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

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.

Published at http://dx.doi.org/10.1214/14-AOP914 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Random walks on supercritical percolation clusters
• Recent progress on the Random Conductance Model
• Invariance principle for the random conductance model with unbounded conductances
• Discrete approximations to reflected Brownian motion
• Reflecting random walk in fractal domains

Cited by in corpus (9)

• Invariance principle for the random conductance model in a degenerate ergodic environment
• Hydrodynamics for the partial exclusion process in random environment
• Moduli of continuity of local times of random walks on graphs in terms of the resistance metric
• Random Conductance Models with Stable-like Jumps: Quenched Invariance Principle
• Symmetric simple exclusion process in dynamic environment: hydrodynamics
• Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha$-stable processes
• From quenched invariance principle to semigroup convergence with applications to exclusion processes
• H-compactness of elliptic operators on weighted Riemannian Manifolds
• Effective resistances for supercritical percolation clusters in boxes