Homeomorphism of the Revuz correspondence for finite energy integrals

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.