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The additive structure of elliptic homogenization

Armstrong, Scott N.; Kuusi, Tuomo; Mourrat, Jean-Christophe (2017), The additive structure of elliptic homogenization, Inventiones Mathematicae, 208, 3, p. 999–1154. 10.1007/s00222-016-0702-4

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Type
Article accepté pour publication ou publié
Date
2017
Journal name
Inventiones Mathematicae
Volume
208
Number
3
Publisher
Springer
Pages
999–1154
Publication identifier
10.1007/s00222-016-0702-4
Metadata
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Author(s)
Armstrong, Scott N.

Kuusi, Tuomo

Mourrat, Jean-Christophe
Abstract (EN)
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.
Subjects / Keywords
stochastic homogenization; error estimates; regularity theory; renormalization; scaling limits; Gaussian free field

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