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 fields $$\Xi ^{b,\textbf{a}}$$ Ξ b , a . They can be characterized as follows: for $$f=\delta $$ f = δ the solution u of $$\nabla \cdot \textbf{a} \nabla u =f$$ ∇ · a ∇ u = f , $$\textbf{a}$$ a is a uniformly elliptic random environment, is the covariance of $$\Xi ^{g,\textbf{a}}$$ Ξ g , a . When f is the white noise, the field $$\Xi ^{b,\textbf{a}}$$ 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 $$D\subset \mathbb {R}^d$$ D ⊂ R d , 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).

The maximum of log‐correlated Gaussian fields in random environment

“…For these, we rely on recently developed versions of homogenization theory, such as those in [6,7,38]. We note that convergence at the level of the associated Gaussian fields is proved (in the continuous setup) in [27]. Further details are provided below in Section 4.…”

“…i.i.d standard Gaussians) under the operator −∇ ⋅ 𝐚∇. This is the approach used in [27]. Again, we would expect that the analogue of Theorem 1.9 holds in this context.…”