{
  "id": "2201.12013",
  "version": "v1",
  "published": "2022-01-28T09:48:50.000Z",
  "updated": "2022-01-28T09:48:50.000Z",
  "title": "Stochastic homogenization of Gaussian fields on random media",
  "authors": [
    "Leandro Chiarini",
    "Wioletta M. Ruszel"
  ],
  "comment": "19 pages, 4 figures",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "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}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-28T09:48:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60G15",
      "60G60",
      "35B27",
      "60J60",
      "60G20"
    ],
    "keywords": [
      "gaussian fields",
      "random media",
      "uniformly elliptic random environment",
      "stochastic homogenization techniques",
      "non-homogeneous gaussian free fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}