On the exit time and stochastic homogenization of isotropic diffusions in large domains

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 (Invent. Math. 164 (2006) 455–567), upon whose results the present work relies strongly.