Random Walk in Dynamic Markovian Random Environment

TL;DR

This paper studies random walks in a Markovian dynamic environment on integer lattices, proving strong laws and invariance principles using probabilistic methods, and extends results to higher dimensions with weaker assumptions.

Contribution

It introduces a probabilistic approach based on regeneration times for analyzing random walks in dynamic environments, providing quenched invariance principles for dimensions greater than 7.

Findings

Proves annealed strong law of large numbers and invariance principle in all dimensions.

Establishes quenched invariance principle for dimensions d > 7.

Offers an alternative, weaker-assumption method compared to previous analytical approaches.

Abstract

We consider a model, introduced by Boldrighini, Minlos and Pellegrinotti, of random walks in dynamical random environments on the integer lattice Z^d with d>=1. In this model, the environment changes over time in a Markovian manner, independently across sites, while the walker uses the environment at its current location in order to make the next transition. In contrast with the cluster expansions approach of Boldrighini, Minlos and Pellegrinotti, we follow a probabilistic argument based on regeneration times. We prove an annealed SLLN and invariance principle for any dimension, and provide a quenched invariance principle for dimension d > 7, providing for d>7 an alternative to the analytical approach of Boldrighini, Minlos and Pellegrinotti, with the added benefit that it is valid under weaker assumptions. The quenched results use, in addition to the regeneration times already mentioned, a technique introduced by Bolthausen and Sznitman.