Einstein relation for random walks in random environment

TL;DR

This paper extends the Einstein relation to random walks in random environments by deriving the speed's derivative under perturbations, using new regeneration structures and conditions for ballisticity.

Contribution

It introduces a generalized Einstein relation for RWRE under specific ballisticity conditions, with a novel approach involving regeneration structures and environment perturbations.

Findings

Derived the derivative of RWRE speed with respect to perturbation.

Established conditions under which the generalized Einstein relation holds.

Developed a new regeneration structure for balanced environments.

Abstract

In this article, we consider the speed of the random walks in a (uniformly elliptic and i.i.d.) random environment (RWRE) under perturbation. We obtain the derivative of the speed of the RWRE w.r.t. the perturbation, under the assumption that one of the following holds: (i) the environment is balanced and the perturbation satisfies a Kalikow-type ballisticity condition, (ii) the environment satisfies Sznitman's ballisticity condition. This is a generalized version of the Einstein relation for RWRE. Our argument is based on a modification of Lebowitz-Rost's argument developed in [Stochastic Process. Appl. 54 (1994) 183-196] and a new regeneration structure for the perturbed balanced environment.