{
  "id": "1706.05260",
  "version": "v1",
  "published": "2017-06-15T08:19:08.000Z",
  "updated": "2017-06-15T08:19:08.000Z",
  "title": "On the domain of elliptic operators defined in subsets of Wiener spaces",
  "authors": [
    "D. Addona",
    "G. Cappa",
    "S. Ferrari"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1609.07337",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\\mu$. The associated Cameron-Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\\rightarrow\\mathbb{R}$ and $G:X\\rightarrow \\mathbb{R}$. We let $\\nu=e^{-U}\\mu$ and $\\Omega=G^{-1}(-\\infty,0]$. In this paper we are interested in the domain of the the self-adjoint operator associated with the quadratic form \\begin{gather} (\\psi,\\varphi)\\mapsto \\int_\\Omega\\langle\\nabla_H\\psi,\\nabla_H\\varphi\\rangle_Hd\\nu\\qquad\\psi,\\varphi\\in W^{1,2}(\\Omega,\\nu).\\qquad\\qquad (\\star) \\end{gather} In particular we obtain a complete characterization of the Ornstein-Uhlenbeck operator on half-spaces, namely if $U\\equiv 0$ and $G$ is an affine function, then the domain of the operator defined via $(\\star)$ is the space \\[\\{u\\in W^{2,2}(\\Omega,\\mu)\\,|\\, \\langle\\nabla_H u(x),\\nabla_H G(x)\\rangle_H=0\\text{ for }\\rho\\text{-a.e. }x\\in G^{-1}(0)\\},\\] where $\\rho$ is the Feyel-de La Pradelle Hausdorff-Gauss surface measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-15T08:19:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28C20",
      "35J15",
      "46G12",
      "47A07",
      "47A30"
    ],
    "keywords": [
      "elliptic operators",
      "wiener spaces",
      "pradelle hausdorff-gauss surface measure",
      "sufficiently regular convex functions",
      "non-degenerate centered gaussian measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}