INVARIANT MEASURE FOR RANDOM WALKS ON ERGODIC ENVIRONMENTS ON A STRIP

• 成果类型: Article
• 署名作者: Dolgopyat, Dmitry; Goldsheid, Ilya
• 署名单位: University System of Maryland; University of Maryland College Park; University System of Maryland; University of Maryland College Park; University of London; Queen Mary University London
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1313
• 发表日期: 2019
• 页码: 2494-2528
• 关键词: dimensional random-walk; transient random-walks; local limit-theorem; quenched limits; hitting times; of-view; particle; LAW

摘要

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.