Periodic homogenization with an interface: The one-dimensional case

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 (1993) [6] 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.