Stochastic Homogenization of Gaussian Fields on Random Media

Abstract

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields Ξ g,a and bi-Laplacian fields Ξ b,a . 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 Ξ g,a . When f is the white noise, the field Ξ b,a can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain D⊂ Rd , or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator Δ , 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 , with constant a¯ depending on the law of the environment a . The proofs are based on the results found in Armstrong et al. (in: Grundlehren der mathematischen Wissenschaften, Springer International Publishing, Cham, 2019) and Gloria et al. (ESAIM Math Model Numer Anal 48(2):325-346, 2014).