choquet capacity in Sect. 2.3. Furthermore, we establish a relation between the family of exceptional sets and the family of zero capacity sets in Sect. 2.4. In Sect. 2.5, we consider positive measures of hyperﬁnite energy integrals and the associated theory. That is, we establish connections among hyperﬁnite excessive functions and hyperﬁnite potentials. Zero capacity subsets will be characterized by positive measures of hyperﬁnite energy integrals. In Sect. 2.6, we introduce internal additive functionals. The relationship between hyperﬁnite measures and additive functionals will be considered. Moreover, we shall obtain a positive hyperﬁnite measure μ associated with an internal function u. In Sect. 2.7, we get Fukushima’s decomposition theorem under individual probability measures. This extends the work of Albeverio et al. [25] and Fan [166]. In Sect. 2.8, we shall discuss the properties of internal multiplicative functionals, subordinate semigroups, subprocesses,

S. Albeverio et al., Hyperfinite Dirichlet Forms and Stochastic Processes, Lecture Notes of the Unione Matematica Italiana 10, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19659-1_2,

65

66

2 Potential Theory

and a Feynman-Kac formula. The motivation for this work is given by the corresponding standard theory developed by Blumenthal and Getoor [96]. This serves as a basis for the development of a perturbation theory of hyperﬁnite Dirichlet forms characterized by internal additive functionals. The content of Sect. 2.9 is the nonstandard version of time change of standard Dirichlet forms. In standard Dirichlet space theory, the question how to change a nonconservative symmetric Markov process into a conservative one has received an answer in Fukushima and Takeda [184] in terms of the Girsanov transformation. In Sect. 2.10, we show that this problem is quite simple in the hyperﬁnite setting (Theorem 2.10.1). In this chapter, we shall use the notations developed in Sect. 1.5. However, we need not assume all the conditions required in Sect. 1.5. In Sect. 2.1, we will deﬁne exceptional sets for a hyperﬁnite Markov chain X. We do not need anything about the dual hyperﬁnite Markov chain of X. Starting from Sect. 2.2, we shall work under the setting of Sect. 1.5. That is, we shall assume the conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14) in Sect. 1.5, and the related notations.

2.1 Exceptional Sets Albeverio et al. [25] has given us a deﬁnition of exceptional sets in the framework of the theory of symmetric hyperﬁnite Dirichlet forms, which is too restrictive for certain cases. In Fan [166], we extended this concept in the same symmetric setting by removing a lot of unnecessary assumptions. In this section, we shall deﬁne exceptional sets for non-symmetric hyperﬁnite Markov chains.

2.1.1 Exceptional Sets We take an inﬁnitesimal Δt such that Δt > 0. Set T = {kΔt | k ∈ ∗ N0 } .

(2.1.1)

Let Y be a Hausdorﬀ space and let ∗ Y be the nonstandard extension of Y . Let S = {s0 , s1 , . . . , sN } be an S-dense subset of ∗ Y for some N ∈ ∗ N − N and let m be a hyperﬁnite measure on S. Denote by S the internal algebra of subsets of S. Let Q = {qij } be an (N + 1) × (N + 1) matrix with nonnegative entries. Assume that

2.1 Exceptional Sets

67 N

qij = 1 for all i = 0, 1, . . . , N,

(2.1.2)

j=0

and assume that the state s0 is a trap, i.e., q0i = 0

for all

i = 0.

(2.1.3)

If (Ω, P ) is an internal measure space, and X : Ω × T −→ S is an internal process, let [ω]t = {ω ∈ Ω | X(ω , s) = X(ω, s) for all s ≤ t}.

(2.1.4)

For each t ∈ T, let Ft be the internal algebra on Ω generated by the sets [ω]t . Assume that for all ω ∈ Ω, we have P ([ω]0 ) = m{X(ω, 0)},

(2.1.5)

and whenever X(ω, t) = si , we have P {ω ∈ [ω]t | X(t + Δt, ω ) = sj } = qij P ([ω]t ).

(2.1.6)

That is, X is a hyperﬁnite Markov chain with the initial distribution m and transition matrix Q. The family (Ω, Ft , Pi , i ∈ S) of internal probability spaces is deﬁned by k−1 qω(nΔt),ω((n+1)Δt) , Pi [ω]kΔt = δiω(0) n=0

for each i ∈ S, where δij is the Kronecker symbol as in Chap. 1. As in Sect. 1.5, let δ ∈ T and let the sub-line Tδ = {kδ | k ∈ ∗ N0 } . Set T fin = {t ∈ T | t is ﬁnite} , Tδfin = {t ∈ Tδ | t is ﬁnite} . In addition, for r ∈ T, we let T r = {t ∈ T | t ≤ r} , Tδr = {t ∈ Tδ | t ≤ r} . Moreover, we know that X (δ) is the restriction X|Tδ .

(2.1.7)

68

2 Potential Theory

For every y ∈ Y, let us deﬁne the monad μ(y) of y by μ(y) =

{∗ O | O is open such that y ∈ O} .

We call a point y ∈ ∗ Y nearstandard if and only if y ∈ μ(x) for some x ∈ Y. Denote by N s(∗ Y ) the set of all nearstandard points in ∗ Y . Since Y is a Hausdorﬀ topological space, each element y ∈ N s(∗ Y ) is nearstandard to exactly one element x in Y (we refer to page 48, 2.1.6. Proposition, [25]). We call x the standard part of y and denote it by ◦y or st(y). In particular, we can take Y = R and use this notation also. Definition 2.1.1. (i) A subset B of S0 is called δ-exceptional for X if L(P ) ω ∃t ∈ Tδfin X(ω, t) ∈ B = 0,

(2.1.8)

where L(P ) is the Loeb measure of P . (ii) A subset B of S0 is called exceptional for X if it is δ-exceptional for some inﬁnitesimal δ ∈ T . Remark 2.1.1. For symmetric hyperﬁnite Markov chains, δ-exceptional sets are deﬁned in Albeverio et al. [25] in the following way fin L(P ) ω (X(ω, 0) ∈ S 0 ) ∧ ∃t ∈ Tδ X(ω, t) ∈ B = 0,

(2.1.9)

where S 0 = S0 ∩ N s(∗ Y ). Therefore, if a subset B is δ-exceptional in the sense of our Deﬁnition 2.1.1, it is δ-exceptional in the sense of (2.1.9). We have noticed this result in Fan [166] for the symmetric case. Here we deal with the general hyperﬁnite non-symmetric Markov chain X(t). Remark 2.1.2. From (2.1.8), we see that for every exceptional set B L(P ) {ω | X(ω, 0) ∈ B} = 0. This implies that L(m)(B) = 0, where L(m) is the Loeb measure of m. We have the following lemma, whose proof is easy and therefore will be omitted. Lemma 2.1.1. (i) All internal subsets B ⊂ S0 with m(B) ≈ 0 are exceptional. (ii) The families of exceptional and δ-exceptional sets are closed under countable unions.

2.1 Exceptional Sets

69

Definition 2.1.2. (i) A δ-exceptional subset A of S0 is called properly δexceptional for X if there is a family {Bm,n | m, n ∈ N} of internal subsets such that

Bm,n A= m∈N n∈N

and for all si ∈ / A, L(Pi ) ω ∃t ∈ Tδfin X(ω, t) ∈ A = 0, where L(Pi ) is the Loeb measure of Pi . (ii) A subset A of S0 is called properly exceptional for X if it is properly δ-exceptional for some δ ≈ 0, δ ∈ T. Proposition 2.1.1. If A ⊂ S0 is a δ-exceptional set, there is a properly δ-exceptional set B ⊃ A. Proof. Since A is δ-exceptional, there is an internal subset Bmn for each pair (m, n) of natural numbers such that A ⊂ Bmn ,

≤ L(P ) ω ∃t ∈ Tδfin X(ω, t) ∈ Bmn

1 . (2.1.10) n2 m

Deﬁne Cmn

1 = i ∈ S Pi ∃t ∈ Tδm X(t) ∈ Bmn ≥ nm

and A¯ =

Cmn .

m∈N n∈N

Let σ(ω) be a stopping time deﬁned by 1 ≥ σ(ω) = min t ∈ Tδ PX(ω,t) ∃s ∈ Tδm X(s) ∈ Bmn . mn Then, we have 1 fin L(P ) ω ∃t ∈ Tδ X(ω, t) ∈ Cmn mn

1 1 fin m ≥ = L(P ) ω ∃t ∈ Tδ PX(ω,t) ∃s ∈ Tδ X(s) ∈ Bmn mn mn

70

2 Potential Theory

1 = L(P ) 1(◦σ(ω) 0 (δ) (δ) ∗ ◦ n ∈ NCapα (An ) ≥ inf Capα (Al )l ∈ N − ε . By saturation, there is an inﬁnite M belonging to the above internal set. Hence, we have ◦

Cap(δ) α

∞

Al

≥ ◦ Cap(δ) α (AM )

l=1

≥ inf

◦

Cap(δ) (A ) l ∈ N − ε. l α

82

2 Potential Theory

By letting ε ↓ 0, we get ◦

Cap(δ) α

∞

Al

≥ inf

◦

l=1

Cap(δ) α (Al )l

∈N .

On the other hand, it is easy to see that ◦

Cap(δ) α

∞

Al

l=1

≤ inf

l∈N

◦

Cap(δ) (A ) . l α

If inf ◦ Cap(δ) (A ) l l ∈ N = ∞, we consider the following internal subset α for N0 ∈ N (A ) ≥ N n ∈ ∗ NCap(δ) n 0 . α By saturation and letting N0 ↑ ∞, we can show that (ii) holds.

Lemma 2.3.3. If {An | n ∈ N} is a sequence of internal subsets of S0 , then we have

(δ) ◦ ◦ Capα An ≤ Cap(δ) (2.3.2) α (An ), n∈N

n∈N

for all δ ∈ T, α ≥ 0. Proof. Set b = n∈N ◦ Cap(δ) α (An ). If b = ∞, the inequality (2.3.2) holds. In the following proof, we shall assume b < ∞. Let {An | n ∈ ∗ N} be an internal extension of {An | n ∈ N} . For every ε > 0, it follows from Lemma 2.3.1 (ii) that n n

(δ) Capα Al ≤ Cap(δ) for all n ∈ N. α (Al ) ≤ ε + b l=1

l=1

Consider the following internal set

n

n

n (δ) (δ) n ∈ N Al is internal and Capα Al ≤ Capα (Al ) ≤ b + ε . ∗

l=1

l=1

l=1

By saturation, there is an inﬁnite element M = M (ε) belonging to the above internal set. Hence, we obtain

2.3 Capacity Theory

83

Cap(δ) α

≤ Cap(δ) α

Al

M

l∈N

Al

l=1

≤

M

Cap(δ) α (Al )

l=1

≤ b + ε. By letting ε ↓ 0, we have proved the inequality (2.3.2).

Proposition 2.3.1. For all δ ∈ T, α ≥ 0, we have (i) If A and B are two subsets of S0 , A ⊂ B, then (δ) Cap(δ) α (A) ≤ Capα (B)

(2.3.3)

(ii) Let {An | n ∈ N} be a sequence of subsets of S0 , then ◦

Cap(δ) α

An

≤

n∈N

◦

Cap(δ) α (An ).

(2.3.4)

n∈N

(iii) Let {An | n ∈ N} be an increasing sequence of subsets of S0 , then ◦

Cap(δ) α

An

= sup

n∈N

◦

Cap(δ) α (An )n

∈N .

Proof. (i) The proof is immediate, using the deﬁnition. (ii) Set b = n∈N ◦ Cap(δ) α (An ). We can assume that b < ∞. Given ε > 0, for every n ∈ N, let us take an internal subset Bn such that An ⊂ Bn and (δ) Cap(δ) α (An ) ≤ Capα (Bn )

≤ Cap(δ) α (An ) +

ε . 2n+1

Therefore, we have from (i) and Lemma 2.3.3 that ◦ Cap(δ) α

An

≤

n∈N

≤

◦ Cap(δ) α

Bn

n∈N ◦

Cap(δ) α (Bn )

◦

Cap(δ) α (An ) + ε.

n∈N

≤

n∈N

By letting ε ↓ 0, we get the inequality (2.3.4).

84

2 Potential Theory

(iii) We may assume that for all n ∈ N, ◦ Cap(δ) α (An ) < ∞. Given ε > 0, for each n ∈ N, let Bn be an internal subset of S0 such that (δ) Cap(δ) α (Bn ) ≤ Capα (An ) + ε.

An ⊂ Bn ,

Then, we have from (2.3.3) and Lemma 2.3.2 (i) that ◦

Cap(δ) α

An

≤

◦

Cap(δ) α

n∈N

= sup n

≤ sup

◦

Bn

n∈N

Cap(δ) (B ) n α

◦

Cap(δ) α (An )

+ε ≤ sup ◦ Cap(δ) α (An ) + ε. n

n

By letting ε ↓ 0, we get ◦

Cap(δ) α

≤ sup

An

◦

n∈N

n ∈ N . Cap(δ) (A ) n α

(2.3.5)

On the other hand, it is easy to see that ◦

Cap(δ) α

An

≥ sup

n∈N

◦

Cap(δ) α (An )n

∈N .

(2.3.6)

From the inequalities (2.3.5) and (2.3.6), we have proved Proposition 2.3.1 (iii). For the purpose of explaining our Theorem 2.3.1 in the following, we ﬁrst introduce some notations in capacity theory (referring to, e.g., [282]). Let G be a set, G be a family of some subsets of G. Denote by Gσ (respectively, Gδ ) the closure of a collection of subsets of G under countable union (respectively, countable intersection). That is, Gσ =

∞

n=1

An An ∈ G ,

Gδ =

∞ n=1

An An ∈ G .

Moreover, we shall write Gσδ = (Gσ )δ . Definition 2.3.1. Let G be a set. A paving G on G is a family of subsets of G such that the empty set ∅ is contained in G. The pair (G, G) consisting of a set G and a paving G on G is called a paved set.

2.3 Capacity Theory

85

Definition 2.3.2. Let (G, G) be a paved set. The paving G is said to be semi-compact if every countable family of elements of G, which has the ﬁnite intersection property, has a nonempty intersection. It is easy to see that (S0 , S0 ) is a semi-compact paved set. Moreover, S0 is closed under the complement, ﬁnite union, and ﬁnite intersection operations. Definition 2.3.3. A subset A of S0 is said to be S0 -analytic if there exists an auxiliary set G with a semi-compact paving G, and a subset B ⊂ G × S0 belonging to (G × S0 )σδ such that A is the projection of B on S0 . We denote by A(S0 ) all the S0 -analytic sets (we notice that G × S0 = {G1 × S1 | G1 ∈ G and S1 ∈ S0 }). Lemma 2.3.4. The σ-ﬁeld σ(S0 ) generated by S0 is contained in A(S0 ). Proof. For every F ∈ S0 , S0 −F belongs to S0 also. By Meyer [282], Chap. III T12 Theorem, we know σ(S0 ) ⊂ A(S0 ). Definition 2.3.4. An extended real valued set function I : 2S0 → [−∞, +∞], deﬁned on all subsets 2S0 of S0 , is called a Choquet S0 -capacity if it satisﬁes the following properties: (i) I is increasing, i.e., A ⊂ B =⇒ I(A) ≤ I(B). (ii) For every increasing sequence {An | n ∈ N} of subsets of S0 , we have I

= sup I(An ).

An

n∈N

n∈N

(iii) For every decreasing sequence {An | n ∈ N} of elements of S0 , we have I

An

= inf I(An ). n∈N

n∈N

We have reached one of our main results. Theorem 2.3.1. For each δ ∈ T and α ≥ 0, α ∈ ∗ R, we have the following results: (i) ◦ Cap(δ) α (·) is a Choquet S0 -capacity. (ii) Every S0 -analytic set is capacitable with respect to capacity ◦ Cap(δ) α (·). That is, for every A ∈ A(S0 ), we have ◦

Cap(δ) α (A)

= sup

◦

Cap(δ) α (B)B

=

Bm , Bm ∈ S0 and B ⊂ A .

m∈N

(iii) Every subset A of S0 belonging to σ(S0 ) is capacitable with respect to the capacity ◦ Cap(δ) α (·) whenever 0 < st(α) < ∞.

86

2 Potential Theory

Proof. By Lemma 2.3.1 and Proposition 2.3.1, we know that ◦ Cap(δ) α (·) is a Choquet S0 -capacity. Therefore, (ii) holds by Meyer [282], Chap. III T19 Theorem. (iii) is the consequence of (ii) and Lemma 2.3.4. Definition 2.3.5. (i) A subset B of S0 is said to be of δ-zero capacity, if (δ) Cap1 (B) ≈ 0. (δ)

(ii) A subset B of S0 is said to be of zero capacity if Cap1 (B) ≈ 0 for some inﬁnitesimal δ ∈ T. (δ)

Remark 2.3.1. For any B ∈ S0 and δ ∈ T, we have m(B) ≤ Cap1 (B). Therefore, for any zero capacity subset B of S0 , we have L(m)(B) = 0.

2.4 Relation of Exceptionality and Capacity Theory In regular Dirichlet space theory, we know that the concepts of exceptional sets and zero capacity sets are equivalent, see Fukushima [175], Theorem 4.3.1. As the fourth section of this chapter, we will discuss the corresponding problem in our hyperﬁnite Dirichlet space theory. We shall continue the discussion of Sect. 2.3. Hence, we assume that all ˆ conditions in Sect. 2.2 are satisﬁed in this section as well, i.e., X and X ˆ are dual hyperﬁnite Markov chains, and E(·, ·) and E(·, ·) are the hyperﬁˆ respectively. Let nite quadratic form and co-form associated with X and X, H be the hyperﬁnite dimensional space with an inner product ·, · deﬁned by (2.2.3) in Sect. 2.2 or (1.5.15) in Sect. 1.5, Chap. 1. Lemma 2.4.1. Let {Bn | n ∈ N} be a sequence of internal subsets of S0 . If ∞ (δ) n limn→∞ ◦ Cap1 ( m=1 Bm ) = 0, then n=1 Bn is a δ-exceptional set, where δ ∈ T, δ ≈ 0. Proof. Since S0 is closed under ﬁnite intersection, we may assume that {Bn | n ∈ N} is a decreasing sequence. Deﬁne a stopping time for each n ∈ N, (δ)

σBn (ω) = min {t ∈ Tδ | X(ω, t) ∈ Bn } . We have 1 L(P ) ω ∃t ∈ Tδ X(ω, t) ∈ Bn ◦ 1 = P ω ∃t ∈ Tδ X(ω, t) ∈ Bn

◦ 1 = Ei ω ∃t ∈ Tδ X(ω, t) ∈ Bn dm(i) S0

2.4 Relation of Exceptionality and Capacity Theory

=

87

◦

Ei 1(σ(δ) ≤1) dm(i) Bn ◦ (δ) −σ /δ −1 = Ei ω (1 + δ) Bn ≥ (1 + δ) δ dm(i) S0 ⎧ ⎫ ◦ ⎨ (1 + δ)−σB(δ)n /δ ⎬ Ei ≤ dm(i) ⎩ (1 + δ)− 1δ ⎭ S0 ◦ (δ) −σ /δ Ei (1 + δ) Bn dm(i) =e· S0

S0

=e·

◦

(δ)

e1 (Bn )(i) dm(i) S0 (δ) (δ) ≤ e · ◦ E1 e1 (Bn ), e1 (Bn ) (δ)

= e · ◦ Cap1 (Bn ) −→ 0,

(2.4.1)

where the last inequality comes from (2.3.1) in Sect. 2.3. Then, we have ∞ 1 L(P ) ω ∃t ∈ Tδ X(ω, t) ∈ Bn = 0.

(2.4.2)

n=1

By the dual property of the Markov process X(t) and (2.4.2), we also have ∞ L(P ) ω ∃t ∈ Tδfin X(ω, t) ∈ Bn = 0. n=1

Therefore, the set

∞

n=1

Bn is δ-exceptional.

Theorem 2.4.1. If a subset A of S0 is of δ-zero capacity, it is δ-exceptional. (δ)

Proof. Since Cap1 (A) ≈ 0, we can take a sequence of internal subsets {Bn | n ∈ N} satisfying A⊂

∞ n=1

Bn ,

(δ)

lim ◦ Cap1 (

n→∞

n

Bm ) = 0.

m=1

Using Lemma 2.4.1, we know that ∞ n=1 Bn is δ-exceptional. Hence, A is δ-exceptional also. This completes the proof of Theorem 2.4.1. Lemma 2.4.2. Let δ1 ∈ T, δ1 ≈ 0. Assume that for all f ∈ H, if (δ ) (E1 1 (f, f )) < ∞ and f (s) ≈ 0 for all s ∈ / B, where B is a δ1 -exceptional set, (δ ) then we have E1 1 (f, f ) ≈ 0. Let A be a subset of S0 . If A is δ1 -exceptional and there exists an internal subset B of S0 such that

◦

88

2 Potential Theory

A ⊂ B,

◦

(δ )

Cap1 1 (B) < ∞,

(2.4.3)

then A is of δ1 -zero capacity. Proof. By using Proposition 2.1.1, there exists a properly δ1 -exceptional set

Bm,n ⊃ A.

m∈N n∈N

For simplicity, we assume that Bm,n ⊂ B for all n, m ∈ N, and for each m, the sequence {Bm,n | n ∈ N} is decreasing withrespect to n. In order to show that A has zero capacity, we ﬁrst prove that n∈N Bm,n has zero capacity for every m. From now on, we ﬁx an m ∈ N. (δ )

By the assumption (2.4.3), we know that ◦ Cap1 1 (Bm,n ) < ∞ for every n. (δ ) Moreover, Cap1 1 (Bm,n ) is decreasing with n. Let {Bm,n | n ∈ ∗ N} be a decreasing extension of {Bm,n | n ∈ N} . By saturation, there exists an inﬁnite element nm ∈ ∗ N − N such that (δ1 )

lim ◦ [E1

n→∞

(δ1 )

= ◦ [E1

(δ )

(δ )

(e1 1 (Bm,n ), e1 1 (Bm,n ))] (δ )

(δ )

(e1 1 (Bm,nm ), e1 1 (Bm,nm ))].

Therefore, we have ◦ (δ ) (δ ) (δ ) e1 1 (Bm,nm ), e1 1 (Bm,nm ) = Cap1 1 (Bm,nm ) < ∞. (δ ) Besides, for every i ∈ S0 , it is easy to see that e1 1 (Bm,n )(i) | n ∈ N is decreasing with respect to n. Since m∈N n∈N Bm,n is properly δ1 exceptional, we can show ◦

(δ1 )

E1

(δ )

e1 1 (Bm,nm )(i) ≈ 0

for every i ∈ /

Bm,n .

(2.4.4)

m∈N n∈N

In fact, for every M0 ∈ [0, ∞), we have (δ ) −σ(δ1 ) /δ e1 1 (Bm,n )(i) = Ei (1 + δ1 ) m,n 1 −σ(δ1 ) /δ = Ei (1 + δ1 ) m,n 1 1(σBm,n ≥M0 ) + 1(σm,n <M0 ) ≤ (1 + δ1 )−M0 /δ1 + Ei 1(σm,n <M0 ) . By letting M0 be suﬃciently large, we know that (1 + δ1 )−M0 /δ1 will be very small. Taking n suﬃciently large, we see that the approximation (2.4.4) holds.

2.4 Relation of Exceptionality and Capacity Theory

89

The assumption in the Lemma implies that (δ1 )

E1

(δ ) (δ ) e1 1 (Bm,nm ), e1 1 (Bm,nm ) ≈ 0.

Therefore, we get ◦

(δ1 )

E1

(δ ) (δ ) e1 1 (Bm,n ), e1 1 (Bm,n ) −→ 0.

This says that (δ ) Cap1 1

≈ 0.

Bm,n

(2.4.5)

n∈N

By Proposition 2.3.1 (i) and (ii) and the approximation (2.4.5), we obtain ◦

(δ ) Cap1 1 (A)

≤

◦

(δ ) Cap1 1

Bm,n

m∈N n∈N

≤

◦

(δ ) Cap1 1

m∈N

Bm,n

n∈N

= 0.

Thus, the set A has a δ1 -zero capacity. (δ )

Theorem 2.4.2. Let δ1 ∈ T, δ1 ≈ 0. Assume for all f ∈ H, if ◦ (E1 1 (f, f )) < ∞ and f (s) ≈ 0 for all s ∈ / B, where B is a δ1 -exceptional set, then we (δ1 ) have E1 (f, f ) ≈ 0. Let A be a subset of S0 . If A is δ1 -exceptional and there exists a sequence of internal subsets {Bn | n ∈ N} of S0 such that A⊂

Bn

and

◦

(δ )

Cap1 1 (Bn ) < ∞,

∀n ∈ N,

n∈N

then A is of δ1 -zero capacity. Proof. The proof follows easily from Lemma 2.4.2 and Proposition 2.3.1 (ii). In Lemma 2.4.2 and Theorem 2.4.2, we talk about the hyperﬁnite quadratic (δ ) form and co-form and make one assumption: for all f ∈ H, if ◦ (E1 1 (f, f )) < ∞ and f (s) ≈ 0 for all s ∈ / B, where B is a δ1 -exceptional set, then we have (δ1 ) E1 (f, f ) ≈ 0. This assumption is somewhat equivalent to say that for all (δ )

(δ)

f , if ◦ (E1 1 (f, f )) < ∞, then f ∈ D(E ). The assumption, however, is not always easy to verify. In the following, we will give results for hyperﬁnite

90

2 Potential Theory

weak coercive quadratic forms, for which we do not make the assumption. The results were ﬁrst proved for hyperﬁnite symmetric Dirichlet forms in Fan [166]. The following results extend those of Fan [166]. Lemma 2.4.3. Assume that E (δ) (·, ·) is a hyperﬁnite weak coercive quadratic form on the space H for all inﬁnitesimal δ ∈ T. Let δ1 ∈ T, δ1 ≈ 0. Let A be a subset of S0 . If A is δ1 -exceptional and there exists an internal subset B of S0 which satisﬁes condition (2.4.3), then there is an inﬁnitesimal δ0 ∈ T which is larger than δ1 such that A is of δ-zero capacity for all δ ≥ δ0 , δ ≈ 0. Proof. In the proof of Lemma 2.4.2, we have ◦

(δ1 )

E1

◦ (δ ) (δ ) (δ ) e1 1 (Bm,nm ), e1 1 (Bm,nm ) = Cap1 1 (Bm,nm ) < ∞.

By Corollary 1.2.4 and Theorem 1.4.2, there is a δm ≈ 0 such that (δ ) e1 1 (Bm,nm ) ∈ D(E (δ) ) for all inﬁnitesimal δ ≥ δm . Moreover, we know from the proof of Lemma 2.4.2 that (δ )

e1 1 (Bm,nm )(i) ≈ 0

for every i ∈ /

Bm,n .

m∈N n∈N (δ )

(δ )

Since e1 1 (Bm,nm ) ∈ D(E (δ) ) for all δ ≥ δm , δ ≈ 0, e1 1 (Bm,nm ) is S 2 integrable in the sense of Albeverio et al. [25], page 77, Chap. 3. Hence, we (δ ) have e1 1 (Bm,nm ) ≈ 0 in the hyperﬁnite dimensional space H because L(m) Bm,n = 0. m∈N m∈N (δ)

(δ )

(δ )

Therefore, we have E1 (e1 1 (Bm,nm ), e1 1 (Bm,nm )) ≈ 0 for all δ ≥ δm , δ ≈ 0. This implies that ◦

(δ)

E1

(δ) (δ) e1 (Bm,n ), e1 (Bm,n ) ↓ 0, n −→ ∞,

for all δ ≥ δm , δ ≈ 0.

Hence, we have (δ) Cap1

Bm,n

≈0

for all

δ ≥ δm , δ ≈ 0.

(2.4.6)

n∈N

By saturation, there is a δ0 ≈ 0 larger than all δm , m ∈ N. Therefore, it follows from the approximation (2.4.6) that for δ ≥ δ0 , δ ≈ 0,

2.5 Measures of Hyperfinite Energy Integrals

(δ)

Cap1

91

≈ 0 for all

Bm,n

m ∈ N.

(2.4.7)

n∈N

By Proposition 2.3.1 (i) and (ii) and the approximation (2.4.7), we obtain ◦

(δ) Cap1 (A)

≤

◦

(δ) Cap1

Bm,n

m∈N n∈N

≤

∞ m=1

◦

(δ) Cap1

Bm,n

n∈N

= 0.

Therefore, the set A has δ-zero capacity. (δ)

Theorem 2.4.3. Let E (·, ·) be a hyperﬁnite weak coercive quadratic form on the space H for all inﬁnitesimal δ ∈ T. Let δ1 ∈ T, δ1 ≈ 0. Let A be a subset of S0 . If A is δ1 -exceptional and there exists a sequence of internal subsets {Bn | n ∈ N} of S0 such that A⊂

Bn

and

◦

(δ )

Cap1 1 (Bn ) < ∞,

∀n ∈ N,

n∈N

then there is an inﬁnitesimal δ0 ∈ T which is larger than δ1 such that A is of δ-zero capacity for all δ ≥ δ0 , δ ≈ 0. Proof. The proof follows easily from Lemma 2.4.3, Proposition 2.3.1 (ii), and saturation.

2.5 Measures of Hyperfinite Energy Integrals We have deﬁned an inner product ·, · on the hyperﬁnite dimensional space H by (2.2.3) in Sect. 2.2 or (1.5.15) in Sect. 1.5, Chap. 1. Let || · || be the norm generated by ·, ·. Denote by Fin(H) the set of all elements in H with ﬁnite norm || · ||. By deﬁning u ≈ v if ||u − v|| ≈ 0, we recall that the space ◦

H = Fin(H)/ ≈

is a Hilbert space with the inner product (◦u, ◦v) = st(u, v), where ◦u denotes the equivalence class of u. In this section, we shall continue our discussion of Sects. 2.2, 2.3, and 2.4. ˆ are dual hyperﬁnite Markov chains. Let E(·, ·) and We assume that X and X ˆ E(·, ·) be the hyperﬁnite quadratic form and co-form associated with X and ˆ respectively. X,

92

2 Potential Theory

We know that for α ∈ ∗ R, α ≥ 0 and δ ∈ T, Eα(δ) (u, v) = E (δ) (u, v) + αu, v. Each of these forms generates a norm (possibly a semi-norm in the case α 1 (δ) (δ) = 0): |u|α = [Eα (u, u)] 2 . Similarly, we denote by Fin(δ) α (H) the set of all (δ) (δ) (δ) elements in H with ﬁnite norm | · |α . Deﬁne u ≈α v if |u − v|α ≈ 0. The space (δ) Hα(δ) = Fin(δ) α (H)/ ≈α

◦

is a Hilbert space if ◦ α > 0 with respect to the inner product (δ) (δ) ([u](δ) α , [v]α )α =

◦

Eα(δ) (u, v) ,

(δ)

(δ)

where [u]α denotes the equivalence class of u under the norm | · |α , and (δ) (·, ·)α denotes the related inner product. Definition 2.5.1. Let μ be a hyperﬁnite positive measure on S0 . For δ ∈ T , if there exists a constant K ∈ ∗ [0, ∞) = ∗ R+ such that S0

|u(s)| μ(ds) =

N

|u(si )|μ(i)

i=1

12 (δ) ≤ K E1 (u, u)

(2.5.1)

for every u ∈ H, we say that μ is of δ-hyperﬁnite energy integral. Moreover, if there exists K ∈ ∗ R+ satisfying (2.5.1) and ◦K < ∞, μ is said to be of δ-ﬁnite energy integral. Henceforth, we will identify a hyperﬁnite measure μ on S0 with the measure μ ˜ on S deﬁned by μ ˜(s0 ) = 0, μ ˜ (si ) = μ(si ) for all si ∈ S0 . Theorem 2.5.1. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form. Then (i) A positive hyperﬁnite measure μ on S0 is of δ-hyperﬁnite energy integral if and only if for each α ∈ ∗ R, 0 < st(α) < ∞, there exists an element (δ) Uα μ ∈ H such that for every v ∈ H, v(s) μ(ds). (2.5.2) Eα(δ) (Uα(δ) μ, v) = S0

(δ)

Moreover, if μ is of δ-ﬁnite energy integral, we have Uα μ ∈ Fin(δ) α (H).

2.5 Measures of Hyperfinite Energy Integrals

93

(ii) A positive hyperﬁnite measure μ on S0 is of δ-hyperﬁnite energy integral if and only if for each α ∈ ∗ R, 0 < st(α) < ∞, there exists an element ˆα(δ) μ ∈ H such that for every v ∈ H, U ˆα(δ) μ) = v(s) μ(ds). Eα(δ) (v, U S0

ˆα(δ) μ ∈ Fin(δ) (H). Moreover, if μ is of δ-ﬁnite energy integral, we have U α Proof. The theorem is an easy consequence of Riesz’s representation theorem. (δ) ˆα(δ) μ in Theorem 2.5.1 hyperﬁnite αRemark 2.5.1. We call Uα μ and U potential and hyperﬁnite α-co-potential of μ associated with E (δ) (·, ·), respectively. An internal element u ∈ H is called a hyperﬁnite α-potential (or (δ) ˆα(δ) μ) for some positive hyperﬁnite α-co-potential) if u = Uα μ (or u = U measure μ of δ-hyperﬁnite energy integral.

Definition 2.5.2. Fix α ∈ ∗ R, α ≥ 0 and δ ∈ T. (i) An element u ∈ H is called hyperﬁnite pre-α-excessive associated with E (δ) (·, ·), if u(i) ≥ 0, Qδ u(i) ≤ (1 + αδ)u(i) for every i ∈ S0 such that m(i) = 0. (ii) An element u ∈ H is called hyperﬁnite pre-α-co-excessive associated with ˆ δ u(i) ≤ (1 + αδ)u(i) for every i ∈ S0 such that Eˆ(δ) (·, ·), if u(i) ≥ 0, Q m(i) = 0. (δ)

In order to develop our theory, we shall denote by {Gβ | β ∈ ∗ ( − ∞, 0)} ˆ (δ) | β ∈ ∗ ( − ∞, 0)} the resolvent and co-resolvent of E (δ) (·, ·), and {G β respectively. That is, they are deﬁned by (δ)

Gβ = (A(δ) − β)−1 , ˆ (δ) = (Aˆ(δ) − β)−1 . G β Hence, we have for α ∈ ∗ R+ , (δ) (δ) 1 + αδ − Qδ G−α = δ(α + A(δ) )G−α =δ and ˆ (δ) ˆ(δ) )G ˆ (δ) ˆδ G 1 + αδ − Q −α = δ(α + A −α = δ.

(2.5.3)

94

2 Potential Theory

Theorem 2.5.2. For δ ∈ T, α ∈ conditions are equivalent:

∗

R, α ≥ 0, and u ∈ H, the following

(i) u is hyperﬁnite pre-α-excessive associated with E (δ) (·, ·). (ii) There exists a hyperﬁnite positive measure μ on S0 such that Eα(δ) (u, v) =

v(s) μ(ds)

for all

S0

v ∈ H.

(δ)

(iii) Eα (u, v) ≥ 0 for all v ∈ H, v ≥ 0. If E (δ) (·, ·) is a hyperﬁnite weak coercive quadratic form, the above statements are equivalent to the following: (iv) u is a hyperﬁnite α-potential of E (δ) (·, ·). Proof. (i) =⇒ (ii). Assume that u is hyperﬁnite pre-α-excessive associated with E (δ) (·, ·). Deﬁne a hyperﬁnite positive measure μ on S0 by μ(s0 ) = 0, 1 μ(si ) = (1 + αδ)u(i) − Qδ u(i) m(i) for i ∈ S0 . δ Since μ(si ) =

1 (1 + αδ)u(i) − Qδ u(i) m(i) δ

= A(δ) u(i) + αu(i) m(i), we see that for every v ∈ H, Eα(δ) (u, v) =

N A(δ) u(i) + αu(i) v(i)m(i)

i=1 =

v(s) μ(ds). S0

(ii) =⇒ (iii). This is easy, and thus we omit it! (iii) =⇒ (i). Such as in Sect. 1.5 of Chap. 1, let u+ = u ∨ 0. Since u+ − u ≥ 0, (δ) we have Eα (u, u+ − u) ≥ 0. Taking into account Corollary 1.5.2, we have Eα(δ) (u+ − u, u+ − u) = Eα(δ) (u+ , u+ − u) − Eα(δ) (u, u+ − u) ≤ Eα(δ) (u+ , u+ − u) = −E (δ) ((−u) ∧ 0, −u − (−u) ∧ 0)

≤ 0.

2.5 Measures of Hyperfinite Energy Integrals

95

This implies that Eα(δ) (u+ − u, u+ − u) = 0. Therefore, u(i) = u(i) ∨ 0 ≥ 0 for every i ∈ S0 such that m(i) = 0. Furthermore, it follows from (2.5.3) that for any v ∈ H, ˆ δ )v (1 + αδ − Qδ )u, v = u, (1 + αδ − Q ˆ δ )G ˆ (δ) v = Eα(δ) u, (1 + αδ − Q −α

=

Eα(δ) (u, δv).

(2.5.4)

Fix i ∈ S0 . Let v ∈ H be an internal function deﬁned by v(l) = δil , l ∈ S. Then, we have from (2.5.4) that 1 + αδ − Qδ u(i)m(i) = Eα(δ) (u, δv) ≥ 0. Therefore, the internal function u is hyperﬁnite pre-α-excessive associated with E (δ) (·, ·). If E (δ) (·, ·) is a hyperﬁnite weak coercive quadratic form, (ii) is equivalent to (iv) by Theorem 2.5.1 (i). Similarly, we have Theorem 2.5.3. For δ ∈ T, α ∈ conditions are equivalent:

∗

R, α ≥ 0, and u ∈ H, the following

(i) u is hyperﬁnite pre-α-co-excessive associated with E (δ) (·, ·). (ii) There exists a hyperﬁnite positive measure μ on S0 such that Eα(δ) (v, u) =

v(s) μ(ds) S0

for all

v ∈ H.

(δ)

(iii) Eα (v, u) ≥ 0 for all v ∈ H, v ≥ 0. If E (δ) (·, ·) is a hyperﬁnite weak coercive quadratic form, the above statements are equivalent to the following: (iv) u is a hyperﬁnite α-co-potential of E (δ) (·, ·). We denote by τ0 (δ) the family of all internal positive measures on S0 of δ-hyperﬁnite energy integrals. Denote τ0 = ∪{τ0 (δ) | δ is inﬁnitesimal,δ ∈ T }. Proposition 2.5.1. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form. For δ ∈ T, a hyperﬁnite positive measure μ on S0 is of δ-hyperﬁnite

96

2 Potential Theory

energy integral if and only if for any α ∈ ∗ R+ , there exists a hyperﬁnite pre-α-excessive function u associated with E (δ) (·, ·) such that μ(i) =

1 (1 + αδ)u(i) − Qδ u(i) m(i) δ

for all

i ∈ S0 .

(2.5.5)

Moreover, if u¯ ∈ H satisﬁes above equation also, then u(i) = u ¯(i) for all i ∈ S0 with m(i) = 0. Proof. ⇐= Let u satisfy the condition (2.5.5). Then, we have Eα(δ) (u, v) =

for all

v(s) μ(ds) S0

v ∈ H. (δ)

=⇒ Assume that μ is of δ-hyperﬁnite energy integral. Let u = U1 μ ∈ H satisfy the condition (2.5.2). Then, we have −1 (δ) μ(i) = G−1 u(i)m(i) 1 (1 + δ)u(i) − Qδ u(i) m(i). = δ Therefore, we have for any α ∈ ∗ R+ , v ∈ H,

S

(δ)

−1 (δ) v(i) G−1 u(i) dm(i) S (δ) (δ) = Eα(δ) G−α ((G−1 )−1 u), v .

v(s) dμ(s) =

(δ)

Hence, w = G−α ((G−1 )−1 u) is hyperﬁnite pre-α-excessive associated with E (δ) (·, ·) by Theorem 2.5.2. Furthermore, we have μ(i) =

1 (1 + αδ)w(i) − Qδ w(i) m(i) for all i ∈ S0 . δ

Proposition 2.5.2. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form. For δ ∈ T, let ν be a hyperﬁnite positive measure on S0 . Deﬁne a measure μ on S0 by μ(s) = ν(s)1(m(s) =0) Then μ is of δ-hyperﬁnite energy integral.

for

s ∈ S0 .

2.5 Measures of Hyperfinite Energy Integrals

97

Proof. Deﬁne f (s) =

ν(s) 1(m(s) =0) m(s)

for

s ∈ S0 ,

f (s0 ) = 0.

For any u ∈ H, we have u(s) dμ(s) = u(s)1(m(s) =0) dν(s) S S = u(s)f (s) dm(s) =

S (δ) (δ) E1 (G−1 f, u).

Hence, μ is of δ-hyperﬁnite energy integral by Theorem 2.5.2.

We recall that in Sect. 1.4 of Chap. 1, we have introduced the standard part (E (δ) (·, ·), D(E (δ) )) on ◦H for a hyperﬁnite weak coercive quadratic form E (δ) (·, ·). Theorem 2.5.4. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form. For α ∈ ∗ R, 0 < st(α) < ∞ and δ ∈ T , let μ be a hyperﬁnite positive measure of δ-ﬁnite energy integral, and let u be an α-potential of μ associated with E (δ) (·, ·). Deﬁne (δ) gn (i) = n u(i) − nG−n−α u(i) , n ∈ N. Then for every v ∈ Fin(δ) α (H), we have (i) (ii)

(δ) ◦ (δ) Eα (G−α gn , v) (δ)

−→ Eα(δ) (◦u, ◦v) ≈δα

as

n → ∞.

(2.5.6)

(δ)

If u ∈ D(E ), let v˜ v and v˜ ∈ D(E ), ◦ ◦ gn (i)v(i) dm(i) = v˜(i) dμ(i). limn→∞ S0

S0

Proof. (i) First of all, we have (δ) ◦ (δ) Eα (G−α gn , v)

= ◦(gn , v) (δ)

= ◦(nu − n2 G−n−α u, v) ◦ (δ) (2.5.7) = (n + α)u − (n + α)2 G−n−α u, v ◦ (δ) (δ) + −αu + α2 G−n−α u + 2nαG−n−α u, v . It follows from Theorem 1.4.1 that

98

2 Potential Theory ◦

(δ) (n + α)u − (n + α)2 G−n−α u, v + αu, v

−→ Eα(δ) (◦u, ◦v) as n → ∞.

(2.5.8)

By Lemma 1.4.4, we have ◦

(δ) α2 G−n−α u, v −→ 0 as n −→ ∞.

(2.5.9)

Besides, it follows from Theorem 1.4.1 ◦

−2αu + 2nαG(δ) −n−α u, v ◦ 2αn (n + α)2 ≤ − u + (n + α)u, v (n + α)2 n (δ) +−(n + α)u + (n + α)2 G−n−α u, v −→ 0 as

n → ∞.

(2.5.10)

From the relations (2.5.7), (2.5.8), (2.5.9), and (2.5.10), we know that the approximation (2.5.6) holds. (ii) From the proof of (i), we get lim

◦

gn (s)v(s) m(ds) = ◦ Eα(δ) (u, v˜)

n→∞ S 0

=

◦ S0

v˜(s) μ(ds).

Proposition 2.5.3. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. For δ ∈ T , let μ be a positive measure of δ-hyperﬁnite energy integral. Then for every L(μ) measurable subset A of S0 , we have ◦'

L(μ)(A) ≤

C

(

)(

(δ) (δ) (δ) E1 (U1 μ, U1 μ) (δ)

◦ Cap(δ) (A). 1

(2.5.11)

Proof. For simplicity, we assume that ◦ Cap1 (A) < ∞. If A is internal, we have ◦ ◦ μ(A) = 1A (s) μ(ds) S0

≤

◦

S0

(δ)

e1 (A)(s) μ(ds)

2.5 Measures of Hyperfinite Energy Integrals (δ)

(δ)

99

(δ)

= ◦ E1 (U1 μ, e1 (A)) ) ( ◦' ( (δ) (δ) (δ) (δ) (δ) (δ) ≤ C E1 (U1 μ, U1 μ) E1 (e1 (A), e1 (A)) ◦'

=

(

C

) ( (δ) Cap1 (A) .

(δ) (δ) (δ) E1 (U1 μ, U1 μ)

Now it is easy to see that the inequality (2.5.11) holds for all L(μ) measurable subsets A. Corollary 2.5.1. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form. Let μ be a positive measure of δ-ﬁnite energy integral on S0 . Then L(μ) charges no set of δ-zero capacity.

Proof. The proof follows easily from Proposition 2.5.3. We shall now state and prove the following characterization theorem.

Theorem 2.5.5. Let A ⊂ S0 be an A(S0 )-measurable set (in particular, A ∈ σ(S0 )) and let δ ∈ T. We have (δ)

(i) For any μ ∈ τ0 (δ), L(μ)(A) = 0 =⇒ ◦ Cap1 (A) = 0. (δ) (ii) For any μ ∈ τ00 (δ), L(μ)(A) = 0 =⇒ ◦ Cap1 (A) = 0, where (δ) τ00 (δ) = μ ∈ τ0 (δ)μ(S0 ) = 1, ◦||U1 μ||∞ < ∞ and (δ) (δ) ||U1 μ||∞ = max |U1 μ(s)|s ∈ S0 . (δ)

(iii) For any μ ∈ τˆ00 (δ), L(μ)(A) = 0 =⇒ ◦ Cap1 (A) = 0, where ˆ (δ) μ||∞ < ∞ τˆ00 (δ) = μ ∈ τ0 (δ)μ(S0 ) = 1, ◦||U 1 and ˆ (δ) μ(s)|s ∈ S0 . ˆ (δ) μ ∞ = max |U U 1 1 (δ)

Proof. (i) Assume that ∞ ≥ ◦ Cap1 (A) = α > 0. Since A ∈ A(S0 ), it follows from Theorem 2.3.1 (ii) that A is capacitable with respect to the capacity (δ) ◦ Cap1 (·). That is ◦

(δ) Cap1 (A)

= sup

◦

(δ) Cap1 (B)B

=

n∈N

Bn , Bn ∈ S0

and

B⊂A .

100

2 Potential Theory

Therefore, there exists a sequence {Bn | n ∈ N} of decreasing internal subsets of S0 such that ∞ ∞ α (δ) ◦ Bn ⊂ A, ∞ ≥ Cap1 Bn ≥ > 0. 2 n=1 n=1 Let {Bn | n ∈ ∗ N} be an internal decreasing extension of {Bn | n ∈ N}. Then, there exists an inﬁnite γ ∈ ∗ N − N such that ∞ α ◦ (δ) (δ) Bn = ◦ Cap1 (Bγ ) ≤ ∞. Bγ ⊂ A, 0 < ≤ Cap1 2 n=1 (δ)

Set B = Bγ . Consider the internal function e1 (B). Notice that if i ∈ / B, then (δ) (δ) −σ /δ (1 + δ)e1 (B)(i) = (1 + δ)Ei (1 + δ) B =

N j=1

(δ)

(δ)

e1 (B)(j)qij (δ)

= Qδ e1 (B)(i),

(2.5.12)

(δ)

where σB = min{t ∈ Tδ | X(ω, t) ∈ B}. If i ∈ B, then we have (δ)

(1 + δ)e1 (B)(i) = (1 + δ) ≥

N j=1

(δ)

(δ)

e1 (B)(j)qij (δ)

= Qδ e1 (B)(i).

(2.5.13)

It follows from the relations (2.5.12) and (2.5.13) that the function (δ) e1 (B) is hyperﬁnite 1-excessive associated with E (δ) (·, ·). Deﬁne a hyperﬁnite positive measure μ on S by μ(s0 ) = 0, 1 (δ) (δ) μ(si ) = (1 + δ)e1 (B)(i) − Qδ e1 (B)(i) m(i), i ∈ S0 . δ Then, we have μ(B) = = =

μ(ds) S0 (δ) (δ) E1 (e1 (B), 1) (δ) Cap1 (B).

2.5 Measures of Hyperfinite Energy Integrals

101

This contradicts the assumption ◦ L(μ)(A) = 0. Hence, we must have (δ) ◦ Cap1 (A) = 0. (δ)

(ii) Assume that 0 < ◦ Cap1 (A) = α ≤ ∞. By the proof (i), we see that there exists an element ν ∈ τ0 (δ) such that (replacing μ in the proof of (i) by ν) (δ)

(δ)

0 < ◦ Cap1 (B) ≤ ∞, (δ)

Cap1 (B) = ν(B) = ν(S0 )

(δ)

and U1 ν(i) = e1 (B)(i) ≤ 1 for any i ∈ S0 , where B is an internal set contained in A. Deﬁne μ(·) by ν(·)

μ(·) =

(δ)

Cap1 (B)

.

Then, we have μ(S0 ) = 1, μ ∈ τ0 (δ) and ◦ (δ) |U1 ν(i)| (δ) max |U1 μ(i)| = max i∈S0 i∈S0 Cap(δ) (B) 1 ◦ 1 ≤ (δ) Cap1 (B) < ∞.

◦

But 1 = ◦μ(B) ≤ ◦μ(A) ≤ 1, i.e., L(μ)(A) = 1. This contradicts our assumption of L(μ)(A) = 0. We have proved that ◦

(δ)

Cap1 (A) = 0.

(iii) Recalling Corollary 2.2.1, we can show this result in the same way as that of above (ii). Theorem 2.5.6. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form, which has the following property: ∀A ⊂ S0 ,

◦

(δ)

Cap1 (A) = ∞

=⇒ ∃B ⊂ A

such that

(δ)

0 < ◦ Cap1 (B) < ∞. (2.5.14)

102

2 Potential Theory

Let τ0f (δ) be the family of all positive measures of δ-ﬁnite energy integrals. Then the following statements are equivalent for A ∈ A(S0 ) (in particular, A ∈ σ(S0 )): (δ)

(i) A is of zero δ-capacity, i.e., ◦ Cap1 (A) = 0. (ii) For any μ ∈ τ0f (δ), L(μ)(A) = 0. (iii) For any μ ∈ τ00f (δ), L(μ)(A) = 0, where (δ) τ00f (δ) = μ ∈ τ0f (δ)μ(S0 ) = 1, ◦ ||U1 μ||∞ < ∞ . (iv) For any μ ∈ τˆ00f (δ), L(μ)(A) = 0, where ˆ (δ) μ||∞ < ∞ . τˆ00f (δ) = μ ∈ τ0f (δ)μ(S0 ) = 1, ◦ ||U 1 Proof. (i) =⇒ (ii) =⇒ (iii) and (ii) =⇒ (iv) are clear by Corollary 2.5.1. We can show (ii) =⇒ (i) and (iii) =⇒ (i) and (iv) =⇒ (i) in the same way as the proof of Theorem 2.5.5.

2.6 Internal Additive Functionals and Associated Measures ˆ be dual hyperﬁnite Markov chains, and let E(·, ·) and E(·, ˆ ·) be Let X and X ˆ the hyperﬁnite quadratic form and co-form of X and X, respectively. Again, let H be the hyperﬁnite dimensional space with an inner product ·, · deﬁned by (2.2.3) in Sect. 2.2 or (1.5.15) in Sect. 1.5, Chap. 1. We recall that in Sect. 1.5 of Chap. 1, we have introduced the dual (δ) hyperﬁnite Markov chains (Ω, X (δ) , {Ft | t ∈ Tδ }, {Pi | i ∈ S}) and ˆ X ˆ (δ) , {Fˆ (δ) | t ∈ Tδ }, {Pˆi | i ∈ S}) for δ ∈ T . For simplicity, we assume (Ω, t ˆ X (δ) = X ˆ (δ) , F (δ) = Fˆ (δ) , t ∈ Tδ . The hyperﬁnite quadratic that Ω = Ω, t t ˆ {Pˆi | i ∈ S})) are given by form associated with (X, {Pi | i ∈ S}) (or (X, the expression (1.5.19) (or (1.5.22)) in Sect. 1.5. As in the study of standard Markov processes, we deﬁne a family of translation operators {θt | t ∈ T } of Ω. That is, for each t ∈ T, θt is a map from Ω to Ω deﬁned by ω ∈ Ω =⇒ θt ω ∈ Ω

and for any s ∈ T, θt ω(s) = ω(s + t). (δ)

Hence for each δ ∈ T , we have a family of translation operators {θt | t ∈ Tδ } (δ) induced by {θt | t ∈ T }, θt = θt for any t ∈ Tδ . In other words, for each (δ) t ∈ Tδ , θt is a map from Ω to Ω given by (δ)

ω ∈ Ω =⇒ θt ω ∈ Ω

and for any

(δ)

s ∈ Tδ , θt ω(s) = ω(s + t).

2.6 Internal Additive Functionals and Associated Measures

103

Definition 2.6.1. For any δ ∈ T, we call an internal ∗ R-valued function A(ω, t) or At (ω), t ∈ Tδ , ω ∈ Ω, δ-internal additive functional (abbreviated by δ-IAF) if it satisﬁes the following two conditions: (1) For each t ∈ Tδ , At (ω) is non-anticipating with respect to the ﬁltration (δ) (δ) (Ω, {Ft t ∈ Tδ }), i.e., At (·) is Ft -measurable, ∀t ∈ Tδ . (2) For each ω ∈ Ω, we have A(ω, 0) = 0, A(ω, t + s) = A(ω, s) + A(θs(δ) ω, t) for any t, s ∈ Tδ . Proposition 2.6.1. If A(ω, t) is a δ-internal additive functional, then there exists a hyperﬁnite measure μ on S0 (not necessarily positive) such that μ (i) = 0 whenever m(i) = 0 and for all n ∈ ∗ N,f, h ∈ H, S0

h(i)Ei

=

n k=0

n

S0

k=0

f (X (δ) (kδ)) A(ω, (k + 1)δ) − A(ω, kδ) dm(i)

ˆi h(X (δ) (kδ)) dμ(i)δ. f (i)E

(2.6.1)

Proof. Deﬁne μ (0) = 0, 1 μ (i) = Ei A(ω, δ)m(i), 1 ≤ i ≤ N. δ

(2.6.2)

Then, we have S0

h(i)Ei

= = = =

n

n k=0

S0

k=0 n

k=0 S0 n S0

k=0 n k=0

f (X (δ) (kδ)) A(ω, (k + 1)δ) − A(ω, kδ) dm(i)

S0

(δ)

h(i)Ei f (X (δ) (kδ, ω))A(θkδ ω, δ) dm(i) h(i)Ei f (X(kδ))EX(kδ) A(ω, δ) dm(i) f (i)Ei A(ω, δ)Eˆi h(X(kδ)) dm(i) ˆi h(X(kδ)) dμ(i)δ. f (i)E

104

2 Potential Theory

Proposition 2.6.2. Let μ be a hyperﬁnite measure on S0 satisfying μ(i) = 0 whenever m(i) = 0. Then for each δ ∈ T , there exists a δ-IAF A(ω, t) such that (2.6.1) holds. Proof. First of all, let f (s) =

μ(s) m(s) 1(m(s) =0) .

For each u ∈ H, we have

S0

u(s) dμ(s) = =

u(s)f (s) dm(s)

S0 (δ) ˆ (δ) f ). E1 (u, G −1

Deﬁne A(ω, 0) = 0, A(ω, kδ) = δ

k

f (X (δ) (ω, (l − 1)δ))

for k ∈ ∗ N, k ≥ 1.

l=1

It is easy to verify that A(ω, t) is a δ-IAF. Moreover, we have for each i ∈ S0 , 1 1 Ei A(ω, δ)m(i) = Ei f (X (δ) (ω, 0))δm(i) δ δ = f (i)m(i) = μ(i). Therefore, it follows from the proof of Proposition 2.6.1 that (2.6.1) holds. For δ ∈ T , let A(ω, t) be a δ-IAF. Deﬁne 1 E(A(ω, δ))2 2δ 1 = Ei A2δ dm(i). 2δ S0

e(A) =

(2.6.3)

We call e(A) the energy of A. Furthermore, we deﬁne the mutual energy e(A, B) for δ-internal additive functionals A and B by e(A, B) =

1 E A(ω, δ)B(ω, δ) . 2δ

Let ΔA(ω, kδ) be the forward increment of A(ω, t) at time kδ, i.e., ΔA(ω, kδ) = A(ω, (k + 1)δ) − A(ω, kδ) for k ∈ ∗ N. We deﬁne the quadratic variation [A] : Ω × Tδ −→ ∗ R by

2.6 Internal Additive Functionals and Associated Measures

105

[A](ω, 0) = 0, [A](ω, nδ) =

n−1

(ΔA(ω, kδ))

2

for n ∈ ∗ N, n > 0.

k=0

Because [A](ω, (n + m)δ) =

n−1

2

n+m−1

2

m−1

(ΔA(ω, kδ)) +

k=0

= =

n−1

(ΔA(ω, kδ))

2

k=n

(ΔA(ω, kδ)) +

k=0

k=0

n−1

m−1

(ΔA(ω, kδ))2 +

k=0

2

(ΔA(ω, (k + n)δ)) (ΔA(θnδ ω, kδ))2 ,

k=0

[A] is a positive δ-IAF. By Proposition 2.6.1 and its proof, we know that μ (i) = 1δ Ei (A(ω, δ))2 m(i) is the hyperﬁnite positive measure associated with [A] in the sense (2.6.1). We call μ the energy measure of A. It is obviously from the relations (2.6.2) and (2.6.3) that e(A) =

1 μ (S0 ). 2

Let u ∈ H. For δ ∈ T , deﬁne a δ-IAF A[u] (ω, t) by A[u] (ω, t) = u(X (δ) (ω, t)) − u(X (δ) (ω, 0)) for t ∈ Tδ . Then, we have 2 1 [u] E A (ω, δ) 2δ 2 1 E u(X (δ) (ω, δ)) − u(X (δ) (ω, 0)) = 2δ N 2 1 = Ei u(X (δ) (ω, δ)) − u(i) m(i) 2δ i=0

e(A[u] ) =

=

N N 1 (δ) (u(j) − u(i))2 qij m(i) 2δ i=0 j=0

=

N N N 1 1 (δ) 2 (δ) (u(j) − u(i)) qij m(i) + (u(i))2 qi0 m(i) 2δ i=1 j=1 2δ i=1

(2.6.4)

106

2 Potential Theory

=

N N N 1 1 (δ) (δ) (u(i) − u(j)) u(i)qij m(i) + (u(i))2 qi0 m(i) 2δ i=1 j=1 2δ i=1

+

N N N 1 1 (δ) (δ) (u(j) − u(i)) u(j)ˆ qji m(j) + (u(i))2 qˆi0 m(i) 2δ i=1 j=1 2δ i=1

−

N 1 (δ) (u(i))2 qˆi0 m(i) 2δ i=1

= E (δ) (u, u) −

N 1 (δ) (u(i))2 qˆi0 m(i), 2δ i=1

(2.6.5)

where we have used Lemma 1.5.1 (i) and (ii) in the last equation of (2.6.5). Theorem 2.6.1. For u ∈ H, f ∈ H, let μ (·) be the energy measure of A[u] . We have (1)

2 1 [u] Ei A (ω, δ) m(i) δ N 1 (δ) = (u(j) − u(i))2 qij m(i). δ j=0

μ (i) =

(2)

S0

f (s) μ (ds) = 2E (δ) (u, uf ) − E (δ) (u2 , f ).

(2.6.6)

Proof. (1) The proof is quite immediate, and hence we omit it! (2) On the one hand, we have

N N 2 1 (δ) f (s) μ (ds) = f (i) u(j) − u(i) qij m(i). (2.6.7) δ S0 i=1 j=0

On the other hand, we have 2E (δ) (u, f u) − E (δ) (u2 , f ) ⎡ ⎤ N N 2 ⎣ (δ) (uf )(i)u(j)qij m(i)⎦ = (u(i))2 f (i)m(i) − δ i=1 j=1 ⎡ ⎤ N N 1 ⎣ (δ) (u(i))2 f (i)m(i) − (u(j))2 f (i)qij m(i)⎦ − δ i=1 j=1 N

=

N

N

1 1 (δ) (u(i))2 f (i)m(i) − (2u(i) − u(j))u(j)f (i)qij m(i) δ i=1 δ i=1 j=1

2.7 Fukushima’s Decomposition Theorem

⎛

107

⎞

=

N N 1 (δ) f (i)m(i) ⎝(u(i))2 − (2u(i) − u(j))u(j)qij ⎠ δ i=1 j=0

=

1 (δ) f (i)(u(j) − u(i))2 qij m(i). δ i=1 j=0

N

N

By (2.6.7) and (2.6.8), we get (2.6.6).

(2.6.8)

2.7 Fukushima’s Decomposition Theorem As before, we shall work under the conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14) of Sect. 1.5, Chap. 1, in this section.

2.7.1 Decomposition Under the Individual Probability Measures Pi Lemma 2.7.1. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Let ν be a positive measure on S0 of δ-hyperﬁnite energy integral. For any u ∈ H, t ∈ Tδ , and ε > 0, we have

t (δ) Pν ω ∃s ∈ Tδ (|u(X (ω, s))| ≥ ε) ≤ .

where Pν (·) =

S0

12 2C 2 (1 + δ)t/δ (δ) (δ) (δ) (δ) E1 (U1 ν, U1 ν)E1 (u, u) , ε Pi (·)dν(i).

Proof. Let A = {i ∈ S0 | u(i) ≥ ε} . Deﬁne (δ)

σA (ω) = min{t ∈ Tδ | X (δ) (ω, t) ∈ A}. Then, we have

Pν ω ∃s ∈ Tδ , s ≤ t u(X (δ) (ω, s)) ≥ ε (δ) Pi ω (1 + δ)−σA /δ ≥ (1 + δ)−t/δ dν(i) = S 0 (δ) Ei (1 + δ)−σA /δ (1 + δ)t/δ dν(i) ≤ S0

108

2 Potential Theory

= (1 + δ)t/δ

(δ)

e1 (A)(i) dν(i) S0 (δ) (δ) t/δ (δ) = (1 + δ) E1 U1 ν, e1 (A) 12 (δ) (δ) (δ) (δ) (δ) (δ) t/δ E1 U1 ν, U1 ν E1 e1 (A), e1 (A) ≤ C (1 + δ) ≤

t/δ

C 2 (1 + δ) ε

12 (δ) (δ) (δ) (δ) E1 U1 ν, U1 ν E1 (u, u) ,

where the reason for the last step holding is (δ)

E1

|u| |u| (δ) (δ) (δ) e1 (A), e1 (A) ≤ C 2 E1 , ε ε 2 C (δ) ≤ 2 E1 (u, u), ε

by Corollary 2.2.3. Hence, we can prove Lemma 2.7.1 by applying the same argument to −u. Proposition 2.7.1. For δ ∈ T, let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Assume that (E (δ) (·, ·), D(E (δ) )) satisﬁes condition (2.5.14) of Theorem 2.5.6. Let u, {un | n ∈ N} be the elements in H and let δ ∈ T. Suppose that ◦ (δ) E1 (un

− u, un − u) −→ 0

as

n → ∞.

Then there exist a subsequence {unk | k ∈ N} and a δ-exceptional set B such that for all i ∈ S0 − B, t ∈ Tδfin , L(Pi ) ◦unk (X (δ) (ω, s)) converges uniformly to in s on

Tδt

as

◦

u(X (δ) (ω, s)) n −→ ∞ = 1.

Proof. Let {nk | k ∈ N} be a subsequence satisfying ◦ (δ) E1 (unk

− u, unk − u) ≤ 2−4k .

Set t (δ) (δ) −k . Λk (t) = ω ∃s ∈ Tδ |unk (X (ω, s)) − u(X (ω, s))| ≥ 2 For ν ∈ τ00f (δ), we have from Lemma 2.7.1 that

2.7 Fukushima’s Decomposition Theorem

109

◦ 12 t (δ) (δ) (δ) ◦ 2 −k+1 δ E1 (U1 ν, U1 ν) Pν (Λk (t)) ≤ C (1 + δ) 2 .

Hence, we have ∞

L(Pν )(Λk (t)) < ∞.

k=1

By the Borel-Cantelli lemma, we get L(Pν )

∞ ∞

Λl (t)

= 0.

(2.7.1)

k=1 l=k

∞ ∞ Set Λ(t) = k=1 l=k Λl (t). From Theorem 2.5.6, Theorem 2.4.1, and (2.7.1), there exists a δ-exceptional set B(t) such that L(Pi )(Λ(t)) = 0

for any

i ∈ S0 − B(t).

Now let us select subset {tn | n ∈ N} ⊂ Tδ such that tn ≈ n. We ∞ a countable ∞ deﬁne Λ = n=1 Λ(tn ), B = n=1 B(tn ). It is easy to see that Proposition 2.7.1 holds. Lemma 2.7.2. For δ ∈ T, let A be a δ-IAF and let μ (i) be the hyperﬁnite measure deﬁned by (2.6.2) in Sect. 2.6. Then for any v ∈ H, we have E (δ) (ft , v) =

S0

v(i) − Eˆi v(X (δ) (t)) dμ (i)

for all

t ∈ Tδ , (2.7.2)

where ft (i) = Ei A(ω, t). Proof. If k = 1, then we have 1 ˆ δ v(i) fδ (i) dm(i) v(i) − Q E (δ) (fδ , v) = δ S0 ˆi v(X (δ) (δ)) dμ (i). = v(i) − E S0

Assume that (2.7.2) holds whenever k ≤ n. Then, we have E

(δ)

1 ˆ δ v(i) Ei A(ω, (n + 1)δ) dm(i) v(i) − Q (f(n+1)δ , v) = δ S0 1 ˆ δ v(i) Ei A(ω, nδ) + Ei A(θ(δ) ω, δ) dm(i) v(i) − Q = nδ δ S0

110

2 Potential Theory

ˆi v(X (δ) (nδ)) dμ (i) v(i) − E S0 1 ˆi v(X (δ) ((n + 1)δ)) + Eˆi v(X (δ) (nδ)) − E δ S0 ×Ei A(ω, δ) dm(i) ˆi v(X (δ) ((n + 1)δ)) dμ (i). v(i) − E = =

S0

Lemma 2.7.3. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Let A(ω, t) be a positive δ-IAF. For all positive hyperﬁnite measures ν on S0 of δ-hyperﬁnite energy integral, we have ˆ ν||∞ μ (S0 ) Eν (At ) ≤ (1 + t)||U 1 (δ)

for all

t ∈ Tδ .

(2.7.3)

Proof. It follows from Theorem 2.5.1 and Lemma 2.7.2 that (δ) ˆ (δ) ν) Ei A(ω, t) dν(i) = E1 (ft , U Eν A(ω, t) = 1 S 0 ˆ (δ) ν(i) − E ˆi (U ˆ (δ) ν(X (δ) (t))) dμ (i) U = 1 1 S0 ˆ (δ) ν(i) dm(i) + ft (i)U 1 S0 ) ' ˆ (δ) ν||∞ μ (S0 ) + f (i) dm(i) , (2.7.4) ≤ ||U t 1 S0

where we have used Theorem 2.5.3 in the latter inequality above. Notice that f(k+1)δ (i) dm(i) = Ei A(ω, (k + 1)δ) dm(i) S0 S 0 Ei A(ω, kδ) dm(i) + Ei A(θkδ ω, δ) dm(i) = S S 0 0 ˆi 1(X(kδ)∈S ) = E Ei A(ω, kδ) dm(i) + 0 S0

S0

×Ei A(δ) dm(i) ≤ Ei A(ω, kδ) dm(i) + S0

S0

Ei A(ω, δ) dm(i).

2.7 Fukushima’s Decomposition Theorem

111

We can show S0

ft (i) dm(i) ≤ tμ (S0 ).

(2.7.5)

From the relations (2.7.4) and (2.7.5), we obtain the inequality (2.7.3).

∗

Definition 2.7.1. We call an internal process A : Ω × Tδ → R a martingale (δ) (δ) with respect to (Ω, Ft , Pi , i ∈ S0 ) if ω → A(ω, t) is Ft measurable for all (δ) t ∈ Tδ , and for all s, t ∈ Tδ , s < t, and all B ∈ Fs , Ei (1B (At − As )) = 0. (δ)

It is easy to see that if [ω]t is the equivalence class of ω deﬁned by (1.5.35) in Sect. 1.5 of Chap. 1, then a non-anticipating process A(ω, t) is a martingale if and only if

ΔA(˜ ω , t)Pi {˜ ω } = 0.

(2.7.6)

(δ)

ω ˜ ∈[ω]t

In the following, we shall use “δ-quasi-everywhere” or “δ-q.e.” to mean “except for a δ-exceptional set”. A statement depending on i ∈ S0 is said to be “δ-quasi-everywhere” or “δ-q.e.” if there exists a set B ⊂ S0 of δ-exceptional such that the statement is true for every i ∈ S0 − B. Let us introduce the following set of internal martingale additive functionals (abbreviated by δ-IMAF) by M(δ) = M | M is a δ-IAF, ◦ (Ei M 2 (t)) < ∞, δ-q.e.i ∈ S0 for all ﬁnite t ∈ Tδ and Ei M (t) = 0, ∀t ∈ Tδ , ∀i ∈ S0 } . For M ∈ M(δ) , we have for all s, t ∈ Tδ (δ)

(δ)

(δ)

Ei (Ms+t | Ft ) = Ei (Mt + Ms (θt ) | Ft ) = Mt + EX(t) Ms = Mt , for all i ∈ S0 . Hence, M ∈ M(δ) is a martingale with respect to Pi for all i ∈ S0 . Meantime, M ∈ M(δ) is also square integrable with respect to Pi for δ-q.e. i ∈ S0 . In the following, let ˚ (δ) = {M ∈ M(δ) | ◦e(M ) < ∞}. M ˚ (δ) is called a δ-internal martingale additive functional of Any element in M ﬁnite energy.

112

2 Potential Theory

Proposition 2.7.2. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Let {An | n ∈ N} be δ-internal additive functionals. Assume that ◦e(An − Am ) → 0 as n, m → ∞, and for (δ) each i ∈ S0 , {Ω, Ft , An (t), Pi } is a martingale for all n ≥ 1. Then there ˚ (δ) with inner product ◦e(·, ·) such that ◦e(An − A) → 0 exists a unique A ∈ M as n → ∞. Moreover, there exist a subsequence {Ank (ω, t) | k ∈ N} and a δ-exceptional set B such that for all i ∈ S0 − B, t ∈ Tδfin,

L(Pi ) ω ◦Ank (ω, s) −→ ◦A(ω, s)

uniformly on

Tδt

= 1.

Proof. By the equality (2.6.4) in Sect. 2.6, we know that ◦

μ (S0 ) =

◦

2e(An − Am )

−→ 0 as n, m → ∞. Hence, there exists a subsequence {nk | k ∈ N} such that ◦μ (S0 ) < 2−3k . Since An , n ≥ 1 are martingales with respect to Pi , i ∈ S0 , we get from Lemma 2.7.3 that for all ν ∈ τˆ00 (δ), t ∈ Tδfin : ◦

2 ◦ / 0 Eν Ank+1 (t) − Ank (t) = Eν ([Ank+1 − Ank ](t)) ≤

◦

ˆ (δ) ν||∞ 2−3k . (1 + t)||U 1

(2.7.7)

From the relation (2.7.7) and Doob’s inequality (for which we refer to [25] 4.2.8), we get ◦'

) Pν max |Ank+1 (s) − Ank (s)| ≥ 2−k s≤t 1 2

◦ 22k Eν

≤ ≤ ≤

2

max |Ank+1 (s) − Ank (s)| s≤t

◦

2 22k+2 Eν Ank+1 (t) − Ank (t)

◦

ˆ (δ) ν||∞ 2−k . 4(1 + t)||U 1

(2.7.8)

It is easy to see from the inequality (2.7.8), by using Borel-Cantelli lemma, that L(Pν )(Λ) = 1, for all ν ∈ τˆ00 ,

2.7 Fukushima’s Decomposition Theorem

113

where Λ = {ω | ◦Ank+1 (ω, ·) converges uniformly on each ﬁnite interval Tδt }. By Theorem 2.5.5 and Theorem 2.4.1, we have L(Pi )(Λ) = 1, δ-q.e.i ∈ S0 . Let {Ank | k ∈ ∗ N} be an internal extension of {Ank | k ∈ N}. Deﬁne A(ω, t) = AnK (ω, t) for some K ∈ ∗ N − N. Then, A(ω, t) is a Pi -martingale δ-IAF for all i ∈ S0 . Since An (t) converges in L2 (Pν ), we have (◦ Eν A2 (t)) < ∞, ∀t ∈ Tδfin . Therefore, we have shown A ∈ M(δ) . For ε > 0, choose N such that e(An −Am ) < ε, n, m > N. By Fatou’s lemma, we have e(An −A) ≤ ε, n > N. ˚ (δ) and ◦e(An − A) −→ 0 as n −→ ∞. Thus, we have A ∈ M Definition 2.7.2. (1) We call an internal function u ∈ H S-bounded if there is a positive constant C ∈ [0, ∞) such that |u(i)| ≤ C for all i ∈ S0 . (2) An internal function f : Tδ −→ ∗ R is called S-continuous if f (s) ≈ f (t) whenever s ≈ t and s and t are nearstandard. Lemma 2.7.4. For δ ∈ T, u ∈ Fin(H), deﬁne a δ-IAF Aδ by Aδ (ω, 0) = 0, Aδ (ω, nδ) = δ

n

u(X(ω, (k − 1)δ)), n ∈ N, n ≥ 1.

k=1

Then, there exists a properly δ-exceptional set B ⊂ S0 such that for all i ∈ S0 − B, δ L(Pi ) ω A (ω, ·) is S-continuous = 1. Proof. First of all, we assume that u is S-bounded. We have for any ω ∈ Ω,

|A (ω, t) − A (ω, s)| ≤ |t − s| max |u(i)| + 1 . δ

δ

i∈S0

This implies that Aδ (ω, t) is S-continuous. Next, we suppose that u ∈ Fin(H). For each n, set Bn = {s ∈ S0 | |u(s)| ≥ n} . In addition, we deﬁne σBn (ω) = min {t ∈ Tδ | X(t) ∈ Bn } . Then, we have

1 P ω ∃t ∈ Tδ (X(ω, t) ∈ Bn )

= P ω σBn (ω) ≤ 1

114

2 Potential Theory

≤ n−2 ≤ n−2

S0

S0

Ei [u(X(σBn ))]2 dm(i) [u(i)]2 dm(i).

Thus, we get ∞ 1 Bn ) = 0. L(P ) ω ∃t ∈ Tδ (X(ω, t) ∈ n=1

This implies that {s ∈ S0 | ◦|u(s)| = ∞} is a δ-exceptional set. Let B be a properly δ-exceptional set containing {s ∈ S0 | ◦|u(s)| = ∞} (Proposition 2.1.1). For each n ∈ N, deﬁne σn (ω) = min {t ∈ Tδ | |u(X(ω, t))| ≥ n} . Then for each ω ∈ Ω, Aδ (ω, ·) is continuous in [0, σn (ω)) for every n. Moreover, for each i ∈ S0 − B, we have L(Pi ) {ω | ◦σn (ω) ↑ ∞ as n → ∞} = 1,

which proves Lemma 2.7.4. Set (δ)

NC = {N | L(Pi ){ω | N (ω, ·) is an S-continuous δ-IAF} = 1, q.e.i ∈ S0 and e(N ) ≈ 0} . (δ)

For N ∈ NC , the variation vanishes in the following sense E([N ](t)) ≈ 0 for all t ∈ Tδfin. In fact, we have for all ﬁnite t ∈ Tδ , E([N ](t)) =

0<s 0, we have

2C 2 (1 + δ)t/δ (δ) P ω ∃s ∈ Tδt (|u(X (δ) (ω, s))| ≥ ε) ≤ E1 (u, u). ε2 Proof. Let A = {i ∈ S0 | u(i) ≥ ε}, and (δ)

σA (ω) = min{t ∈ Tδ | X (δ) (t) ∈ A}. Then, we have P

ω ∃s ∈ Tδ , s ≤ t u(X (δ) (ω, s)) ≥ ε (δ) Pi ω (1 + δ)−σA /δ ≥ (1 + δ)−t/δ dm(i) = S 0 (δ) Ei (1 + δ)−σA /δ (1 + δ)t/δ dm(i) ≤ S0 (δ) t/δ e1 (A)(i) dm(i) = (1 + δ) S0 (δ) (δ) t/δ (δ) ≤ (1 + δ) E1 e1 (A), e1 (A) ,

2.7 Fukushima’s Decomposition Theorem

119

where the reason for the last step holding is (2.3.1) in Sect. 2.3. Since E (δ) (·, ·) has the Markov property, we get from Corollary 2.2.3 that (δ) E1

|u| |u| (δ) (δ) 2 (δ) e1 (A), e1 (A) ≤ C E1 , ε ε C 2 (δ) ≤ 2 E1 (u, u). ε

Hence, we have

P

C 2 (1 + δ)t/δ (δ) ω ∃s ∈ Tδt (u(X (δ) (ω, s)) ≥ ε) ≤ E1 (u, u). ε2

Applying the same argument to −u, the lemma follows.

Corollary 2.7.1. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Let u, un , n ∈ N be elements in H, and assume that ◦ (δ) E1 (u

− un , u − un ) −→ 0 as n −→ ∞.

There is a subsequence {unk } such that for almost every ω, ◦unk (X (δ) (ω, ·)) converges uniformly to ◦u(X (δ) (ω, ·)) on all S-bounded subsets of Tδ . Proof. The result follows from Lemma 2.7.5 and basic measure theory. (δ) , P ) is called Definition 2.7.3. A martingale M with respect to (Ω, Ft a λ2 -martingale if ◦E(Mt2 ) < ∞ for all t ∈ Tδfin . Theorem 2.7.2. For inﬁnitesimal δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. For any u ∈ D(E (δ) ), there are two δ-internal additive functionals M [u] (ω, t) and N [u] (ω, t) such that (i) A[u] (ω, t) = M [u] (ω, t) + N [u] (ω, t). (δ) , P ). (ii) M [u] is a λ2 -martingale with respect to (Ω, Ft (iii) N [u] (ω, ·) is S-continuous for almost all paths ω ∈ Ω under L(P ), and E[N [u] ](t) ≈ 0 for all t ∈ Tδfin. Proof. The proof of this result is similar to the one of Theorem 2.7.1. We also refer the reader to the detailed proof of the corresponding result in the symmetric case given in Albeverio et al. [25].

120

2 Potential Theory

2.8 Internal Multiplicative Functionals 2.8.1 Internal multiplicative functionals Definition 2.8.1. For δ ∈ T, an internal function M (ω, t), t ∈ Tδ , ω ∈ Ω, is said to be a δ-internal multiplicative functional (abbreviated by δ-IMF) of X (δ) (ω, t) if and only if (δ)

(i) For each t ∈ Tδ , Mt (·) is Ft -measurable. (ii) For each t ∈ Tδ , ω ∈ Ω, we have M (ω, t) ∈ ∗ [0, 1]. (iii) For each ω ∈ Ω, we have M (ω, 0) = 1, M (ω, t + s) = M (ω, s)M (θs(δ) ω, t), ∀t, s ∈ Tδ . Remark 2.8.1. Let A(ω, t) be a nonnegative δ-internal additive functional. We can deﬁne a δ-IMF M (ω, t) by M (ω, t) = exp (−A(ω, t))

for all ω ∈ Ω, t ∈ Tδ .

If M (ω, t) is a δ-internal multiplicative functional, let us deﬁne a family of operators {P t | t ∈ Tδ } on H by P t f (i) = Ei f (X (δ) (ω, t))M (ω, t) . Then, we have for all t, s ∈ Tδ , i ∈ S, P t+s f (i) = Ei f (X (δ) (ω, t + s))M (ω, t + s) = Ei f (X (δ) (θs(δ) ω, t))M (ω, s)M (θs(δ) ω, t) = Ei M (ω, s)EX (δ) (ω,s) [f (X (δ) (t))Mt ] = P s P t f (i).

(2.8.1)

Hence, {P t | t ∈ Tδ } is a semigroup. We call it the semigroup generated by (X (δ) , M ). Moreover, we have for all f ∈ H, P 0 f (i) = Ei [f (X (δ) (ω, 0))M (ω, 0)] = f (i). In particular, we have P 0 1 = 1.

(2.8.2)

2.8 Internal Multiplicative Functionals

121

(δ)

Since M (ω, δ) is Fδ -measurable, we have M (ω, δ) =

N

1[ˆω](ij) (ω)Mij ,

i,j=0

where [ˆ ω ](ij) = {ω | ω(0) = i and ω(δ) = j} and {Mij | i, j = 0, 1, 2, . . . , N } is a family of positive hyperreals. Moreover, Mij ∈ ∗ [0, 1]. Therefore, the (δ) transition matrix {pij | i, j = 0, 1, 2, . . . , N } of {P t | t ∈ Tδ } is given by (δ)

(δ)

pij = qij Mij ,

Mij ∈ ∗ [0, 1],i, j = 0, 1, 2, . . . , N.

(2.8.3)

From the relation (2.8.3), we know that for all nonnegative internal functions f ∈ H, P t f (i) ≤ Qt f (i), ∀i ∈ S, t ∈ Tδ .

(2.8.4)

2.8.2 Subordinate Semigroups Definition 2.8.2. A semigroup {P t | t ∈ Tδ } of positive linear operators from H to H is said to be subordinate to {Qt | t ∈ Tδ } if and only if P t f (i) ≤ Qt f (i) for all t ∈ Tδ , f ∈ H, f (i) ≥ 0, i ∈ S and P 0 = I. Theorem 2.8.1. Let {P t | t ∈ Tδ } be a semigroup on H. Then the following two conditions are equivalent: (1) {P t | t ∈ Tδ } is subordinate to {Qt | t ∈ Tδ }. (2) There exists a δ-IMF M (ω, t) of X (δ) (ω, t) generating {P t | t ∈ Tδ }. Proof. (2) =⇒ (1). It follows from the relations (2.8.2) and (2.8.4). (δ)

(1) =⇒ (2). Let {pij | i, j = 0, 1, 2, . . . , N } be the transition matrix of the semigroup {P t | t ∈ Tδ }. Then for each f ∈ H, we have Qδ f (i) =

N j=1

P δ f (i) =

N j=1

(δ)

qij f (j), (δ)

pij f (j).

Since {P t | t ∈ Tδ } is subordinate to {Qt | t ∈ Tδ }, we see that (δ)

(δ)

pij ≤ qij , ∀i, j = 0, 1, 2, . . . , N.

122

2 Potential Theory

Deﬁne ˆ (i, j) = M where we set

a 0

(δ)

pij

1

(δ)

(δ) (qij =0)

qij

,

= 0, a ∈ ∗ [0, ∞). Put M (ω, 0) = 1,

ˆ X (δ) (ω, 0), X (δ) (ω, δ) . M (ω, δ) = M

(2.8.5)

For all i ∈ S, we have for any f ∈ H, Ei [f (X (δ) (ω, δ))M (ω, δ)] =

N j=1

ˆ (i, j)q f (j)M ij

(δ)

= P δ f (i). By using mathematical induction, we deﬁne (δ)

M (ω, (k + 1)δ) = M (ω, kδ)M (θkδ ω, δ)

for all k ∈ ∗ N.

It is then easy to show that M (ω, t) is a δ-IMF generating {P t | t ∈ Tδ }.

2.8.3 Subprocesses In Sect. 1.5 of Chap. 1, we deﬁned a hyperﬁnite Markov chain X (δ) (ω, t) associated with the hyperﬁnite Dirichlet form E (δ) (·, ·). Let us denote (δ)

(δ)

Ω, F t , Y (δ) (ω, t), θt , P i

t∈Tδ

a hyperﬁnite Markov chain with state

(δ) (δ) space (S, S). We call Y (ω, t) a subprocess of X (ω, t) if and only if the semigroup P t | t ∈ Tδ of Y (δ) (ω, t) is subordinate to {Qt | t ∈ Tδ } .

Let Y (δ) (ω, t) be a subprocess of X (δ) (ω, t). From Theorem 2.8.1, there exists a δ-IMF M (ω, t) of X (δ) (ω, t) such that EP i f (Y (δ) (ω, t)) = Ei [f (X (δ) (ω, t))M (ω, t)] for all t ∈ Tδ ,

(2.8.6)

where EP i (·) is the expectation operator corresponding to Y (δ) (ω, t). We are interested in the following question. Given a δ-IMF M (ω, t), could we construct a subprocess Y (δ) (ω, t) such that the relation (2.8.6) holds? (δ) (δ) The answer is yes! In fact, let Ω = Ω, F t = Ft , Y (δ) (ω, t) = X (δ) (ω, t),

2.8 Internal Multiplicative Functionals (δ)

123

(δ)

θt = θt . Furthermore, let {P t | t ∈ Tδ } be the semigroup given by the relation (2.8.1), and let {p(δ) | i, j = 0, 1, 2, . . .} be the transition matrix of ij {P t | t ∈ Tδ }. Deﬁne for ω ∈ Ω = Ω, k ∈ ∗ N, P i ([ω]kδ ) = δiω(0)

k−1

p ω(nδ), ω((n + 1)δ) .

n=0

It is obvious that the relation (2.8.6) holds with respect to Y (δ) (ω, t). We call Y (δ) (ω, t) the canonical subprocess associated with (X (δ) , M ). We notice | i, j = 0, 1, 2, . . . , N } need not to have the regularities (1.5.1) and that {p(δ) ij (1.5.2) in Sect. 1.5 of Chap. 1.

2.8.4 Feynman-Kac Formulae (δ)

Let {qi

| i = 1, 2, . . . , N } be a 1 × N matrix satisfying (δ)

0 ≤ qi

(δ)

≤ qii , i = 1, 2, . . . , N. (δ)

Deﬁne a transition matrix P (δ) = {pij | i, j = 0, 1, 2, . . . , N } by (δ)

(δ)

(δ)

pij = qij − qi δij (δ) pi0 (δ) p00

= =

(δ) (δ) qi0 + qi (δ) 1, pi0 = 0

for for

i, j = 1, 2, . . . , N,

i = 1, 2, . . . , N, and

for i = 1, 2, . . . , N.

(δ) Similarly, deﬁne the transition matrix Pˆ (δ) = {ˆ pij | i, j = 0, 1, 2, . . . , N } by (δ)

(δ)

(δ)

pˆij = qˆij − qi δij (δ) pˆi0 (δ) pˆ00

= =

(δ) (δ) qˆi0 + qi (δ) 1, pˆi0 = 0

for for

i, j = 1, 2, . . . , N,

i = 1, 2, . . . , N and

for i = 1, 2, . . . , N. (δ)

Let {P t | t ∈ Tδ } be the semigroup with P (δ) = {pij | i, j = 0, 1, 2, . . . , N } as its transition matrix, and let {Pˆ t | t ∈ Tδ } be the semigroup with Pˆ (δ) = (δ) {ˆ pij | i, j = 0, 1, 2, . . . , N } as its transition matrix. Then, we have

124

2 Potential Theory

Theorem 2.8.2. We have (i) The semigroups {P t | t ∈ Tδ } and {Pˆ t | t ∈ Tδ } are dual with respect to m. (ii) The Dirichlet form associated with P (δ) and m is given by E

(δ)

(u, v) = E

(δ)

(u, v) +

N i=1

(δ)

u(i)v(i)qi m(i).

(iii) There exists a δ-IMF M (ω, t) such that for any f ∈ H, i ∈ S, t ∈ Tδ , P t f (i) = Ei [f (X (δ) (ω, t))M (ω, t)]. Proof. (i) and (ii) are obvious. (iii) follows easily from Theorem 2.8.1.

2.9 Alternative Expression of Hyperfinite Dirichlet Forms Suppose that μ(·) is a positive internal measure on S. In this section, we shall ﬁnd conditions on μ and μ-dual transition matrices P = {pij | i, j = 0, 1, . . . , N } and Pˆ = {ˆ pij | i, j = 0, 1, 2, . . . , N } with the regularities (1.5.1) and (1.5.2) in Sect. 1.5 such that E(·, ·) is the hyperﬁnite Dirichlet form associated with m and Q. First we observe that if μ and P and Pˆ have been found, then for all u, v ∈ H, we have ⎡ ⎤ N N 1 ⎣u(i)v(i)μ(i) − E(u, v) = u(j)v(i)pij μ(i)⎦ Δt i=1 j=1

(2.9.1)

and ⎡ ⎤ N N 1 ⎣ E(u, v) = u(i)v(i)μ(i) − u(j)v(i)ˆ pji μ(j)⎦ . Δt i=1 j=1

(2.9.2)

Therefore, we have from the expression (1.5.19) in Sect. 1.5 and the expression (2.9.1) that m(i)[1 − qii ] = μ(i)[1 − pii ] m(i)qij = μ(i)pij

i = 1, 2, . . . , N,

(2.9.3)

for all i, j ∈ {1, 2, . . . , N } , i = j.

(2.9.4)

for all

2.10 Transformations of Symmetric Dirichlet Forms

125

Similarly, we have m(i)[1 − qˆii ] = μ(i)[1 − pˆii ] for all i = 1, 2, . . . , N, pij for all i, j ∈ {1, 2, . . . , N } , i = j. m(i)ˆ qij = μ(i)ˆ

(2.9.5) (2.9.6)

On the other hand, if the pairs of (P, μ) and (Q, m) satisfy the conditions ˆ m) satisfy the conditions (2.9.3) and (2.9.4) (or the pairs of (Pˆ , μ) and (Q, (2.9.5) and (2.9.6)), then the expression (2.9.1) (or (2.9.2)) holds also. Hence, we get Theorem 2.9.1. Let μ(·) be a positive internal measure on S, and let P and Pˆ be μ-dual transition matrices. Then the conditions (2.9.3), (2.9.4), (2.9.5), and (2.9.6) are suﬃcient and necessary conditions such that the expression (1.5.19) in Sect. 1.5 and the expressions (2.9.1) and (2.9.2) hold.

2.10 Transformations of Symmetric Dirichlet Forms In this section, we assume that q ii = 0 for all i = 1, 2, . . . , N.

(2.10.1)

In fact, this assumption will not aﬀect our theory. The reason comes from the proof of Proposition 1.5.1. Actually, for the general hyperﬁnite quadratic form E(·, ·) of the expression (1.5.19) in Sect. 1.5, we deﬁne m(i) ˜ = (1 − qii )m(i), q˜ii = 0

for all

i = 1, 2, . . . , N.

Moreover, if i = 1, 2, . . . , N and qii < 1, deﬁne q˜ij =

qij 1 − qii

for j = i, j ∈ {0, 1, 2, . . . , N } ;

if i = 1, 2, . . . , N and qii = 1, deﬁne q˜ij = 0

for j = 0, i and q˜i0 = 1.

Besides, let q˜00 = 1, q˜0j = 0, j = 1, 2, . . . , N. It is very easy to verify from the Beurling–Deny formulae, Lemma 1.5.1 that the hyperﬁnite quadratic form ˜ = {˜ associated with m ˜ and Q qij | i, j = 0, 1, 2, . . . , N } is E(·, ·). Since qii m(i) = qˆii m(i), ∀i = 1, 2, . . . , N,

126

2 Potential Theory

we may suppose qˆii = 0, ∀i = 1, 2, . . . , N . Hence, we may assume 1 (qii + qˆii ) 2 = 0, i = 1, 2, . . . , N.

q ii =

Let Φ be an internal nonnegative function in H. We deﬁne the following quadratic form Φ

E (u, v) =

1 Δt

u(i) − u(j) v(i) − v(j) Φ(i)Φ(j)q ij m(i).

1≤i<j≤N

(2.10.2) Φ

Φ

It is easy to see that E (·, ·) satisﬁes Proposition 1.5.1 (iii). Moreover, E (·, ·) is symmetric. Thus by Proposition 1.5.1, there exists a transition matrix P = {pij | 0 ≤ i, j ≤ N } and a symmetric measure m(·) such that

Φ

1 u(i) v(i) − P Δt v(i) dm(i). Δt S0

E (u, v) =

(2.10.3)

In the following, we will ﬁnd the P and m. From the expression (2.10.3) and the Beurling–Deny formulae (Lemma 1.5.1), we have Φ

E (u, v) =

1 Δt

1≤i<j≤N

+

N i=1

u(i) − u(j) v(i) − v(j) pij m(i)

u(i)v(i)pi0 m(i) .

(2.10.4)

Comparing the expressions (2.10.2) and (2.10.4), we must have Φ(i)Φ(j)q ij m(i) = pij m(i) pi0 m(i) = 0 for all

for all

1 ≤ i, j ≤ N,

(2.10.5)

i = 1, 2, . . . , N.

Therefore, we get m(i) =

N j=0

=

j =i

pij m(i) Φ(j)q ij Φ(i)m(i) + pii m(i)

= E i [Φ(X(Δt))]Φ(i)m(i) + pii m(i).

(2.10.6)

2.10 Transformations of Symmetric Dirichlet Forms

127

Deﬁne p00 = 1, p0i = 0 for pii = 0 for

i = 1, 2, . . . , N, i = 1, 2, . . . , N.

Then from (2.10.6), we obtain m(i) = E i [Φ(X(Δt))]Φ(i)m(i). For i = 1, 2, . . . , N, if E i Φ(X(Δt)) = 0, we see from the relation (2.10.5) that for all j = 1, 2, . . . , N, j = i, q ij Φ(j)

pij =

E i [Φ(X(Δt))] q ij Φ(j) = . l =i q il Φ(l)

(2.10.7)

For i = 1, 2, . . . , N , if E i [Φ(X(Δt))] = 0, we can deﬁne pij , 1 ≤ j ≤ N, j = i arbitrarily such that N j=1

pij = 1.

(2.10.8)

From (2.10.7) and (2.10.8), we get pi0 = 0 for all i = 0. In the following discussion, we suppose that E i [Φ(X(Δt))] > 0

for all

i = 1, 2, . . . , N.

(2.10.9)

Hence, we have from (2.10.7) that for all f ∈ H, P f (i) =

E i [(f Φ)(X(Δt))] . E i [Φ(X(Δt))]

Let {P t | t ∈ T } be the semigroup generated by P . Then it is easy to see that {P t | t ∈ T } is subordinate to {Qt | t ∈ T }. By using Theorem 2.8.1, there exists a Δt-IMF M (ω, t) of X(ω, t) generating {P t | t ∈ T }. From the relation (2.8.5) in Sect. 2.8, we know that M (ω, Δt) =

Φ(X(ω, Δt)) . E X(ω,0) [Φ(X((Δt))]

128

2 Potential Theory

Therefore, we have for all k ∈ ∗ N, M (ω, (k + 1)Δt) =

k

Φ(ω, (l + 1)Δt) . E X(ω,lΔt) [Φ(X(Δt))] l=0

(2.10.10)

On the other hand, let Φ be an internal function in H satisfying the hypothesis (2.10.9). We deﬁne M (ω, t) directly by the relation (2.10.10). Then, we have the following: Theorem 2.10.1. Assume that the hypotheses (2.10.1) and (2.10.9) hold. Then Φ

(1) The semigroup {(Q )t | t ∈ T } generated by M (ω, t) is symmetric with respect to the measure E i [Φ(X(Δt))]Φ(i)m(i) = m(i). Φ Φ (2) E (·, ·) is the hyperﬁnite Dirichlet form associated with {(Q )t | t ∈ T } and m(·). Proof. The proof is straightforward.

Chapter 3

Standard Representation Theory

The purpose of this chapter is to study the standard projection of hyperﬁnite Markov chains, and the standard projection of hyperﬁnite Markov chains associated with hyperﬁnite quadratic forms. At ﬁrst, we shall introduce in Sect. 3.1 a concept of irregularity; and then we shall prove that if a hyperﬁnite Markov chain X(·, t) has a set of irregularities, its standard part x(·, t) is a strong Markov process. In Sect. 3.2, we shall ﬁnd conditions on hyperﬁnite quadratic forms which guarantee that the modiﬁed standard parts of associated hyperﬁnite Markov chains are strong Markov processes. The old version of the materials of this chapter can be found in Chap. 5, Albeverio et al. [25]. However, Chap. 5 of Albeverio et al. [25] only handles symmetric hyperﬁnite Dirichlet form. Moreover, many restrictions in Albeverio et al. [25] are not necessary. For instance, we simplify the deﬁnition of exceptional sets in Chap. 2 and remove the unnecessary assumptions. Using the updated version of exceptional sets, we will deﬁne a concept of exceptional irregularities in Sect. 3.1. We will show that a hyperﬁnite Markov chain with exceptional irregularities has a standard part, which is a strong Markov process. In Sect. 3.2, we shall impose conditions directly on hyperﬁnite quadratic forms to achieve our goal. Again, many assumptions of Albeverio et al. [25] will be simpliﬁed. For instance, the separation of compacts of Albeverio et al. [25] will be replaced by separation of points; all concepts involving exceptional sets will be updated using our deﬁnition in Chap. 2. It is our hope that the materials of this chapter update and signiﬁcantly improve the related materials of Chap. 5, Albeverio et al. [25]. One may want to notice that we will impose conditions directly on a standard Dirichlet form in Chap. 4 to get related strong Markov processes. The materials of Chaps. 1, 2, and 3 can be thought as the theory of hyperﬁnite Dirichlet forms and hyperﬁnite Markov chains. With the solid work of these three chapters, we will be able to attack the important construction of strong Markov processes associated with standard Dirichlet forms in Chap. 4.

S. Albeverio et al., Hyperfinite Dirichlet Forms and Stochastic Processes, Lecture Notes of the Unione Matematica Italiana 10, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19659-1_3,

129

130

3 Standard Representation Theory

3.1 Standard Parts of Hyperfinite Markov Chains In this section, we shall study standard parts of hyperﬁnite Markov chains. In order to take standard parts we need a topology. We shall assume that with the exception of the trap s0 , the state space S is embedded in the nonstandard version ∗ Y of some Hausdorﬀ space Y . If X is a hyperﬁnite Markov chain taking values in S, we want to ﬁnd conditions that guarantee that the standard part of X exists and is a Y -valued Markov process. It turns out that there are two diﬃculties we shall have to overcome. The ﬁrst is that the paths of X may be so irregular that no natural standard part process exists. The second is that even when a standard part does exist, there is no reason why it should automatically be a Markov process – taking standard parts we may lump together states that should be kept apart. By using the theory of right standard parts developed by Albeverio et al. [25], Chap. 4, we can solve the ﬁrst of these problems. Thus, most of our work will be directed to the second problem under discussion. Before we delve into the technicalities, we shall discuss the problem informally in some more details. Assume that x : Ω × R+ −→ Y is the standard part of X. We would like to prove that x is a Markov process with respect to the ﬁltration it generates. Given that x(t) = y, in general there will be several states s ∈ S0 such that y = st(s), and X may be in any one of them. From the nonstandard point of view, these states are totally unrelated, and hence the past and the future of the process may diﬀer widely from one state to the next. Observation of the whole past may indicate which states are more likely to occur, and thus inﬂuence our prediction of the future. This explains why in general x is not a Markov process. Note, however, that if the process started at si and the process started at sj have the “same” future whenever si ≈ sj , the above argument breaks down, and it is reasonable to expect that x is Markov. One way of formulating this condition is to demand that L(Pi ){ω | x(ω, t) ∈ B} = L(Pj ){ω | x(ω, t) ∈ B}

(3.1.1)

for all t ∈ R+ and all Borel sets B. But the above relation turns out to be too strict for the applications. Instead of demanding that it holds for all inﬁnitely close si , sj , we shall only demand that it holds for all such si , sj outside an exceptional set (i.e., a set which the process hits with probability zero). It may at ﬁrst seem that little is achieved by allowing a condition to fail on an exceptional set, but in fact the extra freedom and ﬂexibility we gain will turn out to be very useful. The present section falls into two halves. In the ﬁrst part, we develop the necessary theory for the inner standard part of sets and a general theory of

3.1 Standard Parts of Hyperfinite Markov Chains

131

exceptional sets. In the second part, we shall show that the standard part of hyperﬁnite Markov chain is a strong Markov process under some conditions.

3.1.1 Inner Standard Part of Sets By a hyperﬁnite Markov chain X in this section we shall understand a stationary Markov chain as described in Sect. 1.5, (1.5.1)–(1.5.7). If X has a ˆ as introduced in Sect. 1.5, we have for all dual hyperﬁnite Markov chain X ∗ k∈ N P {ω | X(ω, kΔt) = si } =

N

P {ω | X(ω, 0) = sj , X(ω, kΔt) = si }

j=0

=

N j=0

=

N j=0

(kΔt)

qji

(kΔt)

qˆij

m(j)

m(i)

= m(i). However, we shall NOT assume the duality of X in the following discussion, i.e., we do not need condition (1.5.13) in Sect. 1.5. We shall instead assume that for each i ∈ S0 , the function t → P {ω | X(ω, t) = si } is decreasing. We shall further assume that S0 ⊂ ∗ Y for some topological space Y , but that the trap s0 is not an element of ∗ Y . Similarly as in Sect. 2.1, we deﬁne the exceptional sets and properly exceptional sets of X using Deﬁnition 2.1.1 and Deﬁnition 2.1.2. Furthermore, Lemma 2.1.1 and Proposition 2.1.1 hold for X. Most of the exceptional sets we encounter in the theory of Markov processes are sets we would like to avoid. From the standard point of view, this means that a point y in Y should be avoided if all its nonstandard representations si ∈ st−1 (y) ∩ S0 are in the exceptional set. To study this relationship closely, we deﬁne the inner standard part A◦ of a subset A of S by A◦ = y ∈ Y | st−1 (y) ∩ S0 ⊂ A .

132

3 Standard Representation Theory

The standard part of a set A is deﬁned by ◦

A = st(A) = {y ∈ Y | ∃s ∈ A(st(s) = y)} .

The inner standard part can also be deﬁned in terms of the standard part operation: A◦ = C(st(CA)),

(3.1.2)

where the outer complement is with respect to Y , and the inner complement is with respect to S0 . It is trivial to check that standard parts commute with arbitrary unions. Using the operation (3.1.2), we get that inner standard parts commute with arbitrary intersections. The other way around is less nice, the standard parts and intersections do not commute, and neither do inner standard parts and unions. All we can say is the following: Lemma 3.1.1. (1) If {An | n ∈ N} is a decreasing family of internal sets, then we have ◦ ◦ An = An . (3.1.3) n∈N

n∈N

(2) If {Bn | n ∈ N} is an increasing family of internal sets, then we have

◦ Bn

=

n∈N

Bn◦ .

(3.1.4)

n∈N

Proof. Obviously, the left hand of the property (3.1.3) is contained in the set on the right. To prove the converse, let x∈

◦

An .

n∈N

For each n ∈ N, pick yn ∈ An such that x = st(yn ). Extend {An | n ∈ N} to be a decreasing internal family {An | n ∈ ∗ N} and {yn | n ∈ N} to be an internal sequence {yn | n ∈ ∗ N} such that yn ∈ An for all n ∈ ∗ N. For each neighborhood O of x, consider the set NO = {n ∈ ∗ N | yn ∈ ∗ O}.

3.1 Standard Parts of Hyperfinite Markov Chains

133

All these sets are internal and contain N. Hence, we can ﬁnd an inﬁnite integer η that is in all of them. But then, we must have x = st(yη ). Since the family {An | n ∈ ∗ N} is decreasing, we have yη ∈ Aη ⊂ ∩n∈N An . This proves the property (3.1.3). We now get the property (3.1.4) from the property (3.1.3) by using the operation (3.1.2):

◦

= C st C

Bn

n∈N

= C st =C =

Bn

n∈N

(CBn )

n∈N

(st (CBn ))

n∈N

C (st(CBn ))

n∈N

=

Bn◦ .

n∈N

Since in general

◦ Bm,n

=

m∈N n∈N

◦ Bm,n ,

m∈N n∈N

there is no obvious reason to believe that the inner standard part of a properly exceptional set is always Borel. However, we shall now prove that it must at least be universally measurable. Lemma 3.1.2. Assume A = ∪m∈N ∩n∈N Bm,n for a family {B}m,n∈N of internal sets. For any completed Borel probability measure μ on Y , the inner standard part A◦ is μ measurable. Moreover, there exists a family {Dm,n | m, n ∈ N} of internal sets such that A⊂

n∈N m∈N

Dm,n

134

3 Standard Representation Theory

and

μ

◦ Dm,n

◦

−A

= 0.

n∈N m∈N

Proof. We may assume that the family {Bm,n } is increasing in m and decreasing in n. Note that A=

Bm,n

m∈N n∈N

=

Bm,f (m) .

f ∈NN m∈N

Let f¯(m) be the sequence f (0), f (1), · · · , f (m), and deﬁne Cf¯(m) =

Bk,f (k) .

k≤m

For ﬁxed f , the sequence {Cf¯(m) }m∈N is increasing. Hence by Lemma 3.1.1 and the fact that inner standard parts and arbitrary intersections commute, we have Cf◦¯(m) . (3.1.5) A◦ = f ∈NN m∈N

Since CCf◦¯(m) = st(CCf¯(m) ) is closed, the complement

CA◦ =

f ∈NN m∈N

CCf◦¯(m)

of A can be derived from the closed sets by the Souslin operation, and hence A◦ is measurable with respect to any completed Borel measure (see [321], page 50, for an easy proof). It only remains to ﬁnd the family {Dm,n }. From (3.1.5) it follows that for each ε ∈ R+ , there is a function fε : N −→ N such that ⎛ μ⎝

⎞ Cg¯◦(m) − A◦ ⎠ < ε,

g≤fε m∈N

where g ≤ fε means that g(n) ≤ fε (n) for all n ∈ N. Since our original sequence {Bm,n | n ∈ N} is decreasing, we have

3.1 Standard Parts of Hyperfinite Markov Chains

Cg¯◦(m) =

g≤fε m∈N

135

m∈N

Cf◦¯ε (m) .

Putting Dm,n = Cf¯1/n (m) , the lemma follows.

Remark 3.1.1. The argument above shows that a subset of a Hausdorﬀ space can be derived from the closed sets by using the Souslin operation if and only if it is the standard part of a set derived from the internal sets by the same operation. This result and the proof we have given are due to Henson [201]. We have now reached the last lemma we shall need before we can return to our Markov processes. It will be used to pick hyperﬁnite representations of measures avoiding properly exceptional sets.

Lemma 3.1.3. Let D be a subset of S0 of the form D = n∈N m∈N Dm,n for a family {Dm,n } of internal sets. If μ is a Radon probability measure on Y with μ(D◦ ) = 0, there exists an internal probability measure ν on S0 such that μ = L(ν) ◦ st−1 and L(ν)(D) = 0. Proof. We may obviously assume that the family {Dm,n } is increasing in m ◦ and decreasing in n. Deﬁne a new measure μm,n by μm,n (B) = μ(B − Dm,n ). Then, we have μ(B) = sup inf μm,n (B). n∈N m∈N

(3.1.6)

◦ . Hence, there is an internal measure The set S0 −Dm,n is S-dense in Y −Dm,n νm,n concentrated on S0 − Dm,n such that

μm,n = L(νm,n ) ◦ st−1 .

(3.1.7)

We may choose these measures such that the family {νm,n } is decreasing in m and increasing in n. Extending to an internal sequence {νm,n | m, n ∈ ∗ N}, we ﬁrst pick a γ ∈ ∗ N − N such that ◦

νγ,n (S0 ) = lim

m→∞

◦

νm,n (S0 )

for all n ∈ N, and then an η ∈ ∗ N − N such that ◦

νγ,η (S0 ) = lim ◦ νγ,n (S0 ). n→∞

By using equations (3.1.6) and (3.1.7), and the deﬁnitions of γ and η, we see that L(νγ,η )(D) = 0 and μ = L(νγ,η ) ◦ st−1 . The measure νγ,η has all the properties of the desired measure, except that we still have to show that it is a probability measure. However, this is clear since we have νγ,η (S0 ) = 1 + ε for some ε ≈ 0. Putting ν = (1 + ε)−1 νγ,η , the lemma is then proved.

136

3 Standard Representation Theory

By combining Lemmas 3.1.2 and 3.1.3, we get: Corollary 3.1.1. Let A ⊂ S0 be a properly exceptional set, and let μ be a completed Borel probability measure on Y such that μ(A◦ ) = 0. Then there is an internal probability measure ν on S0 such that μ = L(ν) ◦ st−1 and L(ν)(A) = 0.

3.1.2 Strong Markov Processes and Modified Standard Parts Having completed our study of exceptional sets and inner standard part of sets, we now turn to the real subject matter of this section, an investigation of standard parts of hyperﬁnite Markov chains. Let us ﬁrst describe what kind of processes we would like to obtain as standard parts [152]. Let Y be a Hausdorﬀ space and let Δ be an extra element. Set YΔ = Y ∪ {Δ} . Consider the topological Borel ﬁeld B(Y ) on Y . Let YΔ have the σ-algebra B(YΔ ) generated by the Borel sets on Y and the singleton {Δ}. Our standard processes will be YΔ valued, and the new element Δ will serve as a trap. Assume that (Ω, Π) is a measurable space and {Πt | t ∈ R+ } is a family of sub-σ-algebras of Π satisfying Πs ⊂ Πt

for all s ≤ t

and

Πt =

Πs

for all t ∈ R+ .

(3.1.8)

s>t

We call {Πt | t ∈ R+ } a right continuous ﬁltration on Ω if it satisﬁes the condition (3.1.8). Set Π∞ = σ {Πt | t ∈ R+ } . In our situation, we can take Π = Π∞ . A map σ : Ω → [0, ∞) is called a stopping time of {Πt | t ∈ R+ } if for all t {ω | σ(ω) ≤ t} ∈ Πt . For each stopping time σ, we introduce a σ-algebra Πσ by Πσ = {A ∈ Π∞ | ∀t(A ∩ (σ ≤ t) ∈ Πt )} . Denote by M(Y ) the set of all positive Radon measures on (Y, B(Y )) with ﬁnite masses. Given an element ν ∈ M(Y ), the completion of the σ-ﬁeld B(Y ) with respect to ν is denoted by B(Y )ν . A set or a function is called universally measurable if it is measurable with respect to ν∈M(Y ) B(Y )ν .

3.1 Standard Parts of Hyperfinite Markov Chains

137

If for each y ∈ YΔ , we are given a probability measure Θy on (Ω, Π). Then for each ν ∈ M(Y ), we have Θν (A) =

Y

Θy (A) dν(y), A ∈ Π,

provided, of course, that this makes sense, i.e., y → Θy (A) is μ measurable. Finally, if x(t) is a YΔ valued process, its lifetime ζ is deﬁned by ζ(ω) = inf {t ≥ 0 | x(ω, t) = Δ} . Definition 3.1.1. We call (Ω, Π∞ , {Πt | t ∈ R+ } , x(t), Θy ) a strong Markov process, if {Πt | t ∈ R+ } is a right continuous ﬁltration of Π, each Θy is a probability measure on Π∞ , and x : Ω × [0, ∞] −→ YΔ is a stochastic process on (Ω, Π∞ , Θy ) for each y ∈ YΔ . Moreover, the following conditions are satisﬁed: (i) For all t ≥ 0 and all measurable E ⊂ Y, the map y → Θy (x(t) ∈ E) is universally measurable in y ∈ Y. (ii) x(ω, ∞) = Δ, ∀ω ∈ Ω. x(ω, t) = Δ, ∀t ≥ ζ(ω). (iii) For each y ∈ YΔ , the process x is adapted to (Ω, {Πt } , Θy ), t → xt (ω) is right continuous from [0, ∞) to YΔ , Θy -a.e.ω and lims↑t xs (ω) exists in Y for all t ∈ (0, ζ(ω)), Θy -a.e.ω (a.e. represents almost every). (iv) ΘΔ (x(t) = Δ) = 1, ∀t ≥ 0; Θy (x(0) = y) = 1, ∀y ∈ Y. (v) For all {Πt } stopping times σ, all measurable E ⊂ YΔ , all μ ∈ M(Y ), and all s ∈ R+ , we have Θμ (x(σ + s) ∈ E | Πσ ) = Θx(σ) (x(s) ∈ E), Θμ -a.e.

(3.1.9)

In order to prove that a class of hyperﬁnite Markov chains have standard parts that are strong Markov processes, we shall have to overcome two main diﬃculties: the construction of the family of measures {Θy }, and the proof of the strong Markov property (3.1.9). But ﬁrst we must introduce the necessary regularity conditions in our nonstandard processes. Here, we would remind the reader that we use Pi to mean the internal probability measure on (Ω, Ft ), i ∈ S, t ∈ T , and we use Θy to mean the standard probability measure on (Ω, Π∞ ), y ∈ Y. We hope this will not cause confusion. Definition 3.1.2. Let f : T −→ ∗ R be internal. We say that r ∈ R is the S-right limit of f at t ∈ [0, ∞) if for any standard ε > 0, there is a standard δ > 0 such that if s ∈ T and t < ◦s < t + δ, then |f (s) − r| < ε, we write r = S-lim f (s). The S-left limit, S-lim f (s) is deﬁned analogously. s↓t

s↑t

138

3 Standard Representation Theory

Recall that S 0 = S0 ∩ N s(∗ Y ) and that X (δ) = X|Tδ . The lifetime ζδ of X is deﬁned by (δ)

ζδ (ω) = inf

◦

t | X (δ) (ω, t) ∈ / S0 .

We deﬁne the right standard part ◦X (δ)+ as follows. If t < ζδ (ω), let ◦

X (δ)+ (ω, t) = S- lim X (δ) (ω, s) s↓t

if this limit exists and for all t1 < t, the S-lims↓t1 X (δ) (ω, s) exists also, and ◦

X (δ)+ (ω, t) = Δ

else. If t ≥ ζδ (ω), we always put ◦X (δ)+ (ω, t) = Δ. Definition 3.1.3. A subset A of S0 is called a set of irregularities of X if there is a positive inﬁnitesimal δ0 ∈ T satisfying: (i) For all si ∈ S 0 − A and L(Pi )-a.e. ω, the path X (δ0 ) (ω, ·) has S-right and S-left limits at all t < ζδ0 (ω). (ii) For all si ∈ S 0 − A, the set ◦ (δ0 ) ω ∃t ∈ Tδfin ( t > ζ (ω) ∧ X (ω, t) ∈ S ) δ0 0 0 has L(Pi ) measure zero. (iii) For all inﬁnitely close si , sj ∈ S 0 − A, L(Pi ){◦X (δ0 )+ (ω, t) ∈ B} = L(Pj ){◦X (δ0 )+ (ω, t) ∈ B} for all ﬁnite t ∈ [0, ∞) and all Borel sets B. The hyperﬁnite Markov chain X has δ0 -exceptional irregularities if the set A of irregularities of X in the Deﬁnition 3.1.3 can be taken as a δ0 -exceptional set. Hence, X has exceptional irregularities if it has δ0 exceptional irregularities for some δ0 ≈ 0, δ0 ∈ T. The ﬁrst condition of the above Deﬁnition 3.1.3 guarantees that the hyperﬁnite Markov chainX has a reasonable standard part. The second says that “inﬁnite” is a trap. The third one is a version of the condition (3.1.1). Putting st(si ) = Δ when si ∈ S − S 0 , we have the following deﬁnition of the modiﬁed standard part of X(ω, t). Definition 3.1.4. Assume that X has exceptional irregularities, and let A be a properly δ0 -exceptional set of irregularities, where δ0 is as in Deﬁnition 3.1.3. Let x : Ω × R+ −→ YΔ be deﬁned by

3.1 Standard Parts of Hyperfinite Markov Chains

139

(i) if X(ω, 0) ∈ / A, then x(ω, t) = ◦X (δ0 )+ (ω, t). (ii) if X(ω, 0) ∈ A, then x(ω, t) = st(X(ω, 0)) for all t ∈ R+ . We call x modiﬁed standard part of X(ω, t). Our aim is to prove that if X has exceptional irregularities, then with an appropriate deﬁnition of the family {Θy } of measures, the modiﬁed standard parts of X are strong Markov processes. The ﬁrst step toward the deﬁnition of {Θy } is the following version of condition (iii) in Deﬁnition 3.1.3. If ν is an internal probability measure on S, let Pν be the measure on Ω deﬁned by (3.1.10) Pν (C) = Pi (C) dν(si ). Lemma 3.1.4. For δ0 ≈ 0, δ0 ∈ T , let A be a properly δ0 -exceptional set of irregularities of X, and let x be the corresponding modiﬁed standard part of X. Let ν1 , ν2 be two internal probability measures on S0 such that L(ν1 )(A) = L(ν2 )(A) = 0 and L(ν1 ) ◦ st−1 = L(ν2 ) ◦ st−1 , then for all t ∈ R+ and all Borel sets B L(Pν1 ){x(ω, t) ∈ B} = L(Pν2 ){x(ω, t) ∈ B}. Proof. Let μ = L(ν1 ) ◦ st−1 = L(ν2 ) ◦ st−1 . Choose t˜ ≈ t so large that X (δ0 ) (ω, t˜) = x(ω, t), L(Pν1 ) and L(Pν2 )-a.e.ω ∈ Ω

◦

˜ such that and pick an internal set B ◦ (δ0 ) ˜ X (ω, t˜) ∈ B}) = 0 L(Pνi )({X (δ0 ) (ω, t˜) ∈ B}Δ{

for i = 1, 2. Deﬁne a function f : Y −→ R as follows: (1) If y ∈ / A◦ , let f (y) = L(Pi ){x(ω, t) ∈ B} for some (then for all) si ∈ −1 st (y) ∩ S0 − A. (2) If y ∈ A◦ , deﬁne f (y) arbitrarily. ˜ is a lifting of f with respect to The function si → Pi {X (δ0 ) (ω, t˜) ∈ B} both ν1 and ν2 . Therefore, we have ˜ L(Pν1 ){x(ω, t) ∈ B} = ◦P ν1 {X (δ0 ) (ω, t˜) ∈ B} ◦ ˜ dν1 (si ) = Pi {X (δ0 ) (ω, t˜) ∈ B} = f (y) dμ(y)

140

3 Standard Representation Theory

=

◦

˜ dν2 (si ) Pi {X (δ0 ) (ω, t˜) ∈ B}

˜ = ◦P ν2 {X (δ0 ) (ω, t˜) ∈ B} = L(P ν2 ){x(ω, t) ∈ B},

and the lemma is proved.

Lemma 3.1.5. For δ0 ≈ 0, δ0 ∈ T , assume that X has a properly δ0 exceptional set A of irregularities, and let x be the modiﬁed standard part. For all inﬁnitely close si , sj ∈ S 0 − A, all ﬁnite sequences t1 < t2 < · · · < tn from R+ and all Borel sets B1 , B2 , · · · , Bn , we have L(Pi )

n

{x(tl ) ∈ Bl }

l=1

= L(Pj )

n

{x(tl ) ∈ Bl } .

l=1

Proof. We shall prove this by induction on the length n of the sequences t1 < t2 < · · · < tn , B1 , B2 , · · · , Bn . The case of n = 1 is part of Deﬁnition 3.1.3. Assume that the lemma holds for all sequences of length n − 1, and pick t˜1 , t˜, · · · , t˜n ∈ Tδ0 such that t˜1 ≈ t1 , t˜2 ≈ t2 , · · · , t˜n ≈ tn and X (δ0 ) (ω, t˜l ) = x(ω, tl ), L(Pk )-a.e.

◦

˜2 , · · · , B ˜n such that ˜1 , B for l = 1, 2, · · · , n and k = i, j. Choose internal sets B ˜l }Δ{◦X (δ0 ) (ω, t˜l ) ∈ Bl } = 0 L(Pk ) {X (δ0 ) (ω, t˜l ) ∈ B for l = 1, 2, · · · , n and k = i, j. We deﬁne two measures νi , νj on S by putting ˜l } νk (s) = Pk {ω | X (δ0 ) (ω, t˜n−1 ) = s and for all l < n − 1, X (δ0 ) (ω, t˜l ) ∈ B for all s ∈ S and k = i, j. By the induction hypothesis L(νi ) ◦ st−1 = L(νj ) ◦ st−1 , and since A is properly δ0 -exceptional, we have L(νi )(A) = L(νj )(A) = 0. Applying Lemma 3.1.4 with t = tn − tn−1 , we get L(Pνi ){x(tn − tn−1 ) ∈ Bn } = L(Pνj ){x(tn − tn−1 ) ∈ Bn }. Since X is Markov and time homogeneous, the lemma follows.

In the following, we will deﬁne {Πt } and {Θy } of our standard Markov processes. For t ∈ R+ , let Πt◦ be the σ-algebra generated by the sets

3.1 Standard Parts of Hyperfinite Markov Chains

141

ω x(ω, t1 ) ∈ B1 ∧ · · · ∧ x(ω, tn ) ∈ Bn , where 0 ≤ t1 < t2 < · · · < tn ≤ t and B1 , · · · , Bn are Borel sets in Y. Deﬁne Πt =

s>t

Πs◦ .

The ﬁltration {Πt | t ∈ [0, ∞)} is right continuous. Let us set Π∞ = ∨ {Πt | t ∈ [0, ∞)} . It follows from Lemma 3.1.5 that for all C ∈ Π∞ and all inﬁnitely close si , sj in S 0 − A L(Pi )(C) = L(Pj )(C). This observation makes it possible to give the following deﬁnition of a family {Θy | y ∈ YΔ } of measures on Π∞ . If y ∈ / A◦ , let for all C ∈ Π∞ Θy (C) = L(Pi )(C)

for all si ∈ st−1 (y) − A.

If y ∈ A◦ ∪ {Δ} , C ∈ Π∞ , let Θy (C) =

1, if C contains all constant paths x(t) = y, 0, else.

Observe that since a set C ∈ Π∞ contains either all or none of the constant paths x(ω, t) = y, the set function Θy is a measure. We have reached our goal: Theorem 3.1.1. Assume that S0 is a hyperﬁnite subset of ∗ Y for some Hausdorﬀ space. Let X : Ω × T −→ S be a hyperﬁnite Markov chain with exceptional irregularities. Assume that for each i ∈ S0 , the function t → P {ω | X(ω, t) = si } is decreasing. If x is a modiﬁed standard part of X, then (Ω, {Πt | t ∈ R+ }, {Θy | y ∈ YΔ }, x(t)) is a strong Markov process. Proof. Since we already know that {Πt } is right continuous, all we have to do is to check conditions (i)-(v) of Deﬁnition 3.1.1. Obviously, the (ii), (iii) and (iv) of Deﬁnition 3.1.1 are satisﬁed by our construction. We shall prove (i) and (v). Given a Radon probability measure μ on Y , we deﬁne two new measures μ0 and μ1 on Y by

142

3 Standard Representation Theory

μ0 (B) = μ(B − A◦ ), μ1 (B) = μ(B ∩ A◦ ), where A is the properly δ0 -exceptional set of irregularities used in the construction of x. By Corollary 3.1.1, we can ﬁnd an internal measure ν on S0 satisfying μ0 = L(ν) ◦ st−1 , L(ν)(A) = 0.

(3.1.11)

We ﬁrst prove that (i) in Deﬁnition 3.1.1 is satisﬁed: (i) Given α ∈ [0, 1], a Borel set E ⊂ Y, and t ∈ R+ , we must show that {y ∈ Y | Θy (x(t) ∈ E) > α} is μ-measurable for all ﬁnite Radon measures μ. Since

◦

{y ∈ A | Θy {x(t) ∈ E} > α} =

A◦ ∩ E, α ∈ [0, 1) ∅, α=1

is universally measurable by Lemma 3.1.2, it suﬃces to show that {y ∈ / A◦ | Θy (x(t) ∈ E) > α}

(3.1.12)

is μ-measurable. In fact, we only have to show that the set deﬁned by (3.1.12) is μ0 measurable by the deﬁnition of μ0 . Since X (δ0 ) has S-right limits a.e. with respect to the probability measure Pν constructed in (3.1.11) by using the relation (3.1.10), we can choose tˆ ≈ t so large that L(Pν )({x(t) ∈ E}Δ{◦X (δ0 ) (t˜) ∈ E}) = 0. There must be an internal set E˜ such that ˜ = 0. L(Pν ){X (δ0 ) (t˜) ∈ (st−1 (E)ΔE)} Hence, we have ˜ L(Pi ){x(t) ∈ E} = ◦Pi {X (δ0 ) (t˜) ∈ E} for L(ν)-a.e.si . Combining this with the deﬁnition of Θy , we see that ˜ as a ν-lifting. Hence, it is y → Θy {x(t) ∈ E} has i → Pi {X (δ0 ) (t˜) ∈ E} a μ0 measurable function. This proves that the set deﬁned by (3.1.12) is μ0 measurable, and Deﬁnition 3.1.1 (i) follows.

3.1 Standard Parts of Hyperfinite Markov Chains

143

(v) we must show that for all {Πt } stopping times σ, all sets B ∈ Πσ and all s ∈ [0, ∞), the equation Θμ {ω ∈ B | xσ+s ∈ E} =

B

Θxσ {xs ∈ E} dΘμ

(3.1.13)

holds for all Radon probability measure μ on Y and all Borel sets E. First notice that since the paths of x are constant Θμ1 -a.e., we have Θμ1 {ω ∈ B | xσ+s ∈ E} =

B

Θxσ {xs ∈ E} dΘμ1 .

Equation (3.1.13) will hold if we may prove Θμ0 {ω ∈ B | xσ+s ∈ E} =

B

Θxσ {xs ∈ E} dΘμ0 .

(3.1.14)

In the sequel, we will prove (3.1.14). Let ν be the nonstandard representation of μ0 given in the construction (3.1.11). The hyperﬁnite counterpart of (3.1.14) is

(δ )

0 Pν {ω ∈ C | Xτ +s ∈ F} =

C

PX (δ0 ) {Xs(δ0 ) ∈ F } dPν , τ

(3.1.15)

where C and F are internal sets and C is measurable in the ∗ −algebra generated by the internal stopping time τ . Since X (δ0 ) is a time homogeneous Markov chain, (3.1.15) holds. Our plan is to deduce (3.1.14) from (3.1.15). We ﬁrst pick an internal stopping time τ such that ◦ τ = σ, L(Pν )-a.e., and such that there is a τ measurable set C satisfying L(Pν )(BΔC) = 0.

(3.1.16)

Let P τ be an internal measure on Ω given by P τ (D) = PX (δ0 ) (D) dPν . τ

Since A is properly exceptional, we observe that (δ0 )

L(P τ ){Xt

∈ A} = 0

(where δ0 is the inﬁnitesimal used in the construction of x, for all t ∈ Tδfin 0 see Deﬁnition 3.1.4). Hence, we can choose an s˜ ∈ Tδfin , s˜ = s, such that 0

144

3 Standard Representation Theory (δ0 )

◦

Xs˜

= xs , L(P τ )-a.e.

(3.1.17)

and ◦

(δ )

0 X τ +˜ s = xσ+s , L(Pν )-a.e.

(3.1.18)

Finally, we pick an internal set F such that (δ )

0 −1 L(Pν ){Xτ +˜ (E)ΔF } = 0, s ∈ st

(3.1.19)

(δ ) L(P τ ){Xs˜ 0

(3.1.20)

∈ st−1 (E)ΔF } = 0.

We now have Θμ0 {ω ∈ B | xσ+s ∈ E} = Θy {ω ∈ B | xσ+s ∈ E} dμ0 (y) = L(Pi ){ω ∈ B | xσ+s ∈ E} dL(ν)(si )

(by def. of Θy )

= L(Pν ){ω ∈ B | xσ+s ∈ E}

(by def. of Pν )

◦

= Pν {ω ∈ C | Xτ +˜s ∈ F } ◦ PXτ {Xs˜ ∈ F } dPν = C = L(PXτ ){xs ∈ E} dL(Pν ) B = Θxσ {xs ∈ E} dΘμ0

(by def. of Θμ0 )

(by (3.1.16), (3.1.18), (3.1.19)) (by (3.1.15)) (by (3.1.16), (3.1.17), (3.1.20)) (by def. of ν, τ and {Θy }).

B

This proves (3.1.14) and the theorem.

Example 3.1.1. Let Z = {· · · , −2, −1, 0, 1, 2, · · · } (i.e., set of all the integers). Let η ∈ ∗ N − N, and set S0 =

k ∗ √ k ∈ Z, |k| ≤ η . η

Deﬁne transition probabilities qs,s by qs,s =

1 2

if |s − s | =

0

otherwise.

If s, s ∈ S0 , and if s0 is the trap, let

√1 , η

3.2 Hyperfinite Dirichlet Forms and Markov Processes

qs,s =

1 2

√ if s = ± η,

0

otherwise.

145

Let m be an internal measure on S0 given by 1 m(s) = √ η for all s ∈ S0 , and let a time-line be k ∗ T = k ∈ N0 . η The process X is Anderson’s random walk with a uniform initial distribution corresponding to the Lebesgue measure. To prove that the standard part of X is a strong Markov process, we show that the empty set is a set of irregularities of X. Since X is S-continuous L(Pi )-a.e. for all nearstandard si ∈ S0 , the two ﬁrst conditions in Deﬁnition 3.1.3 are obviously satisﬁed. If si ≈ sj , the paths starting at si look exactly like the paths starting at sj except for an inﬁnitesimal translation. Hence, they induce the same standard paths and the property (iii) in Deﬁnition 3.1.3 follows.

3.2 Hyperfinite Dirichlet Forms and Markov Processes In this section, we shall obtain conditions on hyperﬁnite quadratic forms which guarantee that the modiﬁed standard parts of the associated hyperﬁnite Markov chains are strong Markov processes. The method we shall apply is simple, we just use the relationship between forms and processes established in Sect. 1.5 to translate the conditions of Theorem 3.1.1 into the language of hyperﬁnite quadratic forms. One by one we shall reformulate the conditions of Deﬁnition 3.1.3 in terms of hyperﬁnite quadratic forms. In this section, we need a slightly stronger assumption on the state space. Let Y be a regular Hausdorﬀ space (or a T3 space), i.e., for all closed sets F and x ∈ F, there are disjoint open sets O1 , O2 such that x ∈ O1 , F ⊂ O1 . Suppose that S0 = {s1 , s2 , · · · , sN } is a hyperﬁnite subset of ∗ Y and m is a hyperﬁnite positive measure on S0 , N ∈ ∗ N − N. Let H be the linear space of all internal functions u : S0 −→ ∗ R with the inner product

u, v =

N i=1

u(si )v(si )m(si ).

146

3 Standard Representation Theory

Given an inﬁnitesimal Δt ≈ 0, denote by T = {kΔt | k ∈ ∗ N0 } the hyperﬁnite ˆ be the dual hyperﬁnite Markov chains introduced time line. Let X and X in Sect. 1.5 with m as dual measure. That is, we assume that conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14) in Sect. 1.5 are fulﬁlled. Let E(·, ·) be the hyperﬁnite quadratic form on H associated with the transition matrix Q and the dual measure m, ˆ ·) be the co-form of E(·, ·). and E(·, Given x ∈ Y , we recall that the monad1 of x is the set of ∗ Y deﬁned by μ(x) = {∗ Ox ∈ O and O is open}.

3.2.1 Separation of Points The following assumption will take care of Deﬁnition 3.1.3 (i). In Albeverio et al. [25], 5.5.1. Deﬁnition, a deﬁnition of separation of compacts was given for symmetric hyperﬁnite Dirichlet forms. The following deﬁnition simpliﬁes the condition since a single point can be viewed as a compact set. Definition 3.2.1. Let Z be a subset of Y . A hyperﬁnite quadratic form E(·, ·) separates points of Z if there exists a countable family π = {un | n ∈ N} of internal functions such that for all x, y ∈ Z, x = y, there is an element un ∈ π satisfying ◦

un (s1 ) = ◦un (s2 ), s1 ∈ μ(x), s2 ∈ μ(y), si ∈ S0 , i = 1, 2.

(3.2.1)

Moreover, we have ◦

E1 (un , un ) < ∞, ∀un ∈ π.

(3.2.2)

We call π = {un | n ∈ N} a separating family of Z by E(·, ·). We recall from Sect. 2.1 that if δ ∈ T, the sub-line Tδ is deﬁned by Tδ = {kδ | k ∈ ∗ N0 }, X (δ) is the restriction of X|Tδ . Let ζ = ζδ be the lifetime of X (δ) , i.e., ζδ (ω) = inf{◦ t | X (δ) (ω, t) ∈ S 0 }, where S 0 = S0 ∩ N s(∗ Y ). For any subset A ⊂ S0 , let (δ) τA (ω) = inf{◦ tX(t) ∈ st−1 (A◦ ) and t ∈ Tδ }. 1

For the general concept of monad, we refer to Albeverio et al. [25].

3.2 Hyperfinite Dirichlet Forms and Markov Processes

147

Lemma 3.2.1. Let Y be a regular Hausdorﬀ space, and let A be a subset of S0 . Suppose that E(·, ·) separates the points of Y − A◦ and π is the separating family. For ω ∈ Ω, if the path X(·, ω) fails to have an S-left or S-right limit (Δt) at t < ζΔt (ω) ∧ τA (ω), then so does u(X(ω, ·)) for some u ∈ π. (Δt)

Proof. Let ω be a ﬁxed point in Ω. Fix t < ζΔt (ω)∧τA (ω), t ∈ [0, ∞). Given a sequence {tn | n ∈ N} from T such that the standard part ◦ tn increases strictly to t, we shall prove that the sequence ◦ X(ω, tn )n ∈ N has a cluster point in Y − A◦ . Suppose for a contradiction that

X(ω, tn )n ∈ N

◦

has no cluster point in Y . Then for each y ∈ Y , there is a neighborhood Oy and an integer ny ∈ N such that ◦

X(ω, tn ) ∈ / Oy

when n ≥ ny . Since Y is regular, we can ﬁnd a neighborhood Gy of y such that the closure Gy is contained in Oy . Hence, we have / ∗ Gy X(ω, tn ) ∈ when n ≥ ny . Extend {tn | n ∈ N} to be an internal sequence {tn | n ∈ ∗ N} of elements of T less than t. Consider the set Ay = {n ∈ ∗ N | n ≤ ny

or X(ω, tn ) ∈ / ∗ Gn } .

Since Ay is internal and contains N, there is an ηy ∈ ∗ N − N such that all η ≤ ηy are elements of Ay . By saturation, there is an inﬁnite η less than all ηy . But then, we have X(ω, tη ) ∈ / ∗ Gy for all y. This implies that X(ω, tη ) is not nearstandard, contradicting our assumption that t < ζΔt (ω). Suppose that y ∈ Y is a cluster point of {◦X(ω, tn )}n∈N . Since tn ↑ t, (Δt) t < τA (ω), it is easy to see that y ∈ / A◦ . Let x ∈ Y − A◦ =Z ˆ be a cluster point of {◦X(ω, t)} . If x is not the S-left limit of X(ω, ·) at t, there must be another sequence {sn | n ∈ N} increasing

148

3 Standard Representation Theory

to t such that x is not a cluster point of {◦X(ω, sn ) | n ∈ N}. Repeating the argument above, we see that {◦X(ω, sn ) | n ∈ N} must have a cluster point y ∈ Z. Let u ∈ π be the function satisfying the conditions of (3.2.1) and (3.2.2). Obviously, u(X(ω, ·)) does not have an S-left limit at time t. This proves the S-left case of the lemma. The S-right limit case can be treated in a similar way.

We shall use Lemma 3.2.1 and Fukushima’s decomposition Theorem 2.7.2 to show that if E(·, ·) separates points, then the associated Markov chain X satisﬁes Deﬁnition 3.1.3 (i). But we must get some preparation at ﬁrst. It is easy to see that {ω | ∃t ∈ Tδr (X(ω, t) ∈

An )} =

n∈N

{ω | ∃t ∈ Tδr (X(ω, t) ∈ An )},

n∈N

(3.2.3) for all sequences {An | n ∈ N} of subsets of S. However, the corresponding formula for intersections is false in general. In particular, it does hold if the sequence is decreasing and consists of internal sets. This is the observation behind the next lemma. Lemma 3.2.2. Let A ⊂ S0 , and assume that there is a family {Bm,n | m, n ∈ N} of internal sets such that A=

Bm,n

m∈N n∈N

and for each m, the sequence {Bm,n | n ∈ N} is decreasing. Then {ω | ∃t ∈ Tδr (X(ω, t) ∈ A)} =

ω | ∃t ∈ Tδr X(ω, t) ∈ Bm,n . m∈N n∈N

(3.2.4) Proof. It suﬃces to prove that r r Bm,n ω ∃t ∈ Tδ X(ω, t) ∈ = ω ∃t ∈ Tδ X(ω, t) ∈ Bm,n n∈N

n∈N

(3.2.5) for all m, since (3.2.4) then follows from (3.2.3). Also, it is immediately clear that the left hand of (3.2.5) is included in the right hand side. To prove the opposite inclusion, choose

3.2 Hyperfinite Dirichlet Forms and Markov Processes

ω0 ∈

149

{ω | ∃t ∈ Tδr (X(ω, t) ∈ Bm,n )}.

n∈N

Consider the set ˜m,n )}, {n ∈ ∗ N | ∃t ∈ Tδr (X(ω0 , t) ∈ B ˜m,n | n∈∗ N} is some internal, decreasing extension of {Bm,n | where {B n ∈ N}. By the choice of ω0 , this set contains N. Since it is internal, it must have an inﬁnite member η. Thus, we have ˜m,η )} ⊂ {ω | ∃t ∈ Tδr (X(ω, t) ∈ ω0 ∈ {ω | ∃t ∈ Tδr (X(ω, t) ∈ B

Bm,n )}.

n∈N

The lemma is proved. We are now in the position to prove the following:

Proposition 3.2.1. Let Y be a regular Hausdorﬀ space, and let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form for every inﬁnitesimal δ ∈ T . Assume that A is an exceptional set and E(·, ·) separates the points of Y −A◦ . There exists an inﬁnitesimal δ0 ∈ T such that for all inﬁnitesimal δ ∈ Tδ0 , there is a δ-exceptional set A0 (δ) satisfying that for all si ∈ S0 − A0 (δ), the hyperﬁnite Markov chain X (δ) (t) has S-left and S-right limits at all t < ζδ , L(Pi )-a.e. Proof. To ﬁnd δ0 , we look at the property of the separating family π = (δ) {un | n ∈ N}. From Theorem 1.4.2, we know that D(E (δ) ) = D(E ) for all ◦ δ ∈ T, δ ≈ 0. For each u ∈ π, we have E 1 (u, u) < ∞. Hence, we can ﬁnd (δ)

a δu ≈ 0 such that u ∈ D(E ) = D(E (δ) ) for all inﬁnitesimal δ ≥ δu by Corollary 1.2.4. By saturation, there is a δ0 ≈ 0 larger than all δu and A is δ0 -exceptional. Turning to A0 (δ) for inﬁnitesimal δ ∈ Tδ0 , we ﬁrst observe that by Lemma 3.2.1 and the countability of π, it suﬃces to show that for each un ∈ π = {un | n ∈ N}, there is a δ-exceptional set An (δ) such that for all si ∈ S0 − An (δ), the process un (X (δ) (t)) has S-left and S-right limits at all t < ∞, L(Pi )-a.e. Actually, we deﬁne A0 (δ) = A0

∞

An (δ) ,

n=1

where A0 is a properly δ-exceptional set containing A. Then, A0 (δ) is δexceptional by Lemma 2.1.1 and for all si ∈ Si − A0 (δ), un ∈ π, un (X (δ) (t)) has S-right and S-left limits for all t < ∞, L(Pi )-a.e. Therefore, the hyperﬁnite Markov chain X (δ) (t) has S-left and S-right limits at all t < ζδ (ω),

150

3 Standard Representation Theory

L(Pi )-a.e. by Lemma 3.2.1. From this observation, we only need to ﬁnd the exceptional set An (δ) for every un ∈ π. For the simplicity of notation, we write u = un in the following. Given two standard rationals p, q, −∞ < p < q < ∞, we deﬁne a sequence n {τ(p,q) } of stopping times as follows: 0 τ(p,q) = min{t ∈ Tδ | u(X (δ) )(ω, t)) ≤ p}, 2n−1 2n = min{t ∈ Tδ | t > τ(p,q) (ω) ∧ u(X (δ) )(ω, t)) ≤ p}, τ(p,q) 2n+1 2n τ(p,q) = min{t ∈ Tδ | t > τ(p,q) (ω) ∧ u(X (δ) )(ω, t)) ≥ q}.

Let B ⊂ S0 be deﬁned by B=

n {i | Pi {τ(p,q) ≤ m} ≥

(p,q) m∈N k∈N n∈N

1 }. k

If X (δ) fails to have S-left or S-right limits with positive L(Pi ) probability, then we have si ∈ B. We must show that B is δ-exceptional. By Lemma 3.2.2, we have for all r ∈ [0, ∞) {ω | ∃t ∈ Tδr (X(ω, t) ∈ B)} 1 n ≤ m} ≥ = ω ∃t ∈ Tδr PX (δ) (ω,t) {τ(p,q) . k (p,q) m∈N k∈N n∈N

If B is not δ-exceptional, there must be a pair (p, q) of rationals, as well as integers m, k ∈ N, r ∈ [0, ∞) and an inﬁnite number η ∈ ∗ N such that 1 η r L(P ) ω ∃t ∈ Tδ PX (δ) (ω,t) {τ(p,q) ≤ m} ≥ > 0. k This implies that with positive probability, u(X (δ) ) jumps back and forth between p and q more than η times before time m + r. Since u ∈ D(E (δ) ), Fukushima’s decomposition Theorem 2.7.2 tells us that u(X (δ) ) = N [u] + M [u] , where N [u] is S-continuous L(P )-a.e. and M [u] is a λ2 -martingale. If u(X (δ) ) jumps η times between p and q before t = m+r, there must be an inﬁnitesimal interval where it jumps back and forth inﬁnitely many times. Since N is S-continuous – and hence almost constant on inﬁnitesimal intervals – most

3.2 Hyperfinite Dirichlet Forms and Markov Processes

151

of this jumping is done by M . Hence the quadratic variation of M is inﬁnite on a set of positive measure, contradicting the fact that it is a λ2 -martingale. We then conclude that B must be δ-exceptional, and the proposition is thus proved.

3.2.2 Nearstandardly Concentrated Forms Now we are in a position to ﬁnd a condition on E(·, ·) that implies a similar property like (ii) in Deﬁnition 3.1.3. For any internal subset C of S0 , we set HC = {u ∈ H | u(si ) = 0 for all si ∈ S0 − C} . For each u ∈ H and any internal set C ⊂ S0 , δ ∈ T, we deﬁne (δ) (δ) (δ) uC (i) = u(i) − Ei (1 + δ)−σS0 −C /δ u X (δ) (σS0 −C ) ,

(3.2.6)

where (δ) σS0 −C (ω) = min t ∈ Tδ | X (δ) (ω, t) ∈ S0 − C . It is easy to see that (δ)

u C ∈ HC . Definition 3.2.2. (1) For δ ∈ T , a hyperﬁnite quadratic form E(·, ·) is said to be δ-nearstandardly concentrated if there exists an increasing sequence {Bn | n ∈ N} of internal subsets of S0 such that: (i) The set ∪n∈N Bn − S 0 is δ-exceptional. (δ) (ii) For each F ∈ H, if ◦ E1 (F, F ) < ∞, then we have ◦ (δ) E1 (F

(δ)

(δ)

− FBm , F − FBm ) −→ 0, m −→ ∞.

(2) A hyperﬁnite quadratic form E(·, ·) is said to be nearstandardly concentrated if it is δ-nearstandardly concentrated for some inﬁnitesimal δ ∈ T. Notice that above deﬁnition of nearstandardly concentrated is diﬀerent from that of Albeverio et al. [25], 5.5.4 Deﬁnition. Our deﬁnition is motivated by both Albeverio et al. [25], 5.5.4 Deﬁnition and Condition (I) of Theorem 4.1.1 in Chap. 4.

152

3 Standard Representation Theory

Lemma 3.2.3. For δ ∈ T, let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Assume that E(·, ·) is δ-nearstandardly concentrated. Let X be the related hyperﬁnite Markov (δ) chain. Then for each F ∈ H with ◦ E1 (F, F ) < ∞, the set

si ∈ S 0 −

A(F ) =

m∈N

fin (δ) Bn L(Pi ) ω ∃t ∈ Tδ F (X (t)) ≈ 0 >0 (3.2.7)

is δ-exceptional. Proof. Step 1. For ε > 0, set D = si |F (si )| > ε , (δ) σD (ω) = min t ∈ Tδ | X (δ) (ω, t) ∈ D , (δ) (δ) −σ (ω)/δ . e1 (D) = Ei (1 + δ) D From Corollary 2.2.3, we have (δ)

E1

C2 (δ) (δ) (δ) e1 (D), e1 (D) ≤ 2 E1 (F, F ). ε

Hence, we have ◦ (δ) E1

(δ) (δ) e1 (D), e1 (D) < ∞.

This implies that ◦ (δ) E1

(δ) (δ) (δ) (δ) (δ) (δ) e1 (D) − (e1 (D))Bm , e1 (D) − (e1 (D))Bm −→ 0, m −→ 0, (3.2.8)

(δ)

(δ)

where (e1 (D))Bm is deﬁned in the same way as the deﬁnition (3.2.6). Step 2. From the approximation (3.2.8), there exists an increasing sequence {mk }k∈N satisfying δk = ε >0

Ak (F, ε).

k=K

Since δk < 2−k , it follows from the relation (3.2.9) that 1

P {ω | ∃t ∈ Tδ1 (X(ω, t) ∈ A(F, ε))} ≤ 2C 2 (1 + δ) δ

∞

δk

k=K 1

≤ C 2 (1 + δ) δ 2−K+2 . Hence, A(F, ε) is δ-exceptional. it is easy to see that A(F ) is δ-exceptional, since A(F ) = Step 3. Now 1 A(F, ).

m∈N m Proposition 3.2.2. For δ ∈ T, let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Assume that E(·, ·) is δ-nearstandardly concentrated. Let X be the related hyperﬁnite Markov chain. (δ) Then for each F ∈ H with ◦ E1 (F, F ) < ∞, there exists a δ-exceptional set

154

3 Standard Representation Theory

A1 (F, δ) such that for si ∈ S0 − A1 (F, δ) L(Pi ) ω ∃t ∈ Tδfin ◦ t ≥ ζδ (ω) ∧ F (X (δ) (t)) ≈ 0 = 0. Proof. Let {Bn | n ∈ ∗ N} be an increasing extension of {Bn | n ∈ N}, which is the increasing sequence in Deﬁnition 3.2.2 (1). Set B = ∪n∈N Bn . Deﬁne = min t ∈ Tδ X (δ) (ω, t) ∈ / Bn , n ∈ ∗ N, (δ) /B . τ (ω) = inf ◦ tX (δ) (ω, t) ∈

τn(δ) (ω)

It is not hard to check that τ (δ) (ω) = sup

◦

τn (ω)n ∈ N .

Let A1 (F, δ) be a properly δ-exceptional set containing B − S 0 and the set A(F ), where A(F ) is deﬁned in Lemma 3.2.3 by the deﬁnition (3.2.7). Given si ∈ S0 − A1 (F, δ), we can ﬁnd an η ∈ ∗ N − N such that τ (δ) (ω) = ◦τη(δ) (ω), L(Pi )-a.e.

(3.2.10)

By the deﬁnition of A1 (F, δ), we have B − S 0 ⊂ A1 (F, δ). Since A1 (F, δ) is / A1 (F, δ), this implies that properly δ-exceptional and si ∈ τ (ω) = ζδ (ω), L(Pi )-a.e.

(3.2.11)

Combining (3.2.10) and (3.2.11), we have ◦ (δ) τη (ω)

= ζδ (ω), L(Pi )-a.e.

(3.2.12)

/ A1 (F, δ), A1 (F, δ) contains A(F ), and it is properly Notice that si ∈ δ-exceptional. We have L(Pi )(X(ω, τη(δ) (ω)) ∈ A(F )) = 0. Hence, we have by the deﬁnition of A(F ) fin (δ) (δ) L(Pi ) ω ∃t ∈ Tδ t ≥ τη (ω) ∧ F (X (t)) ≈ 0 = 0. The proposition follows from the relation (3.2.12).

3.2 Hyperfinite Dirichlet Forms and Markov Processes

155

3.2.3 Quasi-Continuity Definition 3.2.3. (1) An internal function F : S0 −→ ∗ R is said to be hyperﬁnite δ-quasi-continuous if there is a hyperﬁnite δ-exceptional set A ⊂ S0 such that for all inﬁnitely close si , sj ∈ S 0 − A, we have F (si ) ≈ F (sj ). (2) An internal function F : S0 −→ ∗ R is called hyperﬁnite quasi-continuous if there is an inﬁnitesimal δ such that F is hyperﬁnite δ-quasi-continuous. Remark 3.2.1. We remark that usually one uses zero capacity sets to deﬁne the quasi-continuity in regular Dirichlet spaces (refer to Sect. 4.1 of Chap. 4, or refer to [175]). Nevertheless, we utilize the exceptional sets in the deﬁnition of the hyperﬁnite quasi-continuity. The reason is that we have the equivalence between the concept of exceptional sets and that of zero capacity sets in regular Dirichlet space theory, e.g., Theorem 4.3.1 in [175]). However, we only have Theorem 2.4.1, Theorem 2.4.2, and Theorem 2.4.3 in Sect. 2.4, from which we understand that the conception of exceptional sets is not equivalent to that of zero capacity sets in hyperﬁnite Dirichlet space theory. Given a hyperﬁnite δ-quasi-continuous and S-bounded function F ∈ H, let A be a δ-exceptional set satisfying F (si ) ≈ F (sj ), si ≈ sj , si , sj ∈ S 0 − A. Deﬁne ◦F = f : Y −→ R by: / A◦ and si ∈ st−1 (y) − A; (i) f (y) = ◦F (si ) whenever y ∈ ◦ (ii) f (y) = 0 whenever y ∈ A . In order to get the condition that makes the process X to satisfy Deﬁnition 3.1.3 (iii), we impose the following condition on E(·, ·). Definition 3.2.4. (1) A hyperﬁnite quadratic form E(·, ·) is said to have a hyperﬁnite δ-quasi-continuous core if there are a family of countable functions π = {Fn | n ∈ N} ⊂ H and a δ-exceptional set A such that Fn (si ) ≈ Fn (sj ), si ≈ sj , si , sj ∈ S 0 − A, ∀n ∈ N, and (Y − A◦ ) ∩ B(Y ) ⊂ σ{fn = ◦F n | n ∈ N}.

(3.2.13)

(δ)

Moreover, st{maxsi ∈S0 |Fn (si )|} < ∞, ◦ E1 (Fn , Fn ) < ∞,∀n ∈ N. π is called a hyperﬁnite δ-quasi-continuous core of E(·, ·). (2) A hyperﬁnite quadratic form E(·, ·) is said to have a hyperﬁnite quasicontinuous core if it has a hyperﬁnite δ-quasi-continuous core for some inﬁnitesimal δ ∈ T. Lemma 3.2.4. Let M be a set. Consider a countable subset G of M and a countable collection Λ of maps from M × M into M . Then, there exists a countable set L such that (a)

G ⊂ L ⊂ M.

(b)

λ(L × L) ⊂ L, ∀λ ∈ Λ.

156

3 Standard Representation Theory

Proof. This is proven in Fukushima [175] Lemma 6.1.1.

Lemma 3.2.5. Suppose that E(·, ·) has a hyperﬁnite δ-quasi-continuous core ˆ = {Fˆn | n ∈ N} such π = {Fn | n ∈ N}. Then, there exists a countable set π that ˆ , a ∈ Q =⇒ |Fˆn | ∈ π ˆ , Fˆn + Fˆm ∈ π ˆ , Fˆn Fˆm ∈ π ˆ, (1) Fˆn , Fˆm ∈ π ˆ ˆ aFn ∈ π ˆ , and Fn ∧ a ∈ π ˆ, ˆ = {Fˆn | n ∈ N}, (2) π = {Fn | n ∈ N} ⊂ π (3) every element in π ˆ is hyperﬁnite δ-quasi-continuous, where Q = {a1 , a2 , · · · , an , · · · } is the set of all rational numbers. Proof. Let us denote by Qδ (S0 ) the set of hyperﬁnite δ-quasi-continuous functions on S0 . We deﬁne the maps Λ = {λ−2 , λ−1 , λ0 , λ1 , λ2 , · · · } from [H ∩ Qδ (S0 )] × [H ∩ Qδ (S0 )] into H ∩ Qδ (S0 ) by λ−2 (u, v) = |u|, λ−1 (u, v) = u + v, λ0 (u, v) = uv, λ2i−1 (u, v) = ai u, λ2i (u, v) = u ∧ ai , i = 1, 2, · · · Applying Lemma 3.2.4 to π and Λ, we get a countable set π ˆ satisfying (1), (2), and (3).

Lemma 3.2.6. Assume that the hyperﬁnite quadratic form E(·, ·) has a δ-quasi-continuous core π = {Fn | n ∈ N}. For any two ﬁnite measures ν1 and ν2 on (Y, B(Y )), if ν1 and ν2 satisfy the following condition: Y

◦

F (y) ν1 (dy) =

Y

◦

F (y) ν2 (dy)

for all

F ∈π ˆ,

(3.2.14)

where π ˆ is the countable set obtained in the Lemma 3.2.5, then ν1 and ν2 coincide on (Y, σ(◦F | ◦F ∈ π ˆ )). Proof. For any F ∈ π ˆ , a ∈ Q, Fn = [n(F − F ∧ a)] ∧ 1 ∈ π ˆ and ◦Fn ↑ 1(◦F >a) . This implies that 1(◦F (y)>a) ν1 (dy) = 1(◦F (y)>a) ν2 (dy) Y

Y

3.2 Hyperfinite Dirichlet Forms and Markov Processes

157

from the conditions (3.2.14). Now we can prove our Lemma by using the monotone class theorem.

Lemma 3.2.7. If the hyperﬁnite quadratic form E(·, ·) has a hyperﬁnite δquasi-continuous core π = {Fn | n ∈ N}, then π separates the points of Y −A◦ by E (δ) (·, ·), where A is the δ-exceptional set in the Deﬁnition 3.2.4 (1). Proof. Since Y is a Hausdorﬀ space, the singleton {x} is a closed subset of Y . This implies that {x} ∈ B(Y ) for any x ∈ Y. For x, y ∈ Y − A◦ , let us assume that Fn (s1 ) ≈ Fn (s2 ) for every Fn ∈ π, s1 ∈ μ(x), s2 ∈ μ(y). Then we obtain −1 {x, y} ⊂ ∩ (◦Fn ) ((◦Fn )(x)) | Fn ∈ π . It follows from the condition (3.2.13) and above observation that −1 ∩ (◦Fn ) ((◦Fn )(x)) | Fn ∈ π = {x} . Therefore, we get x = y. This shows that π separates the points of Y − A◦ by E (δ) (·, ·).

Lemma 3.2.8. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Suppose that E(·, ·) has a hyperﬁnite δ-quasi-continuous core π = {Fn | n ∈ N} and is δ-nearstandardly concentrated. For all η ∈ ∗ N − N, si ∈ / A2 (δ), Fˆn ∈ π ˆ , t ∈ Tδfin, we have Ω

Fˆn (X (δ) (ω, t)) Pi (dω) ≈

Ω

Fˆn (X (δ) (ω, t))1(t 0 | x(t) ∈ Y − Yn } and ζ(ω) = inf {t ≥ 0 | x(ω, t) = Δ} .

(4.1.2)

4.1 Main Result

167

ˆy ) (2) Two strong Markov processes (Ω, Π, Πt , x(t), Θy ) and (Ω, Π, Πt , x(t), Θ 2 are said to be ν-dual if for all t ∈ R+ , u, v ∈ L (Y, ν), we have Y

u(y)Θt v(y) ν(dy) =

Y

ˆt u(y) ν(dy). v(y)Θ

Let (F (·, ·), D(F )) be a coercive closed form on L2 (Y, ν) as introduced in Sect. 1.6 by replacing K with L2 (Y, ν). The inner product (·, ·)ν of K = L2 (Y, ν) is deﬁned by (f, f )ν =

Y

f 2 (y)ν(dy), α ∈ [0, ∞).

ˆ α | α ∈ (−∞, 0)} be the resolvent and Let {Rα | α ∈ (−∞, 0)} and {R co-resolvent of (F (·, ·), D(F )), respectively. Let us denote by {Tt | t ∈ (0, ∞)} and {Tˆt | t ∈ (0, ∞)} the strong continuous contraction semigroups correˆ α | α ∈ (−∞, 0)}, respectively. sponding to {Rα | α ∈ (−∞, 0)} and {R Moreover, set Fα (f, f ) = F (f, f ) + α(f, f )ν . A bounded operator R on L2 (Y, ν) is called a sub-Markovian operator if for every u ∈ L2 (Y, ν), 0 ≤ u(x) ≤ 1, ν-a.e.x ∈ Y, we have 0 ≤ Ru(x) ≤ 1, ν-a.e.x ∈ Y. A coercive closed form F (·, ·) on L2 (Y, ν) is called a Dirichlet form if it generates a semigroup {Tt | t ≥ 0} of sub-Markovian operators and a co-semigroup {Tˆt | t ≥ 0} of sub-Markovian operators. Let (F (·, ·), D(F )) be a Dirichlet form on L2 (Y, ν). For an open subset B of Y , let us deﬁne L(B) = {g ∈ D(F ) | g ≥ 1, ν-a.e. on B} .

(4.1.3)

Suppose L(B) = ∅. From Ma and Röckner [270], Proposition 1.5 of Chap. III, there exists a unique γ1 (B), γˆ1 (B) ∈ L(B) such that for all w ∈ L(B) F1 (γ1 (B), w) ≥ F1 (γ1 (B), γ1 (B)), γ1 (B), γˆ1 (B)). F1 (w, γˆ1 (B)) ≥ F1 (ˆ

(4.1.4)

From Ma and Röckner [270], Remark 1.6 and Exercise 1.7 of Chap. III, we know F1 (γ1 (B), γ1 (B)) = F1 (γ1 (B), γˆ1 (B)) = F1 (ˆ γ1 (B), γˆ1 (B)).

(4.1.5)

168

4 Construction of Markov Processes

Set Γ1 (B) = F1 (γ1 (B), γ1 (B)) if L(B) = ∅ and Γ1 (B) = ∞ if L(B) = ∅. (4.1.6) For a general subset A of Y , let us deﬁne Γ1 (A) = inf {Γ1 (B) | A ⊂ B, B is open} .

(4.1.7)

We call Γ1 (·) the 1-capacity of F (·, ·). For a closed subset B ⊂ Y, we set D(F )B = {f ∈ D(F ) | f = 0, ν-a.e. on Y − B} . It is easy to see that D(F )B is a closed set of D(F ) with the inner product F1 (·, ·). Definition 4.1.2. An increasing sequence of closed subsets {Gn | n ∈ N} of Y is called a nest if Γ1 (Y − Gn ) ↓ 0. A subset B ⊂ Y is saidto be of zero capacity if there exists a nest {Gn | n ∈ N} such that B ⊂ ∞ n=1 (Y − Gn ). A function f on Y is said to be quasi-continuous if there exists a nest {Gn | n ∈ N} such that the restriction f |Gn is continuous on Gn for each n ≥ 1. We shall say that two ν-dual strong Markov processes (Ω, Π, Πt , x(t), Θy ) ˆ y ) are properly associated with the Dirichlet form and (Ω, Π, Πt , x(t), Θ (F (·, ·), D(F )) if ˆ t f, ν-a.e.,∀f ∈ L2 (Y, ν), t > 0. Tt f = Θt f and Tˆt f = Θ The main result of this chapter is the following: Theorem 4.1.1. Assume that Y is a regular Hausdorﬀ topological space and ν is a Radon measure on Y . Let (F (·, ·), D(F )) be a Dirichlet form on L2 (Y, ν). Then there are two ν-dual tight strong Markov processes ˆ y ) properly associated with the (Ω, Π, Πt , x(t), Θy ) and (Ω, Π, Πt , x(t), Θ Dirichlet form (F (·, ·), D(F )) if and only if the following three conditions are satisﬁed: (I) There is an increasing sequence of compact subsets {Yn | n ∈ N} of Y such that n∈N D(F )Yn is F1 -dense in D(F ). (II) There exists a subset π0 of D(F ) such that π0 is dense in D(F ) with the inner product F1 (·, ·). In addition, u is quasi-continuous for each u ∈ π0 .

4.1 Main Result

169

(III) There is a countable subset π of D(F ) such that every element u ∈ π is quasi-continuous and bounded. Moreover, we have B(Y )

Yn

⊂ σ(u|u ∈ π),

n∈N

where {Yn | n ∈ N} is a sequence of compact sets in (I). Remark 4.1.1. If (F (·, ·), D(F )) satisﬁes conditions (I), (II) and (III) in Theorem 4.1.1, we call it quasi-regular. Remark 4.1.2. Previous work of Albeverio and Ma [41] and Ma and Röckner [270] has established necessary and suﬃcient conditions for the existence of ν-perfect processes associated with Dirichlet forms. In these references, the state space Y is a metrizable space and ν is a σ-ﬁnite measure on Y. Thus, there is a strictly positive function ψ ∈ L2 (Y, ν), 0 < ψ ≤ 1. Set h = R1 ψ, where {Rα | α > 0} is the resolvent of F (·, ·). The authors deﬁned the hweighted capacity as in Röckner [313] and used it as a useful tool. In our case, Y is a regular Hausdorﬀ space and m is a Radon measure. We do not have an h-weighted capacity, since this concept has not been developed in our context. Instead, we utilize the 1-capacity theory developed in Chap. 2, Fan [166], and Fukushima [175]. In order to prove Theorem 4.1.1, we will construct in Sect. 4.2 a hyperﬁnite quadratic form (E(·, ·), D(E)) from (F (·, ·), D(F )). We obtain a hyperﬁnite ˆ Markov chain X(t) associated with E(·, ·), as well as the dual X(t) of X(t). In Sect. 4.3, we will discuss the relation between the potential theory of (F (·, ·), D(F )) and the counterpart of its hyperﬁnite lifting (E(·, ·), D(E)). On this basis, we will consider the path regularity of X(t) and get its modiﬁed standard part x(t). In fact, x(t) is the wanted process. We shall show that x(t) satisﬁes all the requirements of Theorem 4.1.1 in Sect. 4.6. At last, we prove in Sect. 4.7 that if there are two ν-dual tight Markov ˆy ) properly associated processes (Ω, Π, Πt , x(t), Θy ) and (Ω, Π, Πt , x(t), Θ with F (·, ·), then x(t) satisﬁes the conditions (I), (II) and (III) of Theorem 4.1.1. Combining Propositions 4.6.1 and 4.7.1, we shall then prove the Theorem 4.1.1. Remark 4.1.3. From Albeverio and Ma [41], we know that even if ν is a nowhere Radon measure on a metric space, it is still possible to get the Markov process x(t). In order to include this case by methods of nonstandard analysis, we would have to develop ﬁrst a nonstandard measure theory associated with nowhere Radon measures on general topological spaces. After that, we could study the related stochastic analysis problems associated with a nonstandard version of nowhere Radon measure spaces. This is, however, a program which remains to be fulﬁlled.

170

4 Construction of Markov Processes

4.2 Hyperfinite Lifts of Quasi-Regular Dirichlet Forms Let Y be a Hausdorﬀ topological space. Consider the topological Borel σ-ﬁeld B(Y ) on Y . Assume that ν is a positive Borel measure on (Y, B(Y )) such that ν(K) < ∞ for every compact set K. It is a Radon measure in the sense that for all B ∈ B(Y ), ν(B) = sup {ν(K) | K ⊂ B, K compact}

(4.2.1)

and for all B ∈ B(Y ) with ν(B) < ∞, ν(B) = inf {ν(O) | B ⊂ O, O open} . We denote by (Y, B(Y ), ν) the completion of (Y, B(Y ), ν). For simplicity, we suppose that supp(ν) = Y. Let ∗ Y be the nonstandard extension of Y . Given an element a ∈ Y , we know that the monad of a is the subset of ∗ Y deﬁned by μ(a) =

{∗O | a ∈ O

and O ⊂ Y

is open} .

A point y ∈ ∗ Y is nearstandard if it belongs to μ(a) for some a ∈ Y. We shall say that it is nearstandard to a. The set of all nearstandard points is denoted by N s(∗ Y ). Since Y is Hausdorﬀ, each element y ∈ N s(∗ Y ) is nearstandard to exactly one element a ∈ Y (refer to page 48, 2.1.6. Proposition, [25]). We call a the standard part of y and denote it by st(y) or ◦ y. Let Δ be an element outside Y , YΔ = Y ∪ {Δ}. For all y ∈ ∗ Y − N s(∗ Y ), we deﬁne st(y) = Δ. Now we have got the standard part operation as a map st : ∗ Y −→ YΔ . A subset S0 of ∗ Y is called rich if it is hyperﬁnite and st(S0 ∩ N s(∗ Y )) = Y. In the following, we will construct a rich subset S0 in ∗ Y and an internal measure m on S0 such that ν = L(m) ◦ st−1 . By a ﬁnite B(Y ) partition of Y , we mean a ﬁnite collection

Bi ∈ B(Y ) 1 ≤ i ≤ n of non-empty sets with Y = ni=1 Bi and Bi ∩ Bj = ∅ if i = j. Let us denote by P the collection of all ﬁnite B(Y ) partitions of Y . If P1 and P2 are elements of P, wesay that P2 is ﬁner than P1 and we write P1 ≤ P2 if for each C ∈ P1 , C = {B ∈ P2 | B ⊂ C}.

4.2 Hyperfinite Lifts of Quasi-Regular Dirichlet Forms

171

We have the following result from 1.1 Theorem, Loeb [266]. Proposition 4.2.1. There is a partition P ∈ ∗ P such that ∗P0 ≤ P for any P0 ∈ P. That is, P ∈ ∗ P has the following properties: (i) There are an inﬁnite integer N ∈ ∗ N and an internal bijection from I = {i ∈ ∗ N | 1 ≤ i ≤ N } onto P . Thus we may write P = {Ai | i ∈ I}. (ii) If i and j are in I and i = j, then Ai = ∅, Aj = ∅ and Ai ∩ Aj = ∅. (iii) ∗ Y = {Ai | i ∈ I}. ∗ (iv) For each B ∈ B(Y ), let IB = {i ∈ I | Ai ⊂ B}. Then IB is hyperﬁnite, ∗ and B = i∈IB Ai . Hereafter, let us ﬁx a partition P = {Ai | i ∈ I} of ∗ Y with above properties (i) through (iv). For every i ∈ I, we pick up an element si ∈ Ai and deﬁne S0 = {si | i ∈ I}. Let m be the internal measure on S0 deﬁned by m({si }) = ∗ν(Ai ). Proposition 4.2.2. S0 is rich in ∗ Y and ν = L(m) ◦ st−1 . Proof. We simply refer to Albeverio et al. [25], 3.4.10 Corollary.

Let H be the hyperﬁnite dimensional space of all the internal functions f : S0 −→ ∗ R with the inner product f g dm. f, g = S0

˜ be the subspace of ∗K consisting of all Let us set K = L2 (Y, ν). Let H ˜ are functions which are constant on each class Aj , j ∈ I. Obviously, H and H ˜ By using Theorem 1.6.1, isomorphic. From now on, we will identify H with H. we can prove the following result. Proposition 4.2.3. Let Y be a Hausdorﬀ space, ν be a Radon measure on Y , and F (·, ·) be a coercive closed form on L2 (Y, ν). Then, we have (i) There exist a hyperﬁnite, rich subset S0 of ∗ Y , and an internal measure m on S0 such that H is S-dense in ∗K. (ii) There are a nonnegative quadratic form (E(·, ·), D(E)) on L2 (S0 , m) and an internal time-line T = {kΔt | k ∈ ∗ N} representing F (·, ·) in the following sense: ν = L(m) ◦ st−1 and for all u ∈ L2 (Y, ν), F (u, u) = inf {◦ E(v, v) | v is a 2-lifting of u} . ˆ s | s ∈ T } be the semigroup and Moreover, let {Qs | s ∈ T } and {Q co-semigroup generated by E(·, ·), respectively. Then for all t ∈ [0, ∞), s ∈ T, u ∈ K, v ∈ H such that t = ◦s, u = stK (v), we have

172

4 Construction of Markov Processes

ˆ s v = Tˆt u. stK Qs v = Tt u and stK Q ˆ α | α ∈ ∗ ( − ∞, 0)} On the other hand, let {Gα | α ∈ ∗ ( − ∞, 0)} and {G be the resolvent and co-resolvent generated by E(·, ·), respectively. Then for all β ∈ (−∞, 0), α ∈ ∗ ( − ∞, 0), u ∈ K, v ∈ H such that β = ◦α, u = stK (v), we have ˆαv = R ˆ β u, stK Gα v = Rβ u and stK G ˆ α | α ∈ (−∞, 0)} is where {Rα | α ∈ (−∞, 0)} is the resolvent, and {R the co-resolvent of (F (·, ·), D(F )). Proof. From Sect. 3.2 of Albeverio et al. [25], we know that each function f ˜ such that ◦f˜(x) = f (◦x) in L2 (Y, ν) has an S-square integrable lifting f˜ in H ˜ is for almost all nearstandard x. If we can show that ||∗f − f˜|| ≈ 0, then H ∗ 2 dense in (L (Y, ν)). By (4.2.1), it suﬃces to show this when f is bounded and of compact support. For such functions, the statement is an immediate consequence of Anderson’s nonstandard version of Lusin’s theorem. The proposition follows by Theorem 1.6.1. Corollary 4.2.1. If the form F (·, ·) in Proposition 4.2.3 is a Dirichlet form, we can also take E(·, ·) to be a hyperﬁnite quadratic form. Proof. We observe that from Proposition 1.5.1 it suﬃces to prove that QΔt ˆ Δt are Markov operators, where and Q ˆ Δt v QΔt u, v = u, Q and for all u, v ∈ H 1 (I − QΔt )u, v Δt 1 ˆ Δt )v. u, (I − Q = Δt

E(u, v) =

Let P be the projection of ∗K onto H. We shall write {∗ Rα | α ∈ ∗ (−∞, 0)} ∗ ˆ α | α ∈ ∗ (−∞, 0)} for ({R ˆα | α ∈ for ∗({Rα | α ∈ (−∞, 0)}) and {∗ R ˆ Δt are just (−∞, 0)}), respectively. We see from Sect. 1.6 that QΔt and Q ˆ γ ) for a suitable inﬁnitesimal Δt = − 1 , respectively. −P ∗ (γRγ ) and −P ∗ (γ R γ With the choice of H made in the proof of Proposition 4.2.3, this projection is just the conditional expectation with respect to the algebra generated by the partition P . Since conditional expectations preserve nonnegativity and decrease the supremum norm, the corollary follows. In the rest of this chapter, we shall be interested in quadratic forms generating Markov processes. That is, we shall assume that F (·, ·) is a Dirichlet

4.2 Hyperfinite Lifts of Quasi-Regular Dirichlet Forms

173

form on L2 (Y, ν). We shall use the results obtained above. First, let us introduce some notations. The inﬁnitesimal generator A and co-generator Aˆ of E(·, ·) are given by A=

1 1 ˆ Δt ), 0 < Δt ≤ 1 = 1 . (I − QΔt ) and Aˆ = (I − Q ˆ Δt Δt ||A|| ||A||

ˆ t , t ∈ T, are deﬁned by The semigroup Qt and co-semigroup Q ˆ t = (Q ˆ Δt )k Qt = (QΔt )k and Q

for each t = kΔt.

Moreover, we have the following relation from Proposition 1.6.3, F (u, u) = = = =

lim

α−→−∞

lim

−α ◦

[−α(I + αRα )P ∗u,P ∗u]

◦

[−α(I + αGα )P ∗u,P ∗u]

α→−∞ α≈−∞ α∈∗(−∞,0)

lim

α→−∞ α≈−∞ α∈∗(−∞,0)

lim

α→−∞ α≈−∞ α∈∗(−∞,0)

Y

u(y) u(y) + αRα u(y) ν(dy)

◦ (α)

[

E(u, u)]

(4.2.2)

for every u ∈ D(F ), where P is the projection of ∗K onto H. Therefore, we see from the proof of Theorem 1.6.1 that 1 (I − QΔt )P ∗u, P ∗u ≈ F (u, u), Δt i.e., ◦

E(P ∗u,P ∗u) = F (u, u) < ∞ for each u ∈ D(F ).

(4.2.3)

We recall from Deﬁnition 1.4.2 that the domain D(E) of E(·, ·) is the set of all u ∈ H such that ◦

(i) ◦ E 1 (u, u) = [E(u, u) + u, u] < ∞. (ii) For all inﬁnite α ∈ ∗(−∞, 0), E(u + αGα u, u + αGα u) ≈ 0 and E(u + ˆ α u). ˆ α u, u + αG αG We remark that E(·, ·) does not necessarily satisfy the hyperﬁnite weak sector condition. Hence, the results of Proposition 1.4.2 need not hold. However, it is easy to see from the proof of Proposition 1.4.2 the following: if ◦ E(u, u) < ∞ and for all inﬁnite α < 0, (α) E(u, u) ≈ E(u, u) and (α) ˆ E(u, u) ≈ E(u, u), then we have u ∈ D(E).

174

4 Construction of Markov Processes

Therefore, we have proved the following using (4.2.2) and (4.2.3): Lemma 4.2.1. For every u ∈ D(F ), we have P ∗u ∈ D(E). ˆ Now, we are in the position to construct the dual processes X(t) and X(t) and their transition matrices associated with E(·, ·). Remember that m is an internal measure on the hyperﬁnite set S0 = {s1 , s2 , · · · , sN }. We shall write mi for m(si ) if it is convenient. Fix j ∈ I. Let uj : S0 −→ ∗ R be the function satisfying uj (sl ) = δlj , l ∈ I. We deﬁne ˆ Δt uj (si ), j ∈ I. bij = QΔt uj (si ) and ˆbij = Q It is easy to see that for any internal function u : S0 −→ ∗ R, we have u(·) =

N

u(sj )uj (·).

j=1

Hence, we obtain QΔt u(si ) =

N

u(sj )QΔt uj (si )

j=1

=

N

bij u(sj ),

j=1

⎛ ⎞ N 1 ⎝ Au(si ) = u(si ) − u(sj )bij ⎠ . Δt j=1 Similarly, we have ˆ Δt u(si ) = Q

N

ˆ Δt uj (si ) u(sj )Q

j=1

=

N

ˆbij u(sj ),

j=1

⎛ ⎞ N 1 ˆ i) = ⎝u(si ) − Au(s u(sj )ˆbij ⎠ . Δt j=1 Let us denote u(si ) by u(i). Then, we have

4.2 Hyperfinite Lifts of Quasi-Regular Dirichlet Forms

175

N

1 Au(i)v(i)m(i) Δt i=1 ⎡ ⎤ N N 1 ⎣ u(i) − bij u(j)⎦ v(i)m(i) = Δt i=1 j=1

E(u, v) =

N

N

N

N

=

1 (δij − bij )u(j)v(i)m(i) Δt i=1 j=1

=

1 (δij − ˆbij )v(j)u(i)m(i). Δt i=1 j=1

ˆ Δt , we have for all i, j ∈ I, Since the duality of QΔt and Q m(i)bij = m(j)ˆbji . Recall that s0 is a point outside S0 , and we put S = S0 ∪ {s0 }. We assume m({s0 }) = 0 and st(s0 ) = Δ. Let Q = {qij | 0 ≤ i, j ≤ N } be an (N + 1) × (N + 1) matrix satisfying the following conditions: q00 = 1, q0i = 0 for i ∈ I, qij = bij , i, j ∈ I, qi0 = 1 −

N

bij , i ∈ I.

j=1

It is easy to see that N

qij = 1 for all i ∈ I ∪ {0} .

j=0

We construct a hyperﬁnite Markov chain {X(t) | t ∈ T } associated with Q and m in the following manner. Let Ω be the set of all internal functions ω : T −→ S. Let X be the coordinate function X(ω, t) = ω(t). We take P to be the measure generated by P ([ω]0 ) = m(X(ω, 0)) and P ([ω]kΔt ) = m {X(ω, 0)}

k−1 n=0

q(ω(nΔt), ω((n + 1)Δt)),

176

4 Construction of Markov Processes

where q(si , sj ) is just qij and [ω]t is deﬁned by [ω]t = {ω | X(ω , s) = X(ω, s) for all s ≤ t} . Let Ft be the internal algebra on Ω generated by the sets [ω]t , ω ∈ Ω. Let us deﬁne the probability measure Pi as follows: Pi ([ω]kΔt ) = δiω(0)

k−1

q(ω(nΔt), ω((n + 1)Δt)).

n=0

It is easy to verify that the following Markov property of X(t) holds. If X(ω, t) = si , then we have P

ω ∈ [ω]t X(ω , t + Δt) = sj = qij P ([ω]t ).

ˆ = {ˆ Similarly, let Q qij | 0 ≤ i, j ≤ N } be an (N + 1) × (N + 1) matrix satisfying the following conditions: qˆ00 = 1, qˆ0i = 0 for i ∈ I, qˆij = ˆbij , i, j ∈ I, qˆi0 = 1 −

N

ˆbij , i ∈ I.

j=1

It is easy to see that N

qˆij = 1 for all i ∈ I ∪ {0} .

j=0

ˆ We may construct the dual hyperﬁnite Markov chain {X(t) = X(t) | t ∈ T } and the corresponding internal measure Pˆ , Pˆi , i ∈ I. If δ ∈ T , let Tδ be the sub-line Tδ = {kδ | k ∈ ∗ N0 } . (δ)

We write X (δ) for the restriction X|Tδ . For each t ∈ Tδ , let Ft be the internal algebra on Ω generated by the sets

(δ) [ω]t = ω ∈ Ω X (δ) (ω , s) = X (δ) (ω, s) for all s ∈ Tδ , s ≤ t .

4.3 Relation with Capacities

177 (δ)

In the following of this chapter, we shall use the notations A(δ) , qij , E (δ) (·, ·), and D(E (δ) ) introduced in Sect. 1.5. For δ ∈ T, α ∈ ∗ R, α ≥ 0, let us introduce the notation Eα(δ) (u, u) = E (δ) (u, u) + αu, u.

4.3 Relation with Capacities Let (F (·, ·), D(F )) be a Dirichlet form on L2 (Y, ν). We consider the 1-capacity Γ1 (·) deﬁned in the relations (4.1.3), (4.1.4), (4.1.5), (4.1.6), and (4.1.7) in Sect. 4.1. We get the following results from Ma and Röckner [270], Chap. III. Lemma 4.3.1. Let B be an open subset of Y such that L(B) = ∅, where L(B) is deﬁned by the deﬁnition (4.1.3) in Sect. 4.1. Then we have the following results: (1) There exists a unique element γ1 (B) ∈ L(B) such that Γ1 (B) = F1 (γ1 (B), γ1 (B)).

(4.3.1)

(2) 0 ≤ γ1 (B) ≤ 1, ν-a.e. and γ1 (B) = 1, ν-a.e. on B. Proposition 4.3.1. Γ1 (·) has the properties: (1) (2) (3) (4)

A ⊂ B =⇒ Γ1 (A) ≤ Γ1 (B). An ↑=⇒ Γ1 (∪n∈N An ) = supn∈N Γ1 (An ). A, B open =⇒ Γ 1 (A ∪ B) + Γ1 (A ∩ B) ≤ Γ1 (A) + Γ1 (B). Γ1 (∪n∈N An ) ≤ n∈N Γ1 (An ).

In the same way as in Sects. 2.2 and 2.3, we introduce the capacity theory associated with (E (δ) (·, ·), D(E (δ) )) and state some properties without proof. For δ ∈ T, α ∈ ∗ R+ , and an internal subset A of S0 , we deﬁne (δ)

σA (ω) = min {t ∈ Tδ | X(ω, t) ∈ A} , (δ) −σA /δ e(δ) , (A)(i) = E i (1 + αδ) α L(A) = {G | S0 −→ ∗ R is internal and G|A = 1A } . (δ)

Using the terminology of Sects. 2.2 and 2.3, we call eα (A) the equilibrium α-potential of A associated with E (δ) (·, ·). Let us set (δ) (δ) e(δ) Cap(δ) α (A) = Eα α (A), eα (A) .

178

4 Construction of Markov Processes

For an arbitrary subset B of S0 , we deﬁne

(δ) (δ) Capα (B) = inf Capα (A) B ⊂ A, A is internal . (δ) We call Cap(δ) (·, ·). We have studied α (·) the α-capacity associated with E the potential theory of hyperﬁnite quadratic forms in Chap. 2. We shall use the same notations of Chap. 2 in the following.

We ﬁrst remark that the weak sector condition ! ! |F1 (x, y)| ≤ C F1 (x, x) F1 (y, y) for all x, y ∈ D(F ) is equivalent to the following: ! ! |F (x, y)| ≤ C F1 (x, x) F1 (y, y) for all x, y ∈ D(F ), for some nonnegative constant C ∈ [0, ∞). Obviously, we can always suppose that C = C . Hence, we have ! ! (4.3.2) |F (x, y)| ≤ C F1 (x, x) F1 (y, y) for all x, y ∈ D(F ). Proposition 4.3.2. Let C satisfy the inequality (4.3.2). For all open subset B ⊂ Y, we have for all δ ∈ T, δ ≈ 0 the following (δ)

Cap1 (∗B ∩ S0 ) ≤ (C + 1)2 Γ1 (B). Proof. For simplicity, we may suppose that L(B) = ∅. From Lemma 4.3.1, there is an element γ1 (B) ∈ D(F ) satisfying (4.3.1). For convenience, let us assume that γ1 (B)(x) = 1 for all x ∈ B. For all δ ∈ T, δ ≈ 0, we have from Proposition 2.2.1 (δ)

(δ)

(δ)

(δ)

(δ)

(δ)

Cap1 (∗B ∩ S0 ) = E1 (e1 (1∗B ), e1 (1∗B )) = E1 (e1 (∗B), γ1 (B)).

(4.3.3)

On the other hand, let γ be given by (1.6.5) in the proof of Theorem 1.6.1, Chap. 1. Then, we have (δ)

(δ)

E (δ) (e1 (∗B), γ1 (B)) = −γ(I + ∗(γRγ ))e1 (∗B), γ1 (B) (δ)

= (γ) F (e1 (∗B), γ1 (B)). From the inequality (4.3.2) and Corollary 1.6.1 (i), we get

(4.3.4)

4.3 Relation with Capacities

179

(δ)

(γ)

(δ)

F (e1 (∗B), γ1 (B)) = F ((−γRγ )e1 (∗B), γ1 (B)) " ! (δ) (δ) ≤ C F1 ((−γRγ )e1 (∗B), (−γRγ )e1 (∗B)) F1 (γ1 (B), γ1 (B)).

(4.3.5)

From Corollary 1.6.1 (ii), we have (δ)

(δ)

F1 ((−γRγ )e1 (∗B), (−γRγ )e1 (∗B)) (δ)

(δ)

(δ)

(δ)

= F ((−γRγ )e1 (∗B), (−γRγ )e1 (∗B)) + (−γRγ )e1 (∗B), (−γRγ )e1 (∗B) (δ)

(δ)

(δ)

(δ)

≤ (γ) F (e1 (∗B), e1 (∗B)) + e1 (∗B), e1 (∗B) (δ)

(δ)

(δ)

(δ)

= −γ(I + ∗(γRγ ))e1 (∗B), e1 (∗B) + e1 (∗B), e1 (∗B) (δ)

(δ)

(δ)

= E1 (e1 (∗B), e1 (∗B)).

(4.3.6)

Therefore, we have from the relations (4.3.3), (4.3.4), (4.3.5), and (4.3.6) that ◦

◦ (δ) (δ) (δ) (δ) Cap1 (∗B ∩ S0 ) = E1 (e1 (1∗B ), e1 (1∗B )) ≤ (C + 1)2 F1 (γ1 (B), γ1 (B)).

This ﬁnishes the proof of Proposition 4.3.2.

Corollary 4.3.1. Let A ⊂ Y satisfy the condition Γ1 (A) = 0. Then for all (δ) δ ∈ T, δ ≈ 0, we have Cap1 (∗A ∩ S0 ) ≈ 0.

Proof. The proof follows easily from Proposition 4.3.2.

Corollary 4.3.2. Let {Gn | n ∈ N} be a sequence of increasing closed subsets of Y satisfying Γ1 (Y − Gn ) ↓ 0. Then for all δ ∈ T, δ ≈ 0, we have

◦

(δ) Cap1

S0 −

∞

∗

( Gn ∩ S0 )

= 0.

(4.3.7)

n=1

Proof. From Proposition 4.3.2, we have ◦

(δ) Cap1 (S0 − ∗Gn ∩ S0 ) =

◦

(δ) Cap1 (∗(Y − Gn ) ∩ S0 )

≤ (C + 1)2 Γ1 (Y − Gn ) −→ 0, n → ∞. Therefore, (4.3.7) holds.

Proposition 4.3.3. Let {Gn | n ∈ N} be a sequenceof increasing closed ∞ subsets of Y satisfying Γ1 (Y − Gn ) ↓ 0. Then S0 − n=1 ∗Gn ∩ S0 is δexceptional for all δ ∈ T, δ ≈ 0. Proof. The proof follows easily from Corollary 4.3.2 and Theorem 2.4.1.

180

4 Construction of Markov Processes

4.4 Path Regularity of Hyperfinite Markov Chains Similarly as in Deﬁnition 3.2.1, we introduce the following Definition 4.4.1. Let Z be a subset of Y . We say that a subset π of L2 (Y, ν) separates points of Z, if for any two diﬀerent points x and y in Z, there exists u ∈ π such that u(x) = u(y). Lemma 4.4.1. If the hypothesis (III) in Theorem 4.1.1 holds, then π separates the points of n∈N Yn . Proof. The proof can be carried through in the same way as the proof of Lemma 3.2.7. Lemma 4.4.2. Assume that Y is a regular Hausdorﬀ space and the hypothesis (III) of Theorem 4.1.1 holds. Let δ ∈ T, δ ≈ 0. For every ω ∈ Ω, if the path X (δ) (ω, t) fails to have an S-left or S-right limit at some t < τ (δ) (ω), then so does (P ∗u)(X (δ) (ω, t)) for some u ∈ π, where the stopping time τ (δ) (ω) is deﬁned by # $ / (∗ Yn ∩ S0 ) . τ (δ) (ω) = inf ◦ t X (δ) (ω, t) ∈ n∈N

Proof. Let us set Bn = ∗ Yn ∩ S0 , n ∈ N and B = ∪n∈N Bn . For each n ∈ N, we deﬁne

(δ) (δ) / Bn , τn (ω) = min t ∈ T X (ω, t) ∈

◦ (δ) ζδ (ω) = inf t X (ω, t) ∈ / S0 , i.e., ζδ is the lifetime of X (δ) (ω, t). It is easy to show that τ (δ) (ω) ≤ ζδ (ω),

(δ) ◦ (δ) τ (ω) = sup τn (ω) n ∈ N . Let ω be a ﬁxed point in Ω. Fix t < τ (δ) (ω), t ∈ [0, ∞). Then, there is an (δ) element N ∈ N such that t < τN (ω). Given a sequence {tn | n ∈ N} from T ◦ such that the standard part tn increases strictly to t, we can show that the sequence

◦ (δ) X (ω, tn ) n ∈ N has a cluster point in YN in same way as the proof of Lemma 3.2.1. The rest follows in a similar way.

4.5 Quasi-Continuity and Nearstandard Concentration

181

Proposition 4.4.1. Assume that Y is a regular Hausdorﬀ space and the hypothesis (III) of Theorem 4.1.1 holds. Let δ ∈ T, δ ≈ 0. Then there exists a δ-exceptional set A0 (δ) such that for all si ∈ S0 − A0 (δ), the hyperﬁnite Markov chain X (δ) (t) has S-left and S-right limits at all t < ζδ , L(Pi )-a.e. Proof. The result can be proved by using the method of Proposition 3.2.1, and the results of Lemmas 4.2.1 and 4.4.2.

4.5 Quasi-Continuity and Nearstandard Concentration Let (F (·, ·), D(F )) be a Dirichlet form on L2 (Y, ν). First of all, we present some properties of E(·, ·). Lemma 4.5.1. Let {fk | k ∈ N} be a sequence of quasi-continuous functions on Y . Then there exists a nest {Gn | n ∈ N} such that for all k ∈ N, fk ∈ C({Gn }), where C({Gn }) = {f | f |Gn

is continuous for each

n ∈ N} .

Proof. The proof is similar to the one in Fukushima [175] Theorem 3.1.2. Lemma 4.5.2. (1) If the condition (I) in Theorem 4.1.1 is satisﬁed, we have F1 (u − un , u − un ) −→ 0

as

n −→ ∞, ∀u ∈ D(F ),

where un ∈ D(F )Yn is the projection of u. (2) Suppose that the condition (II) in Theorem 4.1.1 holds. Then, every element f ∈ D(F ) admits a quasi-continuous version. Proof. (1) Let F (·, ·) be the symmetric part of F (·, ·), i.e., F (u, v) =

1 F (u, v) + F (v, u) , u, v ∈ D(F ). 2

Set F 1 (u, v) = F (u, v) + (u, v), u, v ∈ D(F ). For every u ∈ D(F ), we have the following decomposition with respect to the inner product F 1 (·, ·) u = (u − un ) + un , un ∈ D(F )Yn , un − u⊥D(F )Yn , ∀n ∈ N. Let vn = u − un . Then, we have

182

4 Construction of Markov Processes

F 1 (vn , vn+m ) = F 1 (vn+m , vn+m ) + F 1 (vn − vn+m , vn+m ) = F 1 (vn+m , vn+m ) + F 1 (un+m − un , u − un+m ) = F 1 (vn+m , vn+m ) + F 1 (−un , u − un+m ). Since un ∈ D(F )Yn ⊂ D(F )Yn+m , F 1 (−un , u − un+m ) = 0. Hence, we have F 1 (vn , vn+m ) = F 1 (vn+m , vn+m ). Therefore, we have F1 (vn − vn+m , vn − vn+m ) = F1 (vn , vn ) − F1 (vn+m , vn+m ) −→ 0, n → ∞. This means that {vn | n ∈ N} is a F1 (·, ·) Cauchy sequence. Hence, there is an element v∞ ∈ D(F ) such that F1 (vn − v∞ , vn − v∞ ) −→ 0, n −→ ∞. It is easy to see that

F1 (v∞ , v) = 0, ∀v ∈

D(F )Yn .

n∈N

Since n∈N D(F )Yn is F1 -dense in D(F ), we know that v∞ = 0. This proves (1). (2) For this proof, we refer to Albeverio and Ma [41], 4.1 Lemma and Fukushima [175], Theorem 3.1.3. Lemma 4.5.3. Suppose that the condition (I) in Theorem 4.1.1 is satisﬁed. Then (F1 (·, ·), D(F )) is a separable Hilbert space. Proof. Since Yn is compact, (F1 (·, ·), D(F )Yn ) is separable. Hence (F (·, ·), D(F )) is separable. Lemma 4.5.4. Suppose that the hypotheses (I), (II) and (III) in Theorem 4.1.1 hold. Then, there exists a countable set π ˆ such that (1) (2) (3)

π⊂π ˆ ⊂ D(F ) ∩ Qb (Y ), u, v ∈ π ˆ , a ∈ Q =⇒ |u| ∈ π ˆ, u + v ∈ π ˆ , uv ∈ π ˆ,

(4.5.1)

av ∈ π ˆ and u ∧ a ∈ π ˆ,

(4.5.2)

π ˆ is F1 -dense in D(F ),

where Q = {a1 , a2 , · · · , an , · · · } is the set of all rational numbers and Qb (Y ) is the space of all bounded quasi-continuous functions deﬁned on Y. Moreover, ˆ , u ∈ C({Zn }). there is a nest {Zn | n ∈ N} such that for every u ∈ π

4.5 Quasi-Continuity and Nearstandard Concentration

183

Proof. From Lemmas 4.5.2 and 4.5.3, we can assume that π is F1 -dense in D(F ). Deﬁne the maps Λ = {λ−2 , λ−1 , λ0 , λ1 , λ2 , · · · } from [D(F ) ∩ Qb (Y )] × [D(F ) × Qb (Y )] into D(F ) ∩ (Qb (Y ) by λ−2 (u, v) = |u|, λ−1 (u, v) = u + v, λ0 (u, v) = uv, λ2i−1 (u, v) = ai u, λ2i (u, v) = u ∧ ai , i = 1, 2, · · · Applying Lemma 3.2.4 to π and Λ, we get a countable set π ˆ satisfying the relations (4.5.1) and (4.5.2). Obviously, π ˆ is F1 -dense in D(F ). The existence of {Zn | n ∈ N} follows from Lemma 4.5.1. Lemma 4.5.5. Assume that the hypotheses (I), (II) and (III) in Theorem 4.1.1 hold. For any two ﬁnite measures ν1 and ν2 on (Y, B(Y )) satisfying the following conditions: u(y) ν1 (dy) = u(y) ν2 (dy) for all u ∈ π ˆ, Y

Y

where π ˆ is the countable set obtained in the Lemma 4.5.4, we have that ν1 and ν2 coincide on (Y, σ(u|u ∈ π ˆ )). Proof. The proof is similar to the one of Lemma 3.2.6.

We recall that we have introduced the hyperﬁnite quasi-continuity and related concepts of (E(·, ·), D(E)) in Sect. 3.2. We then have Proposition 4.5.1. Let us assume that f : Y −→ R is a quasi-continuous function. Then for all δ ∈ T, δ ≈ 0, P ∗f is hyperﬁnite δ-quasi-continuous. Proof. The proof follows easily from Corollary 4.3.2 and Theorem 2.4.1.

Corollary 4.5.1. Let us assume that the condition (II) in Theorem 4.1.1 holds. Then for every element f ∈ D(F ) and all inﬁnitesimal δ ∈ T , there is a δ-exceptional set A(δ) ⊂ S0 such that (P ∗f )(si ) ≈ (P ∗f )(sj ) for all inﬁnitely close si , sj ∈ S 0 − A(δ). Proof. From Lemma 4.5.2 (2), there is a quasi-continuous version f˜ of f. Hence, P ∗f˜ is hyperﬁnite δ-quasi-continuous, ∀δ ∈ T, δ ≈ 0. Since P ∗f ≈ P ∗f˜, we prove Corollary 4.5.1 by using Lemma 2.1.1 (i).

184

4 Construction of Markov Processes

Proposition 4.5.2. Let Y be a regular Hausdorﬀ space. If the conditions (I), (II), and (III) in Theorem 4.1.1 are satisﬁed, then there exist a δ0 -exceptional set A1 (δ0 ) of X(t) for some inﬁnitesimal δ0 and some β ∈ Tδ0 , 12 < β ≤ 1 such that (i) For all si ∈ S0 − A1 (δ0 ), the path X (δ0 ) (ω, t) has S-right and S-left limits at all t < ζδ0 (ω), L(Pi )-a.e.ω. ˆ β) = ∞ T2−n β . For all inﬁnitely close si , sj ∈ S 0 − (ii) Let us set Q(δ, n=1 ˆ 0 , β), u ∈ π ˆ , we have A1 (δ0 ), s ∈ Q(δ & ◦ (P T st(s) u)(si ) = (P ∗ u)(X (δ0 ) (s)) Pi (dω) Ω ◦ = (P ∗ u)(X (δ0 ) (s)) Pj (dω).

◦%

∗

(4.5.3)

Ω

ˆ , ◦ [P ∗ (T t u)(si )] : [0, ∞) −→ R is continu(iii) For all si ∈ S0 − A1 (δ0 ), u ∈ π ous with respect to t. Moreover, for all si ∈ S0 − A1 (δ0 ), u ∈ π ˆ , s ∈ Tδfin , 0 we have P ∗ (T st(s) u)(si ) ≈ Qs (P ∗ u)(si ). Proof. Step 1. For s ∈ T fin, let t = st(s). If u ∈ π ˆ , P ∗ u is a 2-lifting of u. s ∗ Therefore, Q (P u) is a 2-lifting of Tt u from Proposition 4.2.3. For all k ∈ N, we have

1 1 (4.5.4) < . μ si ∈ S0 |Qs (P ∗ u)(si ) − P ∗ (Tt u)(si )| > k k Therefore, (4.5.4) holds for some inﬁnite k ∈ ∗ N − N also. This means that

si Qs (P ∗ u)(si ) ≈ P ∗ (Tt u)(si ) is contained in an internal set of inﬁnitesimal measure. Having this in mind, we shall show our proposition step by step. Step 2. Suppose that π ˆ = {un | n ∈ N}. Let {P ∗ un | n ∈ ∗ N} be an ∗ internal extension of {P un | n ∈ N}. For ε ∈ T 1 , N ∈ ∗ N, deﬁne B1 (ε) =

% & si ∃s ∈ Tεfin Qs (P ∗ un )(si ) ≈ P ∗ (Tst(s) un )(si ) ,

n∈N

B1 (ε, N ) =

N % & si ∃s ∈ TεN Qs (P ∗ un )(si ) ≈ P ∗ (Tst(s) un )(si ) ,

n=1

ˆ1 (ε, N ) = B

(4.5.5)

N 1 N s ∗ ∗ si ∃s ∈ Tε |Q (P un )(si ) − P (Tst(s) un )(si )| > . N n=1

4.5 Quasi-Continuity and Nearstandard Concentration

185

Fix ε ≈ 0, ε ∈ T 1 . For each N ∈ N, we have ˆ1 (ε, N )) < μ(B Consider the following internal set 1 ˆ1 (ε, N (ε)) < ε ∈ T μ(B

1 N (ε)

1 . N

1 , N (ε) = + 1. ε

By saturation, there is an inﬁnitesimal ε1 ∈ T 1 such that ˆ1 (ε1 , N (ε1 )) < μ B

1 . N (ε1 )

In view of our Lemma 2.1.1, there is an inﬁnitesimal δ1 ∈ Tε1 such that ˆ1 (ε1 , N (ε1 )) is δ-exceptional for all δ ≥ δ1 , δ ∈ Tε1 . Therefore, B ˆ1 (δ, N (δ)) B is δ-exceptional for all δ ∈ Tε1 , δ ≥ δ1 . Noticing that ˆ 1 (δ, N (δ)), B1 (δ) ⊂ B1 (δ, N (δ)) ⊂ B we know that B1 (δ) is δ-exceptional for all δ ∈ Tε1 , δ ≥ δ1 . Step 3. We take n0 ∈ ∗ N − N satisfying ∞ ˆ Q(β) = n=1 T2−n β . Deﬁne B2 =

u∈ˆ π

1 2

< 2n0 δ1 ≤ 1. Let β = 2n0 δ1 and

% s ∗ & ˆ Q (P u)(si ) ≈ Qs (P ∗ u)(sj ) for some sj ≈ si . si ∈ S 0 ∃s ∈ Q(β)

Since P ∗ (Tst(s) u) is hyperﬁnite δ-quasi-continuous for all δ ∈ T1 , δ ≈ 0, and Qs (P ∗ u) ≈ P ∗ (Tst(s) u), it follows from Lemma 2.1.1 that there is an inﬁnitesimal δ2 ∈ T such that B2 is δ-exceptional for all δ ≥ δ2 , δ ∈ T. ˆ n0 −m0 δ1 ≈ 0 Step 4. We pick an inﬁnite number m0 ≤ n0 such that δ0 =2 and δ0 ≥ δ2 . Let A0 (δ0 ) be the δ0 -exceptional set obtained in Proposition 4.4.1. Then, there exists a δ0 -exceptional set A1 (δ0 ) which contains A0 (δ0 ), B1 (δ0 ) and B2 . Obviously, A1 (δ0 ) satisﬁes (i). For si ∈ S0 − A1 (δ0 ), u ∈ π ˆ , s ∈ Tδfin , we have from the relations (4.5.5) that 0 Qs (P ∗ u)(si ) ≈ P ∗ (T st(s) u)(si ), ◦

since Tδ0 ⊂ Tδ1 ⊂ Tε1 . Thus, (P ∗ (Tt u)(si )) : [0, ∞) −→ R is continuous. This completes the proof of Proposition 4.5.2.

186

4 Construction of Markov Processes

Lemma 4.5.6. Suppose that the condition (I) in Theorem 4.1.1 holds. For every u ∈ D(F ), s ∈ [0, ∞), the set # B(u, s) =

si ∈ S0 −

$ ∗ ( Yn ∩ S0 ) P (Ts u)(si ) ≈ 0 ∗

n∈N

is Δt-exceptional. Proof. From Lemma 4.5.2, we have F1 (u − un , u − un ) −→ 0 as

n −→ ∞, ∀u ∈ D(F ),

where un ∈ D(F )Yn is the projection of u. For each k ∈ N, let us take nk ∈ N such that δk =

! F1 (Ts u − (Ts u)nk , Ts u − (Ts u)nk ) < 2−k ,

where (Ts u)n ∈ D(F )Yn is the projection of Ts u. For simplicity, we may suppose that (Ts u)nk |Y −Ynk = 0. Deﬁne ! ∗ ∗ Bk (u, s) = si ∈ S0 − Ynk ∩ S0 |P (Ts u)(si )| ≥ δk , k ∈ N. Noticing that |P ∗ (Ts u)(si ) − P ∗ (Ts u)nk (si )| ≥

! δk , si ∈ Bk (u, s),

we have P ω ∃t ≤ 1 (X(ω, t) ∈ Bk (u, s)) ! ≤ P ω ∃t ≤ 1 |P ∗ (Ts u)(X(ω, t)) − P ∗ (Ts u)nk (X(ω, t))| ≥ δk ! ≤ P ω ∃t ≤ 1 P ∗ (Ts u)(X(ω, t)) − P ∗ (Ts u)nk (X(ω, t)) ≥ δk ! +P ω ∃t ≤ 1 P ∗ (Ts u)(X(ω, t)) − P ∗ (Ts u)nk (X(ω, t)) ≤ − δk . (4.5.6) Let us set A = {i ∈ S0 | P ∗ (Ts u)(si ) − P ∗ (Ts u)nk (si ) ≥ σA (ω) = min{t ∈ T | X(t) ∈ A}, −σ /Δt e1 (A)(i) = Ei (1 + Δ) A .

! δk },

4.5 Quasi-Continuity and Nearstandard Concentration

187

Then, we have ! ∗ ∗ P ω ∃t ≤ 1 P (Ts u)(X(ω, t)) − P (Ts u)nk (X(ω, t)) ≥ δk = P (ω|σA (ω) ≤ 1)

= Pi ω (1 + Δt)−σA /Δt ≥ (1 + Δt)−1/Δt dm(i) S 0 −σ /Δt 1/Δt Ei (1 + Δt) A (1 + Δt) dm(i) ≤ S0 1/Δt e1 (A)(i) dm(i) = (1 + Δt) S0

1/Δt

≤ (1 + Δt)

E1 (e1 (A), e1 (A)),

(4.5.7)

where the reason for the last step holding is (2.3.1) in Sect. 2.3. From Theorem 2.2.1, we have 1 ∗ ∗ E1 (e1 (A), e1 (A)) ≤ E1 e1 (A), √ (P (Ts u) − P (Ts u)nk ) δk 1 (γ) ∗ ∗ =F e1 (A), √ (P (Ts u) − P (Ts u)nk ) δk 1 +√ e1 (A)(i) (P ∗ (Ts u) − P ∗ (Ts u)nk ) (i)dm(i) δk S 0 1 ∗ ∗ = F (−γRγ )e1 (A), √ (P (Ts u) − P (Ts u)nk ) δk 1 +√ e1 (A)(i) (P ∗ (Ts u) − P ∗ (Ts u)nk ) (i)dm(i), δk S 0 (4.5.8)

where γ is given by (1.6.5) in the proof of Theorem 1.6.1. From the inequality (4.3.2) in Sect. 4.3, we have F

1 (−γRγ )e1 (A), √ (P ∗ (Ts u) − P ∗ (Ts u)nk ) δk " 1 ≤ C√ F1 ((−γRγ )e1 (A), (−γRγ )e1 (A)) δk ! × F1 (P ∗ (Ts u) − P ∗ (Ts u)nk , P ∗ (Ts u) − P ∗ (Ts u)nk ).

(4.5.9)

In the same way as for the relation (4.3.6) in Sect. 4.3, we can show that F1 ((−γRγ )e1 (A), (−γRγ )e1 (A)) ≤ E1 (e1 (A), e1 (A)).

188

4 Construction of Markov Processes

Hence, we have from the inequality (4.5.9) that F

1 (−γRγ )e1 (A), √ (P ∗ (Ts u) − P ∗ (Ts u)nk ) δk 1 ! ≤ C√ E1 (e1 (A), e1 (A)) δk ! × F1 (P ∗ (Ts u) − P ∗ (Ts u)nk , P ∗ (Ts u) − P ∗ (Ts u)nk ). (4.5.10)

From the relations (4.5.8) and (4.5.10), we have 1 ! E1 (e1 (A), e1 (A)) ≤ (C + 1) √ E1 (e1 (A), e1 (A)) δk ! × F1 (P ∗ (Ts u) − P ∗ (Ts u)nk , P ∗ (Ts u) − P ∗ (Ts u)nk ). Therefore, we have E1 (e1 (A), e1 (A)) ≤

% & (C + 1)2 F1 P ∗ (Ts u) − P ∗ (Ts u)nk , P ∗ (Ts u) − P ∗ (Ts u)nk δk

≤ (C + 1)2 δk .

(4.5.11)

From the relations (4.5.7) and (4.5.11), we have ! P ω ∃t ≤ 1 P ∗ (Ts u)(X(t)) − P ∗ (Ts u)nk (X(t)) ≥ δk ≤ (1 + Δt)

1/Δt

(C + 1)2 δk .

(4.5.12)

Similarly, we have ! ∗ ∗ P ω ∃t ≤ 1 P (Ts u)(X(ω, t)) − P (Ts u)nk (X(ω, t)) ≤ − δk ≤ (1 + Δt)

1/Δt

(C + 1)2 δk .

(4.5.13)

From the relations (4.5.6), (4.5.12), and (4.5.13), we have 1/Δt (C + 1)2 δk P ω ∃t ≤ 1 X(ω, t) ∈ Bk (u, s) ≤ 2 1 + Δt 1/Δt ≤ 2 1 + Δt (C + 1)2 2−k . (4.5.14)

4.5 Quasi-Continuity and Nearstandard Concentration

189

For each K ∈ N, we have B(u, s) ⊂ ∞ k=K Bk (u, s). It follows from the inequality (4.5.14) that B(u, s) is Δt-exceptional. Lemma 4.5.7. Suppose that the condition (I) in Theorem 4.1.1 is satisﬁed. Then, there exists a Δt-exceptional set A2 containing all B(u, s), u ∈ π ˆ, s ∈ Q+ , where Q+ is the family of all positive rational numbers. Proof. This follows easily from Lemma 2.1.1 and Lemma 4.5.6.

In Sect. 4.4, we have deﬁned the lifetime ζδ of X (δ) (ω, t). That is, for δ ∈ T , we have ζδ (ω) = inf ◦ t | X (δ) (ω, t) ∈ / S0 . Now, we deﬁne the right standard part ◦X (δ)+ of X (δ) as follows. If t < ζδ (ω), let ◦

X (δ)+ (ω, t) = S- lim X (δ) (ω, s) s↓t

whenever this limit exists and for all t1 < t, the S-lim X (δ) (ω, s) exist also, s↓t1

and ◦X (δ)+ (ω, t) = Δ else. If t ≥ ζδ (ω), we deﬁne ◦X (δ)+ (ω, t) = Δ. Proposition 4.5.3. Assume that Y is a regular Hausdorﬀ space. If the conditions (I), (II), and (III) in Theorem 4.1.1 are satisﬁed, then there exists a properly δ0 -exceptional set A(δ0 ) of X(t) for some inﬁnitesimal δ0 and some β ∈ Tδ0 , 12 < β ≤ 1 such that (i) For all si ∈ S0 − A(δ0 ), the path X (δ0 ) (ω, t) has S-right and S-left limits at all t < ζδ0 (ω), L(Pi )-a.e.ω. ˆ , t ∈ Tδfin , we have (ii) For all inﬁnitely close si , sj ∈ S 0 − A(δ0 ), u ∈ π 0

◦

u( X

(δ0 )

Ω

(ω, t)) L(Pi )(dω) = = =

Ω Ω Ω

u(◦X (δ0 ) (ω, t))1(t0 and t ∈ T , ) (

E eε|Xt | = n∈∗ N

≤

eε|Xt | dμ

{2(n−1)k≤|Xt | |Xt |p on {|Xt | > N0 }, we conclude that E[|Xt |p ] is ﬁnite. Applying Lemma (5.2.4) completes the proof of the theorem.

5.2 Characterizing Hyperfinite Lévy Processes

213

We introduce the following quantities, which may be called normalized increment mean and normalized increment variance, respectively. Definition 5.2.3. mX =

1 1 E [ΔX0 ] = apa , Δt Δt

σX 2 =

a∈A

) 1 2 1 ( 2 E |ΔX0 | = |a| pa . Δt Δt a∈A

Lemma 5.2.5. [263, Lemma 1.2, Corollary 2.4] •

For all t ∈ T and arbitrary X, ) ( 2 2 E |Xt | = σX 2 · t + |mX | · t (t − Δt) .

•

Moreover, if X is a hyperﬁnite random walk with ﬁnite increments, then X will be a hyperﬁnite Lévy process if and only if both mX and σX 2 are ﬁnite.

Proof. For the ﬁrst part of the Lemma, observe that whenever s = u, ΔXs and ΔXu are independent and therefore 2

2

E [ΔXs · ΔXu ] = E [ΔXs ] · E [ΔXu ] = |E [ΔX0 ]| = |mX | Δt2 , whence this quantity is independent of u, s. This yields ) ( E |Xt |2 = E =

&

s

For further volumes: http://www.springer.com/series/7172

10

Editorial Board

Franco Brezzi (Editor in Chief) IMATI-CNR Via Ferrata 5a 27100 Pavia, Italy e-mail: [email protected] John M. Ball Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB United Kingdom e-mail: [email protected] Alberto Bressan Department of Mathematics Penn State University University Park State College PA. 16802, USA e-mail: [email protected] Fabrizio Catanese Mathematisches Institut Universitatstraße 30 95447 Bayreuth, Germany e-mail: [email protected]

Persi Diaconis Department of Statistics Stanford University 450 Serra Mall Stanford, CA 94305-4065, USA e-mail: [email protected], [email protected] Nicola Fusco Dipartimento di Matematica e Applicazioni Universitá di Napoli “Federico II”, via Cintia Complesso Universitario di Monte S. Angelo 80126 Napoli, Italy e-mail: [email protected] Carlos E. Kenig Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637-1514 USA e-mail: [email protected] Fulvio Ricci Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7 56126 Pisa, Italy e-mail: [email protected]

Carlo Cercignani Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milano, Italy e-mail: [email protected]

Gerard Van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam, The Netherlands e-mail: [email protected]

Corrado De Concini Dipartimento di Matematica Universitá di Roma “La Sapienza” Piazzale Aldo Moro 2 00185 Roma, Italy e-mail: [email protected]

Cédric Villani Institut Henri Poincaré 11 rue Pierre et Marie Curie 75230 Paris Cedex 05 France e-mail: [email protected]

The Editorial Policy can be found at the back of the volume.

Sergio Albeverio Frederik Herzberg

•

Ruzong Fan

Hyperfinite Dirichlet Forms and Stochastic Processes

123

Sergio Albeverio University of Bonn Institute for Applied Mathematics and HCM Endenicher Allee 60 53115 Bonn Germany [email protected]

Frederik Herzberg Bielefeld University Institute of Mathematical Economics Universitätsstraße 25 33615 Bielefeld Germany [email protected]

Ruzong Fan Texas A and M University Department of Statistics College Station 77843, TX USA [email protected] Current address Biostatistics and Bioinformatics Branch Division of Epidemiology, Statistics & Prevention Eunice Kennedy Shriver National Institute of Child Health & Human Development 6100 Executive Blvd. MSC 7510, Bethesda, MD 20892 United States of America

ISSN 1862-9113 ISBN 978-3-642-19658-4 e-ISBN 978-3-642-19659-1 DOI 10.1007/978-3-642-19659-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011928508 Mathematics Subject Classification (2000): 03H05; 60J45 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

In memory of the dear father, Mr. Decai Li (1926–2003), of the second author.

Preface

The theory of stochastic processes has developed rapidly in the past decades. Martingale theory and the study of smooth diﬀusion processes as solutions of stochastic diﬀerential equations have been extended in several directions, such as the study of inﬁnite dimensional diﬀusion processes, the study of diﬀusion processes with non-smooth unbounded coeﬃcients, diﬀusion processes on manifolds and on singular spaces. The interplay between stochastic analysis and mathematical physics has been one of the most important and exciting research areas. One of the best techniques to deal with the problems of these areas is Dirichlet space theory. In the original framework of this theory, the state space is a locally compact separable metric space, e.g., Rd , or a d-dimensional manifold. This theory has given us a nice understanding about the property of diﬀusion processes with non-smooth unbounded coeﬃcients. Moreover, it has been fruitfully applied to mathematical physics. This framework has been generalized to state spaces which are more general topological spaces or some inﬁnite dimensional vector spaces or manifolds. Several key problems, such as the closability of quadratic forms and the construction of strong Markov processes associated with quasi-regular Dirichlet forms, have been solved. The study of inﬁnite dimensional stochastic analysis as well as the study of processes on singular structures (like fractals, trees, or general metric spaces) has enriched and extended the Dirichlet space theory. In the meantime, a new framework has been introduced into Dirichlet space theory by the development of nonstandard probabilistic analysis [25, 166]. As is well-known, nonstandard analysis is an alternative setting for analysis (and, indeed, all areas of mathematics), namely by enriching the set of real numbers by inﬁnitesimal and inﬁnite elements. It has its origin in seminal work by Schmieden, Laugwitz [325] and most notably Robinson [310]. By now, several textbooks and surveys exist on this theory and its applications (see, e.g. [25, 63, 125, 217]). Nonstandard analysis gives a novel approach to the theory of stochastic processes. In particular, it has led to hyperﬁnite symmetric Dirichlet space theory. Besides being interesting by itself, it has also many applications. In the ﬁrst part of the book, we extend the research to the

vii

viii

Preface

nonsymmetric case, and remove some restrictive conditions in the previous treatment of the subject (Chap. 5 of [25]). In addition, we shall apply the theory to present a new approach to inﬁnite dimensional stochastic analysis. In writing this book we have two main aims: (1) to give a presentation of research on nonsymmetric hyperﬁnite Dirichlet space theory and its applications in (standard) ﬁnite and inﬁnite dimensional stochastic analysis, Chaps. 1–4; (2) to ﬁnd nonstandard representations for a special class of (ﬁnite dimensional) Feller processes and their inﬁnitesimal generators, viz. stochastically continuous processes with stationary and independent increments (i.e., Lévy processes), Chap. 5. Chapter 6 is a complement to illustrate the usefulness of the hyperﬁnite probability spaces. The ﬁrst part (Chaps. 1–4) is based on Chap. 5 of Albeverio et al. [25] and the further in depth research of Sergio and Ruzong; the second part (Chaps. 5–6) is based on results obtained recently by Tom Lindstrøm and their extensions by Sergio and Frederik. As mentioned earlier, the interplay between stochastic analysis and mathematical physics has been one of the most important and exciting themes of research in the last decades. This is already a suﬃcient rationale for the research of the ﬁrst part of the present book. The motivation for including the second part, Chap. 5, into this book is that many of the issues discussed in the more general framework of the ﬁrst part, such as existence of standard parts of hyperﬁnite Markov chains, become much less technical to resolve for hyperﬁnite Lévy processes. Furthermore, the more restrictive setting of the second part also allows one to obtain ﬁner results on the relation between Lévy processes and their hyperﬁnite analogues, one example being a hyperﬁnite version of the Lévy–Khintchine formula. The contents of this book are arranged as follows: In Chap. 1, we introduce the framework of hyperﬁnite Dirichlet forms. We develop the potential theory of hyperﬁnite Dirichlet forms in Chap. 2. In Chap. 3, we consider standard representations of hyperﬁnite Markov chains under certain conditions, and translate the conditions on hyperﬁnite Markov chains into the language of hyperﬁnite Dirichlet forms. As an interesting and important application in classical stochastic analysis, we construct tight dual strong Markov processes associated with quasi-regular Dirichlet forms by using the language of hyperﬁnite Dirichlet forms in Chap. 4. The results show that hyperﬁnite Dirichlet space theory is a powerful tool to study classical problems. In the ﬁrst sections of Chap. 5, the notion of a hyperﬁnite Lévy process is introduced and its relation to hyperﬁnite random walks as well as to standard Lévy processes is investigated. These results can be used to show that the jump part of any Lévy process is essentially a hyperﬁnite convolution of Poisson processes. Finally, Chap. 6 is an epilogue, providing a rigorous motivation for the study of hyperﬁnite Loeb path spaces as generic probability spaces.

Preface

ix

The entire book is based on nonstandard analysis. For the reader’s convenience, we present some basic notions of nonstandard analysis, such as internal sets and saturation, linear spaces, Loeb measure spaces, structure of ∗ R and topology in the appendix. Because of its monographical character centered around the hyperﬁnite approach, the book has by no means the goal of including all aspects of recent developments in the theory of stochastic processes and its connections with Dirichlet forms theory or the theory of Lévy processes. For this, we rather refer to surveys and proceedings like Albeverio [2], Barndorﬀ-Nielsen et al. [73], and Ma et al. [275], respectively. The germ of this book goes back to the year 1989 when the second author, Ruzong Fan, worked on the construction of symmetric Markov processes associated with Dirichlet forms at Peking University, Beijing ([165] and Chap. 4). At that time, Ruzong was unaware that Sergio’s group was working on the same project using standard methods [41]. The second author, Dr. Zhiming Ma, of [41] did privately inquire Ruzong about the progress of Ruzong’s research in 1989 at the Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing. In response to Dr. Ma’s request of a private meeting, Ruzong presented his work to Dr. Ma in a classroom with Dr. Ma as the only audience. Dr. Ma, however, did not mention his ongoing work with Sergio in any way. Thus, Ruzong was totally unaware of Sergio’s research. In the spring of 1990, Ruzong ﬁrst realized this when he saw a manuscript of Albeverio and Ma [41] in Beijing with a surprise. These events notwithstanding, Ruzong continued to work on a “symmetric version” of Chaps. 1–4 using non-standard language when he was at Peking University till 1991 and when he visited the Humboldt-University, Berlin, between 1991 and 1992. Under Sergio’s supervision and encouragement, Ruzong extended the project to the current “nonsymmetric version” from 1992 to 1994 at Ruhr-University, Bochum. In 2006, Frederik kindly joined the project with a contribution on hyperﬁnite Lévy processes (Chap. 5) and the Epilogue (Chap. 6). In the summer of 2006, the three authors gathered at the University of Bonn to ﬁnalize this monograph. We gratefully acknowledge the manifold support of various institutions in the long process of work on this project. In the run-up to its completion, Sergio and Frederik were supported partially by the collaborative research center SFB 611 of the German Research Foundation (DFG), Germany; in addition, Ruzong’s visit to Bonn was partially funded through a research fellowship from the Alexander von Humboldt Foundation, Germany. Over the course of his career, Ruzong has received a lot of generous support from Sergio. As a Ph.D candidate in Beijing around 1987–1988, Ruzong was greatly fascinated by Sergio and Raphael Høegh-Krohn’s novel work on inﬁnite dimensional stochastic analysis, in which Ruzong ﬁnished his Ph.D thesis. Unfortunately, Ruzong got no chance to meet Raphael Høegh-Krohn; right before Ruzong went to Europe, he was shocked to learn that Raphael

x

Preface

Høegh-Krohn died of a heart attack. In a relatively isolated environment, Ruzong mostly worked on himself by reading numerous papers and books of Sergio and Raphael Høegh-Krohn; and many times, Ruzong had to spend a few days on a single equation or lemma to guess and to understand it. Whilst it seemed like a helpless or hopeless situation for Ruzong at that time, Ruzong eventually came to the forefront of research in areas of inﬁnite dimensional stochastic analysis: he studied the hard and central questions regarding Beurling–Deny formulae, representation of martingale additive functionals and absolute continuity of symmetric diﬀusion processes on Banach spaces, potential theory of symmetric hyperﬁnite Dirichlet forms, and construction of the symmetric strong Markov processes associated with quasi-regular Dirichlet forms by using the non-standard analysis language. This direction of research was initiated by Sergio, although Ruzong was unaware that Sergio’s group already worked on the construction of Markov processes using the language of standard stochastic analysis. In early 1989, Ruzong applied for a fellowship from the Alexander von Humboldt Foundation from Peking University, Beijing; soon after a rejection from the Foundation in the fall 1989, Ruzong received a warm letter from Sergio with encouragement and a kind oﬀer to nominate, as an academic host, Ruzong for the fellowship and by writing a strong letter of recommendation. This is just one anecdote to illustrate how Ruzong has constantly been able to count on Sergio’s help via communications by either mail or face-to-face conversations starting from 1989. Between 1992 and 1994, Sergio generously supported Ruzong at Ruhr-University Bochum to complete the main part of Chaps. 1–4 of this monograph, and helped Ruzong to pass the hard period of time in his career. The story of Ruzong is an example how Sergio has helped many young mathematicians to grow and to mature. Quite probably, Ruzong would have disappeared from academia a long time ago without the support of Sergio. In a true sense, Sergio has been an academic father ﬁgure for Ruzong when he desperately needed one. In recent years, after his departure from Sergio’s research group, Ruzong has been mainly working on statistical genetics guided by his beloved American mentor, Dr. Kenneth Lange, at the University of Michigan and UCLA. Nevertheless, Ruzong has fond memories and deep appreciation of numerous communications with his European academic father Sergio; and both Ruzong and Frederik are deeply grateful for Sergio’s mentoring. Thus, especially right after Sergio’s 70th birthday in 2009 – which also marks the 50th anniversary of his remarkable scientiﬁc career –, Ruzong and Frederik are sure that they will be joined by many other young mathematicians in thanking Sergio for his wonderful role in our professional and personal development and in wishing him all the best for the rest of his life: Not just continued productivity, but most of all good health, happiness, joy, and peace.

Preface

xi

We owe a huge debt of gratitude to our families: In the summer of 2006, Dr. Li Zhu (Ruzong’s wife) kindly took care of two young children when her husband was visiting Bonn. Their adorable daughter, Olivia Wenlu Fan, was with the second author in Germany for the “hot and interesting” summer of Bonn, where she liked everything except German milk. Frederik thanks his wife, Angélique Herzberg, for her love and manifold support with the words of Proverbs 31,10–12: “A wife of noble character [. . . ] is worth far more than rubies. Her husband [. . . ] lacks nothing of value. She brings him good [. . . ] all the days of her life.” We are all very grateful to our families for their love and understanding during the entire process of writing this book. Finally, we would like to thank Dr. Catriona Byrne as well as Susanne Denskus and Ute McCrory of Springer Verlag for their kind, unfainting editorial assistance in the long process of publishing this work.

Contents

1

Hyperfinite Dirichlet Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Hyperﬁnite Quadratic Forms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Domain of the Symmetric Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Resolvent of the Symmetric Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Weak Coercive Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Hyperﬁnite Dirichlet Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Hyperﬁnite Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Weak Coercive Quadratic Forms, Revisited . . . . . . . . . . . . . . . . . . . . .

1 2 6 17 26 36 52 61

2

Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Exceptional Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Exceptional Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Co-Exceptional Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Excessive Functions and Equilibrium Potentials . . . . . . . . . . . . . . . . 2.3 Capacity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Relation of Exceptionality and Capacity Theory . . . . . . . . . . . . . . . 2.5 Measures of Hyperﬁnite Energy Integrals . . . . . . . . . . . . . . . . . . . . . . . 2.6 Internal Additive Functionals and Associated Measures . . . . . . . 2.7 Fukushima’s Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Decomposition Under the Individual Probability Measures Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Decomposition Under the Whole Measure P . . . . . . . . . . . 2.8 Internal Multiplicative Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Internal multiplicative functionals . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Subordinate Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 Subprocesses .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Feynman-Kac Formulae .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Alternative Expression of Hyperﬁnite Dirichlet Forms . . . . . . . . . 2.10 Transformations of Symmetric Dirichlet Forms .. . . . . . . . . . . . . . . .

65 66 66 71 73 78 86 91 102 107 107 118 120 120 121 122 123 124 125

xiii

xiv

3

Contents

Standard Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Standard Parts of Hyperﬁnite Markov Chains . . . . . . . . . . . . . . . . . . 3.1.1 Inner Standard Part of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Strong Markov Processes and Modiﬁed Standard Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hyperﬁnite Dirichlet Forms and Markov Processes . . . . . . . . . . . . 3.2.1 Separation of Points .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Nearstandardly Concentrated Forms . . . . . . . . . . . . . . . . . . . . 3.2.3 Quasi-Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Construction of Strong Markov Processes .. . . . . . . . . . . . . .

129 130 131 136 145 146 151 155 160

4

Construction of Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Hyperﬁnite Lifts of Quasi-Regular Dirichlet Forms . . . . . . . . . . . . 4.3 Relation with Capacities.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Path Regularity of Hyperﬁnite Markov Chains .. . . . . . . . . . . . . . . . 4.5 Quasi-Continuity and Nearstandard Concentration . . . . . . . . . . . . 4.6 Construction of Strong Markov Processes .. . . . . . . . . . . . . . . . . . . . . . 4.7 Necessity for Existence of Dual Tight Markov Processes . . . . . .

165 166 170 177 180 181 192 197

5

Hyperfinite Lévy Processes and Applications . . . . . . . . . . . . . . . . . . 5.1 Standard Lévy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Characterizing Hyperﬁnite Lévy Processes .. . . . . . . . . . . . . . . . . . . . . 5.3 Hyperﬁnite Lévy Processes: Standard Parts . . . . . . . . . . . . . . . . . . . . 5.4 Hyperﬁnite Lévy-Khintchine Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Representation Theorem for Lévy Processes . . . . . . . . . . . . . . . . . . . . 5.6 Extensions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 200 203 217 224 232 239

6

Genericity of Loeb Path Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Adapted Probability Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Universality, Saturation, Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Hyperﬁnite Adapted Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Probability Logic and Markov Processes . . . . . . . . . . . . . . . . . . . . . . . .

243 244 245 246 247

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 General Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Structure of ∗ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Internal Sets and Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Loeb Measure.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 249 250 251 252 253

Historical Notes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Chapter 1

Hyperfinite Dirichlet Forms

The interplay between methods from functional analysis and the theory of stochastic processes is one of the most important and exciting aspects of mathematical physics today. It is a highly technical and sophisticated theory based on decades of research in both areas. Numerous papers have been written on the standard theory of Dirichlet forms. Apart from the articles and monographs cited below, other notable contributions to the area include: Albeverio and Bernabei [5], Albeverio, Kondratiev, and Röckner [32], Albeverio and Kondratiev [33], Albeverio and Ma [39], Albeverio, Rüdiger, and Wu [54], Bliedtner [94], Bouleau [98], Bouleau and Hirsch [99], Chen et al. [112], Chen, Ma, and Röckner [116], Eberle [149], Exner [154], Fabes, Fukushima, Gross, Kenig, Röckner, and Stroock [155], Fitzsimmons and Kuwae [172], Fukushima [177,179,180], Fukushima and Tanaka [185], Fukushima and Ying [188, 189], Gesztesy et al. [191, 192], Grothaus et al. [198], Hesse et al. [208], Jacob [218–220], Jacob and Moroz [221], Jacob and Schilling [222], Jost et al. [225], Kassmann [232], Kim et al. [240], Kumagai and Sturm [248], Le Jan [258], Liskevich and Röckner [265], Ma and Röckner [272, 273], Ma et al. [274], Mosco [283], Okura [292], Oshima [294, 295], D.W. Robinson [312], Röckner and Wang [317], Röckner and Zhang [319], Schmuland and Sun [329], Shiozawa and Takeda [331], da Silva et al. [332], Stannat [336, 338], Stroock [340], Sturm [343], Takeda [346, 347], Wu [363], and Yosida [364]. In this monograph, we present the theory of Dirichlet forms from a uniﬁed vantage point, using nonstandard analysis, thus viewing the continuum of the time line as a discrete lattice of inﬁnitesimal spacing. This approach is close in spirit to the discrete classical formulation of Dirichlet space theory in A. Beurling and J. Deny’s seminal article [87]. The discrete setup in this monograph permits to study the diﬀusion and the jump part by essentially the same methods. This setting being independent of special topological properties of the state space, it is also considerably less technical than other approaches. Thus, the theory has found its natural setting and no longer depends on choosing particular topological spaces; in particular, it is valid for both ﬁnite and inﬁnite dimensional spaces.

S. Albeverio et al., Hyperfinite Dirichlet Forms and Stochastic Processes, Lecture Notes of the Unione Matematica Italiana 10, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19659-1_1,

1

2

1 Hyperﬁnite Dirichlet Forms

Whilst Albeverio et al. [25], Chap. 5, only discussed symmetric hyperﬁnite Dirichlet forms and related Markov chains (refer to [165, 166] also), we shall extend the theory to the nonsymmetric case. We shall try to follow as much as possible the path suggested by the work on the symmetric case. An important sub-class of Markov process are Feller processes with stationary and independent increments (Lévy processes), and in recent years, these processes have attracted a lot of interest, including from nonstandard analysts. Initiated by T. Lindstrøm [263], a number of articles have been devoted to the investigation of hyperﬁnite Lévy processes. Chapter 5 of this monograph is a detailed exposition of Lindstrøm’s theory [263] and its subsequent continuation by Albeverio and Herzberg [14]. The book ends with an expository summary (without proofs) of the model theory of stochastic processes as developed by H.J. Keisler and his coauthors, who formulated and proved the “universality” of hyperﬁnite adapted probability spaces in a rigorous manner, and a short description of recent fundamental results about the deﬁnability of nonstandard universes. Meanwhile, our purpose in the ﬁrst chapter is to develop a general theory of hyperﬁnite quadratic forms. We shall set the scene in Sect. 1.1. Sections 1.2 and 1.3 will study the domains of symmetric parts, the standard parts and resolvents. We shall discuss the property of weak coercive quadratic forms in Sects. 1.4 and 1.7. In Sect. 1.5, we shall study Markov forms and begin the analysis of associated Markov chains and get the basic Beurling–Deny formula. We discuss the hyperﬁnite lifting theory of standard Dirichlet forms in Sect. 1.6.

1.1 Hyperfinite Quadratic Forms We shall develop a hyperﬁnite theory of nonnegative quadratic forms on inﬁnite dimensional spaces. It is well-known that in the Hilbert space case the theory of closed forms of this kind is equivalent to the theory of nonnegative operators. In fact, there is a natural correspondence between forms E(·, ·) and operators A given by E(u, u) = Au, u, where ·, · is the scalar product in the Hilbert space. We have chosen to present the theory in terms of forms and not operators for two reasons: partly because forms are real-valued, and this makes it simpler to take standard parts, but also because in most of our applications, the form is what is naturally given. Let H be an internal, hyperﬁnite dimensional linear space1 equipped with an inner product ·, · generating a norm || · ||. Let ∗ R be the nonstandard

1

The notions of hyperﬁnite dimensional linear space are given in Albeverio et al. [25].

1.1 Hyperﬁnite Quadratic Forms

3

real line2 . We call a map E : H × H −→ ∗ R nonnegative quadratic form if and only if for all α ∈ ∗ R, u, v, w ∈ H, E(u, u) ≥ 0, E(αu, v) = αE(u, v), E(u, αv) = αE(u, v), E(u + v, w) = E(u, w) + E(v, w), E(w, u + v) = E(w, u) + E(w, v). Since E(·, ·) is a nonnegative quadratic form on the hyperﬁnite dimensional space H, elementary linear algebra tells us that there is a unique nonnegative deﬁnite operator A : H −→ H such that E(u, v) = Au, v

for all

u, v ∈ H.

(1.1.1)

To see this, let ∗ N0 be the nonstandard integers3. Let {ei | 1 ≤ i ≤ N } be an N orthonormal basis of (H, ·, ·) for an N ∈ ∗ N. We put Aei = j=1 E(ei , ej )ej . Then (1.1.1) follows immediately. Hence, A is given by the matrix A = (E(ei , ej ))1≤i,j≤N , i.e., ⎛

⎞ E(e1 , e1 ) E(e1 , e2 ) . . . E(e1 , eN ) ⎜ E(e2 , e1 ) E(e2 , e2 ) . . . E(e2 , eN ) ⎟ ⎜ ⎟ A=⎜ ⎟. .. .. .. .. ⎝ ⎠ . . . . E(eN , e1 ) E(eN , e2 ) . . . E(eN , eN )

(1.1.2)

Moreover, Au, u ≥ 0 for all u ∈ H. This means that A is a hyperﬁnite dimensional matrix (not necessarily symmetric). Let Aˆ be the adjoint operator of A, that is, ˆ E(u, v) = u, Av

for all

u, v ∈ H.

ˆ are the By (1.1.2), we have that Aˆ is the transpose of A. If ||A|| and ||A|| ˆ ˆ operator norms of A and A, respectively, we have ||A|| = ||A||. We ﬁx an inﬁnitesimal4 Δt such that

2 ∗R

is the standard notation for the nonstandard real line, refer to Appendix, Albeverio et al. [25], Cutland [125], Davis [135], Hurd [216], Hurd and Loeb [217], Lindstrøm [262], Stroyan and Bayod [341], and Stroyan and Luxemburg [342]. 3 ∗ N is the standard notation for the nonstandard integers, refer to Appendix, Albeverio 0 et al. [25], Cutland [125], Davis [135], Hurd [216], Hurd and Loeb [217], Lindstrøm [262], Stroyan and Bayod [341], and Stroyan and Luxemburg [342]. 4

In the sense of nonstandard analysis, refer to Appendix, Albeverio et al. [25], Keisler [237, 238], Stroyan and Bayod [341], and Stroyan and Luxemburg [342].

4

1 Hyperﬁnite Dirichlet Forms

0 < Δt ≤

1 1 = . ˆ ||A|| ||A||

(1.1.3)

ˆ Δt by Let us deﬁne new operators QΔt and Q QΔt = I − ΔtA, ˆ Δt = I − ΔtA. ˆ Q ˆ Δt are nonnegative. The relation (1.1.3) implies that the operators QΔt and Q ˆ Δt are less than Because A is nonnegative, the operator norms of QΔt and Q ˆ ·) or equal to one. Similarly, we deﬁne the nonnegative quadratic co-form E(·, of E(·, ·) by ˆ v) = E(v, u) for all u, v ∈ H. E(u, Introduce a nonstandard time line T by T = {kΔt | k ∈ ∗ N0 }. ˆ t to be the operators For each element t = kΔt in T , deﬁne Qt and Q Qt = (QΔt )k , ˆ t = (Q ˆ Δt )k . Q ˆ t }t∈T are obviously semigroups. We shall call The families {Qt }t∈T and {Q t ˆ t }t∈T the co-semigroup associated with E(·, ·) {Q }t∈T the semigroup and {Q ˆ ·), A, A, ˆ T, Qt and Q ˆ t in and Δt, respectively. Whenever we refer to E(·, ·), E(·, the rest of this book, we shall assume that they are linked by above relations. In applications, the primary objects will often be the semigroup {Qt }t∈T ˆ t }t∈T . We can then deﬁne A and Aˆ (and hence E(·, ·)) by and co-semigroup {Q

1 I − QΔt , Δt 1 ˆ Δt . I −Q Aˆ = Δt

A=

The operator A is called the inﬁnitesimal generator of E(·, ·), and Aˆ is called the inﬁnitesimal co-generator of E(·, ·). For each t ∈ T, we may deﬁne approximations A(t) of A and Aˆ(t) of Aˆ by

1 I − Qt , t 1 ˆt . I −Q = t

A(t) = Aˆ(t)

(1.1.4)

1.1 Hyperﬁnite Quadratic Forms

5

From A(t) and Aˆ(t) , we get the forms E (t) (u, v) = A(t) u, v = u, Aˆ(t) v,

(1.1.5)

and Eˆ(t) (u, v) = E (t) (v, u) = Aˆ(t) u, v = A(t) v, u. ˚ ·) of We deﬁne the symmetric part E(·, ·) and anti-symmetric part E(·, E(·, ·) by 1 E(u, v) + E(v, u) , 2 1 ˚ E(u, v) = E(u, v) − E(v, u) . 2 E(u, v) =

For α ∈ ∗ R, α ≥ 0, we set E α (u, v) = E(u, v) + αu, v. Each of these forms generates a norm (possibly a semi-norm in the case α = 0):

E α (u, u) = Eα (u, u).

|u|α =

We recall that the original Hilbert space norm on H is denoted by || · ||. Similarly, we set for α ∈ ∗ R, α ≥ 0, Eα (u, v) = E(u, v) + αu, v, ˆ v) + αu, v. Eˆα (u, v) = E(u, t

Let A and {Q } be the generator and semigroup of E(·, ·), respectively. Then A=

1 A + Aˆ , 2

Q

Δt

=

1 Δt ˆ Δt and QkΔt = (QΔt )k , ∀k ∈ ∗ N. Q +Q 2

t

Since A and Q are nonnegative, self-adjoint operators, they have unique 1

t

nonnegative square roots, which we denote by A 2 and Q 2 , respectively.

6

1 Hyperﬁnite Dirichlet Forms

In the same manner as (1.1.4) and (1.1.5), we can deﬁne approximations (t)

A

of A and E (t)

A

(t)

=

(·, ·) of E(·, ·) by 1 t I −Q , t

E

(t)

(t)

(u, v) = A u, v,

t ∈ T.

If a nonnegative quadratic form E(·, ·) : H × H −→ ∗ R satisﬁes E(u, v) = E(v, u) for all u, v ∈ H, i.e., E(u, v) = E(u, v), we shall call it a nonnegative symmetric quadratic form. It is easy to see that a nonnegative quadratic form E(u, v) is symmetric if ˆ t , ∀t ∈ T. and only if A = Aˆ or Qt = Q In this book, we shall deal with nonnegative quadratic forms E(·, ·) and the related theory. For the framework, potential theory and applications of nonnegative symmetric quadratic form, we refer the reader to Albeverio et al. [25], Chap. 5, Sect. 5.1 and Fan [165, 166]. We shall utilize the known results of symmetric forms in our study, and extend them to the nonsymmetric case. In particular, we need the notion of the symmetric part E(·, ·) of E(·, ·), and the related notations. In Sect. 1.2, we shall deﬁne the domain D(E) of the t symmetric part E(·, ·) by using the semigroup {Q | t ∈ T }. We shall introduce the resolvent {Gα | α ∈ ∗ (−∞, 0)} of E(·, ·) in Sect. 1.3, and characterize the domain D(E) by this resolvent. In Sect. 1.4, we shall deﬁne the domain D(E) of E(·, ·) by its resolvent {Gα | α ∈ ∗ (−∞, 0)}; under the hyperﬁnite weak sector condition, we shall show that D(E) = D(E). In Sect. 1.5, we shall introduce hyperﬁnite Dirichlet forms and related Markov chains. For standard coercive forms, we shall construct their nonstandard representation in Sect. 1.6.

1.2 Domain of the Symmetric Part In this section, we shall deﬁne the domain D(E) of the symmetric part E(·, ·) for a hyperﬁnite nonnegative quadratic form E(·, ·). Before giving a strict deﬁnition (Deﬁnition 1.2.1), we shall mention an intuitive description. At ﬁrst, let Fin(H) be the set of all elements in H with ﬁnite norm. By deﬁning x ≈ y if ||x − y|| ≈ 0, we know from Proposition A.5.2 in the Appendix that the space5 ◦

H = Fin(H)/ ≈

5

≈ stands for diﬀering by an inﬁnitesimal, in the sense of nonstandard analysis, refer to Albeverio et al. [25], Cutland [125], Davis [135], Hurd [216], Hurd and Loeb [217], and Lindstrøm [262].

1.2 Domain of the Symmetric Part

7

is a Hilbert space with respect to the inner product (◦x, ◦y) = st(x, y), where ◦ x denotes the equivalence class of x and st : ∗ R −→ R is the mapping of standard part6 . We call (◦H, (·, ·)) the hull of (H, ·, ·). Consider the standard part E(·, ·) of the nonnegative symmetric quadratic form E(·, ·). If E(·, ·) is S-bounded, i.e., there exists a constant K ∈ R+ such that |E(u, v)| ≤ K||u||||v||

for all u, v ∈ H,

we can simply deﬁne E(·, ·) by E(◦u, ◦v) = ◦ E(u, v). If E(·, ·) is not S-bounded, we shall meet two diﬃculties. We no longer have that E(u, v) ≈ E(˜ u, v˜) whenever u ≈ u ˜ and v ≈ v˜, and there may be elements v ∈ Fin(H) such that E(˜ v , v˜) is inﬁnite for all v˜ ≈ v. The latter problem should not surprise us. It is an immediate consequence of the fact that unbounded forms on Hilbert spaces cannot be deﬁned everywhere. We v , v˜) is inﬁnite shall solve it by simply letting E(◦u, ◦v) be undeﬁned when E(˜ for all v˜ ∈ ◦v. The most natural solution to the ﬁrst problem may be to deﬁne E(◦u, ◦u) = inf{◦ E(v, v) | v ∈ ◦u},

(1.2.1)

and then extend E(·, ·) to be a bilinear form by the usual trick E(◦u, ◦v) =

1 ◦ E( u + ◦v, ◦u + ◦v) − E(◦u, ◦u) − E(◦v,◦v) . 2

The disadvantage of this approach is that it gives us little understanding of how the inﬁmum in (1.2.1) is obtained. For an easier access to the regularity properties of E(·, ·) and E(·, ·), we prefer a more indirect way of attack. Our plan is to deﬁne a subset D(E) of Fin(H) – we call it the domain of E(·, ·) – satisfying if ◦ E(u, u) < ∞, there is a v ∈ D(E) such that v ≈ u, if u, v ∈ D(E) and u ≈ v, then ◦ E(u, u) = ◦ E(v, v) < ∞.

(1.2.2) (1.2.3)

We then deﬁne E(·, ·) by E(◦u, ◦u) = ◦ E(v, v),

6

Refer to Albeverio et al. [25].

(1.2.4)

8

1 Hyperﬁnite Dirichlet Forms

when v ∈ D(E) ∩ ◦u. It turns out that the two deﬁnitions (1.2.1) and (1.2.4) agree (see Proposition 1.2.4). If we look at the standard nonsymmetric Dirichlet theory, see Albeverio et al. [9], Kim [241] and Ma and Röckner [270], the domain of a quadratic form is given from the very beginning. After that, the authors such as those of Ma and Röckner [270] introduced the symmetric and anti-symmetric parts (see page 15, [270]). This method makes the domains of the quadratic form and its symmetric part coincide. On the other hand, Albeverio et al. [25] has given us a very nice deﬁnition of domain for the symmetric hyperﬁnite quadratic forms by their semigroups. Therefore, we may deﬁne the domain t D(E) of E(·, ·) via the semigroup of {Q | t ∈ T }. In the next section, we shall discuss the property of the resolvent {Gα | α ∈ ∗ (−∞, 0)} of E(·, ·). We can deﬁne the domain of D(E) through {Gα | α ∈ ∗ (−∞, 0)}. Now it is very natural to ask: can we as well deﬁne the domain D(E) of E(·, ·) directly from {Qt | t ∈ T }? Here we would mention that it seems not easy to do the job. In Sect. 1.4, we shall deﬁne D(E) by means of the resolvent {Gα | α < 0} of E(·, ·). Under the hypothesis of weak sector condition, we shall prove D(E) = D(E) by showing that the two deﬁnitions satisfy (1.2.1). This is similar to the procedure in the standard nonsymmetric Dirichlet space theory, see, e.g., Albeverio et al. [9], Albeverio et al. [47], Albeverio and Ugolini [57], Kim [241], and Ma and Röckner [270]. (t)

Notice that even when E(·, ·) is not S-bounded, E (·, ·) is S-bounded for all non-inﬁnitesimal t. One of the motivations behind our deﬁnition of the domain D(E) is that we want to single out the elements where E(·, ·) is really approximated by the bounded forms E that ◦

(t)

E(u, u) = lim ◦ E t↓0 t≈0

(·, ·), t ≈ 0, i.e., those u ∈ H such (t)

(u, u).

(1.2.5)

We could have taken this to be our deﬁnition of D(E), but for technical and expository reasons we have chosen another one which we shall soon show to be equivalent to (1.2.5) (see Proposition 1.2.2). Definition 1.2.1. Let E(·, ·) be a nonnegative quadratic form on a hyperﬁnite dimensional linear space H. The domain D(E) of the symmetric part of E(·, ·) is the set of all u ∈ H satisfying (i) ◦ E1 (u, u) = ◦ E 1 (u, u) < ∞. t t (ii) For all t ≈ 0, E(Q u, Q u) ≈ E(u, u). Let us try to convey the intuition behind this deﬁnition. Thinking of A as a diﬀerential operator, the elements of D(E) are “smooth” functions and

1.2 Domain of the Symmetric Part

9

t

Q is a “smoothing” operator often given by an integral kernel. If an element t u is already smooth, then an inﬁnitesimal amount of smoothing Q , t ≈ 0, t t should not change it noticeably, and hence E(Q u, Q u) ≈ E(u, u). We shall give a partial justiﬁcation of this rather crude image later, when we show t that if ◦ E 1 (u, u) < ∞, then the “smoothed” elements Q u, t ≈ 0, are all in D(E) (Lemma 1.2.3, see also Corollary 1.2.3). Our ﬁrst task will be to establish a list of alternative deﬁnitions of D(E), among them (1.2.5). We begin with the following simple identity giving the relationship between E(·, ·) and E (t) (·, ·), and also the relationship between E(·, ·) and E

(t)

(·, ·) :

Lemma 1.2.1. For all u ∈ H and t ∈ T , we have (i) (ii) (iii)

(t)

E (t) (u, u) ≥ 0 and E (u, u) ≥ 0, Δt Δt ˆ s u), E (t) (u, u) = E(Qs u, u) = E(u, Q t t E

(t)

0≤s 0. For α ∈ ∗ (−∞, 0), we deﬁne for u, v ∈ ∗ K (α)

F (u, v) = −α(u + αRα u, v) ˆ α v, u) = −α(v + αR

and (α)

Fˆ (u, v) = −α(v + αRα v, u) ˆ α u, v). = −α(u + αR

Then ˆ α u, v) for all u ∈ F (u, v) = F (−αRα u, v) and (α) Fˆ (u, v) = F (−αR ∗ ∗ K, v ∈ (D(F )) and α ∈ (−∞, 0). ˆ α u, αR ˆ α u) ≤ (α) F (u, u) for all (ii) F (αRα u, αRα u) ≤ (α) F (u, u) and F (αR ∗ ∗ u ∈ K and α ∈ (−∞, 0). (iii) |(α) F 1 (u, v)| ≤ (C + 1) F1 (u, u) (α) F 1 (v, v) for all u ∈ ∗ (D(F )), v ∈ ∗ K and α ∈ ∗ (−∞, 0). (iv) F1 (αRα u, αRα u) ≤ (C + 1)2 F1 (u, u) for all u ∈ ∗ (D(F )) and α ∈ ∗ (−∞, 0). (i)

(α) ∗

Proof. The proof follows from Proposition 1.6.2.

When are the standard forms generated by two hyperﬁnite forms diﬀerent? The last result in this section we shall prove shows that to answer this question, it is enough to check whether the forms have the same resolvents. We recall that in Theorem 1.4.1 we found a way to construct a form from its resolvent. This representation will be helpful to solve our problem. Theorem 1.6.2. Let K be a Hilbert space and H be an S-dense, hyperﬁnite ˘ ·) be two hyperﬁnite weak dimensional subspace of ∗ K. Let E(·, ·) and E(·, ˘K (·, ·) on K, respeccoercive quadratic forms on H inducing EK (·, ·) and E ˘ α } be the resolvents of E(·, ·) and E(·, ˘ ·). Assume tively. Let {Gα } and {G ∗ that for some ﬁnite, non-inﬁnitesimal α ∈ (−∞, 0), there is a u ∈ H with ◦ ˘ α u are both nearstandard, but E1 (u, u) < ∞ such that v = Gα u, w = G ◦ ˘ ||v − w|| = 0. Then EK (·, ·) = EK (·, ·). Proof. Assume for contradiction that EK (·, ·) = E˘K (·, ·). Pick v˜ ≈ v, w ˜≈w ˘ w such that v˜ ∈ D(E), ˜ ∈ D(E). Notice that by Lemma 1.4.5, v ∈ D(E), w ∈ ˘ We have D(E).

1.7 Weak Coercive Quadratic Forms, Revisited

61

u, v − w ≈ u, v − w ˜ = E−α (v, v − w). ˜

(1.6.8)

Since v, w are nearstandard and EK (·, ·) = E˘K (·, ·), we have ◦

E −α (v, v − w) ˜ = EK (◦ v, ◦ v − ◦ w) − (◦ α)(◦ v, ◦ v − ◦ w) ˘K (◦ v, ◦ v − ◦ w) − (◦ α)(◦ v, ◦ v − ◦ w) =E v , v˜ − w). = ◦ E˘−α (˜

(1.6.9)

On the other hand, we have u, v − w ≈ u, v˜ − w = E˘−α (w, v˜ − w).

(1.6.10)

Combining the relations (1.6.8), (1.6.9), and (1.6.10), we see that v − w, v˜ − w) 0 = ◦ E˘−α (˜ ◦ ◦ ≥ |α| ||v − w||2 > 0.

The theorem is proved.

1.7 Weak Coercive Quadratic Forms, Revisited Let E(·, ·) be a hyperﬁnite weak coercive quadratic form on a hyperﬁnite dimensional space H. Let {Gα | α ∈ ∗ (−∞, 0)} be the resolvent of E(·, ·), and ˆ α | α ∈ ∗ (−∞, 0)} be the co-resolvent of E(·, ·), respectively. Let us still let {G denote by A the inﬁnitesimal generator of E(·, ·), and by Aˆ the inﬁnitesimal co-generator of E(·, ·). Such as in Sect. 1.1, we ﬁx an inﬁnitesimal Δt, and we ˆ Δt by deﬁne new operators QΔt and Q QΔt = I − ΔtA, ˆ Δt = I − ΔtA. ˆ Q Introduce a nonstandard time line T by T = {kΔt | k ∈ ∗ N0 }. For each ˆ t to element t = kΔt ∈ T , deﬁne the semigroup Qt and the co-semigroup Q be the families of operators Qt = (QΔt )k , ˆ t = (Q ˆ Δt )k , t ∈ T. Q

62

1 Hyperﬁnite Dirichlet Forms

For each t ∈ T, we may deﬁne approximations A(t) of A and Aˆ(t) of Aˆ by

1 I − Qt , t 1 ˆt . I −Q = t

A(t) = Aˆ(t)

From A(t) and Aˆ(t) , we get the forms E (t) (u, v) = A(t) u, v = u, Aˆ(t) v. Let E(·, ·) be the standard part of E(·, ·). Then E(·, ·) is closed. In addition, (E(·, ·), D(E)) satisﬁes the weak sector condition by Remark 1.4.1. Hence, (E(·, ·), D(E)) is a coercive closed form on ◦H. Let {Rβ | β ∈ (−∞, 0)} ˆ β | β ∈ (−∞, 0)} be the resolvent and co-resolvent of (E(·, ·), D(E)), and {R respectively. Similarly, let {Tt | t ∈ [0, ∞)} and {Tˆt | t ∈ [0, ∞)} be the semigroup and co-semigroup of (E(·, ·), D(E)), respectively. For t ∈ (0, ∞), we deﬁne E (t) (x, y) =

1 (x − Tt x, y), x, y ∈ ◦H. t

By Albeverio et al. [9], Theorem 3.4 (or referring to Proposition 1.6.4), we have Lemma 1.7.1. (i) Let x ∈ ◦H. Then x ∈ D(E) if and only if supt>0 E (t) (x, x) < ∞. (ii) For all x, y ∈ D(E), we have lim E (t) (x, y) = E(x, y). t↓0

(iii) For all x ∈ D(E), we have lim E1 (x − Tt x, x − Tt x) = 0. t↓0

By applying Lemma 1.7.1, we can see from the proof of Theorem 1.6.1 that there exists an inﬁnitesimal δ ∈ T such that (E(·, ·), D(E)) is the standard part of E (δ) (u, v) (referring to Remark 1.6.1, and replacing −γRγ by Qδ in the proof of Theorem 1.6.1). In addition, for any x ∈ ◦H, u ∈ x, and for all t ∈ [0, ∞), s ∈ {kδ | k ∈ ∗ N0 }, t = ◦ s, we have that Qs u ∈ Tt x. Since Qs1 +s2 ≈ Qs1 if s2 ≈ 0, we have that Qs u ∈ Tt x for all t ∈ [0, ∞), s ∈ T = {kΔt | k ∈ ∗ N0 }, t = ◦ s. Notice that (E(·, ·), D(E)) is the standard part of both E(u, v) and E (δ) (u, v), and so the resolvent of E(u, v) is almost the same as that of E (δ) (u, v) by Theorem 1.6.2. Summarizing above results, we have Theorem 1.7.1. Let E(·, ·) be a hyperﬁnite weak coercive quadratic form on a hyperﬁnite dimensional space H. Then, we have

1.7 Weak Coercive Quadratic Forms, Revisited

63

(i) Let u ∈ H. Then u ∈ D(E) if and only if sups ◦ E (s) (u, u) < ∞. (ii) For all u, v ∈ D(E), we have lim E (s) (u, v) = E(u, v).

◦ s↓0

(iii) For all u ∈ D(E), we have lim E1 (u − Qs u, u − Qs u) ≈ 0.

◦ s↓0

Proof. Let x = ◦ u. Then for all t ∈ [0, ∞), s ∈ T = {kΔt | k ∈ ∗ N0 }, t = ◦ s, we have that Qs u ∈ Tt x. By Lemma 1.7.1, the theorem follows easily.

Proposition 1.7.1. Let E(·, ·) be a hyperﬁnite weak coercive quadratic form on a hyperﬁnite dimensional linear space H. If ◦ E(u, u) < ∞, then for all ﬁnite s > 0, s ∈ T, and ◦ s = 0, we have ◦ [Qs u] ∈ D(E). Proof. Let x = ◦ u. For t = ◦ s, we have that Qs u ∈ Tt x. Therefore, we have [Qs u] = Tt x ∈ D(E).

◦

Chapter 2

Potential Theory of Hyperfinite Dirichlet Forms

Probabilistic potential theory has been a very important component in the study of hyperﬁnite Dirichlet space theory. It provides a probabilistic interpretation of potential theory; and, more generally, it establishes a beautiful bridge between functional analysis and the theory of Markov processes. There are many applications of this theory, especially in the area of inﬁnite dimensional stochastic analysis and mathematical physics. Our purpose in this chapter is to develop the probabilistic potential theory associated with hyperﬁnite Dirichlet forms and the related Markov chains. The motivation is twofold. On the one hand, we want to establish a relationship between the standard Dirichlet space theory and the hyperﬁnite counterpart. On the other hand, we want to provide new methods for the theory of hyperﬁnite Dirichlet forms itself. Inﬁnite dimensional stochastic analysis has been developed extensively in the last decades. We hope to convince the reader that nonstandard analysis can provide a new tool to deal with problems in this exciting area, see particularly Chap. 4, for example. The arrangement of the present chapter is as follows. In Sect. 2.1, we shall deﬁne exceptional sets for non-symmetric hyperﬁnite Markov chains. Sect. 2.2 will discuss excessive functions and equilibrium potentials. Moreover, we introduce a capacity theory for hyperﬁnite quadratic forms and show that it is a Choquet capacity in Sect. 2.3. Furthermore, we establish a relation between the family of exceptional sets and the family of zero capacity sets in Sect. 2.4. In Sect. 2.5, we consider positive measures of hyperﬁnite energy integrals and the associated theory. That is, we establish connections among hyperﬁnite excessive functions and hyperﬁnite potentials. Zero capacity subsets will be characterized by positive measures of hyperﬁnite energy integrals. In Sect. 2.6, we introduce internal additive functionals. The relationship between hyperﬁnite measures and additive functionals will be considered. Moreover, we shall obtain a positive hyperﬁnite measure μ associated with an internal function u. In Sect. 2.7, we get Fukushima’s decomposition theorem under individual probability measures. This extends the work of Albeverio et al. [25] and Fan [166]. In Sect. 2.8, we shall discuss the properties of internal multiplicative functionals, subordinate semigroups, subprocesses,

S. Albeverio et al., Hyperfinite Dirichlet Forms and Stochastic Processes, Lecture Notes of the Unione Matematica Italiana 10, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19659-1_2,

65

66

2 Potential Theory

and a Feynman-Kac formula. The motivation for this work is given by the corresponding standard theory developed by Blumenthal and Getoor [96]. This serves as a basis for the development of a perturbation theory of hyperﬁnite Dirichlet forms characterized by internal additive functionals. The content of Sect. 2.9 is the nonstandard version of time change of standard Dirichlet forms. In standard Dirichlet space theory, the question how to change a nonconservative symmetric Markov process into a conservative one has received an answer in Fukushima and Takeda [184] in terms of the Girsanov transformation. In Sect. 2.10, we show that this problem is quite simple in the hyperﬁnite setting (Theorem 2.10.1). In this chapter, we shall use the notations developed in Sect. 1.5. However, we need not assume all the conditions required in Sect. 1.5. In Sect. 2.1, we will deﬁne exceptional sets for a hyperﬁnite Markov chain X. We do not need anything about the dual hyperﬁnite Markov chain of X. Starting from Sect. 2.2, we shall work under the setting of Sect. 1.5. That is, we shall assume the conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14) in Sect. 1.5, and the related notations.

2.1 Exceptional Sets Albeverio et al. [25] has given us a deﬁnition of exceptional sets in the framework of the theory of symmetric hyperﬁnite Dirichlet forms, which is too restrictive for certain cases. In Fan [166], we extended this concept in the same symmetric setting by removing a lot of unnecessary assumptions. In this section, we shall deﬁne exceptional sets for non-symmetric hyperﬁnite Markov chains.

2.1.1 Exceptional Sets We take an inﬁnitesimal Δt such that Δt > 0. Set T = {kΔt | k ∈ ∗ N0 } .

(2.1.1)

Let Y be a Hausdorﬀ space and let ∗ Y be the nonstandard extension of Y . Let S = {s0 , s1 , . . . , sN } be an S-dense subset of ∗ Y for some N ∈ ∗ N − N and let m be a hyperﬁnite measure on S. Denote by S the internal algebra of subsets of S. Let Q = {qij } be an (N + 1) × (N + 1) matrix with nonnegative entries. Assume that

2.1 Exceptional Sets

67 N

qij = 1 for all i = 0, 1, . . . , N,

(2.1.2)

j=0

and assume that the state s0 is a trap, i.e., q0i = 0

for all

i = 0.

(2.1.3)

If (Ω, P ) is an internal measure space, and X : Ω × T −→ S is an internal process, let [ω]t = {ω ∈ Ω | X(ω , s) = X(ω, s) for all s ≤ t}.

(2.1.4)

For each t ∈ T, let Ft be the internal algebra on Ω generated by the sets [ω]t . Assume that for all ω ∈ Ω, we have P ([ω]0 ) = m{X(ω, 0)},

(2.1.5)

and whenever X(ω, t) = si , we have P {ω ∈ [ω]t | X(t + Δt, ω ) = sj } = qij P ([ω]t ).

(2.1.6)

That is, X is a hyperﬁnite Markov chain with the initial distribution m and transition matrix Q. The family (Ω, Ft , Pi , i ∈ S) of internal probability spaces is deﬁned by k−1 qω(nΔt),ω((n+1)Δt) , Pi [ω]kΔt = δiω(0) n=0

for each i ∈ S, where δij is the Kronecker symbol as in Chap. 1. As in Sect. 1.5, let δ ∈ T and let the sub-line Tδ = {kδ | k ∈ ∗ N0 } . Set T fin = {t ∈ T | t is ﬁnite} , Tδfin = {t ∈ Tδ | t is ﬁnite} . In addition, for r ∈ T, we let T r = {t ∈ T | t ≤ r} , Tδr = {t ∈ Tδ | t ≤ r} . Moreover, we know that X (δ) is the restriction X|Tδ .

(2.1.7)

68

2 Potential Theory

For every y ∈ Y, let us deﬁne the monad μ(y) of y by μ(y) =

{∗ O | O is open such that y ∈ O} .

We call a point y ∈ ∗ Y nearstandard if and only if y ∈ μ(x) for some x ∈ Y. Denote by N s(∗ Y ) the set of all nearstandard points in ∗ Y . Since Y is a Hausdorﬀ topological space, each element y ∈ N s(∗ Y ) is nearstandard to exactly one element x in Y (we refer to page 48, 2.1.6. Proposition, [25]). We call x the standard part of y and denote it by ◦y or st(y). In particular, we can take Y = R and use this notation also. Definition 2.1.1. (i) A subset B of S0 is called δ-exceptional for X if L(P ) ω ∃t ∈ Tδfin X(ω, t) ∈ B = 0,

(2.1.8)

where L(P ) is the Loeb measure of P . (ii) A subset B of S0 is called exceptional for X if it is δ-exceptional for some inﬁnitesimal δ ∈ T . Remark 2.1.1. For symmetric hyperﬁnite Markov chains, δ-exceptional sets are deﬁned in Albeverio et al. [25] in the following way fin L(P ) ω (X(ω, 0) ∈ S 0 ) ∧ ∃t ∈ Tδ X(ω, t) ∈ B = 0,

(2.1.9)

where S 0 = S0 ∩ N s(∗ Y ). Therefore, if a subset B is δ-exceptional in the sense of our Deﬁnition 2.1.1, it is δ-exceptional in the sense of (2.1.9). We have noticed this result in Fan [166] for the symmetric case. Here we deal with the general hyperﬁnite non-symmetric Markov chain X(t). Remark 2.1.2. From (2.1.8), we see that for every exceptional set B L(P ) {ω | X(ω, 0) ∈ B} = 0. This implies that L(m)(B) = 0, where L(m) is the Loeb measure of m. We have the following lemma, whose proof is easy and therefore will be omitted. Lemma 2.1.1. (i) All internal subsets B ⊂ S0 with m(B) ≈ 0 are exceptional. (ii) The families of exceptional and δ-exceptional sets are closed under countable unions.

2.1 Exceptional Sets

69

Definition 2.1.2. (i) A δ-exceptional subset A of S0 is called properly δexceptional for X if there is a family {Bm,n | m, n ∈ N} of internal subsets such that

Bm,n A= m∈N n∈N

and for all si ∈ / A, L(Pi ) ω ∃t ∈ Tδfin X(ω, t) ∈ A = 0, where L(Pi ) is the Loeb measure of Pi . (ii) A subset A of S0 is called properly exceptional for X if it is properly δ-exceptional for some δ ≈ 0, δ ∈ T. Proposition 2.1.1. If A ⊂ S0 is a δ-exceptional set, there is a properly δ-exceptional set B ⊃ A. Proof. Since A is δ-exceptional, there is an internal subset Bmn for each pair (m, n) of natural numbers such that A ⊂ Bmn ,

≤ L(P ) ω ∃t ∈ Tδfin X(ω, t) ∈ Bmn

1 . (2.1.10) n2 m

Deﬁne Cmn

1 = i ∈ S Pi ∃t ∈ Tδm X(t) ∈ Bmn ≥ nm

and A¯ =

Cmn .

m∈N n∈N

Let σ(ω) be a stopping time deﬁned by 1 ≥ σ(ω) = min t ∈ Tδ PX(ω,t) ∃s ∈ Tδm X(s) ∈ Bmn . mn Then, we have 1 fin L(P ) ω ∃t ∈ Tδ X(ω, t) ∈ Cmn mn

1 1 fin m ≥ = L(P ) ω ∃t ∈ Tδ PX(ω,t) ∃s ∈ Tδ X(s) ∈ Bmn mn mn

70

2 Potential Theory

1 = L(P ) 1(◦σ(ω) 0 (δ) (δ) ∗ ◦ n ∈ NCapα (An ) ≥ inf Capα (Al )l ∈ N − ε . By saturation, there is an inﬁnite M belonging to the above internal set. Hence, we have ◦

Cap(δ) α

∞

Al

≥ ◦ Cap(δ) α (AM )

l=1

≥ inf

◦

Cap(δ) (A ) l ∈ N − ε. l α

82

2 Potential Theory

By letting ε ↓ 0, we get ◦

Cap(δ) α

∞

Al

≥ inf

◦

l=1

Cap(δ) α (Al )l

∈N .

On the other hand, it is easy to see that ◦

Cap(δ) α

∞

Al

l=1

≤ inf

l∈N

◦

Cap(δ) (A ) . l α

If inf ◦ Cap(δ) (A ) l l ∈ N = ∞, we consider the following internal subset α for N0 ∈ N (A ) ≥ N n ∈ ∗ NCap(δ) n 0 . α By saturation and letting N0 ↑ ∞, we can show that (ii) holds.

Lemma 2.3.3. If {An | n ∈ N} is a sequence of internal subsets of S0 , then we have

(δ) ◦ ◦ Capα An ≤ Cap(δ) (2.3.2) α (An ), n∈N

n∈N

for all δ ∈ T, α ≥ 0. Proof. Set b = n∈N ◦ Cap(δ) α (An ). If b = ∞, the inequality (2.3.2) holds. In the following proof, we shall assume b < ∞. Let {An | n ∈ ∗ N} be an internal extension of {An | n ∈ N} . For every ε > 0, it follows from Lemma 2.3.1 (ii) that n n

(δ) Capα Al ≤ Cap(δ) for all n ∈ N. α (Al ) ≤ ε + b l=1

l=1

Consider the following internal set

n

n

n (δ) (δ) n ∈ N Al is internal and Capα Al ≤ Capα (Al ) ≤ b + ε . ∗

l=1

l=1

l=1

By saturation, there is an inﬁnite element M = M (ε) belonging to the above internal set. Hence, we obtain

2.3 Capacity Theory

83

Cap(δ) α

≤ Cap(δ) α

Al

M

l∈N

Al

l=1

≤

M

Cap(δ) α (Al )

l=1

≤ b + ε. By letting ε ↓ 0, we have proved the inequality (2.3.2).

Proposition 2.3.1. For all δ ∈ T, α ≥ 0, we have (i) If A and B are two subsets of S0 , A ⊂ B, then (δ) Cap(δ) α (A) ≤ Capα (B)

(2.3.3)

(ii) Let {An | n ∈ N} be a sequence of subsets of S0 , then ◦

Cap(δ) α

An

≤

n∈N

◦

Cap(δ) α (An ).

(2.3.4)

n∈N

(iii) Let {An | n ∈ N} be an increasing sequence of subsets of S0 , then ◦

Cap(δ) α

An

= sup

n∈N

◦

Cap(δ) α (An )n

∈N .

Proof. (i) The proof is immediate, using the deﬁnition. (ii) Set b = n∈N ◦ Cap(δ) α (An ). We can assume that b < ∞. Given ε > 0, for every n ∈ N, let us take an internal subset Bn such that An ⊂ Bn and (δ) Cap(δ) α (An ) ≤ Capα (Bn )

≤ Cap(δ) α (An ) +

ε . 2n+1

Therefore, we have from (i) and Lemma 2.3.3 that ◦ Cap(δ) α

An

≤

n∈N

≤

◦ Cap(δ) α

Bn

n∈N ◦

Cap(δ) α (Bn )

◦

Cap(δ) α (An ) + ε.

n∈N

≤

n∈N

By letting ε ↓ 0, we get the inequality (2.3.4).

84

2 Potential Theory

(iii) We may assume that for all n ∈ N, ◦ Cap(δ) α (An ) < ∞. Given ε > 0, for each n ∈ N, let Bn be an internal subset of S0 such that (δ) Cap(δ) α (Bn ) ≤ Capα (An ) + ε.

An ⊂ Bn ,

Then, we have from (2.3.3) and Lemma 2.3.2 (i) that ◦

Cap(δ) α

An

≤

◦

Cap(δ) α

n∈N

= sup n

≤ sup

◦

Bn

n∈N

Cap(δ) (B ) n α

◦

Cap(δ) α (An )

+ε ≤ sup ◦ Cap(δ) α (An ) + ε. n

n

By letting ε ↓ 0, we get ◦

Cap(δ) α

≤ sup

An

◦

n∈N

n ∈ N . Cap(δ) (A ) n α

(2.3.5)

On the other hand, it is easy to see that ◦

Cap(δ) α

An

≥ sup

n∈N

◦

Cap(δ) α (An )n

∈N .

(2.3.6)

From the inequalities (2.3.5) and (2.3.6), we have proved Proposition 2.3.1 (iii). For the purpose of explaining our Theorem 2.3.1 in the following, we ﬁrst introduce some notations in capacity theory (referring to, e.g., [282]). Let G be a set, G be a family of some subsets of G. Denote by Gσ (respectively, Gδ ) the closure of a collection of subsets of G under countable union (respectively, countable intersection). That is, Gσ =

∞

n=1

An An ∈ G ,

Gδ =

∞ n=1

An An ∈ G .

Moreover, we shall write Gσδ = (Gσ )δ . Definition 2.3.1. Let G be a set. A paving G on G is a family of subsets of G such that the empty set ∅ is contained in G. The pair (G, G) consisting of a set G and a paving G on G is called a paved set.

2.3 Capacity Theory

85

Definition 2.3.2. Let (G, G) be a paved set. The paving G is said to be semi-compact if every countable family of elements of G, which has the ﬁnite intersection property, has a nonempty intersection. It is easy to see that (S0 , S0 ) is a semi-compact paved set. Moreover, S0 is closed under the complement, ﬁnite union, and ﬁnite intersection operations. Definition 2.3.3. A subset A of S0 is said to be S0 -analytic if there exists an auxiliary set G with a semi-compact paving G, and a subset B ⊂ G × S0 belonging to (G × S0 )σδ such that A is the projection of B on S0 . We denote by A(S0 ) all the S0 -analytic sets (we notice that G × S0 = {G1 × S1 | G1 ∈ G and S1 ∈ S0 }). Lemma 2.3.4. The σ-ﬁeld σ(S0 ) generated by S0 is contained in A(S0 ). Proof. For every F ∈ S0 , S0 −F belongs to S0 also. By Meyer [282], Chap. III T12 Theorem, we know σ(S0 ) ⊂ A(S0 ). Definition 2.3.4. An extended real valued set function I : 2S0 → [−∞, +∞], deﬁned on all subsets 2S0 of S0 , is called a Choquet S0 -capacity if it satisﬁes the following properties: (i) I is increasing, i.e., A ⊂ B =⇒ I(A) ≤ I(B). (ii) For every increasing sequence {An | n ∈ N} of subsets of S0 , we have I

= sup I(An ).

An

n∈N

n∈N

(iii) For every decreasing sequence {An | n ∈ N} of elements of S0 , we have I

An

= inf I(An ). n∈N

n∈N

We have reached one of our main results. Theorem 2.3.1. For each δ ∈ T and α ≥ 0, α ∈ ∗ R, we have the following results: (i) ◦ Cap(δ) α (·) is a Choquet S0 -capacity. (ii) Every S0 -analytic set is capacitable with respect to capacity ◦ Cap(δ) α (·). That is, for every A ∈ A(S0 ), we have ◦

Cap(δ) α (A)

= sup

◦

Cap(δ) α (B)B

=

Bm , Bm ∈ S0 and B ⊂ A .

m∈N

(iii) Every subset A of S0 belonging to σ(S0 ) is capacitable with respect to the capacity ◦ Cap(δ) α (·) whenever 0 < st(α) < ∞.

86

2 Potential Theory

Proof. By Lemma 2.3.1 and Proposition 2.3.1, we know that ◦ Cap(δ) α (·) is a Choquet S0 -capacity. Therefore, (ii) holds by Meyer [282], Chap. III T19 Theorem. (iii) is the consequence of (ii) and Lemma 2.3.4. Definition 2.3.5. (i) A subset B of S0 is said to be of δ-zero capacity, if (δ) Cap1 (B) ≈ 0. (δ)

(ii) A subset B of S0 is said to be of zero capacity if Cap1 (B) ≈ 0 for some inﬁnitesimal δ ∈ T. (δ)

Remark 2.3.1. For any B ∈ S0 and δ ∈ T, we have m(B) ≤ Cap1 (B). Therefore, for any zero capacity subset B of S0 , we have L(m)(B) = 0.

2.4 Relation of Exceptionality and Capacity Theory In regular Dirichlet space theory, we know that the concepts of exceptional sets and zero capacity sets are equivalent, see Fukushima [175], Theorem 4.3.1. As the fourth section of this chapter, we will discuss the corresponding problem in our hyperﬁnite Dirichlet space theory. We shall continue the discussion of Sect. 2.3. Hence, we assume that all ˆ conditions in Sect. 2.2 are satisﬁed in this section as well, i.e., X and X ˆ are dual hyperﬁnite Markov chains, and E(·, ·) and E(·, ·) are the hyperﬁˆ respectively. Let nite quadratic form and co-form associated with X and X, H be the hyperﬁnite dimensional space with an inner product ·, · deﬁned by (2.2.3) in Sect. 2.2 or (1.5.15) in Sect. 1.5, Chap. 1. Lemma 2.4.1. Let {Bn | n ∈ N} be a sequence of internal subsets of S0 . If ∞ (δ) n limn→∞ ◦ Cap1 ( m=1 Bm ) = 0, then n=1 Bn is a δ-exceptional set, where δ ∈ T, δ ≈ 0. Proof. Since S0 is closed under ﬁnite intersection, we may assume that {Bn | n ∈ N} is a decreasing sequence. Deﬁne a stopping time for each n ∈ N, (δ)

σBn (ω) = min {t ∈ Tδ | X(ω, t) ∈ Bn } . We have 1 L(P ) ω ∃t ∈ Tδ X(ω, t) ∈ Bn ◦ 1 = P ω ∃t ∈ Tδ X(ω, t) ∈ Bn

◦ 1 = Ei ω ∃t ∈ Tδ X(ω, t) ∈ Bn dm(i) S0

2.4 Relation of Exceptionality and Capacity Theory

=

87

◦

Ei 1(σ(δ) ≤1) dm(i) Bn ◦ (δ) −σ /δ −1 = Ei ω (1 + δ) Bn ≥ (1 + δ) δ dm(i) S0 ⎧ ⎫ ◦ ⎨ (1 + δ)−σB(δ)n /δ ⎬ Ei ≤ dm(i) ⎩ (1 + δ)− 1δ ⎭ S0 ◦ (δ) −σ /δ Ei (1 + δ) Bn dm(i) =e· S0

S0

=e·

◦

(δ)

e1 (Bn )(i) dm(i) S0 (δ) (δ) ≤ e · ◦ E1 e1 (Bn ), e1 (Bn ) (δ)

= e · ◦ Cap1 (Bn ) −→ 0,

(2.4.1)

where the last inequality comes from (2.3.1) in Sect. 2.3. Then, we have ∞ 1 L(P ) ω ∃t ∈ Tδ X(ω, t) ∈ Bn = 0.

(2.4.2)

n=1

By the dual property of the Markov process X(t) and (2.4.2), we also have ∞ L(P ) ω ∃t ∈ Tδfin X(ω, t) ∈ Bn = 0. n=1

Therefore, the set

∞

n=1

Bn is δ-exceptional.

Theorem 2.4.1. If a subset A of S0 is of δ-zero capacity, it is δ-exceptional. (δ)

Proof. Since Cap1 (A) ≈ 0, we can take a sequence of internal subsets {Bn | n ∈ N} satisfying A⊂

∞ n=1

Bn ,

(δ)

lim ◦ Cap1 (

n→∞

n

Bm ) = 0.

m=1

Using Lemma 2.4.1, we know that ∞ n=1 Bn is δ-exceptional. Hence, A is δ-exceptional also. This completes the proof of Theorem 2.4.1. Lemma 2.4.2. Let δ1 ∈ T, δ1 ≈ 0. Assume that for all f ∈ H, if (δ ) (E1 1 (f, f )) < ∞ and f (s) ≈ 0 for all s ∈ / B, where B is a δ1 -exceptional set, (δ ) then we have E1 1 (f, f ) ≈ 0. Let A be a subset of S0 . If A is δ1 -exceptional and there exists an internal subset B of S0 such that

◦

88

2 Potential Theory

A ⊂ B,

◦

(δ )

Cap1 1 (B) < ∞,

(2.4.3)

then A is of δ1 -zero capacity. Proof. By using Proposition 2.1.1, there exists a properly δ1 -exceptional set

Bm,n ⊃ A.

m∈N n∈N

For simplicity, we assume that Bm,n ⊂ B for all n, m ∈ N, and for each m, the sequence {Bm,n | n ∈ N} is decreasing withrespect to n. In order to show that A has zero capacity, we ﬁrst prove that n∈N Bm,n has zero capacity for every m. From now on, we ﬁx an m ∈ N. (δ )

By the assumption (2.4.3), we know that ◦ Cap1 1 (Bm,n ) < ∞ for every n. (δ ) Moreover, Cap1 1 (Bm,n ) is decreasing with n. Let {Bm,n | n ∈ ∗ N} be a decreasing extension of {Bm,n | n ∈ N} . By saturation, there exists an inﬁnite element nm ∈ ∗ N − N such that (δ1 )

lim ◦ [E1

n→∞

(δ1 )

= ◦ [E1

(δ )

(δ )

(e1 1 (Bm,n ), e1 1 (Bm,n ))] (δ )

(δ )

(e1 1 (Bm,nm ), e1 1 (Bm,nm ))].

Therefore, we have ◦ (δ ) (δ ) (δ ) e1 1 (Bm,nm ), e1 1 (Bm,nm ) = Cap1 1 (Bm,nm ) < ∞. (δ ) Besides, for every i ∈ S0 , it is easy to see that e1 1 (Bm,n )(i) | n ∈ N is decreasing with respect to n. Since m∈N n∈N Bm,n is properly δ1 exceptional, we can show ◦

(δ1 )

E1

(δ )

e1 1 (Bm,nm )(i) ≈ 0

for every i ∈ /

Bm,n .

(2.4.4)

m∈N n∈N

In fact, for every M0 ∈ [0, ∞), we have (δ ) −σ(δ1 ) /δ e1 1 (Bm,n )(i) = Ei (1 + δ1 ) m,n 1 −σ(δ1 ) /δ = Ei (1 + δ1 ) m,n 1 1(σBm,n ≥M0 ) + 1(σm,n <M0 ) ≤ (1 + δ1 )−M0 /δ1 + Ei 1(σm,n <M0 ) . By letting M0 be suﬃciently large, we know that (1 + δ1 )−M0 /δ1 will be very small. Taking n suﬃciently large, we see that the approximation (2.4.4) holds.

2.4 Relation of Exceptionality and Capacity Theory

89

The assumption in the Lemma implies that (δ1 )

E1

(δ ) (δ ) e1 1 (Bm,nm ), e1 1 (Bm,nm ) ≈ 0.

Therefore, we get ◦

(δ1 )

E1

(δ ) (δ ) e1 1 (Bm,n ), e1 1 (Bm,n ) −→ 0.

This says that (δ ) Cap1 1

≈ 0.

Bm,n

(2.4.5)

n∈N

By Proposition 2.3.1 (i) and (ii) and the approximation (2.4.5), we obtain ◦

(δ ) Cap1 1 (A)

≤

◦

(δ ) Cap1 1

Bm,n

m∈N n∈N

≤

◦

(δ ) Cap1 1

m∈N

Bm,n

n∈N

= 0.

Thus, the set A has a δ1 -zero capacity. (δ )

Theorem 2.4.2. Let δ1 ∈ T, δ1 ≈ 0. Assume for all f ∈ H, if ◦ (E1 1 (f, f )) < ∞ and f (s) ≈ 0 for all s ∈ / B, where B is a δ1 -exceptional set, then we (δ1 ) have E1 (f, f ) ≈ 0. Let A be a subset of S0 . If A is δ1 -exceptional and there exists a sequence of internal subsets {Bn | n ∈ N} of S0 such that A⊂

Bn

and

◦

(δ )

Cap1 1 (Bn ) < ∞,

∀n ∈ N,

n∈N

then A is of δ1 -zero capacity. Proof. The proof follows easily from Lemma 2.4.2 and Proposition 2.3.1 (ii). In Lemma 2.4.2 and Theorem 2.4.2, we talk about the hyperﬁnite quadratic (δ ) form and co-form and make one assumption: for all f ∈ H, if ◦ (E1 1 (f, f )) < ∞ and f (s) ≈ 0 for all s ∈ / B, where B is a δ1 -exceptional set, then we have (δ1 ) E1 (f, f ) ≈ 0. This assumption is somewhat equivalent to say that for all (δ )

(δ)

f , if ◦ (E1 1 (f, f )) < ∞, then f ∈ D(E ). The assumption, however, is not always easy to verify. In the following, we will give results for hyperﬁnite

90

2 Potential Theory

weak coercive quadratic forms, for which we do not make the assumption. The results were ﬁrst proved for hyperﬁnite symmetric Dirichlet forms in Fan [166]. The following results extend those of Fan [166]. Lemma 2.4.3. Assume that E (δ) (·, ·) is a hyperﬁnite weak coercive quadratic form on the space H for all inﬁnitesimal δ ∈ T. Let δ1 ∈ T, δ1 ≈ 0. Let A be a subset of S0 . If A is δ1 -exceptional and there exists an internal subset B of S0 which satisﬁes condition (2.4.3), then there is an inﬁnitesimal δ0 ∈ T which is larger than δ1 such that A is of δ-zero capacity for all δ ≥ δ0 , δ ≈ 0. Proof. In the proof of Lemma 2.4.2, we have ◦

(δ1 )

E1

◦ (δ ) (δ ) (δ ) e1 1 (Bm,nm ), e1 1 (Bm,nm ) = Cap1 1 (Bm,nm ) < ∞.

By Corollary 1.2.4 and Theorem 1.4.2, there is a δm ≈ 0 such that (δ ) e1 1 (Bm,nm ) ∈ D(E (δ) ) for all inﬁnitesimal δ ≥ δm . Moreover, we know from the proof of Lemma 2.4.2 that (δ )

e1 1 (Bm,nm )(i) ≈ 0

for every i ∈ /

Bm,n .

m∈N n∈N (δ )

(δ )

Since e1 1 (Bm,nm ) ∈ D(E (δ) ) for all δ ≥ δm , δ ≈ 0, e1 1 (Bm,nm ) is S 2 integrable in the sense of Albeverio et al. [25], page 77, Chap. 3. Hence, we (δ ) have e1 1 (Bm,nm ) ≈ 0 in the hyperﬁnite dimensional space H because L(m) Bm,n = 0. m∈N m∈N (δ)

(δ )

(δ )

Therefore, we have E1 (e1 1 (Bm,nm ), e1 1 (Bm,nm )) ≈ 0 for all δ ≥ δm , δ ≈ 0. This implies that ◦

(δ)

E1

(δ) (δ) e1 (Bm,n ), e1 (Bm,n ) ↓ 0, n −→ ∞,

for all δ ≥ δm , δ ≈ 0.

Hence, we have (δ) Cap1

Bm,n

≈0

for all

δ ≥ δm , δ ≈ 0.

(2.4.6)

n∈N

By saturation, there is a δ0 ≈ 0 larger than all δm , m ∈ N. Therefore, it follows from the approximation (2.4.6) that for δ ≥ δ0 , δ ≈ 0,

2.5 Measures of Hyperfinite Energy Integrals

(δ)

Cap1

91

≈ 0 for all

Bm,n

m ∈ N.

(2.4.7)

n∈N

By Proposition 2.3.1 (i) and (ii) and the approximation (2.4.7), we obtain ◦

(δ) Cap1 (A)

≤

◦

(δ) Cap1

Bm,n

m∈N n∈N

≤

∞ m=1

◦

(δ) Cap1

Bm,n

n∈N

= 0.

Therefore, the set A has δ-zero capacity. (δ)

Theorem 2.4.3. Let E (·, ·) be a hyperﬁnite weak coercive quadratic form on the space H for all inﬁnitesimal δ ∈ T. Let δ1 ∈ T, δ1 ≈ 0. Let A be a subset of S0 . If A is δ1 -exceptional and there exists a sequence of internal subsets {Bn | n ∈ N} of S0 such that A⊂

Bn

and

◦

(δ )

Cap1 1 (Bn ) < ∞,

∀n ∈ N,

n∈N

then there is an inﬁnitesimal δ0 ∈ T which is larger than δ1 such that A is of δ-zero capacity for all δ ≥ δ0 , δ ≈ 0. Proof. The proof follows easily from Lemma 2.4.3, Proposition 2.3.1 (ii), and saturation.

2.5 Measures of Hyperfinite Energy Integrals We have deﬁned an inner product ·, · on the hyperﬁnite dimensional space H by (2.2.3) in Sect. 2.2 or (1.5.15) in Sect. 1.5, Chap. 1. Let || · || be the norm generated by ·, ·. Denote by Fin(H) the set of all elements in H with ﬁnite norm || · ||. By deﬁning u ≈ v if ||u − v|| ≈ 0, we recall that the space ◦

H = Fin(H)/ ≈

is a Hilbert space with the inner product (◦u, ◦v) = st(u, v), where ◦u denotes the equivalence class of u. In this section, we shall continue our discussion of Sects. 2.2, 2.3, and 2.4. ˆ are dual hyperﬁnite Markov chains. Let E(·, ·) and We assume that X and X ˆ E(·, ·) be the hyperﬁnite quadratic form and co-form associated with X and ˆ respectively. X,

92

2 Potential Theory

We know that for α ∈ ∗ R, α ≥ 0 and δ ∈ T, Eα(δ) (u, v) = E (δ) (u, v) + αu, v. Each of these forms generates a norm (possibly a semi-norm in the case α 1 (δ) (δ) = 0): |u|α = [Eα (u, u)] 2 . Similarly, we denote by Fin(δ) α (H) the set of all (δ) (δ) (δ) elements in H with ﬁnite norm | · |α . Deﬁne u ≈α v if |u − v|α ≈ 0. The space (δ) Hα(δ) = Fin(δ) α (H)/ ≈α

◦

is a Hilbert space if ◦ α > 0 with respect to the inner product (δ) (δ) ([u](δ) α , [v]α )α =

◦

Eα(δ) (u, v) ,

(δ)

(δ)

where [u]α denotes the equivalence class of u under the norm | · |α , and (δ) (·, ·)α denotes the related inner product. Definition 2.5.1. Let μ be a hyperﬁnite positive measure on S0 . For δ ∈ T , if there exists a constant K ∈ ∗ [0, ∞) = ∗ R+ such that S0

|u(s)| μ(ds) =

N

|u(si )|μ(i)

i=1

12 (δ) ≤ K E1 (u, u)

(2.5.1)

for every u ∈ H, we say that μ is of δ-hyperﬁnite energy integral. Moreover, if there exists K ∈ ∗ R+ satisfying (2.5.1) and ◦K < ∞, μ is said to be of δ-ﬁnite energy integral. Henceforth, we will identify a hyperﬁnite measure μ on S0 with the measure μ ˜ on S deﬁned by μ ˜(s0 ) = 0, μ ˜ (si ) = μ(si ) for all si ∈ S0 . Theorem 2.5.1. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form. Then (i) A positive hyperﬁnite measure μ on S0 is of δ-hyperﬁnite energy integral if and only if for each α ∈ ∗ R, 0 < st(α) < ∞, there exists an element (δ) Uα μ ∈ H such that for every v ∈ H, v(s) μ(ds). (2.5.2) Eα(δ) (Uα(δ) μ, v) = S0

(δ)

Moreover, if μ is of δ-ﬁnite energy integral, we have Uα μ ∈ Fin(δ) α (H).

2.5 Measures of Hyperfinite Energy Integrals

93

(ii) A positive hyperﬁnite measure μ on S0 is of δ-hyperﬁnite energy integral if and only if for each α ∈ ∗ R, 0 < st(α) < ∞, there exists an element ˆα(δ) μ ∈ H such that for every v ∈ H, U ˆα(δ) μ) = v(s) μ(ds). Eα(δ) (v, U S0

ˆα(δ) μ ∈ Fin(δ) (H). Moreover, if μ is of δ-ﬁnite energy integral, we have U α Proof. The theorem is an easy consequence of Riesz’s representation theorem. (δ) ˆα(δ) μ in Theorem 2.5.1 hyperﬁnite αRemark 2.5.1. We call Uα μ and U potential and hyperﬁnite α-co-potential of μ associated with E (δ) (·, ·), respectively. An internal element u ∈ H is called a hyperﬁnite α-potential (or (δ) ˆα(δ) μ) for some positive hyperﬁnite α-co-potential) if u = Uα μ (or u = U measure μ of δ-hyperﬁnite energy integral.

Definition 2.5.2. Fix α ∈ ∗ R, α ≥ 0 and δ ∈ T. (i) An element u ∈ H is called hyperﬁnite pre-α-excessive associated with E (δ) (·, ·), if u(i) ≥ 0, Qδ u(i) ≤ (1 + αδ)u(i) for every i ∈ S0 such that m(i) = 0. (ii) An element u ∈ H is called hyperﬁnite pre-α-co-excessive associated with ˆ δ u(i) ≤ (1 + αδ)u(i) for every i ∈ S0 such that Eˆ(δ) (·, ·), if u(i) ≥ 0, Q m(i) = 0. (δ)

In order to develop our theory, we shall denote by {Gβ | β ∈ ∗ ( − ∞, 0)} ˆ (δ) | β ∈ ∗ ( − ∞, 0)} the resolvent and co-resolvent of E (δ) (·, ·), and {G β respectively. That is, they are deﬁned by (δ)

Gβ = (A(δ) − β)−1 , ˆ (δ) = (Aˆ(δ) − β)−1 . G β Hence, we have for α ∈ ∗ R+ , (δ) (δ) 1 + αδ − Qδ G−α = δ(α + A(δ) )G−α =δ and ˆ (δ) ˆ(δ) )G ˆ (δ) ˆδ G 1 + αδ − Q −α = δ(α + A −α = δ.

(2.5.3)

94

2 Potential Theory

Theorem 2.5.2. For δ ∈ T, α ∈ conditions are equivalent:

∗

R, α ≥ 0, and u ∈ H, the following

(i) u is hyperﬁnite pre-α-excessive associated with E (δ) (·, ·). (ii) There exists a hyperﬁnite positive measure μ on S0 such that Eα(δ) (u, v) =

v(s) μ(ds)

for all

S0

v ∈ H.

(δ)

(iii) Eα (u, v) ≥ 0 for all v ∈ H, v ≥ 0. If E (δ) (·, ·) is a hyperﬁnite weak coercive quadratic form, the above statements are equivalent to the following: (iv) u is a hyperﬁnite α-potential of E (δ) (·, ·). Proof. (i) =⇒ (ii). Assume that u is hyperﬁnite pre-α-excessive associated with E (δ) (·, ·). Deﬁne a hyperﬁnite positive measure μ on S0 by μ(s0 ) = 0, 1 μ(si ) = (1 + αδ)u(i) − Qδ u(i) m(i) for i ∈ S0 . δ Since μ(si ) =

1 (1 + αδ)u(i) − Qδ u(i) m(i) δ

= A(δ) u(i) + αu(i) m(i), we see that for every v ∈ H, Eα(δ) (u, v) =

N A(δ) u(i) + αu(i) v(i)m(i)

i=1 =

v(s) μ(ds). S0

(ii) =⇒ (iii). This is easy, and thus we omit it! (iii) =⇒ (i). Such as in Sect. 1.5 of Chap. 1, let u+ = u ∨ 0. Since u+ − u ≥ 0, (δ) we have Eα (u, u+ − u) ≥ 0. Taking into account Corollary 1.5.2, we have Eα(δ) (u+ − u, u+ − u) = Eα(δ) (u+ , u+ − u) − Eα(δ) (u, u+ − u) ≤ Eα(δ) (u+ , u+ − u) = −E (δ) ((−u) ∧ 0, −u − (−u) ∧ 0)

≤ 0.

2.5 Measures of Hyperfinite Energy Integrals

95

This implies that Eα(δ) (u+ − u, u+ − u) = 0. Therefore, u(i) = u(i) ∨ 0 ≥ 0 for every i ∈ S0 such that m(i) = 0. Furthermore, it follows from (2.5.3) that for any v ∈ H, ˆ δ )v (1 + αδ − Qδ )u, v = u, (1 + αδ − Q ˆ δ )G ˆ (δ) v = Eα(δ) u, (1 + αδ − Q −α

=

Eα(δ) (u, δv).

(2.5.4)

Fix i ∈ S0 . Let v ∈ H be an internal function deﬁned by v(l) = δil , l ∈ S. Then, we have from (2.5.4) that 1 + αδ − Qδ u(i)m(i) = Eα(δ) (u, δv) ≥ 0. Therefore, the internal function u is hyperﬁnite pre-α-excessive associated with E (δ) (·, ·). If E (δ) (·, ·) is a hyperﬁnite weak coercive quadratic form, (ii) is equivalent to (iv) by Theorem 2.5.1 (i). Similarly, we have Theorem 2.5.3. For δ ∈ T, α ∈ conditions are equivalent:

∗

R, α ≥ 0, and u ∈ H, the following

(i) u is hyperﬁnite pre-α-co-excessive associated with E (δ) (·, ·). (ii) There exists a hyperﬁnite positive measure μ on S0 such that Eα(δ) (v, u) =

v(s) μ(ds) S0

for all

v ∈ H.

(δ)

(iii) Eα (v, u) ≥ 0 for all v ∈ H, v ≥ 0. If E (δ) (·, ·) is a hyperﬁnite weak coercive quadratic form, the above statements are equivalent to the following: (iv) u is a hyperﬁnite α-co-potential of E (δ) (·, ·). We denote by τ0 (δ) the family of all internal positive measures on S0 of δ-hyperﬁnite energy integrals. Denote τ0 = ∪{τ0 (δ) | δ is inﬁnitesimal,δ ∈ T }. Proposition 2.5.1. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form. For δ ∈ T, a hyperﬁnite positive measure μ on S0 is of δ-hyperﬁnite

96

2 Potential Theory

energy integral if and only if for any α ∈ ∗ R+ , there exists a hyperﬁnite pre-α-excessive function u associated with E (δ) (·, ·) such that μ(i) =

1 (1 + αδ)u(i) − Qδ u(i) m(i) δ

for all

i ∈ S0 .

(2.5.5)

Moreover, if u¯ ∈ H satisﬁes above equation also, then u(i) = u ¯(i) for all i ∈ S0 with m(i) = 0. Proof. ⇐= Let u satisfy the condition (2.5.5). Then, we have Eα(δ) (u, v) =

for all

v(s) μ(ds) S0

v ∈ H. (δ)

=⇒ Assume that μ is of δ-hyperﬁnite energy integral. Let u = U1 μ ∈ H satisfy the condition (2.5.2). Then, we have −1 (δ) μ(i) = G−1 u(i)m(i) 1 (1 + δ)u(i) − Qδ u(i) m(i). = δ Therefore, we have for any α ∈ ∗ R+ , v ∈ H,

S

(δ)

−1 (δ) v(i) G−1 u(i) dm(i) S (δ) (δ) = Eα(δ) G−α ((G−1 )−1 u), v .

v(s) dμ(s) =

(δ)

Hence, w = G−α ((G−1 )−1 u) is hyperﬁnite pre-α-excessive associated with E (δ) (·, ·) by Theorem 2.5.2. Furthermore, we have μ(i) =

1 (1 + αδ)w(i) − Qδ w(i) m(i) for all i ∈ S0 . δ

Proposition 2.5.2. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form. For δ ∈ T, let ν be a hyperﬁnite positive measure on S0 . Deﬁne a measure μ on S0 by μ(s) = ν(s)1(m(s) =0) Then μ is of δ-hyperﬁnite energy integral.

for

s ∈ S0 .

2.5 Measures of Hyperfinite Energy Integrals

97

Proof. Deﬁne f (s) =

ν(s) 1(m(s) =0) m(s)

for

s ∈ S0 ,

f (s0 ) = 0.

For any u ∈ H, we have u(s) dμ(s) = u(s)1(m(s) =0) dν(s) S S = u(s)f (s) dm(s) =

S (δ) (δ) E1 (G−1 f, u).

Hence, μ is of δ-hyperﬁnite energy integral by Theorem 2.5.2.

We recall that in Sect. 1.4 of Chap. 1, we have introduced the standard part (E (δ) (·, ·), D(E (δ) )) on ◦H for a hyperﬁnite weak coercive quadratic form E (δ) (·, ·). Theorem 2.5.4. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form. For α ∈ ∗ R, 0 < st(α) < ∞ and δ ∈ T , let μ be a hyperﬁnite positive measure of δ-ﬁnite energy integral, and let u be an α-potential of μ associated with E (δ) (·, ·). Deﬁne (δ) gn (i) = n u(i) − nG−n−α u(i) , n ∈ N. Then for every v ∈ Fin(δ) α (H), we have (i) (ii)

(δ) ◦ (δ) Eα (G−α gn , v) (δ)

−→ Eα(δ) (◦u, ◦v) ≈δα

as

n → ∞.

(2.5.6)

(δ)

If u ∈ D(E ), let v˜ v and v˜ ∈ D(E ), ◦ ◦ gn (i)v(i) dm(i) = v˜(i) dμ(i). limn→∞ S0

S0

Proof. (i) First of all, we have (δ) ◦ (δ) Eα (G−α gn , v)

= ◦(gn , v) (δ)

= ◦(nu − n2 G−n−α u, v) ◦ (δ) (2.5.7) = (n + α)u − (n + α)2 G−n−α u, v ◦ (δ) (δ) + −αu + α2 G−n−α u + 2nαG−n−α u, v . It follows from Theorem 1.4.1 that

98

2 Potential Theory ◦

(δ) (n + α)u − (n + α)2 G−n−α u, v + αu, v

−→ Eα(δ) (◦u, ◦v) as n → ∞.

(2.5.8)

By Lemma 1.4.4, we have ◦

(δ) α2 G−n−α u, v −→ 0 as n −→ ∞.

(2.5.9)

Besides, it follows from Theorem 1.4.1 ◦

−2αu + 2nαG(δ) −n−α u, v ◦ 2αn (n + α)2 ≤ − u + (n + α)u, v (n + α)2 n (δ) +−(n + α)u + (n + α)2 G−n−α u, v −→ 0 as

n → ∞.

(2.5.10)

From the relations (2.5.7), (2.5.8), (2.5.9), and (2.5.10), we know that the approximation (2.5.6) holds. (ii) From the proof of (i), we get lim

◦

gn (s)v(s) m(ds) = ◦ Eα(δ) (u, v˜)

n→∞ S 0

=

◦ S0

v˜(s) μ(ds).

Proposition 2.5.3. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. For δ ∈ T , let μ be a positive measure of δ-hyperﬁnite energy integral. Then for every L(μ) measurable subset A of S0 , we have ◦'

L(μ)(A) ≤

C

(

)(

(δ) (δ) (δ) E1 (U1 μ, U1 μ) (δ)

◦ Cap(δ) (A). 1

(2.5.11)

Proof. For simplicity, we assume that ◦ Cap1 (A) < ∞. If A is internal, we have ◦ ◦ μ(A) = 1A (s) μ(ds) S0

≤

◦

S0

(δ)

e1 (A)(s) μ(ds)

2.5 Measures of Hyperfinite Energy Integrals (δ)

(δ)

99

(δ)

= ◦ E1 (U1 μ, e1 (A)) ) ( ◦' ( (δ) (δ) (δ) (δ) (δ) (δ) ≤ C E1 (U1 μ, U1 μ) E1 (e1 (A), e1 (A)) ◦'

=

(

C

) ( (δ) Cap1 (A) .

(δ) (δ) (δ) E1 (U1 μ, U1 μ)

Now it is easy to see that the inequality (2.5.11) holds for all L(μ) measurable subsets A. Corollary 2.5.1. Let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form. Let μ be a positive measure of δ-ﬁnite energy integral on S0 . Then L(μ) charges no set of δ-zero capacity.

Proof. The proof follows easily from Proposition 2.5.3. We shall now state and prove the following characterization theorem.

Theorem 2.5.5. Let A ⊂ S0 be an A(S0 )-measurable set (in particular, A ∈ σ(S0 )) and let δ ∈ T. We have (δ)

(i) For any μ ∈ τ0 (δ), L(μ)(A) = 0 =⇒ ◦ Cap1 (A) = 0. (δ) (ii) For any μ ∈ τ00 (δ), L(μ)(A) = 0 =⇒ ◦ Cap1 (A) = 0, where (δ) τ00 (δ) = μ ∈ τ0 (δ)μ(S0 ) = 1, ◦||U1 μ||∞ < ∞ and (δ) (δ) ||U1 μ||∞ = max |U1 μ(s)|s ∈ S0 . (δ)

(iii) For any μ ∈ τˆ00 (δ), L(μ)(A) = 0 =⇒ ◦ Cap1 (A) = 0, where ˆ (δ) μ||∞ < ∞ τˆ00 (δ) = μ ∈ τ0 (δ)μ(S0 ) = 1, ◦||U 1 and ˆ (δ) μ(s)|s ∈ S0 . ˆ (δ) μ ∞ = max |U U 1 1 (δ)

Proof. (i) Assume that ∞ ≥ ◦ Cap1 (A) = α > 0. Since A ∈ A(S0 ), it follows from Theorem 2.3.1 (ii) that A is capacitable with respect to the capacity (δ) ◦ Cap1 (·). That is ◦

(δ) Cap1 (A)

= sup

◦

(δ) Cap1 (B)B

=

n∈N

Bn , Bn ∈ S0

and

B⊂A .

100

2 Potential Theory

Therefore, there exists a sequence {Bn | n ∈ N} of decreasing internal subsets of S0 such that ∞ ∞ α (δ) ◦ Bn ⊂ A, ∞ ≥ Cap1 Bn ≥ > 0. 2 n=1 n=1 Let {Bn | n ∈ ∗ N} be an internal decreasing extension of {Bn | n ∈ N}. Then, there exists an inﬁnite γ ∈ ∗ N − N such that ∞ α ◦ (δ) (δ) Bn = ◦ Cap1 (Bγ ) ≤ ∞. Bγ ⊂ A, 0 < ≤ Cap1 2 n=1 (δ)

Set B = Bγ . Consider the internal function e1 (B). Notice that if i ∈ / B, then (δ) (δ) −σ /δ (1 + δ)e1 (B)(i) = (1 + δ)Ei (1 + δ) B =

N j=1

(δ)

(δ)

e1 (B)(j)qij (δ)

= Qδ e1 (B)(i),

(2.5.12)

(δ)

where σB = min{t ∈ Tδ | X(ω, t) ∈ B}. If i ∈ B, then we have (δ)

(1 + δ)e1 (B)(i) = (1 + δ) ≥

N j=1

(δ)

(δ)

e1 (B)(j)qij (δ)

= Qδ e1 (B)(i).

(2.5.13)

It follows from the relations (2.5.12) and (2.5.13) that the function (δ) e1 (B) is hyperﬁnite 1-excessive associated with E (δ) (·, ·). Deﬁne a hyperﬁnite positive measure μ on S by μ(s0 ) = 0, 1 (δ) (δ) μ(si ) = (1 + δ)e1 (B)(i) − Qδ e1 (B)(i) m(i), i ∈ S0 . δ Then, we have μ(B) = = =

μ(ds) S0 (δ) (δ) E1 (e1 (B), 1) (δ) Cap1 (B).

2.5 Measures of Hyperfinite Energy Integrals

101

This contradicts the assumption ◦ L(μ)(A) = 0. Hence, we must have (δ) ◦ Cap1 (A) = 0. (δ)

(ii) Assume that 0 < ◦ Cap1 (A) = α ≤ ∞. By the proof (i), we see that there exists an element ν ∈ τ0 (δ) such that (replacing μ in the proof of (i) by ν) (δ)

(δ)

0 < ◦ Cap1 (B) ≤ ∞, (δ)

Cap1 (B) = ν(B) = ν(S0 )

(δ)

and U1 ν(i) = e1 (B)(i) ≤ 1 for any i ∈ S0 , where B is an internal set contained in A. Deﬁne μ(·) by ν(·)

μ(·) =

(δ)

Cap1 (B)

.

Then, we have μ(S0 ) = 1, μ ∈ τ0 (δ) and ◦ (δ) |U1 ν(i)| (δ) max |U1 μ(i)| = max i∈S0 i∈S0 Cap(δ) (B) 1 ◦ 1 ≤ (δ) Cap1 (B) < ∞.

◦

But 1 = ◦μ(B) ≤ ◦μ(A) ≤ 1, i.e., L(μ)(A) = 1. This contradicts our assumption of L(μ)(A) = 0. We have proved that ◦

(δ)

Cap1 (A) = 0.

(iii) Recalling Corollary 2.2.1, we can show this result in the same way as that of above (ii). Theorem 2.5.6. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form, which has the following property: ∀A ⊂ S0 ,

◦

(δ)

Cap1 (A) = ∞

=⇒ ∃B ⊂ A

such that

(δ)

0 < ◦ Cap1 (B) < ∞. (2.5.14)

102

2 Potential Theory

Let τ0f (δ) be the family of all positive measures of δ-ﬁnite energy integrals. Then the following statements are equivalent for A ∈ A(S0 ) (in particular, A ∈ σ(S0 )): (δ)

(i) A is of zero δ-capacity, i.e., ◦ Cap1 (A) = 0. (ii) For any μ ∈ τ0f (δ), L(μ)(A) = 0. (iii) For any μ ∈ τ00f (δ), L(μ)(A) = 0, where (δ) τ00f (δ) = μ ∈ τ0f (δ)μ(S0 ) = 1, ◦ ||U1 μ||∞ < ∞ . (iv) For any μ ∈ τˆ00f (δ), L(μ)(A) = 0, where ˆ (δ) μ||∞ < ∞ . τˆ00f (δ) = μ ∈ τ0f (δ)μ(S0 ) = 1, ◦ ||U 1 Proof. (i) =⇒ (ii) =⇒ (iii) and (ii) =⇒ (iv) are clear by Corollary 2.5.1. We can show (ii) =⇒ (i) and (iii) =⇒ (i) and (iv) =⇒ (i) in the same way as the proof of Theorem 2.5.5.

2.6 Internal Additive Functionals and Associated Measures ˆ be dual hyperﬁnite Markov chains, and let E(·, ·) and E(·, ˆ ·) be Let X and X ˆ the hyperﬁnite quadratic form and co-form of X and X, respectively. Again, let H be the hyperﬁnite dimensional space with an inner product ·, · deﬁned by (2.2.3) in Sect. 2.2 or (1.5.15) in Sect. 1.5, Chap. 1. We recall that in Sect. 1.5 of Chap. 1, we have introduced the dual (δ) hyperﬁnite Markov chains (Ω, X (δ) , {Ft | t ∈ Tδ }, {Pi | i ∈ S}) and ˆ X ˆ (δ) , {Fˆ (δ) | t ∈ Tδ }, {Pˆi | i ∈ S}) for δ ∈ T . For simplicity, we assume (Ω, t ˆ X (δ) = X ˆ (δ) , F (δ) = Fˆ (δ) , t ∈ Tδ . The hyperﬁnite quadratic that Ω = Ω, t t ˆ {Pˆi | i ∈ S})) are given by form associated with (X, {Pi | i ∈ S}) (or (X, the expression (1.5.19) (or (1.5.22)) in Sect. 1.5. As in the study of standard Markov processes, we deﬁne a family of translation operators {θt | t ∈ T } of Ω. That is, for each t ∈ T, θt is a map from Ω to Ω deﬁned by ω ∈ Ω =⇒ θt ω ∈ Ω

and for any s ∈ T, θt ω(s) = ω(s + t). (δ)

Hence for each δ ∈ T , we have a family of translation operators {θt | t ∈ Tδ } (δ) induced by {θt | t ∈ T }, θt = θt for any t ∈ Tδ . In other words, for each (δ) t ∈ Tδ , θt is a map from Ω to Ω given by (δ)

ω ∈ Ω =⇒ θt ω ∈ Ω

and for any

(δ)

s ∈ Tδ , θt ω(s) = ω(s + t).

2.6 Internal Additive Functionals and Associated Measures

103

Definition 2.6.1. For any δ ∈ T, we call an internal ∗ R-valued function A(ω, t) or At (ω), t ∈ Tδ , ω ∈ Ω, δ-internal additive functional (abbreviated by δ-IAF) if it satisﬁes the following two conditions: (1) For each t ∈ Tδ , At (ω) is non-anticipating with respect to the ﬁltration (δ) (δ) (Ω, {Ft t ∈ Tδ }), i.e., At (·) is Ft -measurable, ∀t ∈ Tδ . (2) For each ω ∈ Ω, we have A(ω, 0) = 0, A(ω, t + s) = A(ω, s) + A(θs(δ) ω, t) for any t, s ∈ Tδ . Proposition 2.6.1. If A(ω, t) is a δ-internal additive functional, then there exists a hyperﬁnite measure μ on S0 (not necessarily positive) such that μ (i) = 0 whenever m(i) = 0 and for all n ∈ ∗ N,f, h ∈ H, S0

h(i)Ei

=

n k=0

n

S0

k=0

f (X (δ) (kδ)) A(ω, (k + 1)δ) − A(ω, kδ) dm(i)

ˆi h(X (δ) (kδ)) dμ(i)δ. f (i)E

(2.6.1)

Proof. Deﬁne μ (0) = 0, 1 μ (i) = Ei A(ω, δ)m(i), 1 ≤ i ≤ N. δ

(2.6.2)

Then, we have S0

h(i)Ei

= = = =

n

n k=0

S0

k=0 n

k=0 S0 n S0

k=0 n k=0

f (X (δ) (kδ)) A(ω, (k + 1)δ) − A(ω, kδ) dm(i)

S0

(δ)

h(i)Ei f (X (δ) (kδ, ω))A(θkδ ω, δ) dm(i) h(i)Ei f (X(kδ))EX(kδ) A(ω, δ) dm(i) f (i)Ei A(ω, δ)Eˆi h(X(kδ)) dm(i) ˆi h(X(kδ)) dμ(i)δ. f (i)E

104

2 Potential Theory

Proposition 2.6.2. Let μ be a hyperﬁnite measure on S0 satisfying μ(i) = 0 whenever m(i) = 0. Then for each δ ∈ T , there exists a δ-IAF A(ω, t) such that (2.6.1) holds. Proof. First of all, let f (s) =

μ(s) m(s) 1(m(s) =0) .

For each u ∈ H, we have

S0

u(s) dμ(s) = =

u(s)f (s) dm(s)

S0 (δ) ˆ (δ) f ). E1 (u, G −1

Deﬁne A(ω, 0) = 0, A(ω, kδ) = δ

k

f (X (δ) (ω, (l − 1)δ))

for k ∈ ∗ N, k ≥ 1.

l=1

It is easy to verify that A(ω, t) is a δ-IAF. Moreover, we have for each i ∈ S0 , 1 1 Ei A(ω, δ)m(i) = Ei f (X (δ) (ω, 0))δm(i) δ δ = f (i)m(i) = μ(i). Therefore, it follows from the proof of Proposition 2.6.1 that (2.6.1) holds. For δ ∈ T , let A(ω, t) be a δ-IAF. Deﬁne 1 E(A(ω, δ))2 2δ 1 = Ei A2δ dm(i). 2δ S0

e(A) =

(2.6.3)

We call e(A) the energy of A. Furthermore, we deﬁne the mutual energy e(A, B) for δ-internal additive functionals A and B by e(A, B) =

1 E A(ω, δ)B(ω, δ) . 2δ

Let ΔA(ω, kδ) be the forward increment of A(ω, t) at time kδ, i.e., ΔA(ω, kδ) = A(ω, (k + 1)δ) − A(ω, kδ) for k ∈ ∗ N. We deﬁne the quadratic variation [A] : Ω × Tδ −→ ∗ R by

2.6 Internal Additive Functionals and Associated Measures

105

[A](ω, 0) = 0, [A](ω, nδ) =

n−1

(ΔA(ω, kδ))

2

for n ∈ ∗ N, n > 0.

k=0

Because [A](ω, (n + m)δ) =

n−1

2

n+m−1

2

m−1

(ΔA(ω, kδ)) +

k=0

= =

n−1

(ΔA(ω, kδ))

2

k=n

(ΔA(ω, kδ)) +

k=0

k=0

n−1

m−1

(ΔA(ω, kδ))2 +

k=0

2

(ΔA(ω, (k + n)δ)) (ΔA(θnδ ω, kδ))2 ,

k=0

[A] is a positive δ-IAF. By Proposition 2.6.1 and its proof, we know that μ (i) = 1δ Ei (A(ω, δ))2 m(i) is the hyperﬁnite positive measure associated with [A] in the sense (2.6.1). We call μ the energy measure of A. It is obviously from the relations (2.6.2) and (2.6.3) that e(A) =

1 μ (S0 ). 2

Let u ∈ H. For δ ∈ T , deﬁne a δ-IAF A[u] (ω, t) by A[u] (ω, t) = u(X (δ) (ω, t)) − u(X (δ) (ω, 0)) for t ∈ Tδ . Then, we have 2 1 [u] E A (ω, δ) 2δ 2 1 E u(X (δ) (ω, δ)) − u(X (δ) (ω, 0)) = 2δ N 2 1 = Ei u(X (δ) (ω, δ)) − u(i) m(i) 2δ i=0

e(A[u] ) =

=

N N 1 (δ) (u(j) − u(i))2 qij m(i) 2δ i=0 j=0

=

N N N 1 1 (δ) 2 (δ) (u(j) − u(i)) qij m(i) + (u(i))2 qi0 m(i) 2δ i=1 j=1 2δ i=1

(2.6.4)

106

2 Potential Theory

=

N N N 1 1 (δ) (δ) (u(i) − u(j)) u(i)qij m(i) + (u(i))2 qi0 m(i) 2δ i=1 j=1 2δ i=1

+

N N N 1 1 (δ) (δ) (u(j) − u(i)) u(j)ˆ qji m(j) + (u(i))2 qˆi0 m(i) 2δ i=1 j=1 2δ i=1

−

N 1 (δ) (u(i))2 qˆi0 m(i) 2δ i=1

= E (δ) (u, u) −

N 1 (δ) (u(i))2 qˆi0 m(i), 2δ i=1

(2.6.5)

where we have used Lemma 1.5.1 (i) and (ii) in the last equation of (2.6.5). Theorem 2.6.1. For u ∈ H, f ∈ H, let μ (·) be the energy measure of A[u] . We have (1)

2 1 [u] Ei A (ω, δ) m(i) δ N 1 (δ) = (u(j) − u(i))2 qij m(i). δ j=0

μ (i) =

(2)

S0

f (s) μ (ds) = 2E (δ) (u, uf ) − E (δ) (u2 , f ).

(2.6.6)

Proof. (1) The proof is quite immediate, and hence we omit it! (2) On the one hand, we have

N N 2 1 (δ) f (s) μ (ds) = f (i) u(j) − u(i) qij m(i). (2.6.7) δ S0 i=1 j=0

On the other hand, we have 2E (δ) (u, f u) − E (δ) (u2 , f ) ⎡ ⎤ N N 2 ⎣ (δ) (uf )(i)u(j)qij m(i)⎦ = (u(i))2 f (i)m(i) − δ i=1 j=1 ⎡ ⎤ N N 1 ⎣ (δ) (u(i))2 f (i)m(i) − (u(j))2 f (i)qij m(i)⎦ − δ i=1 j=1 N

=

N

N

1 1 (δ) (u(i))2 f (i)m(i) − (2u(i) − u(j))u(j)f (i)qij m(i) δ i=1 δ i=1 j=1

2.7 Fukushima’s Decomposition Theorem

⎛

107

⎞

=

N N 1 (δ) f (i)m(i) ⎝(u(i))2 − (2u(i) − u(j))u(j)qij ⎠ δ i=1 j=0

=

1 (δ) f (i)(u(j) − u(i))2 qij m(i). δ i=1 j=0

N

N

By (2.6.7) and (2.6.8), we get (2.6.6).

(2.6.8)

2.7 Fukushima’s Decomposition Theorem As before, we shall work under the conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14) of Sect. 1.5, Chap. 1, in this section.

2.7.1 Decomposition Under the Individual Probability Measures Pi Lemma 2.7.1. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Let ν be a positive measure on S0 of δ-hyperﬁnite energy integral. For any u ∈ H, t ∈ Tδ , and ε > 0, we have

t (δ) Pν ω ∃s ∈ Tδ (|u(X (ω, s))| ≥ ε) ≤ .

where Pν (·) =

S0

12 2C 2 (1 + δ)t/δ (δ) (δ) (δ) (δ) E1 (U1 ν, U1 ν)E1 (u, u) , ε Pi (·)dν(i).

Proof. Let A = {i ∈ S0 | u(i) ≥ ε} . Deﬁne (δ)

σA (ω) = min{t ∈ Tδ | X (δ) (ω, t) ∈ A}. Then, we have

Pν ω ∃s ∈ Tδ , s ≤ t u(X (δ) (ω, s)) ≥ ε (δ) Pi ω (1 + δ)−σA /δ ≥ (1 + δ)−t/δ dν(i) = S 0 (δ) Ei (1 + δ)−σA /δ (1 + δ)t/δ dν(i) ≤ S0

108

2 Potential Theory

= (1 + δ)t/δ

(δ)

e1 (A)(i) dν(i) S0 (δ) (δ) t/δ (δ) = (1 + δ) E1 U1 ν, e1 (A) 12 (δ) (δ) (δ) (δ) (δ) (δ) t/δ E1 U1 ν, U1 ν E1 e1 (A), e1 (A) ≤ C (1 + δ) ≤

t/δ

C 2 (1 + δ) ε

12 (δ) (δ) (δ) (δ) E1 U1 ν, U1 ν E1 (u, u) ,

where the reason for the last step holding is (δ)

E1

|u| |u| (δ) (δ) (δ) e1 (A), e1 (A) ≤ C 2 E1 , ε ε 2 C (δ) ≤ 2 E1 (u, u), ε

by Corollary 2.2.3. Hence, we can prove Lemma 2.7.1 by applying the same argument to −u. Proposition 2.7.1. For δ ∈ T, let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Assume that (E (δ) (·, ·), D(E (δ) )) satisﬁes condition (2.5.14) of Theorem 2.5.6. Let u, {un | n ∈ N} be the elements in H and let δ ∈ T. Suppose that ◦ (δ) E1 (un

− u, un − u) −→ 0

as

n → ∞.

Then there exist a subsequence {unk | k ∈ N} and a δ-exceptional set B such that for all i ∈ S0 − B, t ∈ Tδfin , L(Pi ) ◦unk (X (δ) (ω, s)) converges uniformly to in s on

Tδt

as

◦

u(X (δ) (ω, s)) n −→ ∞ = 1.

Proof. Let {nk | k ∈ N} be a subsequence satisfying ◦ (δ) E1 (unk

− u, unk − u) ≤ 2−4k .

Set t (δ) (δ) −k . Λk (t) = ω ∃s ∈ Tδ |unk (X (ω, s)) − u(X (ω, s))| ≥ 2 For ν ∈ τ00f (δ), we have from Lemma 2.7.1 that

2.7 Fukushima’s Decomposition Theorem

109

◦ 12 t (δ) (δ) (δ) ◦ 2 −k+1 δ E1 (U1 ν, U1 ν) Pν (Λk (t)) ≤ C (1 + δ) 2 .

Hence, we have ∞

L(Pν )(Λk (t)) < ∞.

k=1

By the Borel-Cantelli lemma, we get L(Pν )

∞ ∞

Λl (t)

= 0.

(2.7.1)

k=1 l=k

∞ ∞ Set Λ(t) = k=1 l=k Λl (t). From Theorem 2.5.6, Theorem 2.4.1, and (2.7.1), there exists a δ-exceptional set B(t) such that L(Pi )(Λ(t)) = 0

for any

i ∈ S0 − B(t).

Now let us select subset {tn | n ∈ N} ⊂ Tδ such that tn ≈ n. We ∞ a countable ∞ deﬁne Λ = n=1 Λ(tn ), B = n=1 B(tn ). It is easy to see that Proposition 2.7.1 holds. Lemma 2.7.2. For δ ∈ T, let A be a δ-IAF and let μ (i) be the hyperﬁnite measure deﬁned by (2.6.2) in Sect. 2.6. Then for any v ∈ H, we have E (δ) (ft , v) =

S0

v(i) − Eˆi v(X (δ) (t)) dμ (i)

for all

t ∈ Tδ , (2.7.2)

where ft (i) = Ei A(ω, t). Proof. If k = 1, then we have 1 ˆ δ v(i) fδ (i) dm(i) v(i) − Q E (δ) (fδ , v) = δ S0 ˆi v(X (δ) (δ)) dμ (i). = v(i) − E S0

Assume that (2.7.2) holds whenever k ≤ n. Then, we have E

(δ)

1 ˆ δ v(i) Ei A(ω, (n + 1)δ) dm(i) v(i) − Q (f(n+1)δ , v) = δ S0 1 ˆ δ v(i) Ei A(ω, nδ) + Ei A(θ(δ) ω, δ) dm(i) v(i) − Q = nδ δ S0

110

2 Potential Theory

ˆi v(X (δ) (nδ)) dμ (i) v(i) − E S0 1 ˆi v(X (δ) ((n + 1)δ)) + Eˆi v(X (δ) (nδ)) − E δ S0 ×Ei A(ω, δ) dm(i) ˆi v(X (δ) ((n + 1)δ)) dμ (i). v(i) − E = =

S0

Lemma 2.7.3. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Let A(ω, t) be a positive δ-IAF. For all positive hyperﬁnite measures ν on S0 of δ-hyperﬁnite energy integral, we have ˆ ν||∞ μ (S0 ) Eν (At ) ≤ (1 + t)||U 1 (δ)

for all

t ∈ Tδ .

(2.7.3)

Proof. It follows from Theorem 2.5.1 and Lemma 2.7.2 that (δ) ˆ (δ) ν) Ei A(ω, t) dν(i) = E1 (ft , U Eν A(ω, t) = 1 S 0 ˆ (δ) ν(i) − E ˆi (U ˆ (δ) ν(X (δ) (t))) dμ (i) U = 1 1 S0 ˆ (δ) ν(i) dm(i) + ft (i)U 1 S0 ) ' ˆ (δ) ν||∞ μ (S0 ) + f (i) dm(i) , (2.7.4) ≤ ||U t 1 S0

where we have used Theorem 2.5.3 in the latter inequality above. Notice that f(k+1)δ (i) dm(i) = Ei A(ω, (k + 1)δ) dm(i) S0 S 0 Ei A(ω, kδ) dm(i) + Ei A(θkδ ω, δ) dm(i) = S S 0 0 ˆi 1(X(kδ)∈S ) = E Ei A(ω, kδ) dm(i) + 0 S0

S0

×Ei A(δ) dm(i) ≤ Ei A(ω, kδ) dm(i) + S0

S0

Ei A(ω, δ) dm(i).

2.7 Fukushima’s Decomposition Theorem

111

We can show S0

ft (i) dm(i) ≤ tμ (S0 ).

(2.7.5)

From the relations (2.7.4) and (2.7.5), we obtain the inequality (2.7.3).

∗

Definition 2.7.1. We call an internal process A : Ω × Tδ → R a martingale (δ) (δ) with respect to (Ω, Ft , Pi , i ∈ S0 ) if ω → A(ω, t) is Ft measurable for all (δ) t ∈ Tδ , and for all s, t ∈ Tδ , s < t, and all B ∈ Fs , Ei (1B (At − As )) = 0. (δ)

It is easy to see that if [ω]t is the equivalence class of ω deﬁned by (1.5.35) in Sect. 1.5 of Chap. 1, then a non-anticipating process A(ω, t) is a martingale if and only if

ΔA(˜ ω , t)Pi {˜ ω } = 0.

(2.7.6)

(δ)

ω ˜ ∈[ω]t

In the following, we shall use “δ-quasi-everywhere” or “δ-q.e.” to mean “except for a δ-exceptional set”. A statement depending on i ∈ S0 is said to be “δ-quasi-everywhere” or “δ-q.e.” if there exists a set B ⊂ S0 of δ-exceptional such that the statement is true for every i ∈ S0 − B. Let us introduce the following set of internal martingale additive functionals (abbreviated by δ-IMAF) by M(δ) = M | M is a δ-IAF, ◦ (Ei M 2 (t)) < ∞, δ-q.e.i ∈ S0 for all ﬁnite t ∈ Tδ and Ei M (t) = 0, ∀t ∈ Tδ , ∀i ∈ S0 } . For M ∈ M(δ) , we have for all s, t ∈ Tδ (δ)

(δ)

(δ)

Ei (Ms+t | Ft ) = Ei (Mt + Ms (θt ) | Ft ) = Mt + EX(t) Ms = Mt , for all i ∈ S0 . Hence, M ∈ M(δ) is a martingale with respect to Pi for all i ∈ S0 . Meantime, M ∈ M(δ) is also square integrable with respect to Pi for δ-q.e. i ∈ S0 . In the following, let ˚ (δ) = {M ∈ M(δ) | ◦e(M ) < ∞}. M ˚ (δ) is called a δ-internal martingale additive functional of Any element in M ﬁnite energy.

112

2 Potential Theory

Proposition 2.7.2. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Let {An | n ∈ N} be δ-internal additive functionals. Assume that ◦e(An − Am ) → 0 as n, m → ∞, and for (δ) each i ∈ S0 , {Ω, Ft , An (t), Pi } is a martingale for all n ≥ 1. Then there ˚ (δ) with inner product ◦e(·, ·) such that ◦e(An − A) → 0 exists a unique A ∈ M as n → ∞. Moreover, there exist a subsequence {Ank (ω, t) | k ∈ N} and a δ-exceptional set B such that for all i ∈ S0 − B, t ∈ Tδfin,

L(Pi ) ω ◦Ank (ω, s) −→ ◦A(ω, s)

uniformly on

Tδt

= 1.

Proof. By the equality (2.6.4) in Sect. 2.6, we know that ◦

μ (S0 ) =

◦

2e(An − Am )

−→ 0 as n, m → ∞. Hence, there exists a subsequence {nk | k ∈ N} such that ◦μ (S0 ) < 2−3k . Since An , n ≥ 1 are martingales with respect to Pi , i ∈ S0 , we get from Lemma 2.7.3 that for all ν ∈ τˆ00 (δ), t ∈ Tδfin : ◦

2 ◦ / 0 Eν Ank+1 (t) − Ank (t) = Eν ([Ank+1 − Ank ](t)) ≤

◦

ˆ (δ) ν||∞ 2−3k . (1 + t)||U 1

(2.7.7)

From the relation (2.7.7) and Doob’s inequality (for which we refer to [25] 4.2.8), we get ◦'

) Pν max |Ank+1 (s) − Ank (s)| ≥ 2−k s≤t 1 2

◦ 22k Eν

≤ ≤ ≤

2

max |Ank+1 (s) − Ank (s)| s≤t

◦

2 22k+2 Eν Ank+1 (t) − Ank (t)

◦

ˆ (δ) ν||∞ 2−k . 4(1 + t)||U 1

(2.7.8)

It is easy to see from the inequality (2.7.8), by using Borel-Cantelli lemma, that L(Pν )(Λ) = 1, for all ν ∈ τˆ00 ,

2.7 Fukushima’s Decomposition Theorem

113

where Λ = {ω | ◦Ank+1 (ω, ·) converges uniformly on each ﬁnite interval Tδt }. By Theorem 2.5.5 and Theorem 2.4.1, we have L(Pi )(Λ) = 1, δ-q.e.i ∈ S0 . Let {Ank | k ∈ ∗ N} be an internal extension of {Ank | k ∈ N}. Deﬁne A(ω, t) = AnK (ω, t) for some K ∈ ∗ N − N. Then, A(ω, t) is a Pi -martingale δ-IAF for all i ∈ S0 . Since An (t) converges in L2 (Pν ), we have (◦ Eν A2 (t)) < ∞, ∀t ∈ Tδfin . Therefore, we have shown A ∈ M(δ) . For ε > 0, choose N such that e(An −Am ) < ε, n, m > N. By Fatou’s lemma, we have e(An −A) ≤ ε, n > N. ˚ (δ) and ◦e(An − A) −→ 0 as n −→ ∞. Thus, we have A ∈ M Definition 2.7.2. (1) We call an internal function u ∈ H S-bounded if there is a positive constant C ∈ [0, ∞) such that |u(i)| ≤ C for all i ∈ S0 . (2) An internal function f : Tδ −→ ∗ R is called S-continuous if f (s) ≈ f (t) whenever s ≈ t and s and t are nearstandard. Lemma 2.7.4. For δ ∈ T, u ∈ Fin(H), deﬁne a δ-IAF Aδ by Aδ (ω, 0) = 0, Aδ (ω, nδ) = δ

n

u(X(ω, (k − 1)δ)), n ∈ N, n ≥ 1.

k=1

Then, there exists a properly δ-exceptional set B ⊂ S0 such that for all i ∈ S0 − B, δ L(Pi ) ω A (ω, ·) is S-continuous = 1. Proof. First of all, we assume that u is S-bounded. We have for any ω ∈ Ω,

|A (ω, t) − A (ω, s)| ≤ |t − s| max |u(i)| + 1 . δ

δ

i∈S0

This implies that Aδ (ω, t) is S-continuous. Next, we suppose that u ∈ Fin(H). For each n, set Bn = {s ∈ S0 | |u(s)| ≥ n} . In addition, we deﬁne σBn (ω) = min {t ∈ Tδ | X(t) ∈ Bn } . Then, we have

1 P ω ∃t ∈ Tδ (X(ω, t) ∈ Bn )

= P ω σBn (ω) ≤ 1

114

2 Potential Theory

≤ n−2 ≤ n−2

S0

S0

Ei [u(X(σBn ))]2 dm(i) [u(i)]2 dm(i).

Thus, we get ∞ 1 Bn ) = 0. L(P ) ω ∃t ∈ Tδ (X(ω, t) ∈ n=1

This implies that {s ∈ S0 | ◦|u(s)| = ∞} is a δ-exceptional set. Let B be a properly δ-exceptional set containing {s ∈ S0 | ◦|u(s)| = ∞} (Proposition 2.1.1). For each n ∈ N, deﬁne σn (ω) = min {t ∈ Tδ | |u(X(ω, t))| ≥ n} . Then for each ω ∈ Ω, Aδ (ω, ·) is continuous in [0, σn (ω)) for every n. Moreover, for each i ∈ S0 − B, we have L(Pi ) {ω | ◦σn (ω) ↑ ∞ as n → ∞} = 1,

which proves Lemma 2.7.4. Set (δ)

NC = {N | L(Pi ){ω | N (ω, ·) is an S-continuous δ-IAF} = 1, q.e.i ∈ S0 and e(N ) ≈ 0} . (δ)

For N ∈ NC , the variation vanishes in the following sense E([N ](t)) ≈ 0 for all t ∈ Tδfin. In fact, we have for all ﬁnite t ∈ Tδ , E([N ](t)) =

0<s 0, we have

2C 2 (1 + δ)t/δ (δ) P ω ∃s ∈ Tδt (|u(X (δ) (ω, s))| ≥ ε) ≤ E1 (u, u). ε2 Proof. Let A = {i ∈ S0 | u(i) ≥ ε}, and (δ)

σA (ω) = min{t ∈ Tδ | X (δ) (t) ∈ A}. Then, we have P

ω ∃s ∈ Tδ , s ≤ t u(X (δ) (ω, s)) ≥ ε (δ) Pi ω (1 + δ)−σA /δ ≥ (1 + δ)−t/δ dm(i) = S 0 (δ) Ei (1 + δ)−σA /δ (1 + δ)t/δ dm(i) ≤ S0 (δ) t/δ e1 (A)(i) dm(i) = (1 + δ) S0 (δ) (δ) t/δ (δ) ≤ (1 + δ) E1 e1 (A), e1 (A) ,

2.7 Fukushima’s Decomposition Theorem

119

where the reason for the last step holding is (2.3.1) in Sect. 2.3. Since E (δ) (·, ·) has the Markov property, we get from Corollary 2.2.3 that (δ) E1

|u| |u| (δ) (δ) 2 (δ) e1 (A), e1 (A) ≤ C E1 , ε ε C 2 (δ) ≤ 2 E1 (u, u). ε

Hence, we have

P

C 2 (1 + δ)t/δ (δ) ω ∃s ∈ Tδt (u(X (δ) (ω, s)) ≥ ε) ≤ E1 (u, u). ε2

Applying the same argument to −u, the lemma follows.

Corollary 2.7.1. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Let u, un , n ∈ N be elements in H, and assume that ◦ (δ) E1 (u

− un , u − un ) −→ 0 as n −→ ∞.

There is a subsequence {unk } such that for almost every ω, ◦unk (X (δ) (ω, ·)) converges uniformly to ◦u(X (δ) (ω, ·)) on all S-bounded subsets of Tδ . Proof. The result follows from Lemma 2.7.5 and basic measure theory. (δ) , P ) is called Definition 2.7.3. A martingale M with respect to (Ω, Ft a λ2 -martingale if ◦E(Mt2 ) < ∞ for all t ∈ Tδfin . Theorem 2.7.2. For inﬁnitesimal δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. For any u ∈ D(E (δ) ), there are two δ-internal additive functionals M [u] (ω, t) and N [u] (ω, t) such that (i) A[u] (ω, t) = M [u] (ω, t) + N [u] (ω, t). (δ) , P ). (ii) M [u] is a λ2 -martingale with respect to (Ω, Ft (iii) N [u] (ω, ·) is S-continuous for almost all paths ω ∈ Ω under L(P ), and E[N [u] ](t) ≈ 0 for all t ∈ Tδfin. Proof. The proof of this result is similar to the one of Theorem 2.7.1. We also refer the reader to the detailed proof of the corresponding result in the symmetric case given in Albeverio et al. [25].

120

2 Potential Theory

2.8 Internal Multiplicative Functionals 2.8.1 Internal multiplicative functionals Definition 2.8.1. For δ ∈ T, an internal function M (ω, t), t ∈ Tδ , ω ∈ Ω, is said to be a δ-internal multiplicative functional (abbreviated by δ-IMF) of X (δ) (ω, t) if and only if (δ)

(i) For each t ∈ Tδ , Mt (·) is Ft -measurable. (ii) For each t ∈ Tδ , ω ∈ Ω, we have M (ω, t) ∈ ∗ [0, 1]. (iii) For each ω ∈ Ω, we have M (ω, 0) = 1, M (ω, t + s) = M (ω, s)M (θs(δ) ω, t), ∀t, s ∈ Tδ . Remark 2.8.1. Let A(ω, t) be a nonnegative δ-internal additive functional. We can deﬁne a δ-IMF M (ω, t) by M (ω, t) = exp (−A(ω, t))

for all ω ∈ Ω, t ∈ Tδ .

If M (ω, t) is a δ-internal multiplicative functional, let us deﬁne a family of operators {P t | t ∈ Tδ } on H by P t f (i) = Ei f (X (δ) (ω, t))M (ω, t) . Then, we have for all t, s ∈ Tδ , i ∈ S, P t+s f (i) = Ei f (X (δ) (ω, t + s))M (ω, t + s) = Ei f (X (δ) (θs(δ) ω, t))M (ω, s)M (θs(δ) ω, t) = Ei M (ω, s)EX (δ) (ω,s) [f (X (δ) (t))Mt ] = P s P t f (i).

(2.8.1)

Hence, {P t | t ∈ Tδ } is a semigroup. We call it the semigroup generated by (X (δ) , M ). Moreover, we have for all f ∈ H, P 0 f (i) = Ei [f (X (δ) (ω, 0))M (ω, 0)] = f (i). In particular, we have P 0 1 = 1.

(2.8.2)

2.8 Internal Multiplicative Functionals

121

(δ)

Since M (ω, δ) is Fδ -measurable, we have M (ω, δ) =

N

1[ˆω](ij) (ω)Mij ,

i,j=0

where [ˆ ω ](ij) = {ω | ω(0) = i and ω(δ) = j} and {Mij | i, j = 0, 1, 2, . . . , N } is a family of positive hyperreals. Moreover, Mij ∈ ∗ [0, 1]. Therefore, the (δ) transition matrix {pij | i, j = 0, 1, 2, . . . , N } of {P t | t ∈ Tδ } is given by (δ)

(δ)

pij = qij Mij ,

Mij ∈ ∗ [0, 1],i, j = 0, 1, 2, . . . , N.

(2.8.3)

From the relation (2.8.3), we know that for all nonnegative internal functions f ∈ H, P t f (i) ≤ Qt f (i), ∀i ∈ S, t ∈ Tδ .

(2.8.4)

2.8.2 Subordinate Semigroups Definition 2.8.2. A semigroup {P t | t ∈ Tδ } of positive linear operators from H to H is said to be subordinate to {Qt | t ∈ Tδ } if and only if P t f (i) ≤ Qt f (i) for all t ∈ Tδ , f ∈ H, f (i) ≥ 0, i ∈ S and P 0 = I. Theorem 2.8.1. Let {P t | t ∈ Tδ } be a semigroup on H. Then the following two conditions are equivalent: (1) {P t | t ∈ Tδ } is subordinate to {Qt | t ∈ Tδ }. (2) There exists a δ-IMF M (ω, t) of X (δ) (ω, t) generating {P t | t ∈ Tδ }. Proof. (2) =⇒ (1). It follows from the relations (2.8.2) and (2.8.4). (δ)

(1) =⇒ (2). Let {pij | i, j = 0, 1, 2, . . . , N } be the transition matrix of the semigroup {P t | t ∈ Tδ }. Then for each f ∈ H, we have Qδ f (i) =

N j=1

P δ f (i) =

N j=1

(δ)

qij f (j), (δ)

pij f (j).

Since {P t | t ∈ Tδ } is subordinate to {Qt | t ∈ Tδ }, we see that (δ)

(δ)

pij ≤ qij , ∀i, j = 0, 1, 2, . . . , N.

122

2 Potential Theory

Deﬁne ˆ (i, j) = M where we set

a 0

(δ)

pij

1

(δ)

(δ) (qij =0)

qij

,

= 0, a ∈ ∗ [0, ∞). Put M (ω, 0) = 1,

ˆ X (δ) (ω, 0), X (δ) (ω, δ) . M (ω, δ) = M

(2.8.5)

For all i ∈ S, we have for any f ∈ H, Ei [f (X (δ) (ω, δ))M (ω, δ)] =

N j=1

ˆ (i, j)q f (j)M ij

(δ)

= P δ f (i). By using mathematical induction, we deﬁne (δ)

M (ω, (k + 1)δ) = M (ω, kδ)M (θkδ ω, δ)

for all k ∈ ∗ N.

It is then easy to show that M (ω, t) is a δ-IMF generating {P t | t ∈ Tδ }.

2.8.3 Subprocesses In Sect. 1.5 of Chap. 1, we deﬁned a hyperﬁnite Markov chain X (δ) (ω, t) associated with the hyperﬁnite Dirichlet form E (δ) (·, ·). Let us denote (δ)

(δ)

Ω, F t , Y (δ) (ω, t), θt , P i

t∈Tδ

a hyperﬁnite Markov chain with state

(δ) (δ) space (S, S). We call Y (ω, t) a subprocess of X (ω, t) if and only if the semigroup P t | t ∈ Tδ of Y (δ) (ω, t) is subordinate to {Qt | t ∈ Tδ } .

Let Y (δ) (ω, t) be a subprocess of X (δ) (ω, t). From Theorem 2.8.1, there exists a δ-IMF M (ω, t) of X (δ) (ω, t) such that EP i f (Y (δ) (ω, t)) = Ei [f (X (δ) (ω, t))M (ω, t)] for all t ∈ Tδ ,

(2.8.6)

where EP i (·) is the expectation operator corresponding to Y (δ) (ω, t). We are interested in the following question. Given a δ-IMF M (ω, t), could we construct a subprocess Y (δ) (ω, t) such that the relation (2.8.6) holds? (δ) (δ) The answer is yes! In fact, let Ω = Ω, F t = Ft , Y (δ) (ω, t) = X (δ) (ω, t),

2.8 Internal Multiplicative Functionals (δ)

123

(δ)

θt = θt . Furthermore, let {P t | t ∈ Tδ } be the semigroup given by the relation (2.8.1), and let {p(δ) | i, j = 0, 1, 2, . . .} be the transition matrix of ij {P t | t ∈ Tδ }. Deﬁne for ω ∈ Ω = Ω, k ∈ ∗ N, P i ([ω]kδ ) = δiω(0)

k−1

p ω(nδ), ω((n + 1)δ) .

n=0

It is obvious that the relation (2.8.6) holds with respect to Y (δ) (ω, t). We call Y (δ) (ω, t) the canonical subprocess associated with (X (δ) , M ). We notice | i, j = 0, 1, 2, . . . , N } need not to have the regularities (1.5.1) and that {p(δ) ij (1.5.2) in Sect. 1.5 of Chap. 1.

2.8.4 Feynman-Kac Formulae (δ)

Let {qi

| i = 1, 2, . . . , N } be a 1 × N matrix satisfying (δ)

0 ≤ qi

(δ)

≤ qii , i = 1, 2, . . . , N. (δ)

Deﬁne a transition matrix P (δ) = {pij | i, j = 0, 1, 2, . . . , N } by (δ)

(δ)

(δ)

pij = qij − qi δij (δ) pi0 (δ) p00

= =

(δ) (δ) qi0 + qi (δ) 1, pi0 = 0

for for

i, j = 1, 2, . . . , N,

i = 1, 2, . . . , N, and

for i = 1, 2, . . . , N.

(δ) Similarly, deﬁne the transition matrix Pˆ (δ) = {ˆ pij | i, j = 0, 1, 2, . . . , N } by (δ)

(δ)

(δ)

pˆij = qˆij − qi δij (δ) pˆi0 (δ) pˆ00

= =

(δ) (δ) qˆi0 + qi (δ) 1, pˆi0 = 0

for for

i, j = 1, 2, . . . , N,

i = 1, 2, . . . , N and

for i = 1, 2, . . . , N. (δ)

Let {P t | t ∈ Tδ } be the semigroup with P (δ) = {pij | i, j = 0, 1, 2, . . . , N } as its transition matrix, and let {Pˆ t | t ∈ Tδ } be the semigroup with Pˆ (δ) = (δ) {ˆ pij | i, j = 0, 1, 2, . . . , N } as its transition matrix. Then, we have

124

2 Potential Theory

Theorem 2.8.2. We have (i) The semigroups {P t | t ∈ Tδ } and {Pˆ t | t ∈ Tδ } are dual with respect to m. (ii) The Dirichlet form associated with P (δ) and m is given by E

(δ)

(u, v) = E

(δ)

(u, v) +

N i=1

(δ)

u(i)v(i)qi m(i).

(iii) There exists a δ-IMF M (ω, t) such that for any f ∈ H, i ∈ S, t ∈ Tδ , P t f (i) = Ei [f (X (δ) (ω, t))M (ω, t)]. Proof. (i) and (ii) are obvious. (iii) follows easily from Theorem 2.8.1.

2.9 Alternative Expression of Hyperfinite Dirichlet Forms Suppose that μ(·) is a positive internal measure on S. In this section, we shall ﬁnd conditions on μ and μ-dual transition matrices P = {pij | i, j = 0, 1, . . . , N } and Pˆ = {ˆ pij | i, j = 0, 1, 2, . . . , N } with the regularities (1.5.1) and (1.5.2) in Sect. 1.5 such that E(·, ·) is the hyperﬁnite Dirichlet form associated with m and Q. First we observe that if μ and P and Pˆ have been found, then for all u, v ∈ H, we have ⎡ ⎤ N N 1 ⎣u(i)v(i)μ(i) − E(u, v) = u(j)v(i)pij μ(i)⎦ Δt i=1 j=1

(2.9.1)

and ⎡ ⎤ N N 1 ⎣ E(u, v) = u(i)v(i)μ(i) − u(j)v(i)ˆ pji μ(j)⎦ . Δt i=1 j=1

(2.9.2)

Therefore, we have from the expression (1.5.19) in Sect. 1.5 and the expression (2.9.1) that m(i)[1 − qii ] = μ(i)[1 − pii ] m(i)qij = μ(i)pij

i = 1, 2, . . . , N,

(2.9.3)

for all i, j ∈ {1, 2, . . . , N } , i = j.

(2.9.4)

for all

2.10 Transformations of Symmetric Dirichlet Forms

125

Similarly, we have m(i)[1 − qˆii ] = μ(i)[1 − pˆii ] for all i = 1, 2, . . . , N, pij for all i, j ∈ {1, 2, . . . , N } , i = j. m(i)ˆ qij = μ(i)ˆ

(2.9.5) (2.9.6)

On the other hand, if the pairs of (P, μ) and (Q, m) satisfy the conditions ˆ m) satisfy the conditions (2.9.3) and (2.9.4) (or the pairs of (Pˆ , μ) and (Q, (2.9.5) and (2.9.6)), then the expression (2.9.1) (or (2.9.2)) holds also. Hence, we get Theorem 2.9.1. Let μ(·) be a positive internal measure on S, and let P and Pˆ be μ-dual transition matrices. Then the conditions (2.9.3), (2.9.4), (2.9.5), and (2.9.6) are suﬃcient and necessary conditions such that the expression (1.5.19) in Sect. 1.5 and the expressions (2.9.1) and (2.9.2) hold.

2.10 Transformations of Symmetric Dirichlet Forms In this section, we assume that q ii = 0 for all i = 1, 2, . . . , N.

(2.10.1)

In fact, this assumption will not aﬀect our theory. The reason comes from the proof of Proposition 1.5.1. Actually, for the general hyperﬁnite quadratic form E(·, ·) of the expression (1.5.19) in Sect. 1.5, we deﬁne m(i) ˜ = (1 − qii )m(i), q˜ii = 0

for all

i = 1, 2, . . . , N.

Moreover, if i = 1, 2, . . . , N and qii < 1, deﬁne q˜ij =

qij 1 − qii

for j = i, j ∈ {0, 1, 2, . . . , N } ;

if i = 1, 2, . . . , N and qii = 1, deﬁne q˜ij = 0

for j = 0, i and q˜i0 = 1.

Besides, let q˜00 = 1, q˜0j = 0, j = 1, 2, . . . , N. It is very easy to verify from the Beurling–Deny formulae, Lemma 1.5.1 that the hyperﬁnite quadratic form ˜ = {˜ associated with m ˜ and Q qij | i, j = 0, 1, 2, . . . , N } is E(·, ·). Since qii m(i) = qˆii m(i), ∀i = 1, 2, . . . , N,

126

2 Potential Theory

we may suppose qˆii = 0, ∀i = 1, 2, . . . , N . Hence, we may assume 1 (qii + qˆii ) 2 = 0, i = 1, 2, . . . , N.

q ii =

Let Φ be an internal nonnegative function in H. We deﬁne the following quadratic form Φ

E (u, v) =

1 Δt

u(i) − u(j) v(i) − v(j) Φ(i)Φ(j)q ij m(i).

1≤i<j≤N

(2.10.2) Φ

Φ

It is easy to see that E (·, ·) satisﬁes Proposition 1.5.1 (iii). Moreover, E (·, ·) is symmetric. Thus by Proposition 1.5.1, there exists a transition matrix P = {pij | 0 ≤ i, j ≤ N } and a symmetric measure m(·) such that

Φ

1 u(i) v(i) − P Δt v(i) dm(i). Δt S0

E (u, v) =

(2.10.3)

In the following, we will ﬁnd the P and m. From the expression (2.10.3) and the Beurling–Deny formulae (Lemma 1.5.1), we have Φ

E (u, v) =

1 Δt

1≤i<j≤N

+

N i=1

u(i) − u(j) v(i) − v(j) pij m(i)

u(i)v(i)pi0 m(i) .

(2.10.4)

Comparing the expressions (2.10.2) and (2.10.4), we must have Φ(i)Φ(j)q ij m(i) = pij m(i) pi0 m(i) = 0 for all

for all

1 ≤ i, j ≤ N,

(2.10.5)

i = 1, 2, . . . , N.

Therefore, we get m(i) =

N j=0

=

j =i

pij m(i) Φ(j)q ij Φ(i)m(i) + pii m(i)

= E i [Φ(X(Δt))]Φ(i)m(i) + pii m(i).

(2.10.6)

2.10 Transformations of Symmetric Dirichlet Forms

127

Deﬁne p00 = 1, p0i = 0 for pii = 0 for

i = 1, 2, . . . , N, i = 1, 2, . . . , N.

Then from (2.10.6), we obtain m(i) = E i [Φ(X(Δt))]Φ(i)m(i). For i = 1, 2, . . . , N, if E i Φ(X(Δt)) = 0, we see from the relation (2.10.5) that for all j = 1, 2, . . . , N, j = i, q ij Φ(j)

pij =

E i [Φ(X(Δt))] q ij Φ(j) = . l =i q il Φ(l)

(2.10.7)

For i = 1, 2, . . . , N , if E i [Φ(X(Δt))] = 0, we can deﬁne pij , 1 ≤ j ≤ N, j = i arbitrarily such that N j=1

pij = 1.

(2.10.8)

From (2.10.7) and (2.10.8), we get pi0 = 0 for all i = 0. In the following discussion, we suppose that E i [Φ(X(Δt))] > 0

for all

i = 1, 2, . . . , N.

(2.10.9)

Hence, we have from (2.10.7) that for all f ∈ H, P f (i) =

E i [(f Φ)(X(Δt))] . E i [Φ(X(Δt))]

Let {P t | t ∈ T } be the semigroup generated by P . Then it is easy to see that {P t | t ∈ T } is subordinate to {Qt | t ∈ T }. By using Theorem 2.8.1, there exists a Δt-IMF M (ω, t) of X(ω, t) generating {P t | t ∈ T }. From the relation (2.8.5) in Sect. 2.8, we know that M (ω, Δt) =

Φ(X(ω, Δt)) . E X(ω,0) [Φ(X((Δt))]

128

2 Potential Theory

Therefore, we have for all k ∈ ∗ N, M (ω, (k + 1)Δt) =

k

Φ(ω, (l + 1)Δt) . E X(ω,lΔt) [Φ(X(Δt))] l=0

(2.10.10)

On the other hand, let Φ be an internal function in H satisfying the hypothesis (2.10.9). We deﬁne M (ω, t) directly by the relation (2.10.10). Then, we have the following: Theorem 2.10.1. Assume that the hypotheses (2.10.1) and (2.10.9) hold. Then Φ

(1) The semigroup {(Q )t | t ∈ T } generated by M (ω, t) is symmetric with respect to the measure E i [Φ(X(Δt))]Φ(i)m(i) = m(i). Φ Φ (2) E (·, ·) is the hyperﬁnite Dirichlet form associated with {(Q )t | t ∈ T } and m(·). Proof. The proof is straightforward.

Chapter 3

Standard Representation Theory

The purpose of this chapter is to study the standard projection of hyperﬁnite Markov chains, and the standard projection of hyperﬁnite Markov chains associated with hyperﬁnite quadratic forms. At ﬁrst, we shall introduce in Sect. 3.1 a concept of irregularity; and then we shall prove that if a hyperﬁnite Markov chain X(·, t) has a set of irregularities, its standard part x(·, t) is a strong Markov process. In Sect. 3.2, we shall ﬁnd conditions on hyperﬁnite quadratic forms which guarantee that the modiﬁed standard parts of associated hyperﬁnite Markov chains are strong Markov processes. The old version of the materials of this chapter can be found in Chap. 5, Albeverio et al. [25]. However, Chap. 5 of Albeverio et al. [25] only handles symmetric hyperﬁnite Dirichlet form. Moreover, many restrictions in Albeverio et al. [25] are not necessary. For instance, we simplify the deﬁnition of exceptional sets in Chap. 2 and remove the unnecessary assumptions. Using the updated version of exceptional sets, we will deﬁne a concept of exceptional irregularities in Sect. 3.1. We will show that a hyperﬁnite Markov chain with exceptional irregularities has a standard part, which is a strong Markov process. In Sect. 3.2, we shall impose conditions directly on hyperﬁnite quadratic forms to achieve our goal. Again, many assumptions of Albeverio et al. [25] will be simpliﬁed. For instance, the separation of compacts of Albeverio et al. [25] will be replaced by separation of points; all concepts involving exceptional sets will be updated using our deﬁnition in Chap. 2. It is our hope that the materials of this chapter update and signiﬁcantly improve the related materials of Chap. 5, Albeverio et al. [25]. One may want to notice that we will impose conditions directly on a standard Dirichlet form in Chap. 4 to get related strong Markov processes. The materials of Chaps. 1, 2, and 3 can be thought as the theory of hyperﬁnite Dirichlet forms and hyperﬁnite Markov chains. With the solid work of these three chapters, we will be able to attack the important construction of strong Markov processes associated with standard Dirichlet forms in Chap. 4.

S. Albeverio et al., Hyperfinite Dirichlet Forms and Stochastic Processes, Lecture Notes of the Unione Matematica Italiana 10, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-19659-1_3,

129

130

3 Standard Representation Theory

3.1 Standard Parts of Hyperfinite Markov Chains In this section, we shall study standard parts of hyperﬁnite Markov chains. In order to take standard parts we need a topology. We shall assume that with the exception of the trap s0 , the state space S is embedded in the nonstandard version ∗ Y of some Hausdorﬀ space Y . If X is a hyperﬁnite Markov chain taking values in S, we want to ﬁnd conditions that guarantee that the standard part of X exists and is a Y -valued Markov process. It turns out that there are two diﬃculties we shall have to overcome. The ﬁrst is that the paths of X may be so irregular that no natural standard part process exists. The second is that even when a standard part does exist, there is no reason why it should automatically be a Markov process – taking standard parts we may lump together states that should be kept apart. By using the theory of right standard parts developed by Albeverio et al. [25], Chap. 4, we can solve the ﬁrst of these problems. Thus, most of our work will be directed to the second problem under discussion. Before we delve into the technicalities, we shall discuss the problem informally in some more details. Assume that x : Ω × R+ −→ Y is the standard part of X. We would like to prove that x is a Markov process with respect to the ﬁltration it generates. Given that x(t) = y, in general there will be several states s ∈ S0 such that y = st(s), and X may be in any one of them. From the nonstandard point of view, these states are totally unrelated, and hence the past and the future of the process may diﬀer widely from one state to the next. Observation of the whole past may indicate which states are more likely to occur, and thus inﬂuence our prediction of the future. This explains why in general x is not a Markov process. Note, however, that if the process started at si and the process started at sj have the “same” future whenever si ≈ sj , the above argument breaks down, and it is reasonable to expect that x is Markov. One way of formulating this condition is to demand that L(Pi ){ω | x(ω, t) ∈ B} = L(Pj ){ω | x(ω, t) ∈ B}

(3.1.1)

for all t ∈ R+ and all Borel sets B. But the above relation turns out to be too strict for the applications. Instead of demanding that it holds for all inﬁnitely close si , sj , we shall only demand that it holds for all such si , sj outside an exceptional set (i.e., a set which the process hits with probability zero). It may at ﬁrst seem that little is achieved by allowing a condition to fail on an exceptional set, but in fact the extra freedom and ﬂexibility we gain will turn out to be very useful. The present section falls into two halves. In the ﬁrst part, we develop the necessary theory for the inner standard part of sets and a general theory of

3.1 Standard Parts of Hyperfinite Markov Chains

131

exceptional sets. In the second part, we shall show that the standard part of hyperﬁnite Markov chain is a strong Markov process under some conditions.

3.1.1 Inner Standard Part of Sets By a hyperﬁnite Markov chain X in this section we shall understand a stationary Markov chain as described in Sect. 1.5, (1.5.1)–(1.5.7). If X has a ˆ as introduced in Sect. 1.5, we have for all dual hyperﬁnite Markov chain X ∗ k∈ N P {ω | X(ω, kΔt) = si } =

N

P {ω | X(ω, 0) = sj , X(ω, kΔt) = si }

j=0

=

N j=0

=

N j=0

(kΔt)

qji

(kΔt)

qˆij

m(j)

m(i)

= m(i). However, we shall NOT assume the duality of X in the following discussion, i.e., we do not need condition (1.5.13) in Sect. 1.5. We shall instead assume that for each i ∈ S0 , the function t → P {ω | X(ω, t) = si } is decreasing. We shall further assume that S0 ⊂ ∗ Y for some topological space Y , but that the trap s0 is not an element of ∗ Y . Similarly as in Sect. 2.1, we deﬁne the exceptional sets and properly exceptional sets of X using Deﬁnition 2.1.1 and Deﬁnition 2.1.2. Furthermore, Lemma 2.1.1 and Proposition 2.1.1 hold for X. Most of the exceptional sets we encounter in the theory of Markov processes are sets we would like to avoid. From the standard point of view, this means that a point y in Y should be avoided if all its nonstandard representations si ∈ st−1 (y) ∩ S0 are in the exceptional set. To study this relationship closely, we deﬁne the inner standard part A◦ of a subset A of S by A◦ = y ∈ Y | st−1 (y) ∩ S0 ⊂ A .

132

3 Standard Representation Theory

The standard part of a set A is deﬁned by ◦

A = st(A) = {y ∈ Y | ∃s ∈ A(st(s) = y)} .

The inner standard part can also be deﬁned in terms of the standard part operation: A◦ = C(st(CA)),

(3.1.2)

where the outer complement is with respect to Y , and the inner complement is with respect to S0 . It is trivial to check that standard parts commute with arbitrary unions. Using the operation (3.1.2), we get that inner standard parts commute with arbitrary intersections. The other way around is less nice, the standard parts and intersections do not commute, and neither do inner standard parts and unions. All we can say is the following: Lemma 3.1.1. (1) If {An | n ∈ N} is a decreasing family of internal sets, then we have ◦ ◦ An = An . (3.1.3) n∈N

n∈N

(2) If {Bn | n ∈ N} is an increasing family of internal sets, then we have

◦ Bn

=

n∈N

Bn◦ .

(3.1.4)

n∈N

Proof. Obviously, the left hand of the property (3.1.3) is contained in the set on the right. To prove the converse, let x∈

◦

An .

n∈N

For each n ∈ N, pick yn ∈ An such that x = st(yn ). Extend {An | n ∈ N} to be a decreasing internal family {An | n ∈ ∗ N} and {yn | n ∈ N} to be an internal sequence {yn | n ∈ ∗ N} such that yn ∈ An for all n ∈ ∗ N. For each neighborhood O of x, consider the set NO = {n ∈ ∗ N | yn ∈ ∗ O}.

3.1 Standard Parts of Hyperfinite Markov Chains

133

All these sets are internal and contain N. Hence, we can ﬁnd an inﬁnite integer η that is in all of them. But then, we must have x = st(yη ). Since the family {An | n ∈ ∗ N} is decreasing, we have yη ∈ Aη ⊂ ∩n∈N An . This proves the property (3.1.3). We now get the property (3.1.4) from the property (3.1.3) by using the operation (3.1.2):

◦

= C st C

Bn

n∈N

= C st =C =

Bn

n∈N

(CBn )

n∈N

(st (CBn ))

n∈N

C (st(CBn ))

n∈N

=

Bn◦ .

n∈N

Since in general

◦ Bm,n

=

m∈N n∈N

◦ Bm,n ,

m∈N n∈N

there is no obvious reason to believe that the inner standard part of a properly exceptional set is always Borel. However, we shall now prove that it must at least be universally measurable. Lemma 3.1.2. Assume A = ∪m∈N ∩n∈N Bm,n for a family {B}m,n∈N of internal sets. For any completed Borel probability measure μ on Y , the inner standard part A◦ is μ measurable. Moreover, there exists a family {Dm,n | m, n ∈ N} of internal sets such that A⊂

n∈N m∈N

Dm,n

134

3 Standard Representation Theory

and

μ

◦ Dm,n

◦

−A

= 0.

n∈N m∈N

Proof. We may assume that the family {Bm,n } is increasing in m and decreasing in n. Note that A=

Bm,n

m∈N n∈N

=

Bm,f (m) .

f ∈NN m∈N

Let f¯(m) be the sequence f (0), f (1), · · · , f (m), and deﬁne Cf¯(m) =

Bk,f (k) .

k≤m

For ﬁxed f , the sequence {Cf¯(m) }m∈N is increasing. Hence by Lemma 3.1.1 and the fact that inner standard parts and arbitrary intersections commute, we have Cf◦¯(m) . (3.1.5) A◦ = f ∈NN m∈N

Since CCf◦¯(m) = st(CCf¯(m) ) is closed, the complement

CA◦ =

f ∈NN m∈N

CCf◦¯(m)

of A can be derived from the closed sets by the Souslin operation, and hence A◦ is measurable with respect to any completed Borel measure (see [321], page 50, for an easy proof). It only remains to ﬁnd the family {Dm,n }. From (3.1.5) it follows that for each ε ∈ R+ , there is a function fε : N −→ N such that ⎛ μ⎝

⎞ Cg¯◦(m) − A◦ ⎠ < ε,

g≤fε m∈N

where g ≤ fε means that g(n) ≤ fε (n) for all n ∈ N. Since our original sequence {Bm,n | n ∈ N} is decreasing, we have

3.1 Standard Parts of Hyperfinite Markov Chains

Cg¯◦(m) =

g≤fε m∈N

135

m∈N

Cf◦¯ε (m) .

Putting Dm,n = Cf¯1/n (m) , the lemma follows.

Remark 3.1.1. The argument above shows that a subset of a Hausdorﬀ space can be derived from the closed sets by using the Souslin operation if and only if it is the standard part of a set derived from the internal sets by the same operation. This result and the proof we have given are due to Henson [201]. We have now reached the last lemma we shall need before we can return to our Markov processes. It will be used to pick hyperﬁnite representations of measures avoiding properly exceptional sets.

Lemma 3.1.3. Let D be a subset of S0 of the form D = n∈N m∈N Dm,n for a family {Dm,n } of internal sets. If μ is a Radon probability measure on Y with μ(D◦ ) = 0, there exists an internal probability measure ν on S0 such that μ = L(ν) ◦ st−1 and L(ν)(D) = 0. Proof. We may obviously assume that the family {Dm,n } is increasing in m ◦ and decreasing in n. Deﬁne a new measure μm,n by μm,n (B) = μ(B − Dm,n ). Then, we have μ(B) = sup inf μm,n (B). n∈N m∈N

(3.1.6)

◦ . Hence, there is an internal measure The set S0 −Dm,n is S-dense in Y −Dm,n νm,n concentrated on S0 − Dm,n such that

μm,n = L(νm,n ) ◦ st−1 .

(3.1.7)

We may choose these measures such that the family {νm,n } is decreasing in m and increasing in n. Extending to an internal sequence {νm,n | m, n ∈ ∗ N}, we ﬁrst pick a γ ∈ ∗ N − N such that ◦

νγ,n (S0 ) = lim

m→∞

◦

νm,n (S0 )

for all n ∈ N, and then an η ∈ ∗ N − N such that ◦

νγ,η (S0 ) = lim ◦ νγ,n (S0 ). n→∞

By using equations (3.1.6) and (3.1.7), and the deﬁnitions of γ and η, we see that L(νγ,η )(D) = 0 and μ = L(νγ,η ) ◦ st−1 . The measure νγ,η has all the properties of the desired measure, except that we still have to show that it is a probability measure. However, this is clear since we have νγ,η (S0 ) = 1 + ε for some ε ≈ 0. Putting ν = (1 + ε)−1 νγ,η , the lemma is then proved.

136

3 Standard Representation Theory

By combining Lemmas 3.1.2 and 3.1.3, we get: Corollary 3.1.1. Let A ⊂ S0 be a properly exceptional set, and let μ be a completed Borel probability measure on Y such that μ(A◦ ) = 0. Then there is an internal probability measure ν on S0 such that μ = L(ν) ◦ st−1 and L(ν)(A) = 0.

3.1.2 Strong Markov Processes and Modified Standard Parts Having completed our study of exceptional sets and inner standard part of sets, we now turn to the real subject matter of this section, an investigation of standard parts of hyperﬁnite Markov chains. Let us ﬁrst describe what kind of processes we would like to obtain as standard parts [152]. Let Y be a Hausdorﬀ space and let Δ be an extra element. Set YΔ = Y ∪ {Δ} . Consider the topological Borel ﬁeld B(Y ) on Y . Let YΔ have the σ-algebra B(YΔ ) generated by the Borel sets on Y and the singleton {Δ}. Our standard processes will be YΔ valued, and the new element Δ will serve as a trap. Assume that (Ω, Π) is a measurable space and {Πt | t ∈ R+ } is a family of sub-σ-algebras of Π satisfying Πs ⊂ Πt

for all s ≤ t

and

Πt =

Πs

for all t ∈ R+ .

(3.1.8)

s>t

We call {Πt | t ∈ R+ } a right continuous ﬁltration on Ω if it satisﬁes the condition (3.1.8). Set Π∞ = σ {Πt | t ∈ R+ } . In our situation, we can take Π = Π∞ . A map σ : Ω → [0, ∞) is called a stopping time of {Πt | t ∈ R+ } if for all t {ω | σ(ω) ≤ t} ∈ Πt . For each stopping time σ, we introduce a σ-algebra Πσ by Πσ = {A ∈ Π∞ | ∀t(A ∩ (σ ≤ t) ∈ Πt )} . Denote by M(Y ) the set of all positive Radon measures on (Y, B(Y )) with ﬁnite masses. Given an element ν ∈ M(Y ), the completion of the σ-ﬁeld B(Y ) with respect to ν is denoted by B(Y )ν . A set or a function is called universally measurable if it is measurable with respect to ν∈M(Y ) B(Y )ν .

3.1 Standard Parts of Hyperfinite Markov Chains

137

If for each y ∈ YΔ , we are given a probability measure Θy on (Ω, Π). Then for each ν ∈ M(Y ), we have Θν (A) =

Y

Θy (A) dν(y), A ∈ Π,

provided, of course, that this makes sense, i.e., y → Θy (A) is μ measurable. Finally, if x(t) is a YΔ valued process, its lifetime ζ is deﬁned by ζ(ω) = inf {t ≥ 0 | x(ω, t) = Δ} . Definition 3.1.1. We call (Ω, Π∞ , {Πt | t ∈ R+ } , x(t), Θy ) a strong Markov process, if {Πt | t ∈ R+ } is a right continuous ﬁltration of Π, each Θy is a probability measure on Π∞ , and x : Ω × [0, ∞] −→ YΔ is a stochastic process on (Ω, Π∞ , Θy ) for each y ∈ YΔ . Moreover, the following conditions are satisﬁed: (i) For all t ≥ 0 and all measurable E ⊂ Y, the map y → Θy (x(t) ∈ E) is universally measurable in y ∈ Y. (ii) x(ω, ∞) = Δ, ∀ω ∈ Ω. x(ω, t) = Δ, ∀t ≥ ζ(ω). (iii) For each y ∈ YΔ , the process x is adapted to (Ω, {Πt } , Θy ), t → xt (ω) is right continuous from [0, ∞) to YΔ , Θy -a.e.ω and lims↑t xs (ω) exists in Y for all t ∈ (0, ζ(ω)), Θy -a.e.ω (a.e. represents almost every). (iv) ΘΔ (x(t) = Δ) = 1, ∀t ≥ 0; Θy (x(0) = y) = 1, ∀y ∈ Y. (v) For all {Πt } stopping times σ, all measurable E ⊂ YΔ , all μ ∈ M(Y ), and all s ∈ R+ , we have Θμ (x(σ + s) ∈ E | Πσ ) = Θx(σ) (x(s) ∈ E), Θμ -a.e.

(3.1.9)

In order to prove that a class of hyperﬁnite Markov chains have standard parts that are strong Markov processes, we shall have to overcome two main diﬃculties: the construction of the family of measures {Θy }, and the proof of the strong Markov property (3.1.9). But ﬁrst we must introduce the necessary regularity conditions in our nonstandard processes. Here, we would remind the reader that we use Pi to mean the internal probability measure on (Ω, Ft ), i ∈ S, t ∈ T , and we use Θy to mean the standard probability measure on (Ω, Π∞ ), y ∈ Y. We hope this will not cause confusion. Definition 3.1.2. Let f : T −→ ∗ R be internal. We say that r ∈ R is the S-right limit of f at t ∈ [0, ∞) if for any standard ε > 0, there is a standard δ > 0 such that if s ∈ T and t < ◦s < t + δ, then |f (s) − r| < ε, we write r = S-lim f (s). The S-left limit, S-lim f (s) is deﬁned analogously. s↓t

s↑t

138

3 Standard Representation Theory

Recall that S 0 = S0 ∩ N s(∗ Y ) and that X (δ) = X|Tδ . The lifetime ζδ of X is deﬁned by (δ)

ζδ (ω) = inf

◦

t | X (δ) (ω, t) ∈ / S0 .

We deﬁne the right standard part ◦X (δ)+ as follows. If t < ζδ (ω), let ◦

X (δ)+ (ω, t) = S- lim X (δ) (ω, s) s↓t

if this limit exists and for all t1 < t, the S-lims↓t1 X (δ) (ω, s) exists also, and ◦

X (δ)+ (ω, t) = Δ

else. If t ≥ ζδ (ω), we always put ◦X (δ)+ (ω, t) = Δ. Definition 3.1.3. A subset A of S0 is called a set of irregularities of X if there is a positive inﬁnitesimal δ0 ∈ T satisfying: (i) For all si ∈ S 0 − A and L(Pi )-a.e. ω, the path X (δ0 ) (ω, ·) has S-right and S-left limits at all t < ζδ0 (ω). (ii) For all si ∈ S 0 − A, the set ◦ (δ0 ) ω ∃t ∈ Tδfin ( t > ζ (ω) ∧ X (ω, t) ∈ S ) δ0 0 0 has L(Pi ) measure zero. (iii) For all inﬁnitely close si , sj ∈ S 0 − A, L(Pi ){◦X (δ0 )+ (ω, t) ∈ B} = L(Pj ){◦X (δ0 )+ (ω, t) ∈ B} for all ﬁnite t ∈ [0, ∞) and all Borel sets B. The hyperﬁnite Markov chain X has δ0 -exceptional irregularities if the set A of irregularities of X in the Deﬁnition 3.1.3 can be taken as a δ0 -exceptional set. Hence, X has exceptional irregularities if it has δ0 exceptional irregularities for some δ0 ≈ 0, δ0 ∈ T. The ﬁrst condition of the above Deﬁnition 3.1.3 guarantees that the hyperﬁnite Markov chainX has a reasonable standard part. The second says that “inﬁnite” is a trap. The third one is a version of the condition (3.1.1). Putting st(si ) = Δ when si ∈ S − S 0 , we have the following deﬁnition of the modiﬁed standard part of X(ω, t). Definition 3.1.4. Assume that X has exceptional irregularities, and let A be a properly δ0 -exceptional set of irregularities, where δ0 is as in Deﬁnition 3.1.3. Let x : Ω × R+ −→ YΔ be deﬁned by

3.1 Standard Parts of Hyperfinite Markov Chains

139

(i) if X(ω, 0) ∈ / A, then x(ω, t) = ◦X (δ0 )+ (ω, t). (ii) if X(ω, 0) ∈ A, then x(ω, t) = st(X(ω, 0)) for all t ∈ R+ . We call x modiﬁed standard part of X(ω, t). Our aim is to prove that if X has exceptional irregularities, then with an appropriate deﬁnition of the family {Θy } of measures, the modiﬁed standard parts of X are strong Markov processes. The ﬁrst step toward the deﬁnition of {Θy } is the following version of condition (iii) in Deﬁnition 3.1.3. If ν is an internal probability measure on S, let Pν be the measure on Ω deﬁned by (3.1.10) Pν (C) = Pi (C) dν(si ). Lemma 3.1.4. For δ0 ≈ 0, δ0 ∈ T , let A be a properly δ0 -exceptional set of irregularities of X, and let x be the corresponding modiﬁed standard part of X. Let ν1 , ν2 be two internal probability measures on S0 such that L(ν1 )(A) = L(ν2 )(A) = 0 and L(ν1 ) ◦ st−1 = L(ν2 ) ◦ st−1 , then for all t ∈ R+ and all Borel sets B L(Pν1 ){x(ω, t) ∈ B} = L(Pν2 ){x(ω, t) ∈ B}. Proof. Let μ = L(ν1 ) ◦ st−1 = L(ν2 ) ◦ st−1 . Choose t˜ ≈ t so large that X (δ0 ) (ω, t˜) = x(ω, t), L(Pν1 ) and L(Pν2 )-a.e.ω ∈ Ω

◦

˜ such that and pick an internal set B ◦ (δ0 ) ˜ X (ω, t˜) ∈ B}) = 0 L(Pνi )({X (δ0 ) (ω, t˜) ∈ B}Δ{

for i = 1, 2. Deﬁne a function f : Y −→ R as follows: (1) If y ∈ / A◦ , let f (y) = L(Pi ){x(ω, t) ∈ B} for some (then for all) si ∈ −1 st (y) ∩ S0 − A. (2) If y ∈ A◦ , deﬁne f (y) arbitrarily. ˜ is a lifting of f with respect to The function si → Pi {X (δ0 ) (ω, t˜) ∈ B} both ν1 and ν2 . Therefore, we have ˜ L(Pν1 ){x(ω, t) ∈ B} = ◦P ν1 {X (δ0 ) (ω, t˜) ∈ B} ◦ ˜ dν1 (si ) = Pi {X (δ0 ) (ω, t˜) ∈ B} = f (y) dμ(y)

140

3 Standard Representation Theory

=

◦

˜ dν2 (si ) Pi {X (δ0 ) (ω, t˜) ∈ B}

˜ = ◦P ν2 {X (δ0 ) (ω, t˜) ∈ B} = L(P ν2 ){x(ω, t) ∈ B},

and the lemma is proved.

Lemma 3.1.5. For δ0 ≈ 0, δ0 ∈ T , assume that X has a properly δ0 exceptional set A of irregularities, and let x be the modiﬁed standard part. For all inﬁnitely close si , sj ∈ S 0 − A, all ﬁnite sequences t1 < t2 < · · · < tn from R+ and all Borel sets B1 , B2 , · · · , Bn , we have L(Pi )

n

{x(tl ) ∈ Bl }

l=1

= L(Pj )

n

{x(tl ) ∈ Bl } .

l=1

Proof. We shall prove this by induction on the length n of the sequences t1 < t2 < · · · < tn , B1 , B2 , · · · , Bn . The case of n = 1 is part of Deﬁnition 3.1.3. Assume that the lemma holds for all sequences of length n − 1, and pick t˜1 , t˜, · · · , t˜n ∈ Tδ0 such that t˜1 ≈ t1 , t˜2 ≈ t2 , · · · , t˜n ≈ tn and X (δ0 ) (ω, t˜l ) = x(ω, tl ), L(Pk )-a.e.

◦

˜2 , · · · , B ˜n such that ˜1 , B for l = 1, 2, · · · , n and k = i, j. Choose internal sets B ˜l }Δ{◦X (δ0 ) (ω, t˜l ) ∈ Bl } = 0 L(Pk ) {X (δ0 ) (ω, t˜l ) ∈ B for l = 1, 2, · · · , n and k = i, j. We deﬁne two measures νi , νj on S by putting ˜l } νk (s) = Pk {ω | X (δ0 ) (ω, t˜n−1 ) = s and for all l < n − 1, X (δ0 ) (ω, t˜l ) ∈ B for all s ∈ S and k = i, j. By the induction hypothesis L(νi ) ◦ st−1 = L(νj ) ◦ st−1 , and since A is properly δ0 -exceptional, we have L(νi )(A) = L(νj )(A) = 0. Applying Lemma 3.1.4 with t = tn − tn−1 , we get L(Pνi ){x(tn − tn−1 ) ∈ Bn } = L(Pνj ){x(tn − tn−1 ) ∈ Bn }. Since X is Markov and time homogeneous, the lemma follows.

In the following, we will deﬁne {Πt } and {Θy } of our standard Markov processes. For t ∈ R+ , let Πt◦ be the σ-algebra generated by the sets

3.1 Standard Parts of Hyperfinite Markov Chains

141

ω x(ω, t1 ) ∈ B1 ∧ · · · ∧ x(ω, tn ) ∈ Bn , where 0 ≤ t1 < t2 < · · · < tn ≤ t and B1 , · · · , Bn are Borel sets in Y. Deﬁne Πt =

s>t

Πs◦ .

The ﬁltration {Πt | t ∈ [0, ∞)} is right continuous. Let us set Π∞ = ∨ {Πt | t ∈ [0, ∞)} . It follows from Lemma 3.1.5 that for all C ∈ Π∞ and all inﬁnitely close si , sj in S 0 − A L(Pi )(C) = L(Pj )(C). This observation makes it possible to give the following deﬁnition of a family {Θy | y ∈ YΔ } of measures on Π∞ . If y ∈ / A◦ , let for all C ∈ Π∞ Θy (C) = L(Pi )(C)

for all si ∈ st−1 (y) − A.

If y ∈ A◦ ∪ {Δ} , C ∈ Π∞ , let Θy (C) =

1, if C contains all constant paths x(t) = y, 0, else.

Observe that since a set C ∈ Π∞ contains either all or none of the constant paths x(ω, t) = y, the set function Θy is a measure. We have reached our goal: Theorem 3.1.1. Assume that S0 is a hyperﬁnite subset of ∗ Y for some Hausdorﬀ space. Let X : Ω × T −→ S be a hyperﬁnite Markov chain with exceptional irregularities. Assume that for each i ∈ S0 , the function t → P {ω | X(ω, t) = si } is decreasing. If x is a modiﬁed standard part of X, then (Ω, {Πt | t ∈ R+ }, {Θy | y ∈ YΔ }, x(t)) is a strong Markov process. Proof. Since we already know that {Πt } is right continuous, all we have to do is to check conditions (i)-(v) of Deﬁnition 3.1.1. Obviously, the (ii), (iii) and (iv) of Deﬁnition 3.1.1 are satisﬁed by our construction. We shall prove (i) and (v). Given a Radon probability measure μ on Y , we deﬁne two new measures μ0 and μ1 on Y by

142

3 Standard Representation Theory

μ0 (B) = μ(B − A◦ ), μ1 (B) = μ(B ∩ A◦ ), where A is the properly δ0 -exceptional set of irregularities used in the construction of x. By Corollary 3.1.1, we can ﬁnd an internal measure ν on S0 satisfying μ0 = L(ν) ◦ st−1 , L(ν)(A) = 0.

(3.1.11)

We ﬁrst prove that (i) in Deﬁnition 3.1.1 is satisﬁed: (i) Given α ∈ [0, 1], a Borel set E ⊂ Y, and t ∈ R+ , we must show that {y ∈ Y | Θy (x(t) ∈ E) > α} is μ-measurable for all ﬁnite Radon measures μ. Since

◦

{y ∈ A | Θy {x(t) ∈ E} > α} =

A◦ ∩ E, α ∈ [0, 1) ∅, α=1

is universally measurable by Lemma 3.1.2, it suﬃces to show that {y ∈ / A◦ | Θy (x(t) ∈ E) > α}

(3.1.12)

is μ-measurable. In fact, we only have to show that the set deﬁned by (3.1.12) is μ0 measurable by the deﬁnition of μ0 . Since X (δ0 ) has S-right limits a.e. with respect to the probability measure Pν constructed in (3.1.11) by using the relation (3.1.10), we can choose tˆ ≈ t so large that L(Pν )({x(t) ∈ E}Δ{◦X (δ0 ) (t˜) ∈ E}) = 0. There must be an internal set E˜ such that ˜ = 0. L(Pν ){X (δ0 ) (t˜) ∈ (st−1 (E)ΔE)} Hence, we have ˜ L(Pi ){x(t) ∈ E} = ◦Pi {X (δ0 ) (t˜) ∈ E} for L(ν)-a.e.si . Combining this with the deﬁnition of Θy , we see that ˜ as a ν-lifting. Hence, it is y → Θy {x(t) ∈ E} has i → Pi {X (δ0 ) (t˜) ∈ E} a μ0 measurable function. This proves that the set deﬁned by (3.1.12) is μ0 measurable, and Deﬁnition 3.1.1 (i) follows.

3.1 Standard Parts of Hyperfinite Markov Chains

143

(v) we must show that for all {Πt } stopping times σ, all sets B ∈ Πσ and all s ∈ [0, ∞), the equation Θμ {ω ∈ B | xσ+s ∈ E} =

B

Θxσ {xs ∈ E} dΘμ

(3.1.13)

holds for all Radon probability measure μ on Y and all Borel sets E. First notice that since the paths of x are constant Θμ1 -a.e., we have Θμ1 {ω ∈ B | xσ+s ∈ E} =

B

Θxσ {xs ∈ E} dΘμ1 .

Equation (3.1.13) will hold if we may prove Θμ0 {ω ∈ B | xσ+s ∈ E} =

B

Θxσ {xs ∈ E} dΘμ0 .

(3.1.14)

In the sequel, we will prove (3.1.14). Let ν be the nonstandard representation of μ0 given in the construction (3.1.11). The hyperﬁnite counterpart of (3.1.14) is

(δ )

0 Pν {ω ∈ C | Xτ +s ∈ F} =

C

PX (δ0 ) {Xs(δ0 ) ∈ F } dPν , τ

(3.1.15)

where C and F are internal sets and C is measurable in the ∗ −algebra generated by the internal stopping time τ . Since X (δ0 ) is a time homogeneous Markov chain, (3.1.15) holds. Our plan is to deduce (3.1.14) from (3.1.15). We ﬁrst pick an internal stopping time τ such that ◦ τ = σ, L(Pν )-a.e., and such that there is a τ measurable set C satisfying L(Pν )(BΔC) = 0.

(3.1.16)

Let P τ be an internal measure on Ω given by P τ (D) = PX (δ0 ) (D) dPν . τ

Since A is properly exceptional, we observe that (δ0 )

L(P τ ){Xt

∈ A} = 0

(where δ0 is the inﬁnitesimal used in the construction of x, for all t ∈ Tδfin 0 see Deﬁnition 3.1.4). Hence, we can choose an s˜ ∈ Tδfin , s˜ = s, such that 0

144

3 Standard Representation Theory (δ0 )

◦

Xs˜

= xs , L(P τ )-a.e.

(3.1.17)

and ◦

(δ )

0 X τ +˜ s = xσ+s , L(Pν )-a.e.

(3.1.18)

Finally, we pick an internal set F such that (δ )

0 −1 L(Pν ){Xτ +˜ (E)ΔF } = 0, s ∈ st

(3.1.19)

(δ ) L(P τ ){Xs˜ 0

(3.1.20)

∈ st−1 (E)ΔF } = 0.

We now have Θμ0 {ω ∈ B | xσ+s ∈ E} = Θy {ω ∈ B | xσ+s ∈ E} dμ0 (y) = L(Pi ){ω ∈ B | xσ+s ∈ E} dL(ν)(si )

(by def. of Θy )

= L(Pν ){ω ∈ B | xσ+s ∈ E}

(by def. of Pν )

◦

= Pν {ω ∈ C | Xτ +˜s ∈ F } ◦ PXτ {Xs˜ ∈ F } dPν = C = L(PXτ ){xs ∈ E} dL(Pν ) B = Θxσ {xs ∈ E} dΘμ0

(by def. of Θμ0 )

(by (3.1.16), (3.1.18), (3.1.19)) (by (3.1.15)) (by (3.1.16), (3.1.17), (3.1.20)) (by def. of ν, τ and {Θy }).

B

This proves (3.1.14) and the theorem.

Example 3.1.1. Let Z = {· · · , −2, −1, 0, 1, 2, · · · } (i.e., set of all the integers). Let η ∈ ∗ N − N, and set S0 =

k ∗ √ k ∈ Z, |k| ≤ η . η

Deﬁne transition probabilities qs,s by qs,s =

1 2

if |s − s | =

0

otherwise.

If s, s ∈ S0 , and if s0 is the trap, let

√1 , η

3.2 Hyperfinite Dirichlet Forms and Markov Processes

qs,s =

1 2

√ if s = ± η,

0

otherwise.

145

Let m be an internal measure on S0 given by 1 m(s) = √ η for all s ∈ S0 , and let a time-line be k ∗ T = k ∈ N0 . η The process X is Anderson’s random walk with a uniform initial distribution corresponding to the Lebesgue measure. To prove that the standard part of X is a strong Markov process, we show that the empty set is a set of irregularities of X. Since X is S-continuous L(Pi )-a.e. for all nearstandard si ∈ S0 , the two ﬁrst conditions in Deﬁnition 3.1.3 are obviously satisﬁed. If si ≈ sj , the paths starting at si look exactly like the paths starting at sj except for an inﬁnitesimal translation. Hence, they induce the same standard paths and the property (iii) in Deﬁnition 3.1.3 follows.

3.2 Hyperfinite Dirichlet Forms and Markov Processes In this section, we shall obtain conditions on hyperﬁnite quadratic forms which guarantee that the modiﬁed standard parts of the associated hyperﬁnite Markov chains are strong Markov processes. The method we shall apply is simple, we just use the relationship between forms and processes established in Sect. 1.5 to translate the conditions of Theorem 3.1.1 into the language of hyperﬁnite quadratic forms. One by one we shall reformulate the conditions of Deﬁnition 3.1.3 in terms of hyperﬁnite quadratic forms. In this section, we need a slightly stronger assumption on the state space. Let Y be a regular Hausdorﬀ space (or a T3 space), i.e., for all closed sets F and x ∈ F, there are disjoint open sets O1 , O2 such that x ∈ O1 , F ⊂ O1 . Suppose that S0 = {s1 , s2 , · · · , sN } is a hyperﬁnite subset of ∗ Y and m is a hyperﬁnite positive measure on S0 , N ∈ ∗ N − N. Let H be the linear space of all internal functions u : S0 −→ ∗ R with the inner product

u, v =

N i=1

u(si )v(si )m(si ).

146

3 Standard Representation Theory

Given an inﬁnitesimal Δt ≈ 0, denote by T = {kΔt | k ∈ ∗ N0 } the hyperﬁnite ˆ be the dual hyperﬁnite Markov chains introduced time line. Let X and X in Sect. 1.5 with m as dual measure. That is, we assume that conditions (1.5.1), (1.5.2), (1.5.4), (1.5.5), (1.5.8), (1.5.9), (1.5.10), (1.5.11), (1.5.13), and (1.5.14) in Sect. 1.5 are fulﬁlled. Let E(·, ·) be the hyperﬁnite quadratic form on H associated with the transition matrix Q and the dual measure m, ˆ ·) be the co-form of E(·, ·). and E(·, Given x ∈ Y , we recall that the monad1 of x is the set of ∗ Y deﬁned by μ(x) = {∗ Ox ∈ O and O is open}.

3.2.1 Separation of Points The following assumption will take care of Deﬁnition 3.1.3 (i). In Albeverio et al. [25], 5.5.1. Deﬁnition, a deﬁnition of separation of compacts was given for symmetric hyperﬁnite Dirichlet forms. The following deﬁnition simpliﬁes the condition since a single point can be viewed as a compact set. Definition 3.2.1. Let Z be a subset of Y . A hyperﬁnite quadratic form E(·, ·) separates points of Z if there exists a countable family π = {un | n ∈ N} of internal functions such that for all x, y ∈ Z, x = y, there is an element un ∈ π satisfying ◦

un (s1 ) = ◦un (s2 ), s1 ∈ μ(x), s2 ∈ μ(y), si ∈ S0 , i = 1, 2.

(3.2.1)

Moreover, we have ◦

E1 (un , un ) < ∞, ∀un ∈ π.

(3.2.2)

We call π = {un | n ∈ N} a separating family of Z by E(·, ·). We recall from Sect. 2.1 that if δ ∈ T, the sub-line Tδ is deﬁned by Tδ = {kδ | k ∈ ∗ N0 }, X (δ) is the restriction of X|Tδ . Let ζ = ζδ be the lifetime of X (δ) , i.e., ζδ (ω) = inf{◦ t | X (δ) (ω, t) ∈ S 0 }, where S 0 = S0 ∩ N s(∗ Y ). For any subset A ⊂ S0 , let (δ) τA (ω) = inf{◦ tX(t) ∈ st−1 (A◦ ) and t ∈ Tδ }. 1

For the general concept of monad, we refer to Albeverio et al. [25].

3.2 Hyperfinite Dirichlet Forms and Markov Processes

147

Lemma 3.2.1. Let Y be a regular Hausdorﬀ space, and let A be a subset of S0 . Suppose that E(·, ·) separates the points of Y − A◦ and π is the separating family. For ω ∈ Ω, if the path X(·, ω) fails to have an S-left or S-right limit (Δt) at t < ζΔt (ω) ∧ τA (ω), then so does u(X(ω, ·)) for some u ∈ π. (Δt)

Proof. Let ω be a ﬁxed point in Ω. Fix t < ζΔt (ω)∧τA (ω), t ∈ [0, ∞). Given a sequence {tn | n ∈ N} from T such that the standard part ◦ tn increases strictly to t, we shall prove that the sequence ◦ X(ω, tn )n ∈ N has a cluster point in Y − A◦ . Suppose for a contradiction that

X(ω, tn )n ∈ N

◦

has no cluster point in Y . Then for each y ∈ Y , there is a neighborhood Oy and an integer ny ∈ N such that ◦

X(ω, tn ) ∈ / Oy

when n ≥ ny . Since Y is regular, we can ﬁnd a neighborhood Gy of y such that the closure Gy is contained in Oy . Hence, we have / ∗ Gy X(ω, tn ) ∈ when n ≥ ny . Extend {tn | n ∈ N} to be an internal sequence {tn | n ∈ ∗ N} of elements of T less than t. Consider the set Ay = {n ∈ ∗ N | n ≤ ny

or X(ω, tn ) ∈ / ∗ Gn } .

Since Ay is internal and contains N, there is an ηy ∈ ∗ N − N such that all η ≤ ηy are elements of Ay . By saturation, there is an inﬁnite η less than all ηy . But then, we have X(ω, tη ) ∈ / ∗ Gy for all y. This implies that X(ω, tη ) is not nearstandard, contradicting our assumption that t < ζΔt (ω). Suppose that y ∈ Y is a cluster point of {◦X(ω, tn )}n∈N . Since tn ↑ t, (Δt) t < τA (ω), it is easy to see that y ∈ / A◦ . Let x ∈ Y − A◦ =Z ˆ be a cluster point of {◦X(ω, t)} . If x is not the S-left limit of X(ω, ·) at t, there must be another sequence {sn | n ∈ N} increasing

148

3 Standard Representation Theory

to t such that x is not a cluster point of {◦X(ω, sn ) | n ∈ N}. Repeating the argument above, we see that {◦X(ω, sn ) | n ∈ N} must have a cluster point y ∈ Z. Let u ∈ π be the function satisfying the conditions of (3.2.1) and (3.2.2). Obviously, u(X(ω, ·)) does not have an S-left limit at time t. This proves the S-left case of the lemma. The S-right limit case can be treated in a similar way.

We shall use Lemma 3.2.1 and Fukushima’s decomposition Theorem 2.7.2 to show that if E(·, ·) separates points, then the associated Markov chain X satisﬁes Deﬁnition 3.1.3 (i). But we must get some preparation at ﬁrst. It is easy to see that {ω | ∃t ∈ Tδr (X(ω, t) ∈

An )} =

n∈N

{ω | ∃t ∈ Tδr (X(ω, t) ∈ An )},

n∈N

(3.2.3) for all sequences {An | n ∈ N} of subsets of S. However, the corresponding formula for intersections is false in general. In particular, it does hold if the sequence is decreasing and consists of internal sets. This is the observation behind the next lemma. Lemma 3.2.2. Let A ⊂ S0 , and assume that there is a family {Bm,n | m, n ∈ N} of internal sets such that A=

Bm,n

m∈N n∈N

and for each m, the sequence {Bm,n | n ∈ N} is decreasing. Then {ω | ∃t ∈ Tδr (X(ω, t) ∈ A)} =

ω | ∃t ∈ Tδr X(ω, t) ∈ Bm,n . m∈N n∈N

(3.2.4) Proof. It suﬃces to prove that r r Bm,n ω ∃t ∈ Tδ X(ω, t) ∈ = ω ∃t ∈ Tδ X(ω, t) ∈ Bm,n n∈N

n∈N

(3.2.5) for all m, since (3.2.4) then follows from (3.2.3). Also, it is immediately clear that the left hand of (3.2.5) is included in the right hand side. To prove the opposite inclusion, choose

3.2 Hyperfinite Dirichlet Forms and Markov Processes

ω0 ∈

149

{ω | ∃t ∈ Tδr (X(ω, t) ∈ Bm,n )}.

n∈N

Consider the set ˜m,n )}, {n ∈ ∗ N | ∃t ∈ Tδr (X(ω0 , t) ∈ B ˜m,n | n∈∗ N} is some internal, decreasing extension of {Bm,n | where {B n ∈ N}. By the choice of ω0 , this set contains N. Since it is internal, it must have an inﬁnite member η. Thus, we have ˜m,η )} ⊂ {ω | ∃t ∈ Tδr (X(ω, t) ∈ ω0 ∈ {ω | ∃t ∈ Tδr (X(ω, t) ∈ B

Bm,n )}.

n∈N

The lemma is proved. We are now in the position to prove the following:

Proposition 3.2.1. Let Y be a regular Hausdorﬀ space, and let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form for every inﬁnitesimal δ ∈ T . Assume that A is an exceptional set and E(·, ·) separates the points of Y −A◦ . There exists an inﬁnitesimal δ0 ∈ T such that for all inﬁnitesimal δ ∈ Tδ0 , there is a δ-exceptional set A0 (δ) satisfying that for all si ∈ S0 − A0 (δ), the hyperﬁnite Markov chain X (δ) (t) has S-left and S-right limits at all t < ζδ , L(Pi )-a.e. Proof. To ﬁnd δ0 , we look at the property of the separating family π = (δ) {un | n ∈ N}. From Theorem 1.4.2, we know that D(E (δ) ) = D(E ) for all ◦ δ ∈ T, δ ≈ 0. For each u ∈ π, we have E 1 (u, u) < ∞. Hence, we can ﬁnd (δ)

a δu ≈ 0 such that u ∈ D(E ) = D(E (δ) ) for all inﬁnitesimal δ ≥ δu by Corollary 1.2.4. By saturation, there is a δ0 ≈ 0 larger than all δu and A is δ0 -exceptional. Turning to A0 (δ) for inﬁnitesimal δ ∈ Tδ0 , we ﬁrst observe that by Lemma 3.2.1 and the countability of π, it suﬃces to show that for each un ∈ π = {un | n ∈ N}, there is a δ-exceptional set An (δ) such that for all si ∈ S0 − An (δ), the process un (X (δ) (t)) has S-left and S-right limits at all t < ∞, L(Pi )-a.e. Actually, we deﬁne A0 (δ) = A0

∞

An (δ) ,

n=1

where A0 is a properly δ-exceptional set containing A. Then, A0 (δ) is δexceptional by Lemma 2.1.1 and for all si ∈ Si − A0 (δ), un ∈ π, un (X (δ) (t)) has S-right and S-left limits for all t < ∞, L(Pi )-a.e. Therefore, the hyperﬁnite Markov chain X (δ) (t) has S-left and S-right limits at all t < ζδ (ω),

150

3 Standard Representation Theory

L(Pi )-a.e. by Lemma 3.2.1. From this observation, we only need to ﬁnd the exceptional set An (δ) for every un ∈ π. For the simplicity of notation, we write u = un in the following. Given two standard rationals p, q, −∞ < p < q < ∞, we deﬁne a sequence n {τ(p,q) } of stopping times as follows: 0 τ(p,q) = min{t ∈ Tδ | u(X (δ) )(ω, t)) ≤ p}, 2n−1 2n = min{t ∈ Tδ | t > τ(p,q) (ω) ∧ u(X (δ) )(ω, t)) ≤ p}, τ(p,q) 2n+1 2n τ(p,q) = min{t ∈ Tδ | t > τ(p,q) (ω) ∧ u(X (δ) )(ω, t)) ≥ q}.

Let B ⊂ S0 be deﬁned by B=

n {i | Pi {τ(p,q) ≤ m} ≥

(p,q) m∈N k∈N n∈N

1 }. k

If X (δ) fails to have S-left or S-right limits with positive L(Pi ) probability, then we have si ∈ B. We must show that B is δ-exceptional. By Lemma 3.2.2, we have for all r ∈ [0, ∞) {ω | ∃t ∈ Tδr (X(ω, t) ∈ B)} 1 n ≤ m} ≥ = ω ∃t ∈ Tδr PX (δ) (ω,t) {τ(p,q) . k (p,q) m∈N k∈N n∈N

If B is not δ-exceptional, there must be a pair (p, q) of rationals, as well as integers m, k ∈ N, r ∈ [0, ∞) and an inﬁnite number η ∈ ∗ N such that 1 η r L(P ) ω ∃t ∈ Tδ PX (δ) (ω,t) {τ(p,q) ≤ m} ≥ > 0. k This implies that with positive probability, u(X (δ) ) jumps back and forth between p and q more than η times before time m + r. Since u ∈ D(E (δ) ), Fukushima’s decomposition Theorem 2.7.2 tells us that u(X (δ) ) = N [u] + M [u] , where N [u] is S-continuous L(P )-a.e. and M [u] is a λ2 -martingale. If u(X (δ) ) jumps η times between p and q before t = m+r, there must be an inﬁnitesimal interval where it jumps back and forth inﬁnitely many times. Since N is S-continuous – and hence almost constant on inﬁnitesimal intervals – most

3.2 Hyperfinite Dirichlet Forms and Markov Processes

151

of this jumping is done by M . Hence the quadratic variation of M is inﬁnite on a set of positive measure, contradicting the fact that it is a λ2 -martingale. We then conclude that B must be δ-exceptional, and the proposition is thus proved.

3.2.2 Nearstandardly Concentrated Forms Now we are in a position to ﬁnd a condition on E(·, ·) that implies a similar property like (ii) in Deﬁnition 3.1.3. For any internal subset C of S0 , we set HC = {u ∈ H | u(si ) = 0 for all si ∈ S0 − C} . For each u ∈ H and any internal set C ⊂ S0 , δ ∈ T, we deﬁne (δ) (δ) (δ) uC (i) = u(i) − Ei (1 + δ)−σS0 −C /δ u X (δ) (σS0 −C ) ,

(3.2.6)

where (δ) σS0 −C (ω) = min t ∈ Tδ | X (δ) (ω, t) ∈ S0 − C . It is easy to see that (δ)

u C ∈ HC . Definition 3.2.2. (1) For δ ∈ T , a hyperﬁnite quadratic form E(·, ·) is said to be δ-nearstandardly concentrated if there exists an increasing sequence {Bn | n ∈ N} of internal subsets of S0 such that: (i) The set ∪n∈N Bn − S 0 is δ-exceptional. (δ) (ii) For each F ∈ H, if ◦ E1 (F, F ) < ∞, then we have ◦ (δ) E1 (F

(δ)

(δ)

− FBm , F − FBm ) −→ 0, m −→ ∞.

(2) A hyperﬁnite quadratic form E(·, ·) is said to be nearstandardly concentrated if it is δ-nearstandardly concentrated for some inﬁnitesimal δ ∈ T. Notice that above deﬁnition of nearstandardly concentrated is diﬀerent from that of Albeverio et al. [25], 5.5.4 Deﬁnition. Our deﬁnition is motivated by both Albeverio et al. [25], 5.5.4 Deﬁnition and Condition (I) of Theorem 4.1.1 in Chap. 4.

152

3 Standard Representation Theory

Lemma 3.2.3. For δ ∈ T, let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Assume that E(·, ·) is δ-nearstandardly concentrated. Let X be the related hyperﬁnite Markov (δ) chain. Then for each F ∈ H with ◦ E1 (F, F ) < ∞, the set

si ∈ S 0 −

A(F ) =

m∈N

fin (δ) Bn L(Pi ) ω ∃t ∈ Tδ F (X (t)) ≈ 0 >0 (3.2.7)

is δ-exceptional. Proof. Step 1. For ε > 0, set D = si |F (si )| > ε , (δ) σD (ω) = min t ∈ Tδ | X (δ) (ω, t) ∈ D , (δ) (δ) −σ (ω)/δ . e1 (D) = Ei (1 + δ) D From Corollary 2.2.3, we have (δ)

E1

C2 (δ) (δ) (δ) e1 (D), e1 (D) ≤ 2 E1 (F, F ). ε

Hence, we have ◦ (δ) E1

(δ) (δ) e1 (D), e1 (D) < ∞.

This implies that ◦ (δ) E1

(δ) (δ) (δ) (δ) (δ) (δ) e1 (D) − (e1 (D))Bm , e1 (D) − (e1 (D))Bm −→ 0, m −→ 0, (3.2.8)

(δ)

(δ)

where (e1 (D))Bm is deﬁned in the same way as the deﬁnition (3.2.6). Step 2. From the approximation (3.2.8), there exists an increasing sequence {mk }k∈N satisfying δk = ε >0

Ak (F, ε).

k=K

Since δk < 2−k , it follows from the relation (3.2.9) that 1

P {ω | ∃t ∈ Tδ1 (X(ω, t) ∈ A(F, ε))} ≤ 2C 2 (1 + δ) δ

∞

δk

k=K 1

≤ C 2 (1 + δ) δ 2−K+2 . Hence, A(F, ε) is δ-exceptional. it is easy to see that A(F ) is δ-exceptional, since A(F ) = Step 3. Now 1 A(F, ).

m∈N m Proposition 3.2.2. For δ ∈ T, let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Assume that E(·, ·) is δ-nearstandardly concentrated. Let X be the related hyperﬁnite Markov chain. (δ) Then for each F ∈ H with ◦ E1 (F, F ) < ∞, there exists a δ-exceptional set

154

3 Standard Representation Theory

A1 (F, δ) such that for si ∈ S0 − A1 (F, δ) L(Pi ) ω ∃t ∈ Tδfin ◦ t ≥ ζδ (ω) ∧ F (X (δ) (t)) ≈ 0 = 0. Proof. Let {Bn | n ∈ ∗ N} be an increasing extension of {Bn | n ∈ N}, which is the increasing sequence in Deﬁnition 3.2.2 (1). Set B = ∪n∈N Bn . Deﬁne = min t ∈ Tδ X (δ) (ω, t) ∈ / Bn , n ∈ ∗ N, (δ) /B . τ (ω) = inf ◦ tX (δ) (ω, t) ∈

τn(δ) (ω)

It is not hard to check that τ (δ) (ω) = sup

◦

τn (ω)n ∈ N .

Let A1 (F, δ) be a properly δ-exceptional set containing B − S 0 and the set A(F ), where A(F ) is deﬁned in Lemma 3.2.3 by the deﬁnition (3.2.7). Given si ∈ S0 − A1 (F, δ), we can ﬁnd an η ∈ ∗ N − N such that τ (δ) (ω) = ◦τη(δ) (ω), L(Pi )-a.e.

(3.2.10)

By the deﬁnition of A1 (F, δ), we have B − S 0 ⊂ A1 (F, δ). Since A1 (F, δ) is / A1 (F, δ), this implies that properly δ-exceptional and si ∈ τ (ω) = ζδ (ω), L(Pi )-a.e.

(3.2.11)

Combining (3.2.10) and (3.2.11), we have ◦ (δ) τη (ω)

= ζδ (ω), L(Pi )-a.e.

(3.2.12)

/ A1 (F, δ), A1 (F, δ) contains A(F ), and it is properly Notice that si ∈ δ-exceptional. We have L(Pi )(X(ω, τη(δ) (ω)) ∈ A(F )) = 0. Hence, we have by the deﬁnition of A(F ) fin (δ) (δ) L(Pi ) ω ∃t ∈ Tδ t ≥ τη (ω) ∧ F (X (t)) ≈ 0 = 0. The proposition follows from the relation (3.2.12).

3.2 Hyperfinite Dirichlet Forms and Markov Processes

155

3.2.3 Quasi-Continuity Definition 3.2.3. (1) An internal function F : S0 −→ ∗ R is said to be hyperﬁnite δ-quasi-continuous if there is a hyperﬁnite δ-exceptional set A ⊂ S0 such that for all inﬁnitely close si , sj ∈ S 0 − A, we have F (si ) ≈ F (sj ). (2) An internal function F : S0 −→ ∗ R is called hyperﬁnite quasi-continuous if there is an inﬁnitesimal δ such that F is hyperﬁnite δ-quasi-continuous. Remark 3.2.1. We remark that usually one uses zero capacity sets to deﬁne the quasi-continuity in regular Dirichlet spaces (refer to Sect. 4.1 of Chap. 4, or refer to [175]). Nevertheless, we utilize the exceptional sets in the deﬁnition of the hyperﬁnite quasi-continuity. The reason is that we have the equivalence between the concept of exceptional sets and that of zero capacity sets in regular Dirichlet space theory, e.g., Theorem 4.3.1 in [175]). However, we only have Theorem 2.4.1, Theorem 2.4.2, and Theorem 2.4.3 in Sect. 2.4, from which we understand that the conception of exceptional sets is not equivalent to that of zero capacity sets in hyperﬁnite Dirichlet space theory. Given a hyperﬁnite δ-quasi-continuous and S-bounded function F ∈ H, let A be a δ-exceptional set satisfying F (si ) ≈ F (sj ), si ≈ sj , si , sj ∈ S 0 − A. Deﬁne ◦F = f : Y −→ R by: / A◦ and si ∈ st−1 (y) − A; (i) f (y) = ◦F (si ) whenever y ∈ ◦ (ii) f (y) = 0 whenever y ∈ A . In order to get the condition that makes the process X to satisfy Deﬁnition 3.1.3 (iii), we impose the following condition on E(·, ·). Definition 3.2.4. (1) A hyperﬁnite quadratic form E(·, ·) is said to have a hyperﬁnite δ-quasi-continuous core if there are a family of countable functions π = {Fn | n ∈ N} ⊂ H and a δ-exceptional set A such that Fn (si ) ≈ Fn (sj ), si ≈ sj , si , sj ∈ S 0 − A, ∀n ∈ N, and (Y − A◦ ) ∩ B(Y ) ⊂ σ{fn = ◦F n | n ∈ N}.

(3.2.13)

(δ)

Moreover, st{maxsi ∈S0 |Fn (si )|} < ∞, ◦ E1 (Fn , Fn ) < ∞,∀n ∈ N. π is called a hyperﬁnite δ-quasi-continuous core of E(·, ·). (2) A hyperﬁnite quadratic form E(·, ·) is said to have a hyperﬁnite quasicontinuous core if it has a hyperﬁnite δ-quasi-continuous core for some inﬁnitesimal δ ∈ T. Lemma 3.2.4. Let M be a set. Consider a countable subset G of M and a countable collection Λ of maps from M × M into M . Then, there exists a countable set L such that (a)

G ⊂ L ⊂ M.

(b)

λ(L × L) ⊂ L, ∀λ ∈ Λ.

156

3 Standard Representation Theory

Proof. This is proven in Fukushima [175] Lemma 6.1.1.

Lemma 3.2.5. Suppose that E(·, ·) has a hyperﬁnite δ-quasi-continuous core ˆ = {Fˆn | n ∈ N} such π = {Fn | n ∈ N}. Then, there exists a countable set π that ˆ , a ∈ Q =⇒ |Fˆn | ∈ π ˆ , Fˆn + Fˆm ∈ π ˆ , Fˆn Fˆm ∈ π ˆ, (1) Fˆn , Fˆm ∈ π ˆ ˆ aFn ∈ π ˆ , and Fn ∧ a ∈ π ˆ, ˆ = {Fˆn | n ∈ N}, (2) π = {Fn | n ∈ N} ⊂ π (3) every element in π ˆ is hyperﬁnite δ-quasi-continuous, where Q = {a1 , a2 , · · · , an , · · · } is the set of all rational numbers. Proof. Let us denote by Qδ (S0 ) the set of hyperﬁnite δ-quasi-continuous functions on S0 . We deﬁne the maps Λ = {λ−2 , λ−1 , λ0 , λ1 , λ2 , · · · } from [H ∩ Qδ (S0 )] × [H ∩ Qδ (S0 )] into H ∩ Qδ (S0 ) by λ−2 (u, v) = |u|, λ−1 (u, v) = u + v, λ0 (u, v) = uv, λ2i−1 (u, v) = ai u, λ2i (u, v) = u ∧ ai , i = 1, 2, · · · Applying Lemma 3.2.4 to π and Λ, we get a countable set π ˆ satisfying (1), (2), and (3).

Lemma 3.2.6. Assume that the hyperﬁnite quadratic form E(·, ·) has a δ-quasi-continuous core π = {Fn | n ∈ N}. For any two ﬁnite measures ν1 and ν2 on (Y, B(Y )), if ν1 and ν2 satisfy the following condition: Y

◦

F (y) ν1 (dy) =

Y

◦

F (y) ν2 (dy)

for all

F ∈π ˆ,

(3.2.14)

where π ˆ is the countable set obtained in the Lemma 3.2.5, then ν1 and ν2 coincide on (Y, σ(◦F | ◦F ∈ π ˆ )). Proof. For any F ∈ π ˆ , a ∈ Q, Fn = [n(F − F ∧ a)] ∧ 1 ∈ π ˆ and ◦Fn ↑ 1(◦F >a) . This implies that 1(◦F (y)>a) ν1 (dy) = 1(◦F (y)>a) ν2 (dy) Y

Y

3.2 Hyperfinite Dirichlet Forms and Markov Processes

157

from the conditions (3.2.14). Now we can prove our Lemma by using the monotone class theorem.

Lemma 3.2.7. If the hyperﬁnite quadratic form E(·, ·) has a hyperﬁnite δquasi-continuous core π = {Fn | n ∈ N}, then π separates the points of Y −A◦ by E (δ) (·, ·), where A is the δ-exceptional set in the Deﬁnition 3.2.4 (1). Proof. Since Y is a Hausdorﬀ space, the singleton {x} is a closed subset of Y . This implies that {x} ∈ B(Y ) for any x ∈ Y. For x, y ∈ Y − A◦ , let us assume that Fn (s1 ) ≈ Fn (s2 ) for every Fn ∈ π, s1 ∈ μ(x), s2 ∈ μ(y). Then we obtain −1 {x, y} ⊂ ∩ (◦Fn ) ((◦Fn )(x)) | Fn ∈ π . It follows from the condition (3.2.13) and above observation that −1 ∩ (◦Fn ) ((◦Fn )(x)) | Fn ∈ π = {x} . Therefore, we get x = y. This shows that π separates the points of Y − A◦ by E (δ) (·, ·).

Lemma 3.2.8. For δ ∈ T , let E (δ) (·, ·) be a hyperﬁnite weak coercive quadratic form with continuity constant C. Suppose that E(·, ·) has a hyperﬁnite δ-quasi-continuous core π = {Fn | n ∈ N} and is δ-nearstandardly concentrated. For all η ∈ ∗ N − N, si ∈ / A2 (δ), Fˆn ∈ π ˆ , t ∈ Tδfin, we have Ω

Fˆn (X (δ) (ω, t)) Pi (dω) ≈

Ω

Fˆn (X (δ) (ω, t))1(t 0 | x(t) ∈ Y − Yn } and ζ(ω) = inf {t ≥ 0 | x(ω, t) = Δ} .

(4.1.2)

4.1 Main Result

167

ˆy ) (2) Two strong Markov processes (Ω, Π, Πt , x(t), Θy ) and (Ω, Π, Πt , x(t), Θ 2 are said to be ν-dual if for all t ∈ R+ , u, v ∈ L (Y, ν), we have Y

u(y)Θt v(y) ν(dy) =

Y

ˆt u(y) ν(dy). v(y)Θ

Let (F (·, ·), D(F )) be a coercive closed form on L2 (Y, ν) as introduced in Sect. 1.6 by replacing K with L2 (Y, ν). The inner product (·, ·)ν of K = L2 (Y, ν) is deﬁned by (f, f )ν =

Y

f 2 (y)ν(dy), α ∈ [0, ∞).

ˆ α | α ∈ (−∞, 0)} be the resolvent and Let {Rα | α ∈ (−∞, 0)} and {R co-resolvent of (F (·, ·), D(F )), respectively. Let us denote by {Tt | t ∈ (0, ∞)} and {Tˆt | t ∈ (0, ∞)} the strong continuous contraction semigroups correˆ α | α ∈ (−∞, 0)}, respectively. sponding to {Rα | α ∈ (−∞, 0)} and {R Moreover, set Fα (f, f ) = F (f, f ) + α(f, f )ν . A bounded operator R on L2 (Y, ν) is called a sub-Markovian operator if for every u ∈ L2 (Y, ν), 0 ≤ u(x) ≤ 1, ν-a.e.x ∈ Y, we have 0 ≤ Ru(x) ≤ 1, ν-a.e.x ∈ Y. A coercive closed form F (·, ·) on L2 (Y, ν) is called a Dirichlet form if it generates a semigroup {Tt | t ≥ 0} of sub-Markovian operators and a co-semigroup {Tˆt | t ≥ 0} of sub-Markovian operators. Let (F (·, ·), D(F )) be a Dirichlet form on L2 (Y, ν). For an open subset B of Y , let us deﬁne L(B) = {g ∈ D(F ) | g ≥ 1, ν-a.e. on B} .

(4.1.3)

Suppose L(B) = ∅. From Ma and Röckner [270], Proposition 1.5 of Chap. III, there exists a unique γ1 (B), γˆ1 (B) ∈ L(B) such that for all w ∈ L(B) F1 (γ1 (B), w) ≥ F1 (γ1 (B), γ1 (B)), γ1 (B), γˆ1 (B)). F1 (w, γˆ1 (B)) ≥ F1 (ˆ

(4.1.4)

From Ma and Röckner [270], Remark 1.6 and Exercise 1.7 of Chap. III, we know F1 (γ1 (B), γ1 (B)) = F1 (γ1 (B), γˆ1 (B)) = F1 (ˆ γ1 (B), γˆ1 (B)).

(4.1.5)

168

4 Construction of Markov Processes

Set Γ1 (B) = F1 (γ1 (B), γ1 (B)) if L(B) = ∅ and Γ1 (B) = ∞ if L(B) = ∅. (4.1.6) For a general subset A of Y , let us deﬁne Γ1 (A) = inf {Γ1 (B) | A ⊂ B, B is open} .

(4.1.7)

We call Γ1 (·) the 1-capacity of F (·, ·). For a closed subset B ⊂ Y, we set D(F )B = {f ∈ D(F ) | f = 0, ν-a.e. on Y − B} . It is easy to see that D(F )B is a closed set of D(F ) with the inner product F1 (·, ·). Definition 4.1.2. An increasing sequence of closed subsets {Gn | n ∈ N} of Y is called a nest if Γ1 (Y − Gn ) ↓ 0. A subset B ⊂ Y is saidto be of zero capacity if there exists a nest {Gn | n ∈ N} such that B ⊂ ∞ n=1 (Y − Gn ). A function f on Y is said to be quasi-continuous if there exists a nest {Gn | n ∈ N} such that the restriction f |Gn is continuous on Gn for each n ≥ 1. We shall say that two ν-dual strong Markov processes (Ω, Π, Πt , x(t), Θy ) ˆ y ) are properly associated with the Dirichlet form and (Ω, Π, Πt , x(t), Θ (F (·, ·), D(F )) if ˆ t f, ν-a.e.,∀f ∈ L2 (Y, ν), t > 0. Tt f = Θt f and Tˆt f = Θ The main result of this chapter is the following: Theorem 4.1.1. Assume that Y is a regular Hausdorﬀ topological space and ν is a Radon measure on Y . Let (F (·, ·), D(F )) be a Dirichlet form on L2 (Y, ν). Then there are two ν-dual tight strong Markov processes ˆ y ) properly associated with the (Ω, Π, Πt , x(t), Θy ) and (Ω, Π, Πt , x(t), Θ Dirichlet form (F (·, ·), D(F )) if and only if the following three conditions are satisﬁed: (I) There is an increasing sequence of compact subsets {Yn | n ∈ N} of Y such that n∈N D(F )Yn is F1 -dense in D(F ). (II) There exists a subset π0 of D(F ) such that π0 is dense in D(F ) with the inner product F1 (·, ·). In addition, u is quasi-continuous for each u ∈ π0 .

4.1 Main Result

169

(III) There is a countable subset π of D(F ) such that every element u ∈ π is quasi-continuous and bounded. Moreover, we have B(Y )

Yn

⊂ σ(u|u ∈ π),

n∈N

where {Yn | n ∈ N} is a sequence of compact sets in (I). Remark 4.1.1. If (F (·, ·), D(F )) satisﬁes conditions (I), (II) and (III) in Theorem 4.1.1, we call it quasi-regular. Remark 4.1.2. Previous work of Albeverio and Ma [41] and Ma and Röckner [270] has established necessary and suﬃcient conditions for the existence of ν-perfect processes associated with Dirichlet forms. In these references, the state space Y is a metrizable space and ν is a σ-ﬁnite measure on Y. Thus, there is a strictly positive function ψ ∈ L2 (Y, ν), 0 < ψ ≤ 1. Set h = R1 ψ, where {Rα | α > 0} is the resolvent of F (·, ·). The authors deﬁned the hweighted capacity as in Röckner [313] and used it as a useful tool. In our case, Y is a regular Hausdorﬀ space and m is a Radon measure. We do not have an h-weighted capacity, since this concept has not been developed in our context. Instead, we utilize the 1-capacity theory developed in Chap. 2, Fan [166], and Fukushima [175]. In order to prove Theorem 4.1.1, we will construct in Sect. 4.2 a hyperﬁnite quadratic form (E(·, ·), D(E)) from (F (·, ·), D(F )). We obtain a hyperﬁnite ˆ Markov chain X(t) associated with E(·, ·), as well as the dual X(t) of X(t). In Sect. 4.3, we will discuss the relation between the potential theory of (F (·, ·), D(F )) and the counterpart of its hyperﬁnite lifting (E(·, ·), D(E)). On this basis, we will consider the path regularity of X(t) and get its modiﬁed standard part x(t). In fact, x(t) is the wanted process. We shall show that x(t) satisﬁes all the requirements of Theorem 4.1.1 in Sect. 4.6. At last, we prove in Sect. 4.7 that if there are two ν-dual tight Markov ˆy ) properly associated processes (Ω, Π, Πt , x(t), Θy ) and (Ω, Π, Πt , x(t), Θ with F (·, ·), then x(t) satisﬁes the conditions (I), (II) and (III) of Theorem 4.1.1. Combining Propositions 4.6.1 and 4.7.1, we shall then prove the Theorem 4.1.1. Remark 4.1.3. From Albeverio and Ma [41], we know that even if ν is a nowhere Radon measure on a metric space, it is still possible to get the Markov process x(t). In order to include this case by methods of nonstandard analysis, we would have to develop ﬁrst a nonstandard measure theory associated with nowhere Radon measures on general topological spaces. After that, we could study the related stochastic analysis problems associated with a nonstandard version of nowhere Radon measure spaces. This is, however, a program which remains to be fulﬁlled.

170

4 Construction of Markov Processes

4.2 Hyperfinite Lifts of Quasi-Regular Dirichlet Forms Let Y be a Hausdorﬀ topological space. Consider the topological Borel σ-ﬁeld B(Y ) on Y . Assume that ν is a positive Borel measure on (Y, B(Y )) such that ν(K) < ∞ for every compact set K. It is a Radon measure in the sense that for all B ∈ B(Y ), ν(B) = sup {ν(K) | K ⊂ B, K compact}

(4.2.1)

and for all B ∈ B(Y ) with ν(B) < ∞, ν(B) = inf {ν(O) | B ⊂ O, O open} . We denote by (Y, B(Y ), ν) the completion of (Y, B(Y ), ν). For simplicity, we suppose that supp(ν) = Y. Let ∗ Y be the nonstandard extension of Y . Given an element a ∈ Y , we know that the monad of a is the subset of ∗ Y deﬁned by μ(a) =

{∗O | a ∈ O

and O ⊂ Y

is open} .

A point y ∈ ∗ Y is nearstandard if it belongs to μ(a) for some a ∈ Y. We shall say that it is nearstandard to a. The set of all nearstandard points is denoted by N s(∗ Y ). Since Y is Hausdorﬀ, each element y ∈ N s(∗ Y ) is nearstandard to exactly one element a ∈ Y (refer to page 48, 2.1.6. Proposition, [25]). We call a the standard part of y and denote it by st(y) or ◦ y. Let Δ be an element outside Y , YΔ = Y ∪ {Δ}. For all y ∈ ∗ Y − N s(∗ Y ), we deﬁne st(y) = Δ. Now we have got the standard part operation as a map st : ∗ Y −→ YΔ . A subset S0 of ∗ Y is called rich if it is hyperﬁnite and st(S0 ∩ N s(∗ Y )) = Y. In the following, we will construct a rich subset S0 in ∗ Y and an internal measure m on S0 such that ν = L(m) ◦ st−1 . By a ﬁnite B(Y ) partition of Y , we mean a ﬁnite collection

Bi ∈ B(Y ) 1 ≤ i ≤ n of non-empty sets with Y = ni=1 Bi and Bi ∩ Bj = ∅ if i = j. Let us denote by P the collection of all ﬁnite B(Y ) partitions of Y . If P1 and P2 are elements of P, wesay that P2 is ﬁner than P1 and we write P1 ≤ P2 if for each C ∈ P1 , C = {B ∈ P2 | B ⊂ C}.

4.2 Hyperfinite Lifts of Quasi-Regular Dirichlet Forms

171

We have the following result from 1.1 Theorem, Loeb [266]. Proposition 4.2.1. There is a partition P ∈ ∗ P such that ∗P0 ≤ P for any P0 ∈ P. That is, P ∈ ∗ P has the following properties: (i) There are an inﬁnite integer N ∈ ∗ N and an internal bijection from I = {i ∈ ∗ N | 1 ≤ i ≤ N } onto P . Thus we may write P = {Ai | i ∈ I}. (ii) If i and j are in I and i = j, then Ai = ∅, Aj = ∅ and Ai ∩ Aj = ∅. (iii) ∗ Y = {Ai | i ∈ I}. ∗ (iv) For each B ∈ B(Y ), let IB = {i ∈ I | Ai ⊂ B}. Then IB is hyperﬁnite, ∗ and B = i∈IB Ai . Hereafter, let us ﬁx a partition P = {Ai | i ∈ I} of ∗ Y with above properties (i) through (iv). For every i ∈ I, we pick up an element si ∈ Ai and deﬁne S0 = {si | i ∈ I}. Let m be the internal measure on S0 deﬁned by m({si }) = ∗ν(Ai ). Proposition 4.2.2. S0 is rich in ∗ Y and ν = L(m) ◦ st−1 . Proof. We simply refer to Albeverio et al. [25], 3.4.10 Corollary.

Let H be the hyperﬁnite dimensional space of all the internal functions f : S0 −→ ∗ R with the inner product f g dm. f, g = S0

˜ be the subspace of ∗K consisting of all Let us set K = L2 (Y, ν). Let H ˜ are functions which are constant on each class Aj , j ∈ I. Obviously, H and H ˜ By using Theorem 1.6.1, isomorphic. From now on, we will identify H with H. we can prove the following result. Proposition 4.2.3. Let Y be a Hausdorﬀ space, ν be a Radon measure on Y , and F (·, ·) be a coercive closed form on L2 (Y, ν). Then, we have (i) There exist a hyperﬁnite, rich subset S0 of ∗ Y , and an internal measure m on S0 such that H is S-dense in ∗K. (ii) There are a nonnegative quadratic form (E(·, ·), D(E)) on L2 (S0 , m) and an internal time-line T = {kΔt | k ∈ ∗ N} representing F (·, ·) in the following sense: ν = L(m) ◦ st−1 and for all u ∈ L2 (Y, ν), F (u, u) = inf {◦ E(v, v) | v is a 2-lifting of u} . ˆ s | s ∈ T } be the semigroup and Moreover, let {Qs | s ∈ T } and {Q co-semigroup generated by E(·, ·), respectively. Then for all t ∈ [0, ∞), s ∈ T, u ∈ K, v ∈ H such that t = ◦s, u = stK (v), we have

172

4 Construction of Markov Processes

ˆ s v = Tˆt u. stK Qs v = Tt u and stK Q ˆ α | α ∈ ∗ ( − ∞, 0)} On the other hand, let {Gα | α ∈ ∗ ( − ∞, 0)} and {G be the resolvent and co-resolvent generated by E(·, ·), respectively. Then for all β ∈ (−∞, 0), α ∈ ∗ ( − ∞, 0), u ∈ K, v ∈ H such that β = ◦α, u = stK (v), we have ˆαv = R ˆ β u, stK Gα v = Rβ u and stK G ˆ α | α ∈ (−∞, 0)} is where {Rα | α ∈ (−∞, 0)} is the resolvent, and {R the co-resolvent of (F (·, ·), D(F )). Proof. From Sect. 3.2 of Albeverio et al. [25], we know that each function f ˜ such that ◦f˜(x) = f (◦x) in L2 (Y, ν) has an S-square integrable lifting f˜ in H ˜ is for almost all nearstandard x. If we can show that ||∗f − f˜|| ≈ 0, then H ∗ 2 dense in (L (Y, ν)). By (4.2.1), it suﬃces to show this when f is bounded and of compact support. For such functions, the statement is an immediate consequence of Anderson’s nonstandard version of Lusin’s theorem. The proposition follows by Theorem 1.6.1. Corollary 4.2.1. If the form F (·, ·) in Proposition 4.2.3 is a Dirichlet form, we can also take E(·, ·) to be a hyperﬁnite quadratic form. Proof. We observe that from Proposition 1.5.1 it suﬃces to prove that QΔt ˆ Δt are Markov operators, where and Q ˆ Δt v QΔt u, v = u, Q and for all u, v ∈ H 1 (I − QΔt )u, v Δt 1 ˆ Δt )v. u, (I − Q = Δt

E(u, v) =

Let P be the projection of ∗K onto H. We shall write {∗ Rα | α ∈ ∗ (−∞, 0)} ∗ ˆ α | α ∈ ∗ (−∞, 0)} for ({R ˆα | α ∈ for ∗({Rα | α ∈ (−∞, 0)}) and {∗ R ˆ Δt are just (−∞, 0)}), respectively. We see from Sect. 1.6 that QΔt and Q ˆ γ ) for a suitable inﬁnitesimal Δt = − 1 , respectively. −P ∗ (γRγ ) and −P ∗ (γ R γ With the choice of H made in the proof of Proposition 4.2.3, this projection is just the conditional expectation with respect to the algebra generated by the partition P . Since conditional expectations preserve nonnegativity and decrease the supremum norm, the corollary follows. In the rest of this chapter, we shall be interested in quadratic forms generating Markov processes. That is, we shall assume that F (·, ·) is a Dirichlet

4.2 Hyperfinite Lifts of Quasi-Regular Dirichlet Forms

173

form on L2 (Y, ν). We shall use the results obtained above. First, let us introduce some notations. The inﬁnitesimal generator A and co-generator Aˆ of E(·, ·) are given by A=

1 1 ˆ Δt ), 0 < Δt ≤ 1 = 1 . (I − QΔt ) and Aˆ = (I − Q ˆ Δt Δt ||A|| ||A||

ˆ t , t ∈ T, are deﬁned by The semigroup Qt and co-semigroup Q ˆ t = (Q ˆ Δt )k Qt = (QΔt )k and Q

for each t = kΔt.

Moreover, we have the following relation from Proposition 1.6.3, F (u, u) = = = =

lim

α−→−∞

lim

−α ◦

[−α(I + αRα )P ∗u,P ∗u]

◦

[−α(I + αGα )P ∗u,P ∗u]

α→−∞ α≈−∞ α∈∗(−∞,0)

lim

α→−∞ α≈−∞ α∈∗(−∞,0)

lim

α→−∞ α≈−∞ α∈∗(−∞,0)

Y

u(y) u(y) + αRα u(y) ν(dy)

◦ (α)

[

E(u, u)]

(4.2.2)

for every u ∈ D(F ), where P is the projection of ∗K onto H. Therefore, we see from the proof of Theorem 1.6.1 that 1 (I − QΔt )P ∗u, P ∗u ≈ F (u, u), Δt i.e., ◦

E(P ∗u,P ∗u) = F (u, u) < ∞ for each u ∈ D(F ).

(4.2.3)

We recall from Deﬁnition 1.4.2 that the domain D(E) of E(·, ·) is the set of all u ∈ H such that ◦

(i) ◦ E 1 (u, u) = [E(u, u) + u, u] < ∞. (ii) For all inﬁnite α ∈ ∗(−∞, 0), E(u + αGα u, u + αGα u) ≈ 0 and E(u + ˆ α u). ˆ α u, u + αG αG We remark that E(·, ·) does not necessarily satisfy the hyperﬁnite weak sector condition. Hence, the results of Proposition 1.4.2 need not hold. However, it is easy to see from the proof of Proposition 1.4.2 the following: if ◦ E(u, u) < ∞ and for all inﬁnite α < 0, (α) E(u, u) ≈ E(u, u) and (α) ˆ E(u, u) ≈ E(u, u), then we have u ∈ D(E).

174

4 Construction of Markov Processes

Therefore, we have proved the following using (4.2.2) and (4.2.3): Lemma 4.2.1. For every u ∈ D(F ), we have P ∗u ∈ D(E). ˆ Now, we are in the position to construct the dual processes X(t) and X(t) and their transition matrices associated with E(·, ·). Remember that m is an internal measure on the hyperﬁnite set S0 = {s1 , s2 , · · · , sN }. We shall write mi for m(si ) if it is convenient. Fix j ∈ I. Let uj : S0 −→ ∗ R be the function satisfying uj (sl ) = δlj , l ∈ I. We deﬁne ˆ Δt uj (si ), j ∈ I. bij = QΔt uj (si ) and ˆbij = Q It is easy to see that for any internal function u : S0 −→ ∗ R, we have u(·) =

N

u(sj )uj (·).

j=1

Hence, we obtain QΔt u(si ) =

N

u(sj )QΔt uj (si )

j=1

=

N

bij u(sj ),

j=1

⎛ ⎞ N 1 ⎝ Au(si ) = u(si ) − u(sj )bij ⎠ . Δt j=1 Similarly, we have ˆ Δt u(si ) = Q

N

ˆ Δt uj (si ) u(sj )Q

j=1

=

N

ˆbij u(sj ),

j=1

⎛ ⎞ N 1 ˆ i) = ⎝u(si ) − Au(s u(sj )ˆbij ⎠ . Δt j=1 Let us denote u(si ) by u(i). Then, we have

4.2 Hyperfinite Lifts of Quasi-Regular Dirichlet Forms

175

N

1 Au(i)v(i)m(i) Δt i=1 ⎡ ⎤ N N 1 ⎣ u(i) − bij u(j)⎦ v(i)m(i) = Δt i=1 j=1

E(u, v) =

N

N

N

N

=

1 (δij − bij )u(j)v(i)m(i) Δt i=1 j=1

=

1 (δij − ˆbij )v(j)u(i)m(i). Δt i=1 j=1

ˆ Δt , we have for all i, j ∈ I, Since the duality of QΔt and Q m(i)bij = m(j)ˆbji . Recall that s0 is a point outside S0 , and we put S = S0 ∪ {s0 }. We assume m({s0 }) = 0 and st(s0 ) = Δ. Let Q = {qij | 0 ≤ i, j ≤ N } be an (N + 1) × (N + 1) matrix satisfying the following conditions: q00 = 1, q0i = 0 for i ∈ I, qij = bij , i, j ∈ I, qi0 = 1 −

N

bij , i ∈ I.

j=1

It is easy to see that N

qij = 1 for all i ∈ I ∪ {0} .

j=0

We construct a hyperﬁnite Markov chain {X(t) | t ∈ T } associated with Q and m in the following manner. Let Ω be the set of all internal functions ω : T −→ S. Let X be the coordinate function X(ω, t) = ω(t). We take P to be the measure generated by P ([ω]0 ) = m(X(ω, 0)) and P ([ω]kΔt ) = m {X(ω, 0)}

k−1 n=0

q(ω(nΔt), ω((n + 1)Δt)),

176

4 Construction of Markov Processes

where q(si , sj ) is just qij and [ω]t is deﬁned by [ω]t = {ω | X(ω , s) = X(ω, s) for all s ≤ t} . Let Ft be the internal algebra on Ω generated by the sets [ω]t , ω ∈ Ω. Let us deﬁne the probability measure Pi as follows: Pi ([ω]kΔt ) = δiω(0)

k−1

q(ω(nΔt), ω((n + 1)Δt)).

n=0

It is easy to verify that the following Markov property of X(t) holds. If X(ω, t) = si , then we have P

ω ∈ [ω]t X(ω , t + Δt) = sj = qij P ([ω]t ).

ˆ = {ˆ Similarly, let Q qij | 0 ≤ i, j ≤ N } be an (N + 1) × (N + 1) matrix satisfying the following conditions: qˆ00 = 1, qˆ0i = 0 for i ∈ I, qˆij = ˆbij , i, j ∈ I, qˆi0 = 1 −

N

ˆbij , i ∈ I.

j=1

It is easy to see that N

qˆij = 1 for all i ∈ I ∪ {0} .

j=0

ˆ We may construct the dual hyperﬁnite Markov chain {X(t) = X(t) | t ∈ T } and the corresponding internal measure Pˆ , Pˆi , i ∈ I. If δ ∈ T , let Tδ be the sub-line Tδ = {kδ | k ∈ ∗ N0 } . (δ)

We write X (δ) for the restriction X|Tδ . For each t ∈ Tδ , let Ft be the internal algebra on Ω generated by the sets

(δ) [ω]t = ω ∈ Ω X (δ) (ω , s) = X (δ) (ω, s) for all s ∈ Tδ , s ≤ t .

4.3 Relation with Capacities

177 (δ)

In the following of this chapter, we shall use the notations A(δ) , qij , E (δ) (·, ·), and D(E (δ) ) introduced in Sect. 1.5. For δ ∈ T, α ∈ ∗ R, α ≥ 0, let us introduce the notation Eα(δ) (u, u) = E (δ) (u, u) + αu, u.

4.3 Relation with Capacities Let (F (·, ·), D(F )) be a Dirichlet form on L2 (Y, ν). We consider the 1-capacity Γ1 (·) deﬁned in the relations (4.1.3), (4.1.4), (4.1.5), (4.1.6), and (4.1.7) in Sect. 4.1. We get the following results from Ma and Röckner [270], Chap. III. Lemma 4.3.1. Let B be an open subset of Y such that L(B) = ∅, where L(B) is deﬁned by the deﬁnition (4.1.3) in Sect. 4.1. Then we have the following results: (1) There exists a unique element γ1 (B) ∈ L(B) such that Γ1 (B) = F1 (γ1 (B), γ1 (B)).

(4.3.1)

(2) 0 ≤ γ1 (B) ≤ 1, ν-a.e. and γ1 (B) = 1, ν-a.e. on B. Proposition 4.3.1. Γ1 (·) has the properties: (1) (2) (3) (4)

A ⊂ B =⇒ Γ1 (A) ≤ Γ1 (B). An ↑=⇒ Γ1 (∪n∈N An ) = supn∈N Γ1 (An ). A, B open =⇒ Γ 1 (A ∪ B) + Γ1 (A ∩ B) ≤ Γ1 (A) + Γ1 (B). Γ1 (∪n∈N An ) ≤ n∈N Γ1 (An ).

In the same way as in Sects. 2.2 and 2.3, we introduce the capacity theory associated with (E (δ) (·, ·), D(E (δ) )) and state some properties without proof. For δ ∈ T, α ∈ ∗ R+ , and an internal subset A of S0 , we deﬁne (δ)

σA (ω) = min {t ∈ Tδ | X(ω, t) ∈ A} , (δ) −σA /δ e(δ) , (A)(i) = E i (1 + αδ) α L(A) = {G | S0 −→ ∗ R is internal and G|A = 1A } . (δ)

Using the terminology of Sects. 2.2 and 2.3, we call eα (A) the equilibrium α-potential of A associated with E (δ) (·, ·). Let us set (δ) (δ) e(δ) Cap(δ) α (A) = Eα α (A), eα (A) .

178

4 Construction of Markov Processes

For an arbitrary subset B of S0 , we deﬁne

(δ) (δ) Capα (B) = inf Capα (A) B ⊂ A, A is internal . (δ) We call Cap(δ) (·, ·). We have studied α (·) the α-capacity associated with E the potential theory of hyperﬁnite quadratic forms in Chap. 2. We shall use the same notations of Chap. 2 in the following.

We ﬁrst remark that the weak sector condition ! ! |F1 (x, y)| ≤ C F1 (x, x) F1 (y, y) for all x, y ∈ D(F ) is equivalent to the following: ! ! |F (x, y)| ≤ C F1 (x, x) F1 (y, y) for all x, y ∈ D(F ), for some nonnegative constant C ∈ [0, ∞). Obviously, we can always suppose that C = C . Hence, we have ! ! (4.3.2) |F (x, y)| ≤ C F1 (x, x) F1 (y, y) for all x, y ∈ D(F ). Proposition 4.3.2. Let C satisfy the inequality (4.3.2). For all open subset B ⊂ Y, we have for all δ ∈ T, δ ≈ 0 the following (δ)

Cap1 (∗B ∩ S0 ) ≤ (C + 1)2 Γ1 (B). Proof. For simplicity, we may suppose that L(B) = ∅. From Lemma 4.3.1, there is an element γ1 (B) ∈ D(F ) satisfying (4.3.1). For convenience, let us assume that γ1 (B)(x) = 1 for all x ∈ B. For all δ ∈ T, δ ≈ 0, we have from Proposition 2.2.1 (δ)

(δ)

(δ)

(δ)

(δ)

(δ)

Cap1 (∗B ∩ S0 ) = E1 (e1 (1∗B ), e1 (1∗B )) = E1 (e1 (∗B), γ1 (B)).

(4.3.3)

On the other hand, let γ be given by (1.6.5) in the proof of Theorem 1.6.1, Chap. 1. Then, we have (δ)

(δ)

E (δ) (e1 (∗B), γ1 (B)) = −γ(I + ∗(γRγ ))e1 (∗B), γ1 (B) (δ)

= (γ) F (e1 (∗B), γ1 (B)). From the inequality (4.3.2) and Corollary 1.6.1 (i), we get

(4.3.4)

4.3 Relation with Capacities

179

(δ)

(γ)

(δ)

F (e1 (∗B), γ1 (B)) = F ((−γRγ )e1 (∗B), γ1 (B)) " ! (δ) (δ) ≤ C F1 ((−γRγ )e1 (∗B), (−γRγ )e1 (∗B)) F1 (γ1 (B), γ1 (B)).

(4.3.5)

From Corollary 1.6.1 (ii), we have (δ)

(δ)

F1 ((−γRγ )e1 (∗B), (−γRγ )e1 (∗B)) (δ)

(δ)

(δ)

(δ)

= F ((−γRγ )e1 (∗B), (−γRγ )e1 (∗B)) + (−γRγ )e1 (∗B), (−γRγ )e1 (∗B) (δ)

(δ)

(δ)

(δ)

≤ (γ) F (e1 (∗B), e1 (∗B)) + e1 (∗B), e1 (∗B) (δ)

(δ)

(δ)

(δ)

= −γ(I + ∗(γRγ ))e1 (∗B), e1 (∗B) + e1 (∗B), e1 (∗B) (δ)

(δ)

(δ)

= E1 (e1 (∗B), e1 (∗B)).

(4.3.6)

Therefore, we have from the relations (4.3.3), (4.3.4), (4.3.5), and (4.3.6) that ◦

◦ (δ) (δ) (δ) (δ) Cap1 (∗B ∩ S0 ) = E1 (e1 (1∗B ), e1 (1∗B )) ≤ (C + 1)2 F1 (γ1 (B), γ1 (B)).

This ﬁnishes the proof of Proposition 4.3.2.

Corollary 4.3.1. Let A ⊂ Y satisfy the condition Γ1 (A) = 0. Then for all (δ) δ ∈ T, δ ≈ 0, we have Cap1 (∗A ∩ S0 ) ≈ 0.

Proof. The proof follows easily from Proposition 4.3.2.

Corollary 4.3.2. Let {Gn | n ∈ N} be a sequence of increasing closed subsets of Y satisfying Γ1 (Y − Gn ) ↓ 0. Then for all δ ∈ T, δ ≈ 0, we have

◦

(δ) Cap1

S0 −

∞

∗

( Gn ∩ S0 )

= 0.

(4.3.7)

n=1

Proof. From Proposition 4.3.2, we have ◦

(δ) Cap1 (S0 − ∗Gn ∩ S0 ) =

◦

(δ) Cap1 (∗(Y − Gn ) ∩ S0 )

≤ (C + 1)2 Γ1 (Y − Gn ) −→ 0, n → ∞. Therefore, (4.3.7) holds.

Proposition 4.3.3. Let {Gn | n ∈ N} be a sequenceof increasing closed ∞ subsets of Y satisfying Γ1 (Y − Gn ) ↓ 0. Then S0 − n=1 ∗Gn ∩ S0 is δexceptional for all δ ∈ T, δ ≈ 0. Proof. The proof follows easily from Corollary 4.3.2 and Theorem 2.4.1.

180

4 Construction of Markov Processes

4.4 Path Regularity of Hyperfinite Markov Chains Similarly as in Deﬁnition 3.2.1, we introduce the following Definition 4.4.1. Let Z be a subset of Y . We say that a subset π of L2 (Y, ν) separates points of Z, if for any two diﬀerent points x and y in Z, there exists u ∈ π such that u(x) = u(y). Lemma 4.4.1. If the hypothesis (III) in Theorem 4.1.1 holds, then π separates the points of n∈N Yn . Proof. The proof can be carried through in the same way as the proof of Lemma 3.2.7. Lemma 4.4.2. Assume that Y is a regular Hausdorﬀ space and the hypothesis (III) of Theorem 4.1.1 holds. Let δ ∈ T, δ ≈ 0. For every ω ∈ Ω, if the path X (δ) (ω, t) fails to have an S-left or S-right limit at some t < τ (δ) (ω), then so does (P ∗u)(X (δ) (ω, t)) for some u ∈ π, where the stopping time τ (δ) (ω) is deﬁned by # $ / (∗ Yn ∩ S0 ) . τ (δ) (ω) = inf ◦ t X (δ) (ω, t) ∈ n∈N

Proof. Let us set Bn = ∗ Yn ∩ S0 , n ∈ N and B = ∪n∈N Bn . For each n ∈ N, we deﬁne

(δ) (δ) / Bn , τn (ω) = min t ∈ T X (ω, t) ∈

◦ (δ) ζδ (ω) = inf t X (ω, t) ∈ / S0 , i.e., ζδ is the lifetime of X (δ) (ω, t). It is easy to show that τ (δ) (ω) ≤ ζδ (ω),

(δ) ◦ (δ) τ (ω) = sup τn (ω) n ∈ N . Let ω be a ﬁxed point in Ω. Fix t < τ (δ) (ω), t ∈ [0, ∞). Then, there is an (δ) element N ∈ N such that t < τN (ω). Given a sequence {tn | n ∈ N} from T ◦ such that the standard part tn increases strictly to t, we can show that the sequence

◦ (δ) X (ω, tn ) n ∈ N has a cluster point in YN in same way as the proof of Lemma 3.2.1. The rest follows in a similar way.

4.5 Quasi-Continuity and Nearstandard Concentration

181

Proposition 4.4.1. Assume that Y is a regular Hausdorﬀ space and the hypothesis (III) of Theorem 4.1.1 holds. Let δ ∈ T, δ ≈ 0. Then there exists a δ-exceptional set A0 (δ) such that for all si ∈ S0 − A0 (δ), the hyperﬁnite Markov chain X (δ) (t) has S-left and S-right limits at all t < ζδ , L(Pi )-a.e. Proof. The result can be proved by using the method of Proposition 3.2.1, and the results of Lemmas 4.2.1 and 4.4.2.

4.5 Quasi-Continuity and Nearstandard Concentration Let (F (·, ·), D(F )) be a Dirichlet form on L2 (Y, ν). First of all, we present some properties of E(·, ·). Lemma 4.5.1. Let {fk | k ∈ N} be a sequence of quasi-continuous functions on Y . Then there exists a nest {Gn | n ∈ N} such that for all k ∈ N, fk ∈ C({Gn }), where C({Gn }) = {f | f |Gn

is continuous for each

n ∈ N} .

Proof. The proof is similar to the one in Fukushima [175] Theorem 3.1.2. Lemma 4.5.2. (1) If the condition (I) in Theorem 4.1.1 is satisﬁed, we have F1 (u − un , u − un ) −→ 0

as

n −→ ∞, ∀u ∈ D(F ),

where un ∈ D(F )Yn is the projection of u. (2) Suppose that the condition (II) in Theorem 4.1.1 holds. Then, every element f ∈ D(F ) admits a quasi-continuous version. Proof. (1) Let F (·, ·) be the symmetric part of F (·, ·), i.e., F (u, v) =

1 F (u, v) + F (v, u) , u, v ∈ D(F ). 2

Set F 1 (u, v) = F (u, v) + (u, v), u, v ∈ D(F ). For every u ∈ D(F ), we have the following decomposition with respect to the inner product F 1 (·, ·) u = (u − un ) + un , un ∈ D(F )Yn , un − u⊥D(F )Yn , ∀n ∈ N. Let vn = u − un . Then, we have

182

4 Construction of Markov Processes

F 1 (vn , vn+m ) = F 1 (vn+m , vn+m ) + F 1 (vn − vn+m , vn+m ) = F 1 (vn+m , vn+m ) + F 1 (un+m − un , u − un+m ) = F 1 (vn+m , vn+m ) + F 1 (−un , u − un+m ). Since un ∈ D(F )Yn ⊂ D(F )Yn+m , F 1 (−un , u − un+m ) = 0. Hence, we have F 1 (vn , vn+m ) = F 1 (vn+m , vn+m ). Therefore, we have F1 (vn − vn+m , vn − vn+m ) = F1 (vn , vn ) − F1 (vn+m , vn+m ) −→ 0, n → ∞. This means that {vn | n ∈ N} is a F1 (·, ·) Cauchy sequence. Hence, there is an element v∞ ∈ D(F ) such that F1 (vn − v∞ , vn − v∞ ) −→ 0, n −→ ∞. It is easy to see that

F1 (v∞ , v) = 0, ∀v ∈

D(F )Yn .

n∈N

Since n∈N D(F )Yn is F1 -dense in D(F ), we know that v∞ = 0. This proves (1). (2) For this proof, we refer to Albeverio and Ma [41], 4.1 Lemma and Fukushima [175], Theorem 3.1.3. Lemma 4.5.3. Suppose that the condition (I) in Theorem 4.1.1 is satisﬁed. Then (F1 (·, ·), D(F )) is a separable Hilbert space. Proof. Since Yn is compact, (F1 (·, ·), D(F )Yn ) is separable. Hence (F (·, ·), D(F )) is separable. Lemma 4.5.4. Suppose that the hypotheses (I), (II) and (III) in Theorem 4.1.1 hold. Then, there exists a countable set π ˆ such that (1) (2) (3)

π⊂π ˆ ⊂ D(F ) ∩ Qb (Y ), u, v ∈ π ˆ , a ∈ Q =⇒ |u| ∈ π ˆ, u + v ∈ π ˆ , uv ∈ π ˆ,

(4.5.1)

av ∈ π ˆ and u ∧ a ∈ π ˆ,

(4.5.2)

π ˆ is F1 -dense in D(F ),

where Q = {a1 , a2 , · · · , an , · · · } is the set of all rational numbers and Qb (Y ) is the space of all bounded quasi-continuous functions deﬁned on Y. Moreover, ˆ , u ∈ C({Zn }). there is a nest {Zn | n ∈ N} such that for every u ∈ π

4.5 Quasi-Continuity and Nearstandard Concentration

183

Proof. From Lemmas 4.5.2 and 4.5.3, we can assume that π is F1 -dense in D(F ). Deﬁne the maps Λ = {λ−2 , λ−1 , λ0 , λ1 , λ2 , · · · } from [D(F ) ∩ Qb (Y )] × [D(F ) × Qb (Y )] into D(F ) ∩ (Qb (Y ) by λ−2 (u, v) = |u|, λ−1 (u, v) = u + v, λ0 (u, v) = uv, λ2i−1 (u, v) = ai u, λ2i (u, v) = u ∧ ai , i = 1, 2, · · · Applying Lemma 3.2.4 to π and Λ, we get a countable set π ˆ satisfying the relations (4.5.1) and (4.5.2). Obviously, π ˆ is F1 -dense in D(F ). The existence of {Zn | n ∈ N} follows from Lemma 4.5.1. Lemma 4.5.5. Assume that the hypotheses (I), (II) and (III) in Theorem 4.1.1 hold. For any two ﬁnite measures ν1 and ν2 on (Y, B(Y )) satisfying the following conditions: u(y) ν1 (dy) = u(y) ν2 (dy) for all u ∈ π ˆ, Y

Y

where π ˆ is the countable set obtained in the Lemma 4.5.4, we have that ν1 and ν2 coincide on (Y, σ(u|u ∈ π ˆ )). Proof. The proof is similar to the one of Lemma 3.2.6.

We recall that we have introduced the hyperﬁnite quasi-continuity and related concepts of (E(·, ·), D(E)) in Sect. 3.2. We then have Proposition 4.5.1. Let us assume that f : Y −→ R is a quasi-continuous function. Then for all δ ∈ T, δ ≈ 0, P ∗f is hyperﬁnite δ-quasi-continuous. Proof. The proof follows easily from Corollary 4.3.2 and Theorem 2.4.1.

Corollary 4.5.1. Let us assume that the condition (II) in Theorem 4.1.1 holds. Then for every element f ∈ D(F ) and all inﬁnitesimal δ ∈ T , there is a δ-exceptional set A(δ) ⊂ S0 such that (P ∗f )(si ) ≈ (P ∗f )(sj ) for all inﬁnitely close si , sj ∈ S 0 − A(δ). Proof. From Lemma 4.5.2 (2), there is a quasi-continuous version f˜ of f. Hence, P ∗f˜ is hyperﬁnite δ-quasi-continuous, ∀δ ∈ T, δ ≈ 0. Since P ∗f ≈ P ∗f˜, we prove Corollary 4.5.1 by using Lemma 2.1.1 (i).

184

4 Construction of Markov Processes

Proposition 4.5.2. Let Y be a regular Hausdorﬀ space. If the conditions (I), (II), and (III) in Theorem 4.1.1 are satisﬁed, then there exist a δ0 -exceptional set A1 (δ0 ) of X(t) for some inﬁnitesimal δ0 and some β ∈ Tδ0 , 12 < β ≤ 1 such that (i) For all si ∈ S0 − A1 (δ0 ), the path X (δ0 ) (ω, t) has S-right and S-left limits at all t < ζδ0 (ω), L(Pi )-a.e.ω. ˆ β) = ∞ T2−n β . For all inﬁnitely close si , sj ∈ S 0 − (ii) Let us set Q(δ, n=1 ˆ 0 , β), u ∈ π ˆ , we have A1 (δ0 ), s ∈ Q(δ & ◦ (P T st(s) u)(si ) = (P ∗ u)(X (δ0 ) (s)) Pi (dω) Ω ◦ = (P ∗ u)(X (δ0 ) (s)) Pj (dω).

◦%

∗

(4.5.3)

Ω

ˆ , ◦ [P ∗ (T t u)(si )] : [0, ∞) −→ R is continu(iii) For all si ∈ S0 − A1 (δ0 ), u ∈ π ous with respect to t. Moreover, for all si ∈ S0 − A1 (δ0 ), u ∈ π ˆ , s ∈ Tδfin , 0 we have P ∗ (T st(s) u)(si ) ≈ Qs (P ∗ u)(si ). Proof. Step 1. For s ∈ T fin, let t = st(s). If u ∈ π ˆ , P ∗ u is a 2-lifting of u. s ∗ Therefore, Q (P u) is a 2-lifting of Tt u from Proposition 4.2.3. For all k ∈ N, we have

1 1 (4.5.4) < . μ si ∈ S0 |Qs (P ∗ u)(si ) − P ∗ (Tt u)(si )| > k k Therefore, (4.5.4) holds for some inﬁnite k ∈ ∗ N − N also. This means that

si Qs (P ∗ u)(si ) ≈ P ∗ (Tt u)(si ) is contained in an internal set of inﬁnitesimal measure. Having this in mind, we shall show our proposition step by step. Step 2. Suppose that π ˆ = {un | n ∈ N}. Let {P ∗ un | n ∈ ∗ N} be an ∗ internal extension of {P un | n ∈ N}. For ε ∈ T 1 , N ∈ ∗ N, deﬁne B1 (ε) =

% & si ∃s ∈ Tεfin Qs (P ∗ un )(si ) ≈ P ∗ (Tst(s) un )(si ) ,

n∈N

B1 (ε, N ) =

N % & si ∃s ∈ TεN Qs (P ∗ un )(si ) ≈ P ∗ (Tst(s) un )(si ) ,

n=1

ˆ1 (ε, N ) = B

(4.5.5)

N 1 N s ∗ ∗ si ∃s ∈ Tε |Q (P un )(si ) − P (Tst(s) un )(si )| > . N n=1

4.5 Quasi-Continuity and Nearstandard Concentration

185

Fix ε ≈ 0, ε ∈ T 1 . For each N ∈ N, we have ˆ1 (ε, N )) < μ(B Consider the following internal set 1 ˆ1 (ε, N (ε)) < ε ∈ T μ(B

1 N (ε)

1 . N

1 , N (ε) = + 1. ε

By saturation, there is an inﬁnitesimal ε1 ∈ T 1 such that ˆ1 (ε1 , N (ε1 )) < μ B

1 . N (ε1 )

In view of our Lemma 2.1.1, there is an inﬁnitesimal δ1 ∈ Tε1 such that ˆ1 (ε1 , N (ε1 )) is δ-exceptional for all δ ≥ δ1 , δ ∈ Tε1 . Therefore, B ˆ1 (δ, N (δ)) B is δ-exceptional for all δ ∈ Tε1 , δ ≥ δ1 . Noticing that ˆ 1 (δ, N (δ)), B1 (δ) ⊂ B1 (δ, N (δ)) ⊂ B we know that B1 (δ) is δ-exceptional for all δ ∈ Tε1 , δ ≥ δ1 . Step 3. We take n0 ∈ ∗ N − N satisfying ∞ ˆ Q(β) = n=1 T2−n β . Deﬁne B2 =

u∈ˆ π

1 2

< 2n0 δ1 ≤ 1. Let β = 2n0 δ1 and

% s ∗ & ˆ Q (P u)(si ) ≈ Qs (P ∗ u)(sj ) for some sj ≈ si . si ∈ S 0 ∃s ∈ Q(β)

Since P ∗ (Tst(s) u) is hyperﬁnite δ-quasi-continuous for all δ ∈ T1 , δ ≈ 0, and Qs (P ∗ u) ≈ P ∗ (Tst(s) u), it follows from Lemma 2.1.1 that there is an inﬁnitesimal δ2 ∈ T such that B2 is δ-exceptional for all δ ≥ δ2 , δ ∈ T. ˆ n0 −m0 δ1 ≈ 0 Step 4. We pick an inﬁnite number m0 ≤ n0 such that δ0 =2 and δ0 ≥ δ2 . Let A0 (δ0 ) be the δ0 -exceptional set obtained in Proposition 4.4.1. Then, there exists a δ0 -exceptional set A1 (δ0 ) which contains A0 (δ0 ), B1 (δ0 ) and B2 . Obviously, A1 (δ0 ) satisﬁes (i). For si ∈ S0 − A1 (δ0 ), u ∈ π ˆ , s ∈ Tδfin , we have from the relations (4.5.5) that 0 Qs (P ∗ u)(si ) ≈ P ∗ (T st(s) u)(si ), ◦

since Tδ0 ⊂ Tδ1 ⊂ Tε1 . Thus, (P ∗ (Tt u)(si )) : [0, ∞) −→ R is continuous. This completes the proof of Proposition 4.5.2.

186

4 Construction of Markov Processes

Lemma 4.5.6. Suppose that the condition (I) in Theorem 4.1.1 holds. For every u ∈ D(F ), s ∈ [0, ∞), the set # B(u, s) =

si ∈ S0 −

$ ∗ ( Yn ∩ S0 ) P (Ts u)(si ) ≈ 0 ∗

n∈N

is Δt-exceptional. Proof. From Lemma 4.5.2, we have F1 (u − un , u − un ) −→ 0 as

n −→ ∞, ∀u ∈ D(F ),

where un ∈ D(F )Yn is the projection of u. For each k ∈ N, let us take nk ∈ N such that δk =

! F1 (Ts u − (Ts u)nk , Ts u − (Ts u)nk ) < 2−k ,

where (Ts u)n ∈ D(F )Yn is the projection of Ts u. For simplicity, we may suppose that (Ts u)nk |Y −Ynk = 0. Deﬁne ! ∗ ∗ Bk (u, s) = si ∈ S0 − Ynk ∩ S0 |P (Ts u)(si )| ≥ δk , k ∈ N. Noticing that |P ∗ (Ts u)(si ) − P ∗ (Ts u)nk (si )| ≥

! δk , si ∈ Bk (u, s),

we have P ω ∃t ≤ 1 (X(ω, t) ∈ Bk (u, s)) ! ≤ P ω ∃t ≤ 1 |P ∗ (Ts u)(X(ω, t)) − P ∗ (Ts u)nk (X(ω, t))| ≥ δk ! ≤ P ω ∃t ≤ 1 P ∗ (Ts u)(X(ω, t)) − P ∗ (Ts u)nk (X(ω, t)) ≥ δk ! +P ω ∃t ≤ 1 P ∗ (Ts u)(X(ω, t)) − P ∗ (Ts u)nk (X(ω, t)) ≤ − δk . (4.5.6) Let us set A = {i ∈ S0 | P ∗ (Ts u)(si ) − P ∗ (Ts u)nk (si ) ≥ σA (ω) = min{t ∈ T | X(t) ∈ A}, −σ /Δt e1 (A)(i) = Ei (1 + Δ) A .

! δk },

4.5 Quasi-Continuity and Nearstandard Concentration

187

Then, we have ! ∗ ∗ P ω ∃t ≤ 1 P (Ts u)(X(ω, t)) − P (Ts u)nk (X(ω, t)) ≥ δk = P (ω|σA (ω) ≤ 1)

= Pi ω (1 + Δt)−σA /Δt ≥ (1 + Δt)−1/Δt dm(i) S 0 −σ /Δt 1/Δt Ei (1 + Δt) A (1 + Δt) dm(i) ≤ S0 1/Δt e1 (A)(i) dm(i) = (1 + Δt) S0

1/Δt

≤ (1 + Δt)

E1 (e1 (A), e1 (A)),

(4.5.7)

where the reason for the last step holding is (2.3.1) in Sect. 2.3. From Theorem 2.2.1, we have 1 ∗ ∗ E1 (e1 (A), e1 (A)) ≤ E1 e1 (A), √ (P (Ts u) − P (Ts u)nk ) δk 1 (γ) ∗ ∗ =F e1 (A), √ (P (Ts u) − P (Ts u)nk ) δk 1 +√ e1 (A)(i) (P ∗ (Ts u) − P ∗ (Ts u)nk ) (i)dm(i) δk S 0 1 ∗ ∗ = F (−γRγ )e1 (A), √ (P (Ts u) − P (Ts u)nk ) δk 1 +√ e1 (A)(i) (P ∗ (Ts u) − P ∗ (Ts u)nk ) (i)dm(i), δk S 0 (4.5.8)

where γ is given by (1.6.5) in the proof of Theorem 1.6.1. From the inequality (4.3.2) in Sect. 4.3, we have F

1 (−γRγ )e1 (A), √ (P ∗ (Ts u) − P ∗ (Ts u)nk ) δk " 1 ≤ C√ F1 ((−γRγ )e1 (A), (−γRγ )e1 (A)) δk ! × F1 (P ∗ (Ts u) − P ∗ (Ts u)nk , P ∗ (Ts u) − P ∗ (Ts u)nk ).

(4.5.9)

In the same way as for the relation (4.3.6) in Sect. 4.3, we can show that F1 ((−γRγ )e1 (A), (−γRγ )e1 (A)) ≤ E1 (e1 (A), e1 (A)).

188

4 Construction of Markov Processes

Hence, we have from the inequality (4.5.9) that F

1 (−γRγ )e1 (A), √ (P ∗ (Ts u) − P ∗ (Ts u)nk ) δk 1 ! ≤ C√ E1 (e1 (A), e1 (A)) δk ! × F1 (P ∗ (Ts u) − P ∗ (Ts u)nk , P ∗ (Ts u) − P ∗ (Ts u)nk ). (4.5.10)

From the relations (4.5.8) and (4.5.10), we have 1 ! E1 (e1 (A), e1 (A)) ≤ (C + 1) √ E1 (e1 (A), e1 (A)) δk ! × F1 (P ∗ (Ts u) − P ∗ (Ts u)nk , P ∗ (Ts u) − P ∗ (Ts u)nk ). Therefore, we have E1 (e1 (A), e1 (A)) ≤

% & (C + 1)2 F1 P ∗ (Ts u) − P ∗ (Ts u)nk , P ∗ (Ts u) − P ∗ (Ts u)nk δk

≤ (C + 1)2 δk .

(4.5.11)

From the relations (4.5.7) and (4.5.11), we have ! P ω ∃t ≤ 1 P ∗ (Ts u)(X(t)) − P ∗ (Ts u)nk (X(t)) ≥ δk ≤ (1 + Δt)

1/Δt

(C + 1)2 δk .

(4.5.12)

Similarly, we have ! ∗ ∗ P ω ∃t ≤ 1 P (Ts u)(X(ω, t)) − P (Ts u)nk (X(ω, t)) ≤ − δk ≤ (1 + Δt)

1/Δt

(C + 1)2 δk .

(4.5.13)

From the relations (4.5.6), (4.5.12), and (4.5.13), we have 1/Δt (C + 1)2 δk P ω ∃t ≤ 1 X(ω, t) ∈ Bk (u, s) ≤ 2 1 + Δt 1/Δt ≤ 2 1 + Δt (C + 1)2 2−k . (4.5.14)

4.5 Quasi-Continuity and Nearstandard Concentration

189

For each K ∈ N, we have B(u, s) ⊂ ∞ k=K Bk (u, s). It follows from the inequality (4.5.14) that B(u, s) is Δt-exceptional. Lemma 4.5.7. Suppose that the condition (I) in Theorem 4.1.1 is satisﬁed. Then, there exists a Δt-exceptional set A2 containing all B(u, s), u ∈ π ˆ, s ∈ Q+ , where Q+ is the family of all positive rational numbers. Proof. This follows easily from Lemma 2.1.1 and Lemma 4.5.6.

In Sect. 4.4, we have deﬁned the lifetime ζδ of X (δ) (ω, t). That is, for δ ∈ T , we have ζδ (ω) = inf ◦ t | X (δ) (ω, t) ∈ / S0 . Now, we deﬁne the right standard part ◦X (δ)+ of X (δ) as follows. If t < ζδ (ω), let ◦

X (δ)+ (ω, t) = S- lim X (δ) (ω, s) s↓t

whenever this limit exists and for all t1 < t, the S-lim X (δ) (ω, s) exist also, s↓t1

and ◦X (δ)+ (ω, t) = Δ else. If t ≥ ζδ (ω), we deﬁne ◦X (δ)+ (ω, t) = Δ. Proposition 4.5.3. Assume that Y is a regular Hausdorﬀ space. If the conditions (I), (II), and (III) in Theorem 4.1.1 are satisﬁed, then there exists a properly δ0 -exceptional set A(δ0 ) of X(t) for some inﬁnitesimal δ0 and some β ∈ Tδ0 , 12 < β ≤ 1 such that (i) For all si ∈ S0 − A(δ0 ), the path X (δ0 ) (ω, t) has S-right and S-left limits at all t < ζδ0 (ω), L(Pi )-a.e.ω. ˆ , t ∈ Tδfin , we have (ii) For all inﬁnitely close si , sj ∈ S 0 − A(δ0 ), u ∈ π 0

◦

u( X

(δ0 )

Ω

(ω, t)) L(Pi )(dω) = = =

Ω Ω Ω

u(◦X (δ0 ) (ω, t))1(t0 and t ∈ T , ) (

E eε|Xt | = n∈∗ N

≤

eε|Xt | dμ

{2(n−1)k≤|Xt | |Xt |p on {|Xt | > N0 }, we conclude that E[|Xt |p ] is ﬁnite. Applying Lemma (5.2.4) completes the proof of the theorem.

5.2 Characterizing Hyperfinite Lévy Processes

213

We introduce the following quantities, which may be called normalized increment mean and normalized increment variance, respectively. Definition 5.2.3. mX =

1 1 E [ΔX0 ] = apa , Δt Δt

σX 2 =

a∈A

) 1 2 1 ( 2 E |ΔX0 | = |a| pa . Δt Δt a∈A

Lemma 5.2.5. [263, Lemma 1.2, Corollary 2.4] •

For all t ∈ T and arbitrary X, ) ( 2 2 E |Xt | = σX 2 · t + |mX | · t (t − Δt) .

•

Moreover, if X is a hyperﬁnite random walk with ﬁnite increments, then X will be a hyperﬁnite Lévy process if and only if both mX and σX 2 are ﬁnite.

Proof. For the ﬁrst part of the Lemma, observe that whenever s = u, ΔXs and ΔXu are independent and therefore 2

2

E [ΔXs · ΔXu ] = E [ΔXs ] · E [ΔXu ] = |E [ΔX0 ]| = |mX | Δt2 , whence this quantity is independent of u, s. This yields ) ( E |Xt |2 = E =

&

s

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close