Homeomorphism of the Revuz correspondence under Dynkin class assumptions

Abstract

This paper investigates the topological properties of the Revuz correspondence between positive continuous additive functionals (PCAFs) and their associated smooth measures. Within the Dynkin, local Dynkin, and Green-tight Dynkin classes, we establish bidirectional equivalences among measure convergence, potential convergence, and PCAF convergence. In the local Dynkin class, weak convergence on compact sets, strong $\mathcal{E}_1$-convergence of potentials, uniform convergence of potentials, and $L^1$-convergence of PCAFs are mutually equivalent; under the Green-tight condition, this equivalence extends to the whole space.