Convergence of local times of stochastic processes associated with resistance forms

Ryoichiro Noda

Published 2023-05-22Version 1

We establish that if a sequence of spaces equipped with resistance metrics and measures converge with respect to the Gromov-Hausdorff-vague topology, and certain non-explosion and metric-entropy conditions are satisfied, then the associated stochastic processes and their local times also converge. The metric-entropy condition can be checked by volume estimates of balls. Whilst similar results have been proved previously, the approach of this article is more widely applicable. Indeed, as well as recovering known conclusions for scaling limits of some deterministic self-similar fractal graphs and Galton-Watson trees, we derive new ones for scaling limits of uniform spanning trees. The metric-entropy condition also implies convergence of Gaussian processes.