# Quenched Central Limit Theorem for Random Walks in Doubly Stochastic Random Environment

## TL;DR

This paper establishes a quenched central limit theorem for random walks in doubly stochastic environments, extending previous results by relaxing integrability conditions and generalizing Nash's moment bounds.

## Contribution

It proves a quenched CLT under weaker integrability assumptions and extends Nash's moment bounds to non-reversible, divergence-free drift cases.

## Key findings

- Quenched CLT holds under $H_{-1}$-condition with $L^{2+ ext{epsilon}}$ integrability.
- Extended Nash's moment bound to non-reversible divergence-free drifts.
- Improved understanding of random walks in complex stochastic environments.

## Abstract

We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger, $L^{2+\varepsilon}$ (rather than $L^2$) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.