Robustness of the pathwise structure of fluctuations in stochastic homogenization
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Robustness of the pathwise structure of fluctuations in stochastic homogenization Mitia Duerinckx1,2 · Antoine Gloria2,3
· Felix Otto4
Received: 31 July 2018 / Revised: 5 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider a linear elliptic system in divergence form with random coefficients and study the random fluctuations of large-scale averages of the field and the flux of the solution operator. In the context of the random conductance model, we developed in a previous work a theory of fluctuations based on the notion of homogenization commutator: we proved that the two-scale expansion of this special quantity is accurate at leading order in the fluctuation scaling when averaged on large scales (as opposed to the two-scale expansion of the solution operator taken separately) and that the largescale fluctuations of the field and the flux of the solution operator can be recovered from those of the commutator. This implies that the large-scale fluctuations of the commutator of the corrector drive all other large-scale fluctuations to leading order, which we refer to as the pathwise structure of fluctuations in stochastic homogenization. In the present contribution we extend this result in two directions: we treat continuum elliptic (possibly non-symmetric) systems and allow for strongly correlated coefficient fields (Gaussian-like with a covariance function that can display an arbitrarily slow algebraic decay at infinity). Our main result shows in this general setting that the two-scale expansion of the homogenization commutator is still accurate to leading order when averaged on large scales, which illustrates the robustness of the pathwise structure of fluctuations.
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Antoine Gloria [email protected] Mitia Duerinckx [email protected] Felix Otto [email protected]
1
Laboratoire de Mathématique d’Orsay, UMR 8628, Université Paris-Sud, 91405 Orsay, France
2
Laboratoire Jacques-Louis Lions (LJLL), CNRS, Université de Paris, Sorbonne Université, 75005 Paris, France
3
Département de Mathématique, Université Libre de Bruxelles, Brussels, Belgium
4
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
123
M. Duerinckx et al.
Mathematics Subject Classification 60F99 · 60H25 · 35B27
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Main results and structure of the proof . . . . . . . . . . . . 2.1 Notation and statement of the main results . . . . . . . . 2.2 Structure of the proof . . . . . . . . . . . . . . . . . . . 3 Proof of the representation formulas and of the main estimates 3.1 Proof of Lemma 2.4: representation formulas . . . . . . 3.2 Proof of Lemma 2.9 . . . . . . . . . . . . . . . . . . . . 3.3 Proof of Proposition 2.5: main estimates . . . . . . . . . 3.4 Proof of Proposition 2.6: main estimates (cont’d) . . . . 4 Proof of the main results . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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