Optimal boundary conditions for three dimensional elliptic random media

We are interested in computing the electrical field generated by a charge distribution localized on scale $\ell$ in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter $L\gg\ell$ around the support of the charge. We propose an artificial boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of $\ell$ and $L$, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and in the regime $1\ll\ell$). The boundary condition is motivated by stochastic homogenization that allows for a multipole expansion [Bella, Giunti, Otto 2020]. This work extends [Lu, Otto] from two to three dimensions, which requires to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of [Gloria, Otto 2015].