Stochastic Homogenization of Gaussian Fields on Random Media

Abstract In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $$\Xi ^{g,\textbf{a}} $$ Ξ g , a and bi-Laplacian ^{b,\textbf{a}}$$ b . They can be characterized as follows: for $$f=\delta f = δ the solution u $$\nabla \cdot \textbf{a} \nabla =f$$ ∇ · u , $$\textbf{a}$$ is a uniformly elliptic random environment, covariance ^{g,\textbf{a}}$$ When f white noise, field viewed distributional same equation. Our results characterize scaling limit such on both, sufficiently regular domain $$D\subset \mathbb {R}^d$$ D ⊂ R d or discrete torus. Based techniques applied to eigenfunction basis Laplace operator $$\Delta Δ will show that families converge an appropriate multiple GFF resp. bi-Laplacian. The limiting are determined by their respective homogenized $${{\,\mathrm{\bar{\textbf{a}}}\,}}\Delta ¯ with constant $${{\,\mathrm{\bar{\textbf{a}}}\,}}$$ depending law environment proofs based found in Armstrong et al. (in: Grundlehren der mathematischen Wissenschaften, Springer International Publishing, Cham, 2019) Gloria (ESAIM Math Model Numer Anal 48(2):325-346, 2014).