TY - JOUR

T1 - The additive structure of elliptic homogenization

AU - Armstrong, Scott

AU - Kuusi, Tuomo

AU - Mourrat, Jean Christophe

N1 - Funding Information:
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. .
Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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.

AB - 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.

KW - 35B27

KW - 35B45

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U2 - 10.1007/s00222-016-0702-4

DO - 10.1007/s00222-016-0702-4

M3 - Article

AN - SCOPUS:84995776311

SN - 0020-9910

VL - 208

SP - 999

EP - 1154

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -