CENTRAL LIMIT THEOREM FOR RANDOM WALKS IN DOUBLY STOCHASTIC RANDOM ENVIRONMENT: H-1 SUFFICES

• 成果类型: Article
• 署名作者: Kozma, Gady; Toth, Balint
• 署名单位: Weizmann Institute of Science; University of Bristol; HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1166
• 发表日期: 2017
• 页码: 4307-4347
• 关键词: tagged particle; homogenization; diffusion

摘要

We prove a central limit theorem under diffusive scaling for the displacement of a random walk on Z(d) in stationary and ergodic doubly stochastic random environment, under the 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 L-max{2+delta,L-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 Oelschlager [Ann. Probab. 16 (1988) 1084-1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes-Time Symmetry and Martingale Approximation (2012) Springer].