Continuity of the Revuz correspondence under the absolute continuity condition

TL;DR

This paper proves the continuity of the Revuz correspondence for processes with transition densities, showing that convergence of potentials implies convergence of associated additive functionals, under certain conditions.

Contribution

It establishes the continuity of the Revuz correspondence under the absolute continuity condition, linking potential convergence to additive functional convergence.

Findings

Convergence of 1-potentials implies convergence of PCAFs.

Sufficient conditions for potential convergence based on measure convergence.

Framework includes symmetric Hunt processes with Dirichlet forms.

Abstract

In this paper, we consider standard processes that admit dual processes and satisfy the absolute continuity condition, i.e., processes possess transition densities. For such processes, the Revuz correspondence relates positive continuous additive functionals (PCAFs) to so-called smooth measures. We show the continuity of this correspondence. Specifically, we show that if the $1$-potentials of smooth measures converge (locally) uniformly as functions, then the associated PCAFs converge. This result is derived by directly estimating the distance between the PCAFs in terms of the distance between the $1$-potentials of the associated smooth measures. Furthermore, in cases where the transition density is jointly continuous, we present sufficient conditions for the convergence of $1$-potentials based on the weak or vague convergence of smooth measures. The framework in this paper contains the class of symmetric Hunt processes that are associated with regular Dirichlet forms and satisfy the absolute continuity condition.