{
  "id": "1811.12757",
  "version": "v1",
  "published": "2018-11-30T12:17:59.000Z",
  "updated": "2018-11-30T12:17:59.000Z",
  "title": "Optimal lower bounds on hitting probabilities for stochastic heat equations in spatial dimension $k \\geq 1$",
  "authors": [
    "Robert Dalang",
    "Fei Pu"
  ],
  "comment": "34 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix $\\gamma_Z$ of $Z := (u(s, y), u(t, x) - u(s, y))$, where $u$ is the solution to system of $d$ non-linear stochastic heat equations in spatial dimension $k \\geq 1$. We also obtain the optimal exponents for the $L^p$-modulus of continuity of the increments of the solution and of its Malliavin derivatives. These lead to optimal lower bounds on hitting probabilities of the process $\\{u(t, x): (t, x) \\in [0, \\infty[ \\times \\mathbb{R}\\}$ in the non-Gaussian case in terms of Newtonian capacity, and improve a result in Dalang, Khoshnevisan and Nualart [\\textit{Stoch PDE: Anal Comp} \\textbf{1} (2013) 94--151].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-30T12:17:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60J45",
      "60H07",
      "60G60"
    ],
    "keywords": [
      "optimal lower bounds",
      "spatial dimension",
      "hitting probabilities",
      "non-linear stochastic heat equations",
      "newtonian capacity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}