Limit behaviour of random walks on ℤ^m with two-sided membrane

¹ National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, ave. Pobedy 37, Kiev 03056, Ukraine
² Institute for Mathematics, Friedrich Schiller University Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany
³ Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str. 3, 01601 Kiev, Ukraine

^* Corresponding author: ilya.pavlyukevich@uni-jena.de

Received: 4 August 2021
Accepted: 25 July 2022

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.