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Approximation of the effective conductivity of ergodic media by periodization

Owhadi, Houman (2003) Approximation of the effective conductivity of ergodic media by periodization. Probability Theory and Related Fields, 125 (2). pp. 225-258. ISSN 0178-8051. https://resolver.caltech.edu/CaltechAUTHORS:20160224-124326510

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Abstract

This paper is concerned with the approximation of the effective conductivity σ(A, μ) associated to an elliptic operator ∇_xA(x,η)∇_x where for xЄℝ^d,d ≥ 1, A(x,η) is a bounded elliptic random symmetric d×d matrix and η takes value in an ergodic probability space (X, μ). Writing A^N (x, η) the periodization of A(x, η) on the torus T^d_N of dimension d and side N we prove that for μ-almost all η lim ^(N→+∞) σ(A^N, η) = σ(A,μ) We extend this result to non-symmetric operators ∇_x (a+E(x, η))∇_x corresponding to diffusions in ergodic divergence free flows (a is d×d elliptic symmetric matrix and E(x, η) an ergodic skew-symmetric matrix); and to discrete operators corresponding to random walks on ℤ^d with ergodic jump rates. The core of our result is to show that the ergodic Weyl decomposition associated to L^2(X,μ) can almost surely be approximated by periodic Weyl decompositions with increasing periods, implying that semi-continuous variational formulae associated to L^2(X,μ) can almost surely be approximated by variational formulae minimizing on periodic potential and solenoidal functions.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00440-002-0240-4DOIArticle
http://link.springer.com/article/10.1007%2Fs00440-002-0240-4PublisherArticle
http://arxiv.org/abs/math/0201062arXivDiscussion Paper
ORCID:
AuthorORCID
Owhadi, Houman0000-0002-5677-1600
Additional Information:© 2003 Springer-Verlag. Received: 10 January 2002. Revised version: 12 August 2002. Published online: 14 November 2002. Part of this work was supported by the Aly Kaufman fellowship. The author would like to thank Dmitry Ioffe for his hospitality during his stay at the Technion, for suggesting this problem and for stimulating and helpful discussions. Thanks are also due to the referee for many useful comments.
Funders:
Funding AgencyGrant Number
Aly Kaufman fellowshipUNSPECIFIED
Subject Keywords:Effective conductivity – periodization of ergodic media – Weyl decomposition
Issue or Number:2
Classification Code:Mathematics Subject Classification (2000): Primary 74Q20, 37A15; Secondary 37A25
Record Number:CaltechAUTHORS:20160224-124326510
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160224-124326510
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:64738
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:24 Feb 2016 21:44
Last Modified:03 Oct 2019 09:40

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