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

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 Poincaré 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.

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References in corpus (5)

• An almost sure invariance principle for random walks in a space-time random environment
• Random Walk in Dynamic Markovian Random Environment
• Random walk driven by the simple exclusion process
• Random walks in space time mixing environments
• Negative differential mobility of weakly driven particles in models of glass formers

Cited by in corpus (4)

• Random walk on the simple symmetric exclusion process
• Random walk on a perturbation of the infinitely-fast mixing interchange process
• Ancestral lineages in spatial population models with local regulation
• Law of large numbers for a finite-range random walk in a dynamic random environment with nonuniform mixing