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

TL;DR

This paper analyzes the scaling limits of Markov chains on multi-dimensional integer lattices with a two-sided membrane, revealing a diffusion process with skew Brownian motion characteristics and singular drift components.

Contribution

It introduces a novel model of Markov chains with a two-sided membrane and derives their diffusion limit involving skew Brownian motion and local time-dependent drift.

Findings

The Markov chains converge to a diffusion with skew Brownian motion in the first coordinate.

The other coordinates are Brownian motions with singular drift influenced by local time.

Effective permeability and slide directions are explicitly characterized.

Abstract

We study Markov chains on $\mathbb Z^m$, $m\geq 2$, that behave like a standard symmetric random walk outside of the hyperplane (membrane) $H=\{0\}\times \mathbb Z^{m-1}$. The transition probabilities on the membrane $H$ 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 transition probabilities.