On the existence of an invariant measure for isotropic diffusions in random environment

Abstract

The results of this paper build upon those first obtained by Sznitman and Zeitouni (Invent Math 164(3), 455–567, 2006). We establish, for spacial dimensions dge 3, the existence of a unique invariant measure for isotropic diffusions in random environment on mathbb {R}^d which are small perturbations of Brownian motion. Furthermore, we establish a general homogenization result for initial data which are locally measurable with respect to the coefficients.