Dirichlet Forms and Symmetric Markov Processes

✍ Scribed by Masatoshi Fukushima, Yoichi Oshima, Masayoshi Takeda

• Publisher: De Gruyter
• Year: 2010
• Tongue: English
• Leaves: 501
• Series: de Gruyter Studies in Mathematics, Vol. 19
• Edition: 2
• Category: Library

✦ Synopsis

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 some new sections as well as several 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......Page 6
Preface to the second edition......Page 7
Contents......Page 8
Notation......Page 10
I Dirichlet Forms......Page 12
1.1 Basic notions......Page 14
1.2 Examples......Page 17
1.3 Closed forms and semigroups......Page 27
1.4 Dirichlet forms and Markovian semigroups......Page 36
1.5 Transience of Dirichlet spaces and extended Dirichlet spaces......Page 48
1.6 Global properties of Markovian semigroups......Page 64
2.1 Capacity and quasi continuity......Page 77
2.2 Measures of finite energy integrals......Page 88
2.3 Reduced functions and spectral synthesis......Page 106
2.4 Capacities and Sobolev type inequalities......Page 112
3.1 Closability and the smallest closed extensions......Page 120
3.2 Formulae of Beurling–Deny and LeJan......Page 131
3.3 Maximum Markovian extensions......Page 141
II Symmetric Markov processes......Page 160
4 Analysis by symmetric Hunt processes......Page 162
4.1 Smallness of sets and symmetry......Page 163
4.2 Identification of potential theoretic notions......Page 171
4.3 Orthogonal projections and hitting distributions......Page 179
4.4 Parts of forms and processes......Page 183
4.5 Continuity, killing, and jumps of sample paths......Page 189
4.6 Quasi notions, fine notions and global properties......Page 200
4.7 Irreducible recurrence and ergodicity......Page 212
4.8 Recurrence and Poincaré type inequalities......Page 218
5 Stochastic analysis by additive functionals......Page 232
5.1 Positive continuous additive functionals and smooth measures......Page 233
5.2 Decomposition of additive functionals of finite energy......Page 252
5.3 Martingale additive functionals and Beurling–Deny formulae......Page 267
5.4 Continuous additive functionals of zero energy......Page 272
5.5 Extensions to additive functionals locally of finite energy......Page 281
5.6 Martingale additive functionals of finite energy and stochastic integrals......Page 297
5.7 Forward and backward martingale additive functionals......Page 306
6 Transformations of forms and processes......Page 318
6.1 Perturbed Dirichlet forms and killing by additive functionals......Page 319
6.2 Traces of Dirichlet forms and time changes by additive functionals......Page 325
6.2.1 Transient case......Page 327
6.2.2 General case......Page 332
6.3 Transformations by supermartingale multiplicative functionals......Page 343
6.4 Donsker–Varadhan type large deviation principle......Page 357
7.1 Construction of a Markovian transition function......Page 380
7.2 Construction of a symmetric Hunt process......Page 384
A.1 Choquet capacities......Page 393
A.2 An introduction to Hunt processes......Page 395
A.3 A summary on martingale additive functionals......Page 417
A.4 Regular representations of Dirichlet spaces......Page 433
A.5 Solutions to Exercises......Page 450
Notes......Page 464
Bibliography......Page 472
Index......Page 496