Markov Processes Associated With Positivity Preserving Coercive Forms

Abstract

Coercive closed forms on L2-spaces are studied whose associated L2-semigroups are positivity preserving. Earlier work by other authors is extended by further developing the potential theory of such forms and completed by proving an analytic characterization of those of these forms which have a probabilistic counterpart, i.e., are associated with (special standard) Markov processes. Examples with finite and infinite dimensional state spaces are discussed.

Fukushima's Decomposition for Diffusions Associated With Semi-Dirichlet Forms

“…Suppose that µ ∈ S 0 . By [14,Proposition 4.13], there exists an E-nest {F k } of compact subsets of E such that Ĝ 1 φ, Û 1 µ ∈ C({F k }) and Ĝ 1 φ > 0 on F k for each k ∈ N. Then, there exists a sequences of positive constants {a k } such that…”

“…In [3], Fitzsimmons extended the smooth measure characterization of PCAFs from the Dirichlet forms setting to the semi-Dirichlet forms setting. Applying [3,Proposition 4.12], following the arguments of [5, Theorems 5.1.3 and 5.1.4] (with slight modifications by virtue of [13,14,10] and [1, Theorem 3.4]), we can also obtain a one to one correspondence between the family of all equivalent classes of PCAFs and the family S. The correspondence, which is referred to as Revuz correspondence, is described in the following lemma.…”

On the structure of bounded smooth measures associated with a quasi-regular Dirichlet form

“…Throughout, we assume that (E, D(E)) is a quasi-regular Dirichlet form on L 2 (E; m) (see [3,12,13] for the definitions). For α ≥ 0 we set…”

Perturbation of symmetric Markov processes

“…By Proposition 1.3(i) in [16], it suffices to show Q(f + , f − ) ≤ 0 for f ∈ F. Let Q (c)+(k) be the sum of continuous part and killing part of Q: (n) ). Let τ n be the first exit time of X from E n .…”