{
  "id": "1604.02681",
  "version": "v1",
  "published": "2016-04-10T12:42:54.000Z",
  "updated": "2016-04-10T12:42:54.000Z",
  "title": "Uniqueness of stable-like processes",
  "authors": [
    "Zhen-Qing Chen",
    "Xicheng Zhang"
  ],
  "comment": "34",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "In this work we consider the following $\\alpha$-stable-like operator (a class of pseudo-differential operator) $$ {\\mathscr L} f(x):=\\int_{\\mathbb R^d}[f(x+\\sigma_x y)-f(x)-1_{\\alpha\\in[1,2)}1_{|y|\\leq 1}\\sigma_x y\\cdot\\nabla f(x)]\\nu_x(d y), $$ where the L\\'evy measure $\\nu_x(d y)$ is comparable with a non-degenerate $\\alpha$-stable-type L\\'evy measure (possibly singular), and $\\sigma_x$ is a bounded and nondegenerate matrix-valued function. Under H\\\"older assumption on $x\\mapsto\\nu_x(d y)$ and uniformly continuity assumption on $x\\mapsto\\sigma_x$, we show the well-posedness of martingale problem associated with the operator $\\mathscr L$. Moreover, we also obtain the existence-uniqueness of strong solutions for the associated SDE when $\\sigma$ belongs to the first order Sobolev space $\\mathbb W^{1,p}(\\mathbb R^d)$ provided $p>d(1+\\alpha\\vee 1)$ and $\\nu_x=\\nu$ is a non-degenerate $\\alpha$-stable-type L\\'evy measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-10T12:42:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "60G52"
    ],
    "keywords": [
      "stable-like processes",
      "stable-type levy measure",
      "uniqueness",
      "first order sobolev space",
      "uniformly continuity assumption"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160402681C"
    }
  }
}