## The Annals of Probability

### Invariant measure for random walks on ergodic environments on a strip

#### Abstract

Environment viewed from the particle is a powerful method of analyzing random walks (RW) in random environment (RE). It is well known that in this setting the environment process is a Markov chain on the set of environments. We study the fundamental question of existence of the density of the invariant measure of this Markov chain with respect to the measure on the set of environments for RW on a strip. We first describe all positive subexponentially growing solutions of the corresponding invariant density equation in the deterministic setting and then derive necessary and sufficient conditions for the existence of the density when the environment is ergodic in both the transient and the recurrent regimes. We also provide applications of our analysis to the question of positive and null recurrence, the study of the Green functions and to random walks on orbits of a dynamical system.

#### Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2494-2528.

Dates
Revised: May 2018
First available in Project Euclid: 4 July 2019

https://projecteuclid.org/euclid.aop/1562205714

Digital Object Identifier
doi:10.1214/18-AOP1313

Mathematical Reviews number (MathSciNet)
MR3980926

#### Citation

Dolgopyat, Dmitry; Goldsheid, Ilya. Invariant measure for random walks on ergodic environments on a strip. Ann. Probab. 47 (2019), no. 4, 2494--2528. doi:10.1214/18-AOP1313. https://projecteuclid.org/euclid.aop/1562205714

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