Semi-Dirichlet Forms and Markov Processes

Description

This book deals with analytic treatments of Markov processes. Symmetric Dirichlet forms and their associated Markov processes are important and powerful tools in the theory of Markov processes and their applications. The theory is well studied and used in various fields. In this monograph, we intend to generalize 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 dual Markov 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. The text is written for graduate students, but also researchers. Preface 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 form 1.5 Examples 1.5.1 Diffusion case 1.5.2 Jump type case 2 Some analytic properties of Dirichlet forms 2.1 Capacity 2.2 Quasi-Continuity 2.3 Potential of measures 2.4 An orthogonal decomposition of the Dirichlet forms 3 Markov processes 3.1 Hunt processes 3.2 Excessive functions and negligible sets 3.3 Hunt processes associated with a regular Dirichlet form 3.4 Negligible sets for Hunt processes 3.5 Decompositions of Dirichlet forms 4 Additive functionals and smooth measures 4.1 Positive continuous additive functionals 4.2 Dual PCAFs and duality relations 4.3 Time changes and killings 5 Martingale AFs and AFs of zero energy 5.1 Fukushima’s decomposition of AFs 5.1.1 AFs generated by functions of ℱ 5.1.2 Martingale additive functionals of finite energy 5.1.3 CAFs of zero energy 5.2 Beurling-Deny type decomposition 5.3 CAFs of locally zero energy in the weak sense 5.4 Martingale AFs of strongly local Dirichlet forms 5.5 Transformations by multiplicative functionals 5.6 Conservativeness and recurrence of Dirichlet forms 6 Time dependent Dirichlet forms 6.1 Time dependent Dirichlet forms and associated resolvents 6.2 A parabolic potential theory 6.3 Associated space-time processes 6.4 Additive functionals and associated measures 6.5 Some stochastic calculus Notes Bibliography Index