Law of large numbers for a finite-range random walk in a dynamic random environment with nonuniform mixing

Abstract: In this paper, we study random walks evolving on Z in a dynamic random environment that we assume to have time correlations that decrease polynomially fast. We show a law of large numbers by generalizing methods already used for the nearest-neighbor framework to the finite-range one. This requires some new ideas to get around the absence of a monotonicity property that was crucial in the proof for the nearest-neighbour case. Our proof works both in discrete and continuous time.