Hairer, Martin and Manson, Charles. (2011) Periodic homogenization with an interface : the multi-dimensional case. Annals of Probability, Vol.39 (No.2). pp. 648-682. ISSN 0091-1798

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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 resealing. 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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Annals of Probability
Publisher:	Institute of Mathematical Statistics
ISSN:	0091-1798
Date:	March 2011
Volume:	Vol.39
Number:	No.2
Page Range:	pp. 648-682
Identification Number:	10.1214/10-AOP564
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	EPSRC (EP/D071593/1)
URI:	http://wrap.warwick.ac.uk/id/eprint/41501

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