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

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  • Avena, L.
  • Blondel, O.
  • Faggionato, A.

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 satisfies 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 finite-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.

Suggested Citation

  • Avena, L. & Blondel, O. & Faggionato, A., 2018. "Analysis of random walks in dynamic random environments via L2-perturbations," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3490-3530.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:10:p:3490-3530
    DOI: 10.1016/j.spa.2017.11.010
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    References listed on IDEAS

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    1. den Hollander, F. & dos Santos, R. & Sidoravicius, V., 2013. "Law of large numbers for non-elliptic random walks in dynamic random environments," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 156-190.
    2. Rassoul-Agha, F. & Seppäläinen, T., 2008. "An almost sure invariance principle for additive functionals of Markov chains," Statistics & Probability Letters, Elsevier, vol. 78(7), pages 854-860, May.
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