Communications in Mathematical Physics

, Volume 274, Issue 2, pp 381–397 | Cite as

Resistivity of an Infinite Three Dimensional Stationary Random Electric Conductor

  • Jérôme DepauwEmail author


The aim of this paper is to prove that the resistivity of an infinite stationary random medium can be computed from one realization. This is accomplished through the application of the pointwise degree 2 ergodic theorem for divergence-free stationary random fields (see [4]).


Compact Support Ergodic Theorem Weak Sense Riesz Operator Duality Formula 
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  1. 1.
    Boivin D. (1993). Weak convergence for reversible random walks in a random environment. Ann. Probab., 21(3): 1427–1440 zbMATHGoogle Scholar
  2. 2.
    Calderon A.P. and Zygmund A. (1952). On the existence of certain singular integrals. Acta Math., 88: 85–139 zbMATHCrossRefGoogle Scholar
  3. 3.
    Depauw J. (1999). Flux moyen d’un courant électrique dans un réseau aléatoire stationnaire de résistances. Ann. Inst. H. Poincaré Probab. Statist. 35(3): 355–370 zbMATHCrossRefGoogle Scholar
  4. 4.
    Depauw J. (2007). Degree two ergodic theorem for divergence-free random fields. Israel J. Math. 157: 283–308 zbMATHCrossRefGoogle Scholar
  5. 5.
    Giaquinta M. (1983). Multiple integrals in the calculus of variations and nonlinear elliptic systems, volume 105 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ: Google Scholar
  6. 6.
    Golden K. and Papanicolaou G. (1983). Bounds for effective parameters of heterogeneous media by analytic continuation. Commun. Math. Phys., 90(4): 473–491 CrossRefADSGoogle Scholar
  7. 7.
    Jikov, V.V., Kozlov, S.M., Oleĭnik, O.A.: Homogenization of differential operators and integral functionals. Berlin:Springer-Verlag, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′ yan]Google Scholar
  8. 8.
    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton, N.J.: Princeton University Press, 1970Google Scholar
  9. 9.
    Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, No. 32. Princeton, N.J.: Princeton University Press, 1971.Google Scholar
  10. 10.
    Wiener N. (1939). The ergodic theorem. Duke Math. J. 5(1): 1–18 zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Physique ThéoriqueCNRS UMR 6083, Université RabelaisToursFrance

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