{
  "id": "2007.06040",
  "version": "v1",
  "published": "2020-07-12T17:04:25.000Z",
  "updated": "2020-07-12T17:04:25.000Z",
  "title": "On strong solutions of Itô's equations with a$\\,\\in W^{1}_{d}$ and b$\\,\\in L_{d}$",
  "authors": [
    "N. V. Krylov"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider It\\^o uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in $W^{1}_{d,loc}$, and the drift in $L_{d}$. We prove the unique strong solvability for any starting point and prove that as a function of the starting point the solutions are H\\\"older continuous with any exponent $<1$. We also prove that if we are given a sequence of coefficients converging in an appropriate sense to the original ones, then the solutions of approximating equations converge to the solution of the original one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-12T17:04:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "60J60"
    ],
    "keywords": [
      "strong solutions",
      "itôs equations",
      "time independent coefficients",
      "unique strong solvability",
      "starting point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}