{
  "id": "1708.04467",
  "version": "v1",
  "published": "2017-08-15T11:55:45.000Z",
  "updated": "2017-08-15T11:55:45.000Z",
  "title": "Well-posedness of the martingale problem for non-local perturbations of Lévy-type generators",
  "authors": [
    "Peng Jin"
  ],
  "comment": "25 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $L$ be a L\\'evy-type generator whose L\\'evy measure is controlled from below by that of a non-degenerate $\\alpha$-stable ($0<\\alpha<2$) process. In this paper, we study the martingale problem for the operator $\\mathcal{L}_{t}=L+K_{t}$, with $K_{t}$ being a time-dependent non-local operator defined by \\[ K_{t}f(x):=\\int_{\\mathbb{R}^{d}\\backslash\\{0\\}}[f(x+y)-f(x)-\\mathbf{1}_{\\alpha>1}\\mathbf{1}_{\\{|y|\\le1\\}}y\\cdot\\nabla f(x)]M(t,x,dy), \\] where $M(t,x,\\cdot)$ is a L\\'evy measure on $\\mathbb{R}^{d}\\backslash\\{0\\}$ for each $(t,x)\\in \\mathbb{R}_+ \\times \\mathbb{R}^{d}$. We show that if \\[ \\sup_{t\\geq0,x\\in\\mathbb{R}^{d}}\\int_{\\mathbb{R}^{d}\\backslash\\{0\\}}1\\wedge|y|^{\\beta}M(t,x,dy)<\\infty \\] for some $0<\\beta<\\alpha$, then the martingale problem for $\\mathcal{L}_{t}$ is well-posed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-15T11:55:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J75",
      "60J35"
    ],
    "keywords": [
      "martingale problem",
      "non-local perturbations",
      "lévy-type generators",
      "well-posedness",
      "levy measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}