| The purpose of this thesis is to describe the author's work [6,7], which builds upon results first obtained by Sznitman and Zeitouni [16]. We establish, for spacial dimensions at least three, the existence of a unique invariant measure for isotropic diffusions in random environment which are small perturbations of Brownian motion. The subsequent analysis is used to obtain a general homogenization result for the related parabolic equation with random, oscillating and locally measurable initial data. Furthermore, we obtain a Liouville property for such environments. It is shown that, on a subset of full probability, the constant functions are the only strictly sub-linear maps left invariant under the evolution of the diffusion. Our methods here are motivated by work in the discrete setting of Benjamini, Duminil-Copin, Kozma and Yadin [2]. |
