Dirichlet Forms and Symmetric Markov Processes

Masatoshi Fukushima, Yoichi Oshima, Masayoshi Takeda

• Publisher: De Gruyter
• Year: 2011
• ISBN:
  9783110889741
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

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Table of contents

• Preface
• Notation
• Part I. Dirichlet forms
• Chapter 1. Basic theory of Dirichlet forms
• 1.1. Basic notions
• 1.2. Examples
• 1.3. Closed forms and semigroups
• 1.4. Dirichlet forms and Markovian semigroups
• 1.5. Transience of Dirichlet spaces and extended Dirichlet spaces
• 1.6. Global properties of Markovian semigroups
• Chapter 2. Potential theory for Dirichlet forms
• 2.1. Capacity and quasi continuity
• 2.2. Measures of finite energy integrals
• 2.3. Reduced functions and spectral synthesis
• Chapter 3. The scope of Dirichlet forms
• 3.1. Closability and the smallest closed extensions
• 3.2. Formulae of Beurling-Deny and LeJan
• 3.3. Maximum Markovian extensions
• Part II. Symmetric Markov processes
• Chapter 4. Analysis by symmetric Hunt processes
• 4.1. Smallness of sets and symmetry
• 4.2. Identification of potential theoretic notions
• 4.3. Orthogonal projections and hitting distributions
• 4.4. Parts of forms and processes
• 4.5. Continuity, killing and jumps of sample paths
• 4.6. Quasi notions, fine notions and global properties
• Chapter 5. Stochastic analysis by additive functionals
• 5.1. Positive continuous additive functionals and smooth measures
• 5.2. Decomposition of additive functionals of finite energy
• 5.3. Martingale additive functionals and Beurling-Deny formulae
• 5.4. Continuous additive functionals of zero energy
• 5.5. Extensions to additive functionals locally of finite energy
• 5.6. Martingale additive functionals of finite energy and stochastic integrals
• 5.7. Forward and backward martingale additive functionals
• Chapter 6. Transformations of forms and processes
• 6.1. Perturbed Dirichlet forms and killing by additive functionals
• 6.2. Traces of Dirichlet forms and time changes by additive functionals
• 6.3. Transformations by supermartingale multiplicative functionals
• Chapter 7. Construction of symmetric Markov processes
• 7.1. Construction of a Markovian transition function
• 7.2. Construction of a symmetric Hunt process
• 7.3. Dirichlet forms and Hunt processes on a Lusin space
• A Appendix
• A.1 Choquet capacities
• A.2 An introduction to Hunt processes
• A.3 A summary on martingale additive functionals
• A.4 Regular representations of Dirichlet spaces
• Notes
• Bibliography
• Index