2016/02/19 by Luca Avena, Avena, L., Oriane Blondel +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Economics, Econometrics and Finance · Mathematics · #60F17 #60K37 #82C22 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.1602.06322
openalex publication_date 2016/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.