Exit Laws of Isotropic Diffusions in Random Environment from Large Domains

TL;DR

This paper proves that in high dimensions, small isotropic perturbations of Brownian motion in random environments have exit laws from large domains that converge to those of standard Brownian motion, with a quantifiable rate.

Contribution

It extends previous work by analyzing the elliptic boundary-value problem to establish convergence of exit laws and provides a convergence rate in large domains.

Findings

Almost sure convergence of exit laws to Brownian motion

Quantitative rate of convergence based on boundary condition modulus

Extension of Sznitman and Zeitouni's results to larger domains

Abstract

This paper studies, in dimensions greater than two, stationary diffusion processes in random environment which are small, isotropic perturbations of Brownian motion satisfying a finite range dependence. Such processes were first considered in the continuous setting by Sznitman and Zeitouni [20]. Building upon their work, it is shown by analyzing the associated elliptic boundary-value problem that, almost surely, the smoothed (in the sense that the boundary data is continuous) exit law of the diffusion from large domains converges, as the domain's scale approaches infinity, to that of a Brownian motion. Furthermore, a rate for the convergence is established in terms of the modulus of the boundary condition.