{
  "id": "2305.18853",
  "version": "v1",
  "published": "2023-05-30T08:49:30.000Z",
  "updated": "2023-05-30T08:49:30.000Z",
  "title": "Polarity of points for systems of nonlinear stochastic heat equations in the critical dimension",
  "authors": [
    "Cheuk Yin Lee",
    "Yimin Xiao"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $u(t, x) = (u_1(t, x), \\dots, u_d(t, x))$ be the solution to the systems of nonlinear stochastic heat equations \\[ \\begin{split} \\frac{\\partial}{\\partial t} u(t, x) &= \\frac{\\partial^2}{\\partial x^2} u(t, x) + \\sigma(u(t, x)) \\dot{W}(t, x),\\\\ u(0, x) &= u_0(x), \\end{split} \\] where $t \\ge 0$, $x \\in \\mathbb{R}$, $\\dot{W}(t, x) = (\\dot{W}_1(t, x), \\dots, \\dot{W}_d(t, x))$ is a vector of $d$ independent space-time white noises, and $\\sigma: \\mathbb{R}^d \\to \\mathbb{R}^{d\\times d}$ is a matrix-valued function. We say that a subset $S$ of $\\mathbb{R}^d$ is polar for $\\{u(t, x), t \\ge 0, x \\in \\mathbb{R}\\}$ if \\[ \\mathbb{P}\\{u(t,x) \\in S \\text{ for some } t>0 \\text{ and } x\\in\\mathbb{R} \\}=0. \\] The main result of this paper shows that, in the critical dimension $d=6$, all points in $\\mathbb{R}^d$ are polar for $\\{u(t, x), t \\ge 0, x \\in \\mathbb{R}\\}$. This solves an open problem of Dalang, Khoshnevisan and Nualart (2009, 2013) and Dalang, Mueller and Xiao (2021). We also provide a sufficient condition for a subset $S$ of $\\mathbb{R}^d$ to be polar.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-30T08:49:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear stochastic heat equations",
      "critical dimension",
      "independent space-time white noises",
      "sufficient condition",
      "main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}