Stochastic Homogenization of Gaussian Fields on Random Media

Abstract

AbstractIn this article, we study stochastic homogenization of non-homogeneous Gaussian free fields$$\Xi ^{g,\textbf{a}} $$Ξg,aand bi-Laplacian fields$$\Xi ^{b,\textbf{a}}$$Ξb,a. They can be characterized as follows: for$$f=\delta $$f=δthe solutionuof$$\nabla \cdot \textbf{a} \nabla u =f$$∇·a∇u=f,$$\textbf{a}$$ais a uniformly elliptic random environment, is the covariance of$$\Xi ^{g,\textbf{a}}$$Ξg,a. Whenfis the white noise, the field$$\Xi ^{b,\textbf{a}}$$Ξb,acan 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$$D\subset \mathbb {R}^d$$D⊂Rd, or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator$$\Delta $$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$${{\,\mathrm{\bar{\textbf{a}}}\,}}\Delta $$a¯Δ, with constant$${{\,\mathrm{\bar{\textbf{a}}}\,}}$$a¯depending on the law of the environment$$\textbf{a}$$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).