Journal of the European Mathematical Society

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Volume 8, Issue 3, 2006, pp. 491–513
DOI: 10.4171/JEMS/64

Random walk in random environment with asymptotically zero perturbation

M.V. Menshikov (1) and Andrew R. Wade (2)

(1) Department of Mathematical Sciences, University of Durham, South Road, DH1 3HP, DURHAM, UNITED KINGDOM
(2) Department of Mathematical Sciences, University of Bath, BA2 7AY, BATH, UNITED KINGDOM

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.

Keywords: Random walk in random environment, perturbation of Sinai's regime, recurrence/transience criteria, Lyapunov functions.