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Homogenization of periodic linear degenerate PDEs
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Hairer, Martin and Pardoux, Etienne (2008) Homogenization of periodic linear degenerate PDEs. Journal of Functional Analysis, Volume 255 (Number 9). pp. 2462-2487. doi:10.1016/j.jfa.2008.04.014 ISSN 0022-1236.
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Official URL: http://dx.doi.org/10.1016/j.jfa.2008.04.014
Abstract
It is well known under the name of 'periodic homogenization' that, under a centering condition of the drift, a periodic diffusion process on R-d converges, under diffusive rescaling, to a d-dimensional Brownian motion. Existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion. In this paper, we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set. As a consequence, it is no longer the case that the effective diffusivity matrix is necessarily non-degenerate. It turns out that, provided that some very weak regularity conditions are met, the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion. In particular, this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank. We also discuss the application of our results to the homogenization of a class of elliptic and parabolic PDEs. (C) 2008 Elsevier Inc. All rights reserved.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Homogenization (Differential equations), Diffusion, Malliavin calculus, Spectral theory (Mathematics), Degenerate differential equations | ||||
Journal or Publication Title: | Journal of Functional Analysis | ||||
Publisher: | Academic Press | ||||
ISSN: | 0022-1236 | ||||
Official Date: | 1 November 2008 | ||||
Dates: |
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Volume: | Volume 255 | ||||
Number: | Number 9 | ||||
Number of Pages: | 26 | ||||
Page Range: | pp. 2462-2487 | ||||
DOI: | 10.1016/j.jfa.2008.04.014 | ||||
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/13071593/1 (EPSRC) |
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