Periodic homogenization with an interface: The multi-dimensional case

TL;DR

This paper studies the long-term behavior of a diffusion process with periodic coefficients interrupted by an interface, revealing a limiting process that behaves like Brownian motion outside the interface and exhibits complex behavior on it.

Contribution

It extends homogenization theory to multi-dimensional diffusions with interfaces, explicitly characterizing the limiting process and its parameters.

Findings

Limiting process is a semimartingale with local time proportionality on the interface.

The limiting diffusion may be non-reversible due to parallel components.

Explicit identification of parameters in terms of original coefficients.

Abstract

We consider a diffusion process with coefficients that are periodic outside of an "interface region" of finite thickness. The question investigated in this article is the limiting long time/large scale behavior 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 behavior 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. The proportionality vector can have nonzero components parallel to the interface, so that the limiting diffusion is not necessarily reversible. We also exhibit an explicit way of identifying its parameters in terms of the coefficients of the original diffusion. Similarly to the one-dimensional case, our method of proof relies on the framework provided by Freidlin and Wentzell [Ann. Probab. 21 (1993) 2215--2245] for diffusion processes on a graph in order to identify the generator of the limiting process.