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Approximation of the effective conductivity of ergodic media by periodization
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  • Published: February 2003

Approximation of the effective conductivity of ergodic media by periodization

  • Houman Owhadi1 

Probability Theory and Related Fields volume 125, pages 225–258 (2003)Cite this article

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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 xℝd, 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

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 ℤd with ergodic jump rates.

The core of our result is to show that the ergodic Weyl decomposition associated to 2(X, μ) can almost surely be approximated by periodic Weyl decompositions with increasing periods, implying that semi-continuous variational formulae associated to 2(X, μ) can almost surely be approximated by variational formulae minimizing on periodic potential and solenoidal functions.

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Authors and Affiliations

  1. LATP, UMR CNRS 6632, CMI, Université de Provence. e-mail: owhadi@cmi.univ-mrs.fr, , , , , , FR

    Houman Owhadi

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  1. Houman Owhadi
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Additional information

Received: 10 January 2002 / Revised version: 12 August 2002 / Published online: 14 November 2002

Mathematics Subject Classification (2000): Primary 74Q20, 37A15; Secondary 37A25

Key words or phrases: Effective conductivity – periodization of ergodic media – Weyl decomposition

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Owhadi, H. Approximation of the effective conductivity of ergodic media by periodization. Probab Theory Relat Fields 125, 225–258 (2003). https://doi.org/10.1007/s00440-002-0240-4

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  • Issue Date: February 2003

  • DOI: https://doi.org/10.1007/s00440-002-0240-4

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Keywords

  • Probability Space
  • Symmetric Matrix
  • Elliptic Operator
  • Free Flow
  • Variational Formula
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