Limit behaviour of random walks on ℤmwith two-sided membrane

Abstract

We study Markov chains on ℤm,m≥ 2, that behave like a standard symmetric random walk outside of the hyperplane (membrane)H= {0} × ℤm−1. The exit probabilities from the membrane (penetration probabilities) H are periodic and also depend on the incoming direction toH, 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 otherm− 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.