Continuity of the Revuz correspondence under the absolute continuity condition
Ryoichiro Noda
TL;DR
This work addresses the continuity of the Revuz correspondence between positive continuous additive functionals (PCAFs) and smooth measures for symmetric Hunt processes with transition densities under the absolute continuity condition. It introduces a central inequality that bounds the distance between PCAFs by the distance between their $1$-potentials, enabling convergence results via killed-process techniques. Under uniform convergence of $1$-potentials, PCAFs converge in the local uniform topology in probability, and with strict-sense convergence plus conservativeness, convergence is uniform on compacts in probability (ucp). The results extend previous compactness-type findings and provide practical criteria for verifying convergence, improving understanding of the topological structure of the Revuz correspondence and its applications to time-changed processes. Collectively, the paper offers a rigorous framework for analyzing convergence of additive functionals in Dirichlet form settings and for constructing stable time-changed diffusions.
Abstract
In this paper, we consider symmetric Hunt processes that correspond to regular Dirichlet forms 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 using 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.