Logarithmic speeds for one-dimensional perturbed random walks in random environments

We study the random walk in a random environment on Z + = { 0 , 1 , 2 , … } , where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) a random walk in a random environment perturbed from Sinai’s regime; (ii) a simple random walk with a random perturbation. We give almost sure results on how far the random walker is from the origin, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order ( log t ) β , for β ∈ ( 1 , ∞ ) , depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution.