EXISTENCE AND APPROXIMATION OF HUNT PROCESSES ASSOCIATED WITH GENERALIZED DIRICHLET FORMS

We show that any strictly quasi-regular generalized Dirichlet form that satisfies the mild structural condition D3 is associated to a Hunt process, and that the associated Hunt process can be approximated by a sequence of multivariate Poisson processes. This also gives a new proof for the existence of a Hunt process associated to a strictly quasi-regular generalized Dirichlet form that satisfies SD3 and extends all previous results.

Symmetric Hunt Processes and Regular Dirichlet Forms

This chapter studies a symmetric Hunt process associated with a regular Dirichlet form. Without loss of generality, the majority of the chapter assumes that E is a locally compact separable metric space, m is a positive Radon measure on E with supp[m] = E, and X = (Xₜ, Pₓ) is an m-symmetric Hunt process on (E,B(E)) whose Dirichlet form (E,F) is regular on L²(E; m). It adopts without any specific notices those potential theoretic terminologies and notations that are formulated in the previous chapter for the regular Dirichlet form (E,F). Furthermore, throughout this chapter, the convention that any numerical function on E is extended to the one-point compactification E ∂ = E ∪ {∂} by setting its value at δ‎ to be zero is adopted.

Reflected Dirichlet Spaces

This chapter turns to reflected Dirichlet spaces. It first introduces the notion of terminal random variables and harmonic functions of finite energy for a Hunt process associated with a transient regular Dirichlet form. The chapter next establishes several equivalent notions of reflected Dirichlet space (ℰ ref,ℱ ref) for a regular transient Dirichlet form (E,F). One of these equivalent notions is then used to define reflected Dirichlet space for a regular recurrent Dirichlet form. Moreover, the chapter gives yet another equivalent definition of reflected Dirichlet space that is invariant under quasi-homeomorphism of Dirichlet forms. Various concrete examples of reflected Dirichlet spaces are also exhibited for regular Dirichlet forms. Finally, the chapter defines a Silverstein extension of a quasi-regular Dirichlet form (E,F) on L²(E; m) and investigates the equivalence of analytic and probabilistic concepts of harmonicity.

Additive Functionals of Symmetric Markov Processes

This chapter is devoted to the study of additive functionals of symmetric Markov processes under the same setting as in the preceding chapter, namely, that E is a locally compact separable metric space, B(E) is the family of all Borel sets of E, and m is a positive Radon measure on E with supp[m] = E, and this chapter considers an m-symmetric Hunt process X = (Ω‎,M,Xₜ,ζ‎,Pₓ) on (E,B(E)) whose Dirichlet form (E,F) on L²(E; m) is regular on L²(E; m). The transition function and the resolvent of X are denoted by {Pₜ; t ≥ 0}, {R α‎, α‎ > 0}, respectively. B*(E) will denote the family of all universally measurable subsets of E. Any numerical function f defined on E will be always extended to E ∂ by setting f(∂) = 0.

ON HUNT PROCESSES AND STRICT CAPACITIES ASSOCIATED WITH GENERALIZED DIRICHLET FORMS

Introducing the corresponding strict capacity, we give necessary and sufficient conditions for a generalized Dirichlet form to be associated with a Hunt process. We also show that Borel measurable sets with strict capacity zero can be checked-out by an appropriate subclass of smooth measures. In the last part of this paper we present applications to three classes of examples.

Dirichlet forms on partial *-algebras

Dirichlet forms and their associated function spaces have been studied by a number of authors [4, 6, 7, 12, 15–18, 22, 25, 26]. Important motivation for the study has been the connection of Dirichlet forms with Markov processes [16–18, 25, 26]: for example, to every regular symmetric Dirichlet form, there is an associated Hunt process [13, 20]. This makes the theory of Dirichlet forms a convenient source of examples of Hunt processes. In the non-commutative setting, Markov fields have been studied by several authors [1–3, 14, 19, 24, 28]. It is therefore interesting to develop a non-commutative extension of the theory of Dirichlet forms and to study their connection with non-commutative Markov processes.