{
  "id": "1609.01005",
  "version": "v1",
  "published": "2016-09-05T00:22:05.000Z",
  "updated": "2016-09-05T00:22:05.000Z",
  "title": "The third moment for the parabolic Anderson model",
  "authors": [
    "Le Chen"
  ],
  "comment": "16 pages, 6 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study the {\\it parabolic Anderson model} starting from the Dirac delta initial data: \\[ \\left(\\frac{\\partial}{\\partial t} -\\frac{\\nu}{2}\\frac{\\partial^2}{\\partial x^2} \\right) u(t,x) = \\lambda u(t,x) \\dot{W}(t,x), \\qquad u(0,x)=\\delta_0(x), \\quad x\\in\\mathbb{R}, \\] where $\\dot{W}$ denotes the space-time white noise. By evaluating the threefold contour integral in the third moment formula by Borodin and Corwin [2], we obtain some explicit formulas for $\\mathbb{E}[u(t,x)^3]$. One application of these formulas is given to show the exact phase transition for the intermittency front of order three.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-05T00:22:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "35R60"
    ],
    "keywords": [
      "parabolic anderson model",
      "dirac delta initial data",
      "threefold contour integral",
      "third moment formula",
      "space-time white noise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}