{
  "id": "2407.09046",
  "version": "v1",
  "published": "2024-07-12T07:15:26.000Z",
  "updated": "2024-07-12T07:15:26.000Z",
  "title": "Weak well-posedness of energy solutions to singular SDEs with supercritical distributional drift",
  "authors": [
    "Lukas Gräfner",
    "Nicolas Perkowski"
  ],
  "comment": "44 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We study stochastic differential equations with additive noise and distributional drift on $\\mathbb{T}^d$ or $\\mathbb{R}^d$ and $d \\geqslant 2$. We work in a scaling-supercritical regime using energy solutions and recent ideas for generators of singular stochastic partial differential equations. We mainly focus on divergence-free drift, but allow for scaling-critical non-divergence free perturbations. In the time-dependent divergence-free case we roughly speaking prove weak well-posedness of energy solutions with initial law $\\mu \\ll \\text{Leb}$ for drift $b \\in L^p_T B^{-\\gamma}_{p, 1}$ with $p \\in (2, \\infty]$ and $p \\geqslant \\frac{2}{1 -\\gamma}$. For time-independent $b$ we show weak well-posedness of energy solutions with initial law $\\mu \\ll \\text{Leb}$ under certain structural assumptions on $b$ which allow local singularities such that $b \\notin B^{-1}_{2 d/(d-2), 2}$, meaning that for any $p > 2$ in sufficiently high dimension there exists $b \\notin B^{-1}_{p, 2}$ such that weak well-posedness holds for energy solutions with drift $b$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-12T07:15:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "energy solutions",
      "weak well-posedness",
      "supercritical distributional drift",
      "singular sdes",
      "singular stochastic partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}