Central Limit Theorem for Random Walks in Doubly Stochastic Random Environment: $\mathcal{H}_{-1}$ Suffices

TL;DR

This paper establishes a central limit theorem for random walks in stationary, ergodic doubly stochastic environments under the minimal  condition, improving previous results by relaxing regularity assumptions on the drift field.

Contribution

It proves a CLT under the  condition on the drift's stream tensor, extending prior work that required stronger integrability conditions, and simplifies the proof methodology.

Findings

Proves CLT under  condition on drift field.

Extends previous results with weaker assumptions.

Provides a simpler proof approach.

Abstract

We prove a central limit theorem under diffusive scaling for the displacement of a random walk on ${\mathbb Z}^d$ in stationary and ergodic doubly stochastic random environment, under the $\mathcal{H}_{-1}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result of Komorowski, Landim and Olla (2012), where it is assumed that the stream tensor is in $\mathcal{L}^{\max\{2+\delta, d\}}$, with $\delta>0$. Our proof relies on an extension of the \emph{relaxed sector condition} of Horv\'ath, T\'oth and Vet\H{o} (2012) and is technically rather simpler than existing earlier proofs of similar results by Oelschl\"ager (1988) and Komorowski, Landim and Olla (2012)