Random walk in random environment and their time-reversed counterpart

TL;DR

This paper investigates the properties of random walks in Dirichlet environments, revealing a statistical invariance under time-reversal and characterizing when such walks are Dirichlet environments.

Contribution

It establishes that under weak conditions, a random walk with independent transition probabilities and a time-reversal that also has independent transitions is necessarily a Dirichlet environment.

Findings

Time-reversal invariance in Dirichlet environments.

Characterization of environments where time-reversal preserves independence.

Applicability to graphs satisfying weak assumptions.

Abstract

The random walk in Dirichlet environment is a random walk in random environment where the transition probabilities are independent Dirichlet random variables. This random walk exhibits a property of statistical invariance by time-reversal which leads to several results. More precisely, a time-reversed random walk in Dirichlet environment (with null divergence) is also a random walk in random environment where the transition probabilities are independent Dirichlet random variables with different parameters. We show that on all graphs that satisfy a few weak assumptions, a random walk in random environment with independent transition probabilities and such that the transition probabilities of the time-reversed random walk in random environment are also independent is a random walk in Dirichlet environment.