Stochastic homogenization of Gaussian fields on random media

Leandro Chiarini, Wioletta M. Ruszel

Published 2022-01-28Version 1

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $\Xi^{g,{\bf a}}_D $ and bi-Laplacian fields $\Xi^{b,{\bf a}}_D$. They can be characterized as follows: for $f=\delta$ the solution $u$ of $\nabla \cdot \mathbf{a} \nabla u =f$, ${\bf a}$ is a uniformly elliptic random environment, is the covariance of $\Xi^{g,{\bf a}}_D$. When $f$ is the white noise, the field $\Xi^{b,{\bf a}}_D$ can be viewed as the distributional solution of the same elliptic equation. Furthermore, we consider fields on domains $D\subset \mathbb{R}^d$ 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 $\Delta$ 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 $\overline{{\bf a}}\Delta$, where $\overline{{\bf a}}$ is a deterministic constant depending on the law of ${\bf a}$. The proofs are based on the techniques introduced by \cite{Armstrong2019} and \cite{gloria2014optimal}.