{
  "abstract": "We study the stochastic differential equation $$d X_t=b(t,X_t)d t+\\sqrt{2}d W_t$$ on $\\mathbb R^d$, where $b$ is a time-dependent, divergence-free distributional drift of critical Hölder--Besov regularity $-1$, strengthened by an iterated-logarithmic correction. For every initial probability law, we construct a weak solution by smooth approximation and realize the singular drift as an additive functional. The main analytic ingredient is the Schauder estimate with a logarithmic smallness factor. Combined with uniform logarithmic Krylov estimates and a stochastic substitution formula for distributional test functions, this estimate allows us to apply a Zvonkin transformation and prove uniqueness in law among weak solutions satisfying the corresponding Krylov bounds. For solutions starting from deterministic points, we further show that their time-marginal distributions admit densities satisfying two-sided Aronson-type Gaussian estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02022",
  "author_total": 3,
  "authors": [
    {
      "name": "Zikai Chen"
    },
    {
      "name": "Zimo Hao"
    },
    {
      "name": "Xicheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9d028hkf3hrb8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02022",
  "fulltext_url": null,
  "id": "9d028hkf3hrb8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02022",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7207746411",
  "pdf_url": "https://arxiv.org/pdf/2609.02022",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9d028hkf3hrb8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02022",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02022"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Well-Posedness for SDEs with Logarithmical Critical Distributional Drifts",
  "title_text": "Well-Posedness for SDEs with Logarithmical Critical Distributional Drifts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/9d028hkf3hrb8",
  "venue": "arXiv",
  "versions": []
}