Stochastic homogenization of Gaussian fields on random media

Abstract

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields ${\Xi }^{g,\mathbf{a}}$ and bi-Laplacian fields ${\Xi }^{b,\mathbf{a}}$. They can be characterized as follows: for $f=\delta$ the solution $u$ of $\nabla \cdot \mathbf{a}\nabla u=f$, $\mathbf{a}$ is a uniformly elliptic random environment, is the covariance of ${\Xi }^{g,\mathbf{a}}$. When $f$ is the white noise, the field ${\Xi }^{b,\mathbf{a}}$ 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 $D\subset {\mathbb{R}}^d$, or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator $\Delta$, 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 $\ahom Δ$, with constant $\ahom$ depending on the law of the environment $\mathbf{a}$. The proofs are based on the results found in \cite{Armstrong2019} and \cite{gloria2014optimal}.

24 pages, 4 figures