Random integrals and correctors in homogenization

Guillaume Bal, Josselin Garnier, Sébastien Motsch, Vincent Perrier

Research output: Contribution to journalArticle

43 Scopus citations

Abstract

This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation.

Original languageEnglish (US)
Pages (from-to)1-26
Number of pages26
JournalAsymptotic Analysis
Volume59
Issue number1-2
DOIs
StatePublished - 2008

Keywords

  • Central limit
  • Gaussian processes
  • Homogenization
  • Long-range memory effects
  • Partial differential equations with random coefficients

ASJC Scopus subject areas

  • Mathematics(all)

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