{
  "id": "2405.12332",
  "version": "v1",
  "published": "2024-05-20T19:03:53.000Z",
  "updated": "2024-05-20T19:03:53.000Z",
  "title": "Feller generators with singular drifts in the critical range",
  "authors": [
    "D. Kinzebulatov",
    "Yu. A. Semenov"
  ],
  "categories": [
    "math.PR",
    "math.AP",
    "math.FA"
  ],
  "abstract": "We consider diffusion operator $-\\Delta + b \\cdot \\nabla$ in $\\mathbb R^d$, $d \\geq 3$, with drift $b$ in a large class of locally unbounded vector fields that can have critical-order singularities. Covering the entire range of admissible magnitudes of singularities of $b$ (but excluding the borderline value), we construct a strongly continuous Feller semigroup on the space of continuous functions vanishing at infinity, thus completing a number of results on well-posedness of SDEs with singular drifts. The previous results on Feller semigroups employed strong elliptic gradient bounds and hence required the magnitude of the singularities to be less than a small dimension-dependent constant. Our approach is different and uses De Giorgi's method ran in $L^p$ for $p$ sufficiently large, hence the gain in the assumptions on singular drift. For the critical borderline value of the magnitude of singularities of $b$, we construct a strongly continuous semigroup in a ``critical'' Orlicz space on $\\mathbb R^d$ whose local topology is stronger than the local topology of $L^p$ for any $2 \\leq p<\\infty$ but is slightly weaker than that of $L^\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-20T19:03:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singular drift",
      "feller generators",
      "critical range",
      "semigroups employed strong elliptic gradient",
      "feller semigroups employed strong elliptic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}