Locality properties of standard homogenization commutator

Article type: Research Article

Authors: Chatzigeorgiou, Georgiana^{; *}

Affiliations: Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany

Correspondence: [*] Corresponding author. E-mail: georgiana.chatzigeorgiou@mis.mpg.de.

Abstract: We study stochastic homogenization for linear elliptic equations in divergence form and focus on the recently developed theory of fluctuations. It has been observed that the fluctuations of averages of the solution are captured by the so-called standard homogenization commutator Ξεo. Our aim is to study how Ξεo (and its higher-order analogs) decorrelates on large scales when averaged on balls which are far enough. Taking advantage of its approximate locality, we give a quantitative characterization of this decorrelation in terms of both the macroscopic scale and the distance between the balls showing that Ξεo inherits the correlation properties of the environment.

Keywords: Stochastic homogenization, fluctuations, homogenization commutator, covariance estimate, higher-order theory

DOI: 10.3233/ASY-221797

Journal: Asymptotic Analysis, vol. 132, no. 3-4, pp. 409-426, 2023