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March 2011 Periodic homogenization with an interface: The multi-dimensional case
Martin Hairer, Charles Manson
Ann. Probab. 39(2): 648-682 (March 2011). DOI: 10.1214/10-AOP564

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.

Citation

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Martin Hairer. Charles Manson. "Periodic homogenization with an interface: The multi-dimensional case." Ann. Probab. 39 (2) 648 - 682, March 2011. https://doi.org/10.1214/10-AOP564

Information

Published: March 2011
First available in Project Euclid: 25 February 2011

zbMATH: 1217.60044
MathSciNet: MR2789509
Digital Object Identifier: 10.1214/10-AOP564

Subjects:
Primary: 60H10 , 60J60

Keywords: Interface , Local time , Periodic homogenization , skew Brownian motion

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 2 • March 2011
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