Stochastic homogenization of Gaussian fields on random media

Abstract

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields Ξg,aD and bi-Laplacian fields Ξb,aD. They can be characterized as follows: for f=δ the solution u of ∇⋅a∇u=f, a is a uniformly elliptic random environment, is the covariance of Ξg,aD. When f is the white noise, the field Ξb,aD can be viewed as the distributional solution of the same elliptic equation. Furthermore, we consider fields on domains D⊂Rd such that the boundary is smooth enough and on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator Δ in a smooth domain D, we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator a¯¯¯Δ, where a¯¯¯ is a deterministic constant depending on the law of a. The proofs are based on the techniques introduced by \cite{Armstrong2019} and \cite{gloria2014optimal}.