{
  "id": "1711.05058",
  "version": "v1",
  "published": "2017-11-14T11:11:03.000Z",
  "updated": "2017-11-14T11:11:03.000Z",
  "title": "On weak solutions of stochastic differential equations with sharp drift coefficients",
  "authors": [
    "Jinlong Wei",
    "Guangying Lv",
    "Jiang-Lun Wu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We extend Krylov and R\\\"{o}ckner's result \\cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\\times{\\mathbb R}^d\\rightarrow{\\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+\\int_0^tb(s,X_s)ds+W_t,\\quad t\\in[0,T], \\, x\\in{\\mathbb R}^d,$$ where $\\{W_t\\}_{t\\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\\cdot)\\in\\mathcal{C}_q^0((0,T];L^p({\\mathbb R}^d))$ with $2/q+d/p=1$ for $p,q\\ge1$ and $\\|b_1(T-\\cdot)\\|_{\\mathcal{C}_q((0,T];L^p({\\mathbb R}^d))}$ is sufficiently small, and that $b_2$ is bounded and Borel measurable, then there exits a unique weak solution to the above equation. Furthermore, we obtain the strong Feller property of the semi-group and existence of density associated with above SDE. Besides, we extend the classical partial differential equations (PDEs) results for $L^q(0,T;L^p({\\mathbb R}^d))$ coefficients to $L^\\infty_q(0,T;L^p({\\mathbb R}^d))$ ones, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Lemma 2.1).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-14T11:11:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "34F05"
    ],
    "keywords": [
      "sharp drift coefficients",
      "stochastic differential equations",
      "second order parabolic pdes",
      "dimensional standard wiener process",
      "unique weak solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}