On the Exit Time and Stochastic Homogenization of Isotropic Diffusions in Large Domains

TL;DR

This paper proves stochastic homogenization for isotropic diffusions in large domains, showing convergence to Brownian motion with an algebraic rate, and introduces new methods to handle unbounded exit times due to singular drift effects.

Contribution

It establishes stochastic homogenization with an algebraic rate for isotropic diffusions in large domains, extending previous results with novel techniques for unbounded exit times.

Findings

Homogenization occurs with an algebraic rate.

Exit times are controlled despite unbounded microscopic expectations.

Diffusions converge to Brownian motion in large domains.

Abstract

Stochastic homogenization is achieved for a class of elliptic and parabolic equations describing the lifetime, in large domains, of stationary diffusion processes in random environment which are small, statistically isotropic perturbations of Brownian motion in dimension at least three. Furthermore, the homogenization is shown to occur with an algebraic rate. Such processes were first considered in the continuous setting by Sznitman and Zeitouni [21], upon whose results the present work relies strongly, and more recently their smoothed exit distributions from large domains were shown to converge to those of a Brownian motion by the author [10]. This work shares in philosophy with [10], but requires substantially new methods in order to control the expectation of exit times which are generically unbounded in the microscopic scale due to the emergence of a singular drift in the asymptotic limit.