Planar Random Walk in a Stratified Quasi-periodic Environment

This article investigates the question of the recurrence of a class of inhomogeneous Markov chains in the plane, assuming the environment invariant under horizontal translations. This type of random walks were first considered by Matheron and de Marsily [20] around 1980, with a motivation coming from hydrology and the modelization of pollutants diffusion in a porous and stratified ground. In 2003, a discrete version was introduced by Campanino and Petritis in [7]. As in [4-6], we consider an extension of the latter, restricting here to the plane and simplifying a little the hypotheses. We shall define a Markov chain (S-k) k >= 0 in Z(2), starting at the origin, such that the transition laws are constant on each stratum Z x {n}, n is an element of Z. The first and second coordinates will be respectively called \horizontal" and \vertical". For each (vertical) n is an element of Z, let positive reals alpha(n); beta(n); gamma(n), with alpha(n) + beta(n) + gamma(n) = 1, and a probability measure mu(n) so that: