We consider random walks in dynamic random environments given by Markovian dynamics on Zd . We assume that the environment has a stationary distribution μ and satis es the Poincaré inequality w.r.t. μ. The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution μ. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be nite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of L2- bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion.

Analysis of random walks in dynamic random environments via L2-perturbations / Avena, L.; Blondel, O.; Faggionato, A.. - In: STOCHASTIC PROCESSES AND THEIR APPLICATIONS. - ISSN 0304-4149. - (2018), pp. 3490-3530. [10.1016/j.spa.2017.11.010]

Analysis of random walks in dynamic random environments via L2-perturbations

A. Faggionato
2018

Abstract

We consider random walks in dynamic random environments given by Markovian dynamics on Zd . We assume that the environment has a stationary distribution μ and satis es the Poincaré inequality w.r.t. μ. The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution μ. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be nite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of L2- bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion.
2018
Perturbations of Markov processes; Poincaré inequality; Dyson–Phillips expansion; Random walk in dynamic random environment; Asymptotic velocity; Invariance principle
01 Pubblicazione su rivista::01a Articolo in rivista
Analysis of random walks in dynamic random environments via L2-perturbations / Avena, L.; Blondel, O.; Faggionato, A.. - In: STOCHASTIC PROCESSES AND THEIR APPLICATIONS. - ISSN 0304-4149. - (2018), pp. 3490-3530. [10.1016/j.spa.2017.11.010]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1511781
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