The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations

Antoine Gloria 1, 2 Felix Otto 3
1 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 consider uniformly elliptic coefficient fields that are randomly distributed according to a stationary ensemble of a finite range of dependence. We show that the gradient and flux (∇ϕ,a(∇ϕ+e)) of the corrector ϕ, when spatially averaged over a scale R≫1 decay like the CLT scaling R−d2. We establish this optimal rate on the level of sub-Gaussian bounds in terms of the stochastic integrability, and also establish a suboptimal rate on the level of optimal Gaussian bounds in terms of the stochastic integrability. The proof unravels and exploits the self-averaging property of the associated semi-group, which provides a natural and convenient disintegration of scales, and culminates in a propagator estimate with strong stochastic integrability. As an application, we characterize the fluctuations of the homogenization commutator, and prove sharp bounds on the spatial growth of the corrector, a quantitative two-scale expansion, and several other estimates of interest in homogenization.
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https://hal.inria.fr/hal-01230985
Contributor : Antoine Gloria <>
Submitted on : Thursday, November 19, 2015 - 1:46:18 PM
Last modification on : Monday, August 20, 2018 - 9:44:02 AM

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  • HAL Id : hal-01230985, version 1
  • ARXIV : 1510.08290

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Antoine Gloria, Felix Otto. The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations. 2016. ⟨hal-01230985⟩

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