{
  "id": "1802.01151",
  "version": "v1",
  "published": "2018-02-04T15:49:13.000Z",
  "updated": "2018-02-04T15:49:13.000Z",
  "title": "Uniqueness in law for stable-like processes of variable order",
  "authors": [
    "Peng Jin"
  ],
  "comment": "26 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $d\\ge1$. Consider a stable-like operator of variable order \\begin{align*} \\mathcal{A}f(x) & =\\int_{\\mathbb{R}^{d} \\backslash\\{0\\}} \\left[f(x+h) -f(x) -\\mathbf{1}_{\\{|h|\\le1\\}}h \\cdot\\nabla f(x)\\right]\\frac{n(x,h)}{|h|^{d+\\alpha(x)}} \\mathrm{d}h, \\end{align*} where $0<\\inf_{x}\\alpha(x) \\le \\sup_{x}\\alpha(x)<2$ and $n(x,h)$ satisfies \\[ n(x,h)=n(x,-h),\\quad0<\\kappa_{1}\\le n(x,h)\\le\\kappa_{2},\\quad\\forall x,h\\in \\mathbb{R}^{d}, \\] with $\\kappa_{1}$ and $\\kappa_{2}$ being some positive constants. Under some further mild conditions on the functions $n(x,h)$ and $\\alpha(x)$, we show the uniqueness of solutions to the martingale problem for $\\mathcal{A}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-04T15:49:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J75",
      "60G52"
    ],
    "keywords": [
      "variable order",
      "stable-like processes",
      "uniqueness",
      "mild conditions",
      "martingale problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}