Semi-Dirichlet Forms and Markov Processes

ISBN-13: 9783110302004 / Angielski / Twarda / 2013 / 294 str.

Yoichi Oshima

Opis

Thisbook deals with analytic treatments of Markov processes. Symmetric Dirichlet forms andtheir associated Markov processes are important and powerful toolsin the theory of Markovprocesses and their applications. The theoryis well studied and used in various fields. In this monograph, we intend togeneralize the theory to non-symmetric and time dependent semi-Dirichlet forms. By this generalization, we can cover the wide class of Markov processes and analytic theory which do not possess the dualMarkov processes. In particular, under the semi-Dirichlet form setting, the stochastic calculus is not well established yet.In this monograph, we intend to give an introduction to such calculus. Furthermore, basic examples different from the symmetric cases are given.Thetext is writtenfor graduate students, but alsoresearchers.

Recenzje

"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

Spis treści

Chapter 1 Dirichlet forms
 1.1 Semi-Dirichlet forms and resolvents
 1.2 Closability and regular Dirichlet forms
 1.3 Transience and recurrence of Dirichlet forms
 1.4 An auxiliary bilinear forms
 1.5 Examples

Chapter 2 Some analytic properties of Dirichlet forms
  2.1 Capacity
  2.2 Qasi-continuity
  2.3 Potential of measures
  2.4 An orthogonal decomposition of Dirichlet forms

Chapter 3 Markov processes
  3.1 Hunt processes
  3.2 Excessive functions and negligible sets
  3.3 Hunt processes associated with Dirichlet forms
  3.4 Negligible sets of Hunt processes
  3.5 Decomposition of Dirichlet forms

Chapter 4 Additive functionals and smooth measures
  4.1 Positive continuous additive functionals
  4.2 Dual PCAFs and duality measures
  4.3 Time changes and killings by PCAFs

Chapter 5 Martingale AFs and AFs of zero energy
  5.1 Decomposition of AFs
  5.2 Beurling-Deny type decompositions
  5.3 CAFs of zero energy
  5.4 Martingale AFs of local Dirichlet forms
  5.5 Transformations by multiplicative functionals
  5.6 Conservativeness and recurrence of Dirichlet forms

Chapter 6 Time dependent Dirichlet forms
  6.1 Time dependent Dirichlet forms and associated resolvents
  6.2 Some parabolic potential theory
  6.3 Associated space-time processes
  6.4 Additive functionals and associated measures
  6.5 Some stochastic calculus

O autorze

Y. Oshima, Kumamoto University.