A Liouville Property for Isotropic Diffusions in Random Environment

TL;DR

This paper proves a Liouville property for certain isotropic diffusions in random environments, showing that only constant functions are invariant under the diffusion's evolution in high dimensions.

Contribution

It establishes a Liouville property for stationary diffusions in random environments that are small isotropic perturbations of Brownian motion, extending previous discrete and continuous results.

Findings

Constant functions are the only bounded, ancient invariant maps.

Constant functions are the only strictly sub-linear invariant maps.

Results hold for a subset of full probability in high spatial dimensions.

Abstract

We obtain a Liouville property for stationary diffusions in random environment which are small, isotropic perturbations of Brownian motion in spacial dimension greater than two. Precisely, we prove that, on a subset of full probability, the constant functions are the only strictly sub-linear maps which are invariant with respect to the evolution of the diffusion. And, we prove that the constant functions are the only bounded, ancient maps which are invariant under the evolution. These results depend upon the previous work of Fehrman [3] and Sznitman and Zeitouni [7] and, in the first case, our methods are motivated by the work, in the discrete setting, of Benjamini, Duminil-Copin, Kozma and Yadin [1].