Characterization of Particular Processes

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


This chapter deals with further sample path properties of Markov processes associated with Dirichlet forms. In Section 1 we prove that the continuity of sample paths (up to lifetime) is equivalent with the “local property” of the Dirichlet form extending M. Fukushima’s result for the locally compact case. On the way we prove a result on the existence of “localizing” functions in the domain of a Dirichlet form. We give applications to the examples of Chap.IV, Sect. 4 at the end. In Section 2 we prove a characterization of those Dirichlet forms associated with Hunt processes (also in non-locally compact cases). In particular, we show that every quasi-regular Dirichlet form which contains the function f ≡ 1 in its domain is associated with a Hunt process provided the cemetery is an isolated point. In all of this chapter we assume E to be a Hausdorff topological space such that B(E) = σ(C(E)). We fix a σ-finite positive measure m on (E,B(E)) and adjoin a point Δ to E as an isolated point of EΔ:= E ∪ {A}. If E is locally compact, EΔ is also taken to be the one point compactification of E (cf. Chap. IV, Sect. 1). We extend m to (EΔ, B(EΔ)) by m({Δ}):= 0. Again every function f on E is considered as a function on EΔ with f(Δ):= 0. For AE we set A c := E\A. Let (ε, D(ε)) be a fixed Dirichlet form on L2(E; m) with continuity constant K ≥ 1 and let (T t )t>0, ( t )t>0 and (G α )α>0, (Ĝ α )α>0 be the associated strongly continuous contraction (co)semigroups and (co)resolvents.


Local Property Dirichlet Form Countable Family Positive Radon Measure Hunt Process 
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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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