论文标题
随机介质上高斯田地的随机均质化
Stochastic homogenization of Gaussian fields on random media
论文作者
论文摘要
在本文中，我们研究了非同质性高斯自由场的随机均质化$ξ^{g，{\ bf a}} $和bi-laplacian fields $ξ^{b，{\ bf a}} $。它们的特征如下：对于$ f =δ$，解决方案$ u $ of $ \ nabla \ cdot \ cdot \ mathbf {a} \ nabla u = f $，$ {\ bf a} $是一个均匀椭圆的随机环境，是$ $ξ^^{g，{当$ f $是白噪声时，字段$ξ^{b，{\ bf a}} $可以将其视为相同的椭圆方程的分布解决方案。我们的结果表征了这两个字段的缩放限制，这是一个足够的常规域$ d \ subset \ mathbb {r}^d $，或在离散的圆环上。基于应用于拉普拉斯操作员$δ$的本征函数基础的随机均质化技术，我们将表明此类田地群体会融合到GFF resp的适当倍数。双拉普拉斯。限制字段取决于其各自的均质操作员$ \ahomδ$，并取决于环境法律$ {\ bf a} $，其常数$ \ ahom $。证明基于\ cite {armstrong2019}和\ cite {gloria2014optimal}中发现的结果。
In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $Ξ^{g,{\bf a}} $ and bi-Laplacian fields $Ξ^{b,{\bf a}}$. They can be characterized as follows: for $f=δ$ the solution $u$ of $\nabla \cdot \mathbf{a} \nabla u =f$, ${\bf a}$ is a uniformly elliptic random environment, is the covariance of $Ξ^{g,{\bf a}}$. When $f$ is the white noise, the field $Ξ^{b,{\bf a}}$ 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 $Δ$, 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 $\ahom Δ$, with constant $\ahom$ depending on the law of the environment ${\bf a}$. The proofs are based on the results found in \cite{Armstrong2019} and \cite{gloria2014optimal}.
