The structure of fluctuations in stochastic homogenization

Mitia Duerinckx 1, 2 Antoine Gloria 1, 2 Felix Otto 3
2 MEPHYSTO - Quantitative methods for stochastic models in physics
Inria Lille - Nord Europe, ULB - Université Libre de Bruxelles [Bruxelles], LPP - Laboratoire Paul Painlevé - UMR 8524
Abstract : We establish a path-wise theory of fluctuations in stochastic homogenization of linear elliptic equations in divergence form. More precisely we consider the model problem of a discrete equation with independent and identically distributed conductances. We identify a single quantity, which we call the homogenization commutator, that drives the fluctuations in stochastic homogenization in the following sense. On the one hand, this tensor-valued stationary random field satisfies a functional central limit theorem, and (when suitably rescaled) converges to a Gaussian white noise. On the other hand, the fluctuations of the gradient of the corrector, the fluctuations of the flux of the corrector, and the fluctuations of any solution of the PDE with random coefficients and localized right-hand side are characterized at leading order by the fluctuations of this homogenization commutator in a path-wise sense. As a consequence, when properly rescaled, the solution satisfies a functional central limit theorem, the gradient of the corrector converges to the Helmholtz projection of a Gaussian white noise, and the flux of the corrector converges to the Leray projection of the same white noise. Compared to previous contributions, our approach, based on the homogenization commutator, unravels the complete structure of fluctuations. It holds in any dimension d≥2, yields the first path-wise results, quantifies the limit theorems in Wasserstein distance, and only relies on arguments that extend to the continuum setting and to the case of systems.
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https://hal.inria.fr/hal-01398515
Contributor : Antoine Gloria <>
Submitted on : Thursday, November 17, 2016 - 1:19:26 PM
Last modification on : Monday, August 20, 2018 - 9:44:02 AM

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  • HAL Id : hal-01398515, version 1
  • ARXIV : 1602.01717

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Mitia Duerinckx, Antoine Gloria, Felix Otto. The structure of fluctuations in stochastic homogenization. 2016. ⟨hal-01398515⟩

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