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The additive structure of elliptic homogenization

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.

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Fig. 1

Notes

  1. Several months after this paper was submitted and posted to arXiv and before it was accepted, Gloria and Otto completed a substantial revision [23] of [22] in which they prove Theorem 1 as well as Theorem 2. Their analysis is based on a quantity they call the “homogenization commutator” which is closely related to the quantity J considered here.

  2. This heuristic derivation was obtained by SA, Yu Gu and JCM. It was the object of a talk given in Banff in July 2015 and reproduced during the Oberwolfach seminar on stochastic homogenization shortly afterwards. The talk can be watched at http://goo.gl/5bgfpR.

  3. We only use the word “distribution” to refer to Schwartz distributions, and call the probability measure associated with a random variable its law.

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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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Armstrong, S., Kuusi, T. & Mourrat, JC. The additive structure of elliptic homogenization. Invent. math. 208, 999–1154 (2017). https://doi.org/10.1007/s00222-016-0702-4

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