The additive structure of elliptic homogenization

By S. Armstrong, Tuomo Kuusi, Jean-Christophe Mourrat. 2016

122 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/8c48594e6feee6df713e1f85a829dc8d0c60b204

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In Computational Mechanics

In Computational Theory and Mathematics

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Citations in the corpus

Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, background): “3 is from [92, 70] and was first made rigorous in [12].”
A higher-order large-scale regularity theory for random elliptic operators (2015, background): “In the context of nonlinear elliptic systems in divergence form, the result of Armstrong and Smart [3] on the large-scale C0,1-estimate was generalized by Armstrong and Mourrat [2] to nonsymmetric coe cients and well beyond nite range, further con rming that the random large-scale regularity theory ”
Quantitative stochastic homogenization and regularity theory of parabolic equations (2017, methodology): “Since the random variables we encounter in this paper are very often the sum of a deterministic quantity and a “small” random part, it is useful to work with the notation introduced in [4] for expressing the sizes of random variables (essentially, an alternative notation for certain Orlicz norms).”
A Regularity Theory for Random Elliptic Operators (2014, methodology): “Compared to the series [4, 6, 7], our approach has advantages both in terms of stochastic results and large-scale regularity.”