Central limit theorem for random walks in doubly stochastic random environment: ${\mathscr{H}_{-1}}$ suffices

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 ${\mathscr{H}_{-1}}$-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 [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer], where it is assumed that the stream tensor is in $\mathscr{L}^{\max\{2+\delta,d\}}$, with $\delta>0$. Our proof relies on an extension of the relaxed sector condition of [Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012) 463–476], and is technically rather simpler than existing earlier proofs of similar results by Oelschläger [Ann. Probab. 16 (1988) 1084–1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer].