{
  "id": "1207.3565",
  "version": "v4",
  "published": "2012-07-16T02:37:35.000Z",
  "updated": "2014-09-03T13:07:00.000Z",
  "title": "Densities for SDEs driven by degenerate $α$-stable processes",
  "authors": [
    "Xicheng Zhang"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/13-AOP900 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2014, Vol. 42, No. 5, 1885-1910",
  "doi": "10.1214/13-AOP900",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "In this work, by using the Malliavin calculus, under H\\\"ormander's condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions. Moreover, in a special degenerate case, we also obtain the smoothness of the density. In particular, we obtain the existence of smooth heat kernels for the following fractional kinetic Fokker-Planck (nonlocal) operator: \\[\\mathscr{L}^{(\\alpha)}_b:=\\Delta^{\\alpha/2}_{\\mathrm{v}}+\\mathrm {v}\\cdot \\nabla_x+b(x,\\mathrm{v})\\cdot \\nabla_{\\mathrm{v}},\\qquad x,\\mathrm{v}\\in\\mathbb{R}^d,\\] where $\\alpha\\in(0,2)$ and $b:\\mathbb{R}^d\\times\\mathbb{R}^d\\to \\mathbb{R}^d$ is smooth and has bounded derivatives of all orders.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-11-02T00:07:59.000Z",
      "abstract": "In this work, by using the Malliavin calculus, under H\\\"ormander's condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions. Moreover, in a special degenerate case, we also obtain the smoothness of the density. In particular, we obtain the existence of smooth heat kernel for the following degenerate fractional order (nonlocal) operator: $$ \\sL^{(\\alpha)}_b:=\\Delta^{\\frac{\\alpha}{2}}_\\mathrm{v}+\\mathrm{v}\\cdot\\nabla_x +b(x,\\mathrm{v})\\cdot\\nabla_\\mathrm{v}, x,\\mathrm{v}\\in\\mR^d, $$ where $\\alpha\\in(0,2)$ and $b:\\mR^d\\times\\mR^d\\to\\mR^d$ is smooth and has bounded derivatives of all orders.",
      "comment": "18pp",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2014-09-03T13:07:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H07",
      "60H10",
      "35K65",
      "35K08"
    ],
    "keywords": [
      "sdes driven",
      "stable processes",
      "stochastic differential equations driven",
      "special degenerate case",
      "degenerate fractional order"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.3565Z"
    }
  }
}