Locality properties of standard homogenization commutator

We focus on the theory of fluctuations in stochastic homogenization for linear uniformly elliptic PDEs in divergence form. More precisely, we study the properties of the so-called standard homogenization commutator, a random field which plays a crucial role in this theory. Main tool in our approach is a covariance estimate -- a consequence of the Hellfer-Sjöstrand representation formula for covariances which holds in the Gaussian setting we adopt.