{
  "id": "1908.01255",
  "version": "v1",
  "published": "2019-08-04T01:52:13.000Z",
  "updated": "2019-08-04T01:52:13.000Z",
  "title": "$L^q(L^p)$-theory of stochastic differential equations",
  "authors": [
    "Pengcheng Xia",
    "Longjie Xie",
    "Xicheng Zhang",
    "Guohuan Zhao"
  ],
  "comment": "22pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we show the weak differentiability of the unique strong solution with respect to the starting point $x$ as well as Bismut-Elworthy-Li's derivative formula for the following stochastic differential equation in $\\mathbb R^d$: $$ {\\rm d} X_t=b(t,X_t){\\rm d} t+\\sigma(t,X_t){\\rm d} W_t,\\ \\ X_0=x\\in\\mathbb R^d, $$ where $\\sigma$ is bounded, uniformly continuous and nondegenerate, $\\nabla\\sigma\\in \\widetilde{\\mathbb L}^{p_1}_{q_1}$ and $b\\in \\widetilde{\\mathbb L}^{p_2}_{q_2}$ for some $p_i,q_i\\in[2,\\infty)$ with $\\frac{d}{p_i}+\\frac{2}{q_i}<1$, $i=1,2$, where $\\widetilde{\\mathbb L}^{p_i}_{q_i}, i=1,2$ are some localized spaces. Moreover, in the endpoint case $b\\in \\widetilde{\\mathbb L}^{d; {\\rm uni}}_\\infty$, we also show the weak well-posedness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-04T01:52:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10"
    ],
    "keywords": [
      "stochastic differential equation",
      "unique strong solution",
      "weak differentiability",
      "bismut-elworthy-lis derivative formula",
      "weak well-posedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}