Random walk in random environment with asymptotically zero perturbation

Mikhail V. Menshikov and Andrew R. Wade

Journal of the European Mathematical Society, 8, no. 3, September 2006, 491–513. DOI: 10.4171/JEMS/64.

Abstract

We give criteria for ergodicity, transience and null recurrence for the random walk in random environment on $\mathbb{Z}^+ =\{0,1,2,\ldots\}$, 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.

Further remarks

In the case in which the environment is generated by a sequence of i.i.d. transition probabilities $p_n$, classical work of Solomon shows that $\zeta_n = \log (p_n/q_n)$ is the key quantity. The critical regime for the recurrence classification in which $\mathbb{E} [ \zeta_n^2 ] \in (0,\infty)$ and $\mathbb{E} [ \zeta_n ] = 0$ is known as Sinai's regime. This paper and its sequel [MW2] study environments that are asymptotically small perturbations of Sinai's regime, in which case the transition probabilities are no longer i.i.d. Questions of interest include: how big does the perturbation have to be in order for the walk to be transient? After a long time, how far, typically, is the particle from where it starts?

In the heavy-tailed case where $\mathbb{E}[ \zeta_n^2] = \infty$, different phenomena are observed. Perturbations of the i.i.d. environment in the heavy-tailed case are studied in [HMW2].