{
  "id": "2206.14139",
  "version": "v1",
  "published": "2022-06-28T16:54:51.000Z",
  "updated": "2022-06-28T16:54:51.000Z",
  "title": "Parabolic Anderson model on Heisenberg groups: the Itô setting",
  "authors": [
    "Fabrice Baudoin",
    "Cheng Ouyang",
    "Samy Tindel",
    "Jing Wang"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group $\\mathbf{H}^{n}$ of order $n$. This equation is written as $\\partial_t u=\\frac{1}{2}\\Delta u+u\\dot{W}_\\alpha$, where $\\Delta$ is the hypoelliptic Laplacian on $\\mathbf{H}^{n}$ and $\\{\\dot{W}_\\alpha; \\alpha>0\\}$ is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by $(-\\Delta)^{-\\alpha}$ in space. Our aim is threefold: (i) Give a proper description of the noise $W_\\alpha$; (ii) Prove that one can solve the stochastic heat equation in the It\\^{o} sense as soon as $\\alpha>\\frac{n}{2}$; (iii) Give some basic moment estimates for the solution $u(t,x)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-28T16:54:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H07",
      "60H15",
      "60D05"
    ],
    "keywords": [
      "parabolic anderson model",
      "heisenberg group",
      "stochastic heat equation",
      "basic moment estimates",
      "gaussian space-time noises"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}