Large time behaviour of symmetric random walk in high-contrast periodic environment

TL;DR

This paper studies the long-term behavior of symmetric random walks in high-contrast periodic environments, revealing non-standard diffusive limits and coupled limit processes that differ from classical Markovian behavior.

Contribution

It introduces a two-component limit process capturing the walk's position and environment, showing non-Markovian limit behavior in high-contrast periodic media.

Findings

Random walk exhibits non-standard diffusive limits

Limit process is a coupled Markov process with non-Markovian coordinate

Convergence in path space established

Abstract

The paper deals with the asymptotic properties of a symmetric random walk in a high contrast periodic medium in $\mathbb Z^d$, $d\geq 1$. We show that under proper diffusive scaling the random walk exhibits a non-standard limit behaviour. In addition to the coordinate of the random walk in $\mathbb Z^d$ we introduce an extra variable that characterizes the position of the random walk in the period and show that this two-component process converges in law to a limit Markov process. The components of the limit process are mutually coupled, thus we cannot expect that the limit behaviour of the coordinate process is Markov. We also prove the convergence in the path space for the said random walk.