Quenched invariance principles for random walks and elliptic diffusions in random media with boundary

TL;DR

This paper proves that certain random walks in complex random environments with boundaries converge to reflecting Brownian motion, advancing understanding of their long-term behavior and solving key open problems.

Contribution

It establishes quenched invariance principles for random walks with boundaries in various random media, including percolation clusters and conductance models, using heat kernel estimates.

Findings

Random walk on percolation clusters converges to reflecting Brownian motion.

Improved asymptotic estimates for mixing times in conductance models.

Quenched invariance principles hold for domains with general boundaries.

Abstract

Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for random walks in random environments with a boundary. In particular, we prove that the random walk on a supercritical percolation cluster or among random conductances bounded uniformly from below in a half-space, quarter-space, etc., converges when rescaled diffusively to a reflecting Brownian motion, which has been one of the important open problems in this area. We establish a similar result for the random conductance model in a box, which allows us to improve existing asymptotic estimates for the relevant mixing time. Furthermore, in the uniformly elliptic case, we present quenched invariance principles for domains with more general boundaries.