{
  "id": "1305.5624",
  "version": "v2",
  "published": "2013-05-24T05:43:04.000Z",
  "updated": "2014-11-18T03:38:00.000Z",
  "title": "Pathwise uniqueness for a SPDE with Hölder continuous coefficient driven by α-stable noise",
  "authors": [
    "Xu Yang",
    "Xiaowen Zhou"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with H\\\"older continuous coefficient: \\begin{eqnarray*} \\frac{\\partial X_t(x)}{\\partial t}=\\frac{1}{2} \\Delta X_t(x) +G(X_t(x))+H(X_{t-}(x)) \\dot{L}_t(x),~~~ t>0, ~x\\in\\mathbb{R}, \\end{eqnarray*} where $\\dot{L}$ denotes an $\\alpha$-stable white noise on $\\mathbb{R}_+\\times \\mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $\\beta$-H\\\"older continuous for $1<\\alpha<2$ and $0<\\beta<1$. For $G\\equiv0$ and $H(x)=x^\\beta$, in Mytnik (2002) a weak solution to the above SPDE was constructedand the pathwise uniqueness of the solution was left as an open problem. In this paper we give an affirmative answer to this problem for certain values of $\\alpha$ and $\\beta$. In particular, for $\\alpha\\beta=1$, where the solution to the above equation is the density of a super-Brownian motion with $\\alpha$-stable branching (see also Mytnik (2002)), our result leads to its pathwise uniqueness for $1<\\alpha<4-2\\sqrt{2}$. The local H\\\"older continuity of the solution is also obtained in this paper for fixed time $t>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-24T05:43:04.000Z",
      "title": "The pathwise uniqueness of solution to a SPDE driven by $α$-stable noise",
      "abstract": "In this paper we study the pathwise uniqueness of solution to a stochastic partial differential equation driven by $\\alpha$-stable noise. In a special case, where solution to this equation is the density of a super-Brownian motion with $\\alpha$-stable branching (see Mytnik (2002)), our result leads to its pathwise uniqueness for $1<\\alpha<4-2\\sqrt{2}$. The local H\\\"older continuity of the solution is also obtained for fixed time $t>0$ and general case",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-18T03:38:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pathwise uniqueness",
      "stable noise",
      "spde driven",
      "stochastic partial differential equation driven",
      "super-brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.5624Y"
    }
  }
}