Periodic homogenization with an interface

TL;DR

This paper studies the long-term behavior of diffusion processes with periodic coefficients interrupted by an interface, revealing that the limit is a semimartingale influenced by local time on the interface, with explicit parameter identification.

Contribution

It introduces a novel analysis of homogenization with an interface, characterizing the limiting process as a semimartingale with explicit interface parameters.

Findings

The limiting process is a semimartingale with local time influence.

Explicit formulas for interface parameters are provided.

The method uses Freidlin and Wentzell's diffusion on a graph framework.

Abstract

We consider a diffusion process with coefficients that are periodic outside of an 'interface region' of finite thickness. The question investigated in the articles [1,2] is the limiting long time / large scale behaviour of such a process under diffusive rescaling. It is clear that outside of the interface, the limiting process must behave like Brownian motion, with diffusion matrices given by the standard theory of homogenization. The interesting behaviour therefore occurs on the interface. Our main result is that the limiting process is a semimartingale whose bounded variation part is proportional to the local time spent on the interface. We also exhibit an explicit way of identifying its parameters 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.