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Higher-order pathwise theory of fluctuations in stochastic homogenization

  • Mitia DuerinckxEmail author
  • Felix Otto
Article

Abstract

We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order \(\frac{d}{2}\), where d is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the “homogenization commutator”, thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (second-order) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calderón–Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates.

Keywords

Stochastic homogenization Higher-order homogenization Linear elliptic equation Fluctuations Homogenization commutator 

Notes

Acknowledgements

The work of MD was supported by F.R.S.-FNRS and by the CNRS-Momentum program.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématique d’Orsay, UMR 8628Université Paris-SudOrsayFrance
  2. 2.Département de MathématiqueUniversité Libre de BruxellesBrusselsBelgium
  3. 3.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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