Random walks in dynamic random environments: A transference principle

TL;DR

This paper establishes a transference principle linking the mixing properties of a Markovian environment to the environment process seen from a random walk, leading to ergodicity, law of large numbers, and CLT results.

Contribution

It introduces a method to transfer mixing rates from the environment to the environment process, enabling analysis of random walks in dynamic environments with polynomial mixing.

Findings

Environment process is uniquely ergodic.

Invariant measure depends continuously on jump rates.

Law of large numbers and CLT hold for the walk.

Abstract

We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties for the environment as seen from the position of the walker, that is, the environment process. We can transfer the rate of mixing in time of the environment to the rate of mixing of the environment process with a loss of at most polynomial order. Therefore the method is applicable to environments with sufficiently fast polynomial mixing. We obtain unique ergodicity of the environment process. Moreover, the unique invariant measure of the environment process depends continuously on the jump rates of the walker. As a consequence we obtain the law of large numbers and a central limit theorem with nondegenerate variance for the position of the walk.