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.