{
  "id": "1809.06111",
  "version": "v1",
  "published": "2018-09-17T10:18:40.000Z",
  "updated": "2018-09-17T10:18:40.000Z",
  "title": "On the homogenization of random stationary elliptic operators in divergence form",
  "authors": [
    "Arianna Giunti",
    "Juan J. L. Velázquez"
  ],
  "comment": "11 pages, 0 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this note we comment on the homogenization of a random elliptic operator in divergence form $-\\nabla \\cdot a\\nabla$, where the coefficient field $a$ is distributed according to a stationary, but not necessarily ergodic, probability measure $P$. We generalize the well-known case for $P$ stationary and ergodic by showing that the operator $-\\nabla \\cdot a(\\frac{\\cdot}{\\varepsilon})\\nabla$ almost surely homogenizes to a constant-coefficient, random operator $-\\nabla \\cdot A_h\\nabla$. Furthermore, we use a disintegration formula for $P$ with respect to a family of ergodic and stationary probability measures to show that the law of $A_h$ may be obtained by using the standard homogenization results on each probability measure of the previous family. We finally provide a more explicit formula for $A_h$ in the case of coefficient fields which are a function of a stationary Gaussian field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-17T10:18:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B27",
      "35J15",
      "37A50",
      "37A20",
      "60G15"
    ],
    "keywords": [
      "random stationary elliptic operators",
      "divergence form",
      "coefficient field",
      "stationary probability measures",
      "stationary gaussian field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}