Vol. 14, No. 8, 2021

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Quantitative estimates in stochastic homogenization for correlated coefficient fields

Antoine Gloria, Stefan Neukamm and Felix Otto

Vol. 14 (2021), No. 8, 2497–2537

This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal growth estimates of the (extended) corrector. In line with the heuristics, there are transitions at dimension d = 2, and for a correlation-decay exponent β = 2 we capture the correct power of logarithms coming from these two sources of criticality.

The decay of correlations is sharply encoded in terms of a multiscale logarithmic Sobolev inequality (LSI) for the ensemble under consideration — the results would fail if correlation decay were encoded in terms of an α-mixing condition. Among other ensembles popular in modeling of random media, this class includes coefficient fields that are local transformations of stationary Gaussian fields.

The optimal growth of the corrector ϕ is derived from bounding the size of spatial averages F =g ϕ of its gradient. This in turn is done by a (deterministic) sensitivity estimate of F, that is, by estimating the functional derivative F a of F with respect to the coefficient field a. Appealing to the LSI in form of concentration of measure yields a stochastic estimate on F. The sensitivity argument relies on a large-scale Schauder theory for the heterogeneous elliptic operator a. The treatment allows for nonsymmetric a and for systems like linear elasticity.

stochastic homogenization, convergence rates, fat tails
Mathematical Subject Classification 2010
Primary: 35J15, 35J47, 60H25, 74Q05
Received: 11 October 2019
Revised: 8 May 2020
Accepted: 16 June 2020
Published: 19 December 2021
Antoine Gloria
Sorbonne Université, CNRS
Université de Paris
Laboratoire Jacques-Louis Lions
Institut Universitaire de France Université Libre de Bruxelles
Département de Mathématique
Stefan Neukamm
Faculty of Mathematics
Technische Universität Dresden
Felix Otto
Max Planck Institute for Mathematics in the Sciences