{
  "id": "1501.04758",
  "version": "v1",
  "published": "2015-01-20T10:42:33.000Z",
  "updated": "2015-01-20T10:42:33.000Z",
  "title": "Stochastic flows for Lévy processes with Hölder drifts",
  "authors": [
    "Zhen-Qing Chen",
    "Renming Song",
    "Xicheng Zhang"
  ],
  "comment": "22pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "In this paper we study the following stochastic differential equation (SDE) in ${\\mathbb R}^d$: $$ \\mathrm{d} X_t= \\mathrm{d} Z_t + b(t, X_t)\\mathrm{d} t, \\quad X_0=x, $$ where $Z$ is a L\\'evy process. We show that for a large class of L\\'evy processes ${Z}$ and H\\\"older continuous drift $b$, the SDE above has a unique strong solution for every starting point $x\\in{\\mathbb R}^d$. Moreover, these strong solutions form a $C^1$-stochastic flow. As a consequence, we show that, when ${Z}$ is an $\\alpha$-stable-type L\\'evy process with $\\alpha\\in (0, 2)$ and $b$ is bounded and $\\beta$-H\\\"older continuous with $\\beta\\in (1- {\\alpha}/{2},1)$, the SDE above has a unique strong solution. When $\\alpha \\in (0, 1)$, this in particular solves an open problem from Priola \\cite{Pr1}. Moreover, we obtain a Bismut type derivative formula for $\\nabla {\\mathbb E}_x f(X_t)$ when ${Z}$ is a subordinate Brownian motion. To study the SDE above, we first study the following nonlocal parabolic equation with H\\\"older continuous $b$ and $f$: $$ \\partial_t u+{\\mathscr L} u+b\\cdot \\nabla u+f=0,\\quad u(1, \\cdot )=0, $$ where $\\mathscr L$ is the generator of the L\\'evy process ${Z}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-20T10:42:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "35K05",
      "60H30",
      "47G20"
    ],
    "keywords": [
      "stochastic flow",
      "lévy processes",
      "hölder drifts",
      "unique strong solution",
      "stochastic differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150104758C"
    }
  }
}