Invariant measure for the process viewed from the particle for 2D random walks in Dirichlet environment

TL;DR

The paper analyzes random walks in Dirichlet environments on ${\mathbb Z}^2$, proving that in the recurrent regime (when the drift vanishes, $d_{\alpha}=0$) there is no invariant measure for the environment viewed from the particle that is absolutely continuous with respect to the static Dirichlet law. It introduces a new identity on finite directed graphs, connecting hitting probabilities to a product–ratio structure via a generalized inverse Gaussian distribution, with links to the star–VRJP Schrödinger representation and a Matsumoto–Yor–type discrete property on the line. In the transient regime under (T′) and with trapping strength $\kappa>1$, an invariant probability with density in $L^p$ for all $p<\kappa$ exists; if $\kappa\le 1$, invariance fails unless one accelerates the walk by a local function of the environment, which can restore existence. The work provides a coherent conjectural classification for invariant-measure existence in 2D RWDE, supported by torus-approximation techniques and finite-graph analyses, and sheds light on trapping phenomena through the parameter $\kappa$. Overall, the results connect probabilistic invariance, trapping, and non-reversible dynamics in RWDE, contributing to the broader understanding of two-dimensional random environments and their long-time behavior.

Abstract

In this paper, we consider random walks in Dirichlet random environment (RWDE) on $\mathbb{Z}^2$. We prove that, if the RWDE is recurrent (which is strongly conjectured when the weights are symmetric), then there does not exist any invariant measure for the process viewed from the particle which is absolutely continuous with respect to the static law of the environment. Besides, if the walk is directional transient and under condition $\mathbf{(T')}$, we prove that there exists such an invariant probability measure if the trapping parameter verifies $κ> 1$ or after acceleration of the process by a local function of the environment. This gives strong credit to a conjectural classification of cases of existence or non-existence of the invariant measure for two dimensional RWDE. The proof is based on a new identity, stated on general finite graphs, which is inspired by the representation of the $\star$-VRJP, a non-reversible generalization of the Vertex reinforced Jump Process, in terms of random Schrödinger operators. In the case of RWDE on 1D graph, the previous identity entails also a discrete analogue of the Matsumoto-Yor property for Brownian motion.

Theorems & Definitions (47)

• Lemma 1.1: Time reversal property, sabot2011randomsabot2011reversed
• Theorem 1.2: Theorem 2 of sabot2011random and Theorem 1 of sabot2013random
• Theorem 1.3: sabot2011reversed and tournier2015asymptotic
• Lemma 2.1: Product-ratio identity for hitting probabilities
• Theorem 2.2
• Theorem 2.3
• Remark 1
• Definition 2.1
• Theorem 2.4
• Conjecture 1