Random Markov property for random walks in random environments

TL;DR

This paper introduces a criterion based on a random Markov property for decomposing trajectories of random walks in dynamic environments, enabling limit theorem proofs, with applications to correlated environments like percolation and renewal chains.

Contribution

It proposes a novel criterion involving a random Markov property and mixing conditions to analyze random walks in dynamic environments, extending previous methods.

Findings

Decomposition of random walk trajectories into i.i.d. increments under the criterion.

Application of the criterion to correlated environments such as Boolean percolation.

Establishment of limit theorems for random walks in these environments.

Abstract

We consider random walks in dynamic random environments and propose a criterion which, if satisfied, allows to decompose the random walk trajectory into i.i.d. increments, and ultimately to prove limit theorems. The criterion involves the construction of a random field built from the environment, that has to satisfy a certain random Markov property along with some mixing estimates. We apply this criterion to correlated environments such as Boolean percolation and renewal chains featuring polynomial decay of correlations.