{
  "id": "2002.08212",
  "version": "v1",
  "published": "2020-02-19T14:38:31.000Z",
  "updated": "2020-02-19T14:38:31.000Z",
  "title": "Polarity of almost all points for systems of non-linear stochastic heat equations in the critical dimension",
  "authors": [
    "Robert C. Dalang",
    "Carl Mueller",
    "Yimin Xiao"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study vector-valued solutions $u(t,x)\\in\\mathbb{R}^d$ to systems of nonlinear stochastic heat equations with multiplicative noise: \\begin{equation*} \\frac{\\partial}{\\partial t} u(t,x)=\\frac{\\partial^2}{\\partial x^2} u(t,x)+\\sigma(u(t,x))\\dot{W}(t,x). \\end{equation*} Here $t\\geq 0$, $x\\in\\mathbb{R}$ and $\\dot{W}(t,x)$ is an $\\mathbb{R}^d$-valued space-time white noise. We say that a point $z\\in\\mathbb{R}^d$ is polar if \\begin{equation*} P\\{u(t,x)=z\\text{ for some $t>0$ and $x\\in\\mathbb{R}$}\\}=0. \\end{equation*} We show that in the critical dimension $d=6$, almost all points in $\\mathbb{R}^d$ are polar.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-19T14:38:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60J45",
      "60G60"
    ],
    "keywords": [
      "non-linear stochastic heat equations",
      "critical dimension",
      "nonlinear stochastic heat equations",
      "valued space-time white noise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}