Valleys and the maximum local time for random walk in random environment
成果类型：
Article
署名作者：
Dembo, Amir; Gantert, Nina; Peres, Yuval; Shi, Zhan
署名单位：
Stanford University; University of Munster; University of California System; University of California Berkeley; Sorbonne Universite; Universite Paris Cite
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-006-0005-6
发表日期：
2007
页码：
443-473
关键词：
摘要：
Let xi (n, x) be the local time at x for a recurrent one-dimensional random walk in random environment after n steps, and consider the maximum xi*(n) = max(x) xi(n, x). It is known that lim sup(n)xi(n)/n is a positive constant a.s. We prove that lim inf(n)(log log log n)xi*(n)/n is a positive constant a.s.;this answers a question of P. Revesz [5]. The proof is based on an analysis of the valleys in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time n large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.
来源URL：