  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.
