{
  "id": "1810.05386",
  "version": "v1",
  "published": "2018-10-12T07:36:51.000Z",
  "updated": "2018-10-12T07:36:51.000Z",
  "title": "Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations",
  "authors": [
    "Robert C. Dalang",
    "Fei Pu"
  ],
  "comment": "48 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a system of $d$ non-linear stochastic fractional heat equations in spatial dimension $1$ driven by multiplicative $d$-dimensional space-time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of $(u(s, y), u (t, x))$. From this result, we deduce 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, which is as sharp as that in the Gaussian case. This also improves the result in Dalang, Khoshnevisan and Nualart [\\textit{Probab. Theory Related Fields} \\textbf{144} (2009) 371--424] for systems of classical stochastic heat equations. We also establish upper bounds on hitting probabilities of the solution in terms of Hausdorff measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-12T07:36:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hitting probabilities",
      "non-linear systems",
      "non-linear stochastic fractional heat equations",
      "deduce optimal lower bounds",
      "two-point probability density function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}