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Periodic homogenization with an interface : the multidimensional case
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Hairer, Martin and Manson, Charles. (2011) Periodic homogenization with an interface : the multidimensional case. Annals of Probability, Volume 39 (Number 2). pp. 648682. ISSN 00911798
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Official URL: http://dx.doi.org/10.1214/10AOP564
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 onedimensional case, our method of proof relies on the framework provided by Freidlin and Wentzell [Ann. Probab. 21 (1993) 22152245] 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  
Library of Congress Subject Headings (LCSH):  Homogenization (Differential equations), Diffusion processes  
Journal or Publication Title:  Annals of Probability  
Publisher:  Institute of Mathematical Statistics  
ISSN:  00911798  
Official Date:  March 2011  
Dates: 


Volume:  Volume 39  
Number:  Number 2  
Page Range:  pp. 648682  
Identifier:  10.1214/10AOP564  
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:  EP/D071593/1 (EPSRC)  
References:  [1] ALLAIRE, G. andAMAR, M. (1999). Boundary layer tails in periodic homogenization. ESAIM 

URI:  http://wrap.warwick.ac.uk/id/eprint/41501 
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