Logarithmic speeds for one-dimensional perturbed random walk in random environment

Details Logarithmic speeds for one-dimensional perturbed random walk in random environment Logarithmic speeds for one-dimensional perturbed random walk in random environment

2006-08-28

arxiv.org/abs/math/0608697v2

Stochastic Processes and their Applications, Vol. 118 (2008), no. 3, p. 389-416

Mathematics

Probability

Revised version

Scientific paper

10.1016/j.spa.2007.04.011

We study the random walk in random environment on {0,1,2,...}, where the environment is subject to a vanishing (random) perturbation. The two particular cases we consider are: (i) random walk in random environment perturbed from Sinai's regime; (ii) simple random walk with random perturbation. We give almost sure results on how far the random walker will be from the origin after a long time t, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order $(\log t)^\beta$, for $\beta \in (1,\infty)$, depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution.

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