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

Mikhail V. Menshikov and Andrew R. Wade

Stochastic Processes and their Applications, 118, no. 3, March 2008, 389–416. DOI: 10.1016/j.spa.2007.04.011.

Abstract

We study the random walk in random environment on $\mathbb{Z}^+ =\{0,1,2,\ldots\}$, 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 $\beta > 1$ depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution.