{
  "id": "1706.09493",
  "version": "v1",
  "published": "2017-06-28T21:45:20.000Z",
  "updated": "2017-06-28T21:45:20.000Z",
  "title": "Berry-Esseen Theorem and Quantitative homogenization for the Random Conductance Model with degenerate Conductances",
  "authors": [
    "Sebastian Andres",
    "Stefan Neukamm"
  ],
  "comment": "38 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We study the random conductance model on the lattice $\\mathbb{Z}^d$, i.e.\\ we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension $d\\geq 3$ quantitative central limit theorems for the random walk in form of a Berry-Esseen estimate with speed $t^{-\\frac 1 5+\\varepsilon}$ for $d\\geq 4$ and $t^{-\\frac{1}{10}+\\varepsilon}$ for $d=3$. In addition, for $d\\geq 3$ we show near-optimal decay estimates on the semigroup associated with the environment process, which plays a central role in quantitative stochastic homogenization. This extends some recent results by Gloria, Otto and the second author to the degenerate elliptic case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-28T21:45:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60F05",
      "35B27",
      "35K65"
    ],
    "keywords": [
      "random conductance model",
      "berry-esseen theorem",
      "degenerate conductances",
      "quantitative homogenization",
      "degenerate elliptic case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}