A Liouville Property for Isotropic Diffusions in Random Environment

We will discuss recent results regarding diffusion processes in random environment which are small, isotropic perturbations of Brownian motion in spacial dimension greater than two. Such processes were originally considered in the continuous setting by Sznitman and Zeitouni [4], where it was shown that, almost surely in the limit with respect to Brownian scaling, the diffusion converges in law to a Brownian motion with deterministic variance. This result was obtained through the construction of an inductive scheme which will be explained in the first portion of the talk.

The remainder will be spent discussing more recent results regarding the existence of an invariant measure and proving a Liouville property for such environments. The invariant measure was constructed in Fehrman [2], and the method of proof relies upon aspects of the inductive scheme from [4]. The Liouville property for the environment was shown in Fehrman [3]. It relies upon the results of [2] and [4], and adapts a technique from the discrete setting of Benjamini, Duminil-Copin, Kozma and Yadin [1]. Finally, we will mention the applications of this work to the corresponding problems in random homogenization.

[1] I. Benjamini, H. Duminil-Copin, G. Kozma, and A. Yadin. Disorder, entropy and harmonic functions. Arxiv:1111.4853, 2011.
[2] B. Fehrman. On the existence of an invariant measure for isotropic diffusions in random environment. Arxiv:1404.5274, 2014.
[3] B. Fehrman. A Liouville property for isotropic diffusions in random environment. Arxiv:1406.1549, 2014.
[4] Alain-Sol Sznitman and Ofer Zeitouni. An invariance principle for isotropic diffusions in random environment. Invent. Math., 164(3):455–567, 2006.