Central limit theorem for random walks in divergence-free random drift field: "H-minus-one" suffices

TL;DR

This paper establishes a central limit theorem for random walks in divergence-free random drift fields under the ${ m H}_{-1}$-condition, improving previous results by relaxing the regularity assumptions on the drift field.

Contribution

It proves a CLT under the ${ m H}_{-1}$-condition, simplifying earlier proofs and extending the class of drift fields for which the CLT holds.

Findings

Proves CLT for random walks in divergence-free fields under ${ m H}_{-1}$-condition.

Shows the ${ m H}_{-1}$-condition is sufficient for diffusive behavior.

Simplifies proof using relaxed sector condition.

Abstract

We prove central limit theorem under diffusive scaling for the displacement of a random walk on ${\mathbb Z}^d$ in stationary divergence-free random drift field, under the ${\mathcal H}_{-1}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor 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 be in ${\mathcal L}^{\max\{2+\delta,d\}}$, with $\delta>0$. Our proof relies on the 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, Olla (2012).