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Abstract: We consider a diffusion process with coefficients that are periodic outsideof an -interface region- of finite thickness. The question investigated in thisarticle is the limiting long time-large scale behavior of such a process underdiffusive rescaling. It is clear that outside of the interface, the limitingprocess must behave like Brownian motion, with diffusion matrices given by thestandard theory of homogenization. The interesting behavior therefore occurs onthe interface. Our main result is that the limiting process is a semimartingalewhose bounded variation part is proportional to the local time spent on theinterface. The proportionality vector can have nonzero components parallel tothe interface, so that the limiting diffusion is not necessarily reversible. Wealso exhibit an explicit way of identifying its parameters in terms of thecoefficients of the original diffusion. Similarly to the one-dimensional case,our method of proof relies on the framework provided by Freidlin and WentzellAnn. Probab. 21 1993 2215-2245 for diffusion processes on a graph in orderto identify the generator of the limiting process.



Author: Martin Hairer, Charles Manson

Source: https://arxiv.org/







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