{
  "id": "1309.6414",
  "version": "v1",
  "published": "2013-09-25T07:26:30.000Z",
  "updated": "2013-09-25T07:26:30.000Z",
  "title": "Uniqueness of Stable Processes with Drift",
  "authors": [
    "Zhen-Qing Chen",
    "Longmin Wang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Suppose that $d\\geq1$ and $\\alpha\\in (1, 2)$. Let $Y$ be a rotationally symmetric $\\alpha$-stable process on $\\R^d$ and $b$ a $\\R^d$-valued measurable function on $\\R^d$ belonging to a certain Kato class of $Y$. We show that $\\rd X^b_t=\\rd Y_t+b(X^b_t)\\rd t$ with $X^b_0=x$ has a unique weak solution for every $x\\in \\R^d$. Let $\\sL^b=-(-\\Delta)^{\\alpha/2} + b \\cdot \\nabla$, which is the infinitesimal generator of $X^b$. Denote by $C^\\infty_c(\\R^d)$ the space of smooth functions on $\\R^d$ with compact support. We further show that the martingale problem for $(\\sL^b, C^\\infty_c(\\R^d))$ has a unique solution for each initial value $x\\in \\R^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-25T07:26:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "47G20",
      "60G52"
    ],
    "keywords": [
      "stable processes",
      "uniqueness",
      "unique weak solution",
      "infinitesimal generator",
      "kato class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.6414C"
    }
  }
}