Time-changes of stochastic processes associated with resistance forms

TL;DR

This paper proves the convergence of Brownian motions and local times associated with resistance forms under certain conditions, leading to new insights into the scaling limits of various stochastic models on fractals and trees.

Contribution

It establishes the convergence of stochastic processes linked to resistance forms under Gromov-Hausdorff-vague topology and applies this to models like Liouville Brownian motion and trap models.

Findings

Convergence of Brownian motions and local times under specified conditions

Scaling limits of models on fractals and trees are characterized as FIN diffusions

Provides a unified framework for analyzing stochastic processes on complex spaces

Abstract

Given a sequence of resistance forms that converges with respect to the Gromov-Hausdorff-vague topology and satisfies a uniform volume doubling condition, we show the convergence of corresponding Brownian motions and local times. As a corollary of this, we obtain the convergence of time-changed processes. Examples of our main results include scaling limits of Liouville Brownian motion, the Bouchaud trap model and the random conductance model on trees and self-similar fractals. For the latter two models, we show that under some assumptions the limiting process is a FIN diffusion on the relevant space.