Symmetric Markov processes, time change, and boundary theory

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetri...

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Hlavní autoři:	Chen, Zhenqing, Fukushima, Masatoshi
Médium:	E-kniha Kniha
Jazyk:	angličtina
Vydáno:	Princeton Princeton University Press 2011
Vydání:	1
Edice:	London Mathematical Society monographs
Témata:	Absolute continuity Bilinear form Borel right process Borel set Boundary value problems Brownian motion Cauchy process Characterization (mathematics) Compact space Constant function Continuous function Continuous function (set theory) Corollary Diffusion process Dimension Dirichlet integral Division by zero Equation Equivalence class Existence theorem Existential quantification Fatou's lemma Fourier transform Fubini's theorem Function (mathematics) Function space Harmonic function Hilbert space Homeomorphism Identity matrix Infinitesimal generator (stochastic processes) Interest Interval (mathematics) Irreducibility (mathematics) Lebesgue measure Linear map Lipschitz continuity Lipschitz domain London Mathematical Society Markov chain Markov process Markov processes Markov property Martingale (probability theory) MATHEMATICS MATHEMATICS / Probability & Statistics / General Measurable function Measure (mathematics) Monotonic function Open set Oxford University Press Poisson point process Potential theory Princeton University Press Probability Probability measure Radon measure Riemannian manifold Rotational symmetry Self-adjoint operator Semigroup Semimartingale Sign (mathematics) Sobolev space Special case Statistics Stochastic calculus Stochastic differential equation Stochastic process Subset Summation Theorem Theory Transition function Constant function Open set Measurable function Subset Identity matrix Function space Fatou's lemma Stochastic process Interest Self-adjoint operator Infinitesimal generator (stochastic processes) Martingale (probability theory) Characterization (mathematics) Hilbert space Semimartingale Corollary Monotonic function Radon measure Absolute continuity Lipschitz domain Existence theorem Semigroup Continuous function Fubini's theorem Riemannian manifold Function (mathematics) Bilinear form Harmonic function Existential quantification Brownian motion Division by zero Poisson point process Homeomorphism Borel right process Special case Summation Probability measure Markov process Sign (mathematics) Continuous function (set theory) Equivalence class Sobolev space Theory Rotational symmetry Dirichlet integral Fourier transform Potential theory Mathematics London Mathematical Society Interval (mathematics) Lipschitz continuity Markov chain Lebesgue measure Linear map Diffusion process Oxford University Press Cauchy process Theorem Stochastic differential equation Transition function Markov property Compact space Borel set Probability Equation Dimension Princeton University Press Stochastic calculus Measure (mathematics) Irreducibility (mathematics)
ISBN:	9780691136059, 069113605X, 9781400840564, 1400840562
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Symmetric Markov processes, time change, and boundary theory -- CONTENTS -- NOTATION -- PREFACE -- CHAPTER ONE: SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS -- CHAPTER TWO: BASIC PROPERTIES AND EXAMPLES OF DIRICHLET FORMS -- CHAPTER THREE: SYMMETRIC HUNT PROCESSES AND REGULAR DIRICHLET FORMS -- CHAPTER FOUR: ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES -- CHAPTER FIVE: TIME CHANGES OF SYMMETRIC MARKOV PROCESSES -- CHAPTER SIX: REFLECTED DIRICHLET SPACES -- CHAPTER SEVEN: BOUNDARY THEORY FOR SYMMETRIC MARKOV PROCESSES -- APPENDIX A: ESSENTIALS OF MARKOV PROCESSES -- APPENDIX B: SOLUTIONS TO EXERCISES -- NOTES -- BIBLIOGRAPHY -- CATALOGUE OF SOME USEFUL THEOREMS -- INDEX
Front Matter Table of Contents Notation Preface Chapter One: SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS Chapter Two: BASIC PROPERTIES AND EXAMPLES OF DIRICHLET FORMS Chapter Three: SYMMETRIC HUNT PROCESSES AND REGULAR DIRICHLET FORMS Chapter Four: ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES Chapter Five: TIME CHANGES OF SYMMETRIC MARKOV PROCESSES Chapter Six: REFLECTED DIRICHLET SPACES Chapter Seven: BOUNDARY THEORY FOR SYMMETRIC MARKOV PROCESSES Appendix A. Appendix B. Notes Bibliography Catalogue of Some Useful Theorems Index
Cover -- Title -- Copyright -- Contents -- Notation -- Preface -- CHAPTER 1. SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS -- 1.1 Dirichlet Forms and Extended Dirichlet Spaces -- 1.2 Excessive Functions and Capacities -- 1.3 Quasi-Regular Dirichlet Forms -- 1.4 Quasi-Homeomorphism of Dirichlet Spaces -- 1.5 Symmetric Right Processes and Quasi-Regular Dirichlet Forms -- CHAPTER 2. BASIC PROPERTIES AND EXAMPLES OF DIRICHLET FORMS -- 2.1 Transience, Recurrence, and Irreducibility -- 2.2 Basic Examples -- 2.3 Analytic Potential Theory for Regular Dirichlet Forms -- 2.4 Local Properties -- CHAPTER 3. SYMMETRIC HUNT PROCESSES AND REGULAR DIRICHLET FORMS -- 3.1 Relations between Probabilistic and Analytic Concepts -- 3.2 Hitting Distributions and Projections I -- 3.3 Quasi Properties, Fine Properties, and Part Processes -- 3.4 Hitting Distributions and Projections II -- 3.5 Transience, Recurrence, and Path Behavior -- CHAPTER 4. ADDITIVE FUNCTIONALS OF SYMMETRIC MARKOV PROCESSES -- 4.1 Positive Continuous Additive Functionals and Smooth Measures -- 4.2 Decompositions of Additive Functionals of Finite Energy -- 4.3 Probabilistic Derivation of Beurling-Deny Formula -- CHAPTER 5. TIME CHANGES OF SYMMETRIC MARKOV PROCESSES -- 5.1 Subprocesses and Perturbed Dirichlet Forms -- 5.2 Time Changes and Trace Dirichlet Forms -- 5.3 Examples -- 5.4 Energy Functionals for Transient Processes -- 5.5 Trace Dirichlet Forms and Feller Measures -- 5.6 Characterization of Time-Changed Processes -- 5.7 Excursions, Exit System, and Feller Measures -- 5.8 More Examples -- CHAPTER 6. REFLECTED DIRICHLET SPACES -- 6.1 Terminal Random Variables and Harmonic Functions -- 6.2 Reflected Dirichlet Spaces: Transient Case -- 6.3 Recurrent Case -- 6.4 Toward Quasi-Regular Cases -- 6.5 Examples -- 6.6 Silverstein Extensions -- 6.7 Equivalent Notions of Harmonicity
CHAPTER 7. BOUNDARY THEORY FOR SYMMETRIC MARKOV PROCESSES -- 7.1 Reflected Dirichlet Space for Part Processes -- 7.2 Douglas Integrals and Reflecting Extensions -- 7.3 Lateral Condition for L2-Generator -- 7.4 Countable Boundary -- 7.5 One-Point Extensions -- 7.6 Examples of One-Point Extensions -- 7.7 Many-Point Extensions -- 7.8 Examples of Many-Point Extensions -- APPENDIX A. ESSENTIALS OF MARKOV PROCESSES -- A.1 Markov Processes -- A.2 Basic Properties of Borel Right Processes -- A.3 Additive Functionals of Right Processes -- A.4 Review of Symmetric Forms -- APPENDIX B. SOLUTIONS TO EXERCISES -- Notes -- Bibliography -- Catalogue of Some Useful Theorems -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- W -- X
Chapter Six. Reflected Dirichlet Spaces
Chapter Two. Basic Properties and Examples of Dirichlet Forms
Chapter Four. Additive Functionals of Symmetric Markov Processes
Index
Appendix A. Essentials of Markov Processes
Notation
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Appendix B. Solutions To Exercises
Contents
Chapter Three. Symmetric Hunt Processes and Regular Dirichlet Forms
Catalogue Of Some Useful Theorems
Chapter Seven. Boundary Theory for Symmetric Markov Processes
Frontmatter --
Preface
Chapter Five. Time Changes of Symmetric Markov Processes
Chapter One. Symmetric Markovian Semigroups and Dirichlet Forms
Notes
Bibliography