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July 2019 Invariant measure for random walks on ergodic environments on a strip
Dmitry Dolgopyat, Ilya Goldsheid
Ann. Probab. 47(4): 2494-2528 (July 2019). DOI: 10.1214/18-AOP1313


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.


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Dmitry Dolgopyat. Ilya Goldsheid. "Invariant measure for random walks on ergodic environments on a strip." Ann. Probab. 47 (4) 2494 - 2528, July 2019.


Received: 1 December 2016; Revised: 1 May 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114722
MathSciNet: MR3980926
Digital Object Identifier: 10.1214/18-AOP1313

Primary: 60K37
Secondary: 60J05, 82C44

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.47 • No. 4 • July 2019
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