Analysis of random walks in dynamic random environments via $L^2$-perturbations

TL;DR

This paper studies the behavior of random walks in dynamic environments modeled by Markov processes, providing limit theorems, series expansions for speed, and conditions for diffusion non-degeneracy using $L^2$-perturbation techniques.

Contribution

It introduces a novel $L^2$-perturbation framework for analyzing random walks in dynamic environments, extending previous methods to non-finite-range cases.

Findings

Law of large numbers for the walk's position

Invariance principle (diffusion limit) established

Series expansion for asymptotic speed derived

Abstract

We consider random walks in dynamic random environments given by Markovian dynamics on $\mathbb{Z}^d$. We assume that the environment has a stationary distribution $\mu$ and satisfies the Poincar\'e inequality w.r.t. $\mu$. The random walk is a perturbation of another random walk (called "unperturbed"). We assume that also the environment viewed from the unperturbed random walk has stationary distribution $\mu$. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of $L^2$-bounded perturbations of Markov processes by means of the so-called Dyson-Phillips expansion.