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

2014/11/15 by Gady Kozma, Kozma, Gady, Bálint Tóth +1
Decision Sciences · Mathematics · #60F05 #60G99 #60K37 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Probability and Risk Models #Statistical Distribution Estimation and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60G99 #msc:60K37

paper · pdf · doi:10.48550/arxiv.1411.4171

33 pages

arxiv created 2014/11/15 · openalex publication_date 2014/11/15 · arxiv updated 2014/11/18 · openalex created_date 2022/09/04 · openalex updated_date 2026/07/28

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+δ,d\, with δ>0. Our proof relies on the relaxed sector condition of Horváth, Tóth and Vető (2012), and is technically rather simpler than existing earlier proofs of similar results by Oelschläger (1988) and Komorowski, Landim, Olla (2012).

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