Dirichlet Forms and Symmetric Markov Processes

✍ Scribed by Masatoshi Fukushima; Yoichi Oshima; Masayoshi Takeda

• Publisher: De Gruyter
• Year: 2010
• Tongue: English
• Leaves: 500
• Series: De Gruyter Studies in Mathematics; 19
• Edition: 2nd rev. and ext. ed.
• Category: Library

✦ Synopsis

This book contains an introductory and comprehensive account of the theory of (symmetric) Dirichlet forms. Moreover this analytic theory is unified with the probabilistic potential theory based on symmetric Markov processes and developed further in conjunction with the stochastic analysis based on additive functional.

Since the publication of the first edition in 1994, this book has attracted constant interests from readers and is by now regarded as a standard reference for the theory of Dirichlet forms. For the present second edition, the authors not only revised the existing text, but also added sections on capacities and Sobolev type inequalities, irreducible recurrence and ergodicity, recurrence and Poincaré type inequalities, the Donsker-Varadhan type large deviation principle, as well as several new exercises with solutions.

The book addresses to researchers and graduate students who wish to comprehend the area of Dirichlet forms and symmetric Markov processes.

✦ Table of Contents

Preface to the first and second edition
Preface to the second edition
Contents
Notation
I Dirichlet Forms
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
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
2.4 Capacities and Sobolev type inequalities
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
II Symmetric Markov processes
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
4.7 Irreducible recurrence and ergodicity
4.8 Recurrence and Poincaré type inequalities
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
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
6.2.1 Transient case
6.2.2 General case
6.3 Transformations by supermartingale multiplicative functionals
6.4 Donsker–Varadhan type large deviation principle
7 Construction of symmetric Markov processes
7.1 Construction of a Markovian transition function
7.2 Construction of a symmetric Hunt process
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
A.5 Solutions to Exercises
Notes
Bibliography
Index