Markov Processes and Dirichlet Forms

  • Zhi-Ming Ma
  • Michael Röckner
Part of the Universitext book series (UTX)


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 (ε,D(ε)) 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 (ε,D(ε)). 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.


State Space Markov Process Dirichlet Form Strong Markov Property Hausdorff Topological Space 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Zhi-Ming Ma
    • 1
  • Michael Röckner
    • 2
  1. 1.Institute of Applied MathematicsAcademia SinicaBeijingPeople’s Republic of China
  2. 2.Institut für Angewandte MathematikUniversität BonnBonn 1Germany

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