Limit behaviour of random walks on Ζm with two-sided membrane

Abstract

We study Markov chains on Z m , m ≥ 2, that behave like a standard symmetric random walk outside of the hyperplane (membrane) H = {0} × Z m-1 . The exit probabilities from the membrane (penetration probabilities) H are periodic and also depend on the incoming direction to H, what makes the membrane H two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a m-dimensional diffusion whose first coordinate is a skew Brownian motion and the other m-1 coordinates is a Brownian motion with a singular drift controlled by the local time of the first coordinate at 0. In the proof we utilize a martingale characterization of the Walsh Brownian motion and determine the effective permeability and slide direction. Eventually, a similar convergence theorem is established for the one-sided membrane without slides and random iid penetration probabilities.