{
  "id": "0802.2743",
  "version": "v2",
  "published": "2008-02-20T02:57:14.000Z",
  "updated": "2009-02-25T23:29:34.000Z",
  "title": "Ellipticity and Ergodicity",
  "authors": [
    "Derek W. Robinson",
    "Adam Sikora"
  ],
  "comment": "8 pages--Replacement, with corrections, of an earlier version",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $S=\\{S_t\\}_{t\\geq0}$ be the submarkovian semigroup on $L_2(\\Ri^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients $c_{ij}$. Further let $\\Omega$ be an open subset of $\\Ri^d$. Under the assumption that $C_c^\\infty(\\Ri^d)$ is a core for $H$ we prove that $S$ leaves $L_2(\\Omega)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $Y_i=\\sum^d_{j=1}c_{ij}\\partial_j$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-02-25T23:29:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J70",
      "35Hxx",
      "35F05",
      "31C15"
    ],
    "keywords": [
      "ellipticity",
      "ergodicity",
      "submarkovian semigroup",
      "elliptic operator",
      "lipschitz continuous coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0802.2743R"
    }
  }
}