Random walk in random environment with asymptotically zero perturbation

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

Abstract

We give criteria for ergodicity, transience and null recurrence for the random walk in random environment on $\Z^+=\{0,1,2,\ldots\}$, with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different to these previously studied cases. Our method is based on a martingale technique - the method of Lyapunov functions.

Citation

Menshikov, M., & Wade, A. (2006). Random walk in random environment with asymptotically zero perturbation. Journal of the European Mathematical Society, 8(3), 491-513. https://doi.org/10.4171/jems/64

• Journal Article Type: Article
• Publication Date: 2006
• Deposit Date: Jan 9, 2009
• Publicly Available Date: Feb 1, 2013
• Journal: Journal of the European Mathematical Society
• Print ISSN: 1435-9855
• Electronic ISSN: 1435-9863
• Publisher: EMS Press
• Peer Reviewed: Peer Reviewed
• Volume: 8
• Issue: 3
• Pages: 491-513
• DOI: https://doi.org/10.4171/jems/64
• Keywords: Random walk in random environment, Perturbation of Sinai's regime, Recurrence/transience criteria, Lyapunov functions.