Simple random walks on higher dimensional tori are mixing and not uniquely ergodic

TL;DR

This paper demonstrates that in higher dimensions, the environment viewed by the particle process can lack unique ergodicity, contrasting with the one-dimensional case, while still maintaining mixing properties under certain conditions.

Contribution

It constructs an analytic quasi-periodic environment on higher dimensional tori where the EVP process is not uniquely ergodic, extending understanding beyond the one-dimensional case.

Findings

Higher dimensional EVP processes can lack unique ergodicity.

The constructed environment has atomless stationary measures.

The EVP process remains mixing under smoothness assumptions.

Abstract

It has been shown that in one dimension the environment viewed by the particle process (EVP process) in quasi periodic random environment is uniquely ergodic and mixing under mild additional assumptions. Here we construct an analytic quasi periodic environment on higher dimensional torus such that the EVP process is not uniquely ergodic. The stationary measures in this counterexample are necessarily atomless. We show also that the EVP process is mixing with respect to any ergodic stationary measure under some smoothness assumption.