{
  "id": "2501.01759",
  "version": "v1",
  "published": "2025-01-03T11:14:24.000Z",
  "updated": "2025-01-03T11:14:24.000Z",
  "title": "Stochastic flows for Hölder drifts and transport/continuity equations with noise",
  "authors": [
    "Magnus C. Ørke"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove existence of a stochastic flow of diffeomorphisms generated by SDEs with drift in $L^q_t C^{0, \\alpha}_x$ for any $q \\in [2, \\infty)$ and $\\alpha \\in (0, 1)$. This result is achieved using a Zvonkin-type transformation for the SDE. As a key intermediate step, well-posedness and optimal regularity for a class of parabolic PDEs related to the transformation is established. Using the existence of a differentiable stochastic flow, we prove well-posedness of $BV_\\text{loc}$-solutions of stochastic transport equations and weak solutions of stochastic continuity equations with so-called transport noise and velocity fields in $L^q_t C^{0, \\alpha}_x$. For both equations, solutions may fail to be unique in the deterministic setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-03T11:14:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "60H15",
      "35B65"
    ],
    "keywords": [
      "transport/continuity equations",
      "hölder drifts",
      "stochastic transport equations",
      "stochastic continuity equations",
      "intermediate step"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}