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