Random Dirichlet environment viewed from the particle in dimension $d\ge 3$

TL;DR

This paper characterizes the conditions under which a random walk in a Dirichlet environment in dimensions three and higher admits an invariant measure from the particle's perspective, advancing understanding of its ballistic behavior.

Contribution

It provides a complete characterization of weights leading to an invariant measure for the process viewed from the particle in higher dimensions.

Findings

Identifies weights with absolutely continuous invariant measures in $d ext{≥}3$

Deduces a full description of ballistic regimes in these dimensions

Advances theoretical understanding of RWDE dynamics

Abstract

We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On ${\mathbb Z}^d$, RWDE are parameterized by a 2d-uplet of positive reals called weights. In this paper, we characterize for $d\ge 3$ the weights for which there exists an absolutely continuous invariant probability for the process viewed from the particle. We can deduce from this result and from [27] a complete description of the ballistic regime for $d\ge 3$.