The additive structure of elliptic homogenization

Scott Armstrong, Tuomo Kuusi, Jean Christophe Mourrat

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1-156
Number of pages156
JournalInventiones Mathematicae
DOIs
StateAccepted/In press - Nov 17 2016

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Additivity
Stochastic Homogenization
Homogenization
Corrector
Energy Density
First-order
Regularity Theory
Sums of Independent Random Variables
Weak Convergence
Central limit theorem
Length Scale
Bootstrap
Elliptic Equations
Integrability
Linear equation
Divergence
Scaling
Gradient
Converge
Coefficient

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

The additive structure of elliptic homogenization. / Armstrong, Scott; Kuusi, Tuomo; Mourrat, Jean Christophe.

In: Inventiones Mathematicae, 17.11.2016, p. 1-156.

Research output: Contribution to journalArticle

Armstrong, Scott ; Kuusi, Tuomo ; Mourrat, Jean Christophe. / The additive structure of elliptic homogenization. In: Inventiones Mathematicae. 2016 ; pp. 1-156.
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