Central limit theorem for random walks in divergence free random drift field -- revisited

TL;DR

This paper revisits the central limit theorem for random walks in divergence-free random fields, offering an alternative proof that relies solely on functional analysis, potentially extending to non-elliptic cases.

Contribution

Provides a simplified, functional-analytic proof of the CLT for divergence-free environments, avoiding heat kernel bounds and enabling extensions to non-elliptic settings.

Findings

Alternative proof relies only on functional analysis.

Proof can be extended to non-elliptic environments.

Revisits and simplifies previous CLT proof for divergence-free fields.

Abstract

In [Kozma-Toth, Ann. Probab. v 45, pp 4307-4347 (2017)] the weak CLT was established for random walks in doubly stochastic (or, divergence-free) random environments, under the following conditions: 1. Strict ellipticity assumed for the symmetric part of the drift field. 2. $H_{-1}$ assumed for the antisymmetric part of the drift field. The proof relied on a martingale approximation (a la Kipnis-Varadhan) adapted to the non-self-adjoint and non-sectorial nature of the problem. The two substantial technical components of the proof were: 1. A functional analytic statement about the unbounded operator formally written as $|L+L^*|^{-1/2}(L-L^*)|L+L^*|^{-1/2}$, where $L$ is the infinitesimal generator of the environment process, as seen from the position of the moving random walker. 2. A diagonal heat kernel upper bound which follows directly from Nash's inequality, or, alternatively, from the "evolving sets" arguments of [Morris-Peres, Probab. Theory Rel. Fields. v. 133 pp 245-266 (2005)] valid only under the assumed strict ellipticity. In this note we present a partly alternative proof of the same result which relies only on functional analytic arguments and not on the diagonal heat kernel upper bound provided by Nash's inequality. This alternative proof is relevant since it can be naturally extended to non-elliptic settings pushed to the optimum, which will be presented in a forthcoming paper. The goal of this note is to present the argument in its simplest and most transparent form.