Semi-Dirichlet Forms and Markov Processes (eBook)

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Y. Oshima, Kumamoto University, Japan.

Y. Oshima, Kumamoto University.

Preface 7
1 Dirichlet forms 11
1.1 Semi-Dirichlet forms and resolvents 11
1.2 Closability and regular Dirichlet forms 19
1.3 Transience and recurrence of Dirichlet forms 22
1.4 An auxiliary bilinear form 30
1.5 Examples 40
1.5.1 Diffusion case 40
1.5.2 Jump type case 46
2 Some analytic properties of Dirichlet forms 53
2.1 Capacity 53
2.2 Quasi-Continuity 61
2.3 Potential of measures 65
2.4 An orthogonal decomposition of the Dirichlet forms 70
3 Markov processes 82
3.1 Hunt processes 82
3.2 Excessive functions and negligible sets 86
3.3 Hunt processes associated with a regular Dirichlet form 90
3.4 Negligible sets for Hunt processes 99
3.5 Decompositions of Dirichlet forms 104
4 Additive functionals and smooth measures 122
4.1 Positive continuous additive functionals 122
4.2 Dual PCAFs and duality relations 138
4.3 Time changes and killings 148
5 Martingale AFs and AFs of zero energy 159
5.1 Fukushima’s decomposition of AFs 159
5.1.1 AFs generated by functions of F 161
5.1.2 Martingale additive functionals of finite energy 162
5.1.3 CAFs of zero energy 164
5.2 Beurling-Deny type decomposition 179
5.3 CAFs of locally zero energy in the weak sense 189
5.4 Martingale AFs of strongly local Dirichlet forms 200
5.5 Transformations by multiplicative functionals 213
5.6 Conservativeness and recurrence of Dirichlet forms 218
6 Time dependent Dirichlet forms 225
6.1 Time dependent Dirichlet forms and associated resolvents 225
6.2 A parabolic potential theory 234
6.3 Associated space-time processes 249
6.4 Additive functionals and associated measures 258
6.5 Some stochastic calculus 269
Notes 277
Bibliography 282
Index 293

lt;P>"The intention of the author is for the book to serve as a self-contained textbook. The results are therefore mostly followed by detailed proofs. [...] This book, the first systematic treatment of lower bounded semi-Dirichlet forms, will be a valuable contribution to the literature and will certainly become a classic reference in the field." Mathematical Reviews

"This new book is a most welcome addition to the existing literature on Dirichlet forms. It is a readily accessible, advanced graduate-level account of analytic and probabilistic potential theory of Hunt processes given by (lower bounded semi-) Dirichlet forms." Zentralblatt für Mathematik