Locality properties of standard homogenization commutator

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 ${\mathrm{\Xi }}_{\mathit{\epsilon }}^{\mathit{o}}$ . Our aim is to study how ${\mathrm{\Xi }}_{\mathit{\epsilon }}^{\mathit{o}}$ (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 ${\mathrm{\Xi }}_{\mathit{\epsilon }}^{\mathit{o}}$ inherits the correlation properties of the environment.