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Inventiones mathematicae

, Volume 208, Issue 3, pp 999–1154 | Cite as

The additive structure of elliptic homogenization

  • Scott Armstrong
  • Tuomo Kuusi
  • Jean-Christophe Mourrat
Article

Abstract

One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in Armstrong et al. (Commun. Math. Phys. 347(2):315–361, 2016): using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

Mathematics Subject Classification

35B27 35B45 

Notes

Acknowledgements

The second author was supported by the Academy of Finland project #258000. We thank Antti Hannukainen (Aalto University) for performing the numerical computations of the corrector and producing Fig. 1.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Scott Armstrong
    • 1
    • 2
  • Tuomo Kuusi
    • 3
  • Jean-Christophe Mourrat
    • 4
  1. 1.Université Paris-DauphinePSL Research University, CNRS, UMR [7534], CEREMADEParisFrance
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Department of Mathematics and Systems AnalysisAalto UniversityEspooFinland
  4. 4.Ecole normale supérieure de Lyon, CNRSLyonFrance

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