Absolute Continuity and Weak Uniform Mixing of Random Walk in Dynamic Random Environment

TL;DR

This paper establishes new results on random walks in dynamic environments without relying on strong mixing assumptions, focusing on the environment process and its invariant law, with implications for CLT and density control.

Contribution

It introduces general conditions for the environment seen from the walker to have an invariant law that is absolutely continuous, extending previous results to broader settings.

Findings

Invariant law exists under general conditions

Mutual absolute continuity with the environment law

Uniform control on density and quenched CLT under stronger assumptions

Abstract

We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the "environment seen from the walker"-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.