{
  "id": "1611.05999",
  "version": "v1",
  "published": "2016-11-18T08:04:32.000Z",
  "updated": "2016-11-18T08:04:32.000Z",
  "title": "Stochastic wave equation in a plane driven by spatial stable noise",
  "authors": [
    "Larysa Pryhara",
    "Georgiy Shevchenko"
  ],
  "comment": "Published at http://dx.doi.org/10.15559/16-VMSTA62 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)",
  "journal": "Modern Stochastics: Theory and Applications 2016, Vol. 3, No. 3, 237-248",
  "doi": "10.15559/16-VMSTA62",
  "categories": [
    "math.PR"
  ],
  "abstract": "The main object of this paper is the planar wave equation \\[\\bigg(\\frac{\\partial^2}{\\partial t^2}-a^2\\varDelta\\bigg)U(x,t)=f(x,t),\\quad t\\ge0, x\\in \\mathbb {R}^2,\\] with random source $f$. The latter is, in certain sense, a symmetric $\\alpha$-stable spatial white noise multiplied by some regular function $\\sigma$. We define a candidate solution $U$ to the equation via Poisson's formula and prove that the corresponding expression is well defined at each point almost surely, although the exceptional set may depend on the particular point $(x,t)$. We further show that $U$ is H\\\"{o}lder continuous in time but with probability 1 is unbounded in any neighborhood of each point where $\\sigma$ does not vanish. Finally, we prove that $U$ is a generalized solution to the equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-18T08:04:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic wave equation",
      "spatial stable noise",
      "plane driven",
      "stable spatial white noise",
      "planar wave equation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}