Law of large numbers for a finite-range random walk in a dynamic random environment with nonuniform mixing

TL;DR

This paper proves a law of large numbers for finite-range random walks in dynamic environments with polynomially decaying correlations, extending previous methods to more general settings without monotonicity.

Contribution

It generalizes the law of large numbers to finite-range walks in dynamic environments with nonuniform mixing, overcoming the lack of monotonicity.

Findings

Law of large numbers established for finite-range walks

Applicable to both discrete and continuous time models

Handles environments with polynomial decay of correlations

Abstract

In this paper, we study random walks evolving on Z in a dynamic random environment that we assume to have time correlations that decrease polynomially fast. We show a law of large numbers by generalizing methods already used for the nearest-neighbor framework to the finite-range one. This requires some new ideas to get around the absence of a monotonicity property that was crucial in the proof for the nearest-neighbour case. Our proof works both in discrete and continuous time.