Markov Processes and Dirichlet Forms

Abstract

In this chapter we develop the probabilistic part of the theory of Dirichlet forms. We start with some basics on Markov processes in Section 1. In particular, we recall some facts on Ray resolvents whose proofs are, however, postponed to the Appendix (cf. A. Sect.3). In Section 2 we explain the “(proper) association” of Dirichlet forms with (a pair of “nice”) Markov processes. In Section 3 we introduce the notion of “quasi-regularity” and prove that a Dirichlet form on an arbitrary topological state space always possesses an associated “nice” Markov process provided it is quasi-regular. This extends M. Fukushima’s fundamental existence theorem for regular (symmetric) Dirichlet forms on locally compact, separable metric state spaces (cf. [F 71, 80]). In Section 4 we present examples of quasi-regular Dirichlet forms including cases with infinite dimensional state spaces. In Section 5 we prove that the quasi-regularity of a Dirichlet form (e,D(e)) is also necessary for the existence of an associated “nice” Markov process M. On the way, we study the essentials of the probabilistic potential theory of M and its relation with the analytic potential theory of (e,D(e)). In all of this chapter we assume E to be a Hausdorff topological space and B(E) denotes its Borel σ-algebra. In Sections 2, 3 and 5 we assume for convenience in addition, that B(E) = σ(C(E)) where C(E) denotes the set of all continuous functions on E.