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

Menshikov, M.V.; Wade, Andrew R.

Abstract

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 (logt)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.

Citation

Menshikov, M., & Wade, A. R. (2008). Logarithmic speeds for one-dimensional perturbed random walks in random environments. Stochastic Processes and their Applications, 118(3), 389-416. https://doi.org/10.1016/j.spa.2007.04.011

• Journal Article Type: Article
• Publication Date: Mar 1, 2008
• Deposit Date: Mar 1, 2011
• Publicly Available Date: Jan 31, 2013
• Journal: Stochastic Processes and their Applications
• Print ISSN: 0304-4149
• Electronic ISSN: 1879-209X
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 118
• Issue: 3
• Pages: 389-416
• DOI: https://doi.org/10.1016/j.spa.2007.04.011
• Keywords: Random walk in perturbed random environment, Logarithmic speeds, Almost sure behaviour, Slow transience.
• Public URL: https://durham-repository.worktribe.com/output/1543085