Periodic homogenization with an interface: the one-dimensional case

TL;DR

This paper studies the long-term behavior of a one-dimensional diffusion process with periodic coefficients outside an interface, showing it converges to a skew Brownian motion with explicitly determined parameters.

Contribution

It provides a rigorous analysis of the limiting process for diffusions with an interface, explicitly characterizing the convergence to skew Brownian motion in one dimension.

Findings

Convergence to skew Brownian motion under diffusive scaling.

Explicit formulas for skew Brownian motion parameters.

Method based on diffusion processes on a graph framework.

Abstract

We consider a one-dimensional diffusion process with coefficients that are periodic outside of a finite 'interface region'. The question investigated in this article is the limiting long time / large scale behaviour of such a process under diffusive rescaling. Our main result is that it converges weakly to a rescaled version of skew Brownian motion, with parameters that can be given explicitly in terms of the coefficients of the original diffusion. Our method of proof relies on the framework provided by Freidlin and Wentzell for diffusion processes on a graph in order to identify the generator of the limiting process. The graph in question consists of one vertex representing the interface region and two infinite segments corresponding to the regions on either side.