Homeomorphism of the Revuz correspondence for finite energy integrals

TL;DR

This paper establishes a precise topological correspondence between finite energy measures and positive continuous additive functionals in the context of Dirichlet forms, clarifying convergence conditions for Revuz measures.

Contribution

It provides necessary and sufficient conditions for the homeomorphic convergence of Revuz measures associated with finite energy integrals in Dirichlet form theory.

Findings

Characterization of Revuz measure convergence conditions

Homeomorphism between measure space and additive functionals

Enhanced understanding of Dirichlet form associated processes

Abstract

We provide necessary and sufficient conditions for the convergence of Revuz measures of finite energy integrals. More precisely, the Revuz map from the set of all smooth measures of finite energy integrals, equipped with the topology induced by the norm given by the sum of the Dirichlet form and the $L^2(m)$-norm, to the space of positive continuous additive functionals, equipped with the topology induced by the $L^2(\mathbb{P}_{m+\kappa+\nu_0})$-norm with the local uniform topology, is a homeomorphism, where $m$ is the underlying measure, $\kappa$ is the killing measure of a Dirichlet form and $\nu_0$ is an energy functional for the part that the process continuously escaping to the cemetery point.