{
  "id": "1706.09050",
  "version": "v1",
  "published": "2017-06-27T21:16:05.000Z",
  "updated": "2017-06-27T21:16:05.000Z",
  "title": "Feynman-Kac representation for the parabolic Anderson model driven by fractional noise",
  "authors": [
    "Kamran Kalbasi",
    "Thomas S. Mountford"
  ],
  "journal": "Journal of Functional Analysis, Volume 269, Issue 5, 2015, Pages 1234-1263",
  "doi": "10.1016/j.jfa.2015.06.003",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We consider the parabolic Anderson model driven by fractional noise: $$ \\frac{\\partial}{\\partial t}u(t,x)= \\kappa \\boldsymbol{\\Delta} u(t,x)+ u(t,x)\\frac{\\partial}{\\partial t}W(t,x) \\qquad x\\in\\mathbb{Z}^d\\;,\\; t\\geq 0\\,, $$ where $\\kappa>0$ is a diffusion constant, $\\boldsymbol{\\Delta}$ is the discrete Laplacian defined by $\\boldsymbol{\\Delta} f(x)= \\frac{1}{2d}\\sum_{|y-x|=1}\\bigl(f(y)-f(x)\\bigr)$, and $\\{W(t,x)\\;;\\;t\\geq0\\}_{x \\in \\mathbb{Z}^d}$ is a family of independent fractional Brownian motions with Hurst parameter $H\\in(0,1)$, indexed by $\\mathbb{Z}^d$. We make sense of this equation via a Stratonovich integration obtained by approximating the fractional Brownian motions with a family of Gaussian processes possessing absolutely continuous sample paths. We prove that the Feynman-Kac representation \\begin{equation} u(t,x)=\\mathbb{E}^x\\Bigl[u_o(X(t))\\exp \\int_0^t W\\bigl(\\mathrm{d}s, X(t-s)\\bigr)\\Bigr]\\,, \\end{equation} is a mild solution to this problem. Here $u_o(y)$ is the initial value at site $y\\in\\mathbb{Z}^d$, $\\{X(t)\\;;\\;t\\geq0\\}$ is a simple random walk with jump rate $\\kappa$, started at $x \\in \\mathbb{Z}^d$ and independent of the family $\\{W(t,x)\\;;\\;t\\geq0\\}_{x\\in\\mathbb{Z}^d}$ and $\\mathbb{E}^x$ is expectation with respect to this random walk. We give a unified argument that works for any Hurst parameter $H\\in (0,1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-27T21:16:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H07",
      "58J35"
    ],
    "keywords": [
      "parabolic anderson model driven",
      "feynman-kac representation",
      "fractional noise",
      "fractional brownian motions",
      "absolutely continuous sample paths"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}