L^2-Perturbed Markov processes and applications to random walks in dynamic random environments

L. Avena, O. Blondel, A. Faggionato

Published 2016-02-19Version 1

We consider a Markov process satisfying a Poincar\'e inequality w.r.t. a stationary, ergodic distribution $\mu$ and perturb its generator by a bounded operator in $L^2(\mu)$, thus leading to a new Markov process. By using a Dyson-Phillips type expansion, we prove that the perturbed Markov process has a unique invariant distribution absolutely continuous w.r.t. $\mu$, which in addition is ergodic. We also derive a law of large numbers and an invariance principle for additive functionals for the perturbed Markov process. We apply these results to continuous-time random walks in dynamic random environments given by Markovian dynamics. Consequently, we prove a series expansion for the asymptotic speed of the random walk and an averaged invariance principle for its position. We 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.