The purpose of this book is to give a stream-lined introduction to the theory of Dirichlet forms on general state spaces. It includes both the analytic and probabilistic components of the theory. A substantial part of the book is designed for a one-year graduate course. It provides a framework which covers both the well-studied "classical" theory of regular Dirichlet forms on locally compact state spaces and all recent extensions to infinite-dimensional state spaces. It also contains a complete proof of an analytic characterization of the class of Dirichlet forms which are associated with right continuous strong Markov processes, i.e. those having a probabilistic counterpart. Finally, a general regularization method is developed which makes it possible to transfer all results known in the classical locally compact regular case to this (in the above sense) most general class of Dirichlet forms.
0 Introduction.- I Functional Analytic Background.- 1 Resolvents, semigroups, generators.- 2 Coercive bilinear forms.- 3 Closability.- 4 Contraction properties.- 5 Notes/References.- II Examples.- 1 Starting point: operator.- 2 Starting point: bilinear form - finite dimensional case.- 3 Starting point: bilinear form - infinite dimensional case.- 4 Starting point: semigroup of kernels.- 5 Starting point: resolvent of kernels.- 6 Notes/References.- III Analytic Potential Theory of Dirichlet Forms.- 1 Excessive functions and balayage.- 2 ?-exceptional sets and capacities.- 3 Quasi-continuity.- 4 Notes/References.- IV Markov Processes and Dirichlet Forms.- 1 Basics on Markov processes.- 2 Association of right processes and Dirichlet forms.- 3 Quasi-regularity and the construction of the process.- 4 Examples of quasi-regular Dirichlet forms.- 5 Necessity of quasi-regularity and some probabilistic potential theory.- 6 One-to-one correspondences.- 7 Notes/References.- V Characterization of Particular Processes.- 1 Local property and diffusions.- 2 A new capacity and Hunt processes.- 3 Notes/References.- VI Regularization.- 1 Local compactification.- 2 Consequences - the transfer method.- 3 Notes/References.- A Some Complements.- 1 Adjoint operators.- 2 The Banach/Alaoglu and Banach/Saks theorems.- 3 Supplement on Ray resolvents and right processes.
Number Of Pages: 209
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6
Weight (kg): 0.32