{
  "abstract": "We study the Euclidean Varadhan formula (EVF) and path laws for diffusions with critical divergence-free drifts in dimension $d\\geq2$. For drifts given by smooth bounded skew-symmetric stream matrices, we prove a quantitative lower bound under a sector condition with a lower-order term. Its constants depend only on the dimension and the sector condition data. By approximation, the locally uniform Euclidean lower bound holds for all divergence-free $L^\\infty_t{\\rm BMO}^{-1}_x$ drifts. However, a time-dependent counterexample shows that EVF can fail even for a drift in $L^\\infty_tL^d_x$. A complementary upper bound is given under a structural decomposition condition. In particular, for every time-independent $L^{d,\\infty}$ drift (without any smallness assumption), and the decoupled diffusion associated with the two-dimensional vorticity equation, the EVF holds. Finally, we construct a time-independent $L^d$ drift whose SDE has a unique strong solution from the origin and whose path law is mutually singular with the corresponding driftless Brownian law. The same diffusion satisfies EVF, showing that Brownian short-time endpoint costs can persist despite singularity of the path law.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06341",
  "author_total": 2,
  "authors": [
    {
      "name": "Jintao Feng"
    },
    {
      "name": "Guohuan Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a97w6e21j7g4p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06341",
  "fulltext_url": null,
  "id": "a97w6e21j7g4p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06341",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7220742117",
  "pdf_url": "https://arxiv.org/pdf/2610.06341",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a97w6e21j7g4p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06341",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06341"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Diffusions with Critical Divergence-Free Drifts: Varadhan Asymptotics and Path-Law Singularity",
  "title_text": "Diffusions with Critical Divergence-Free Drifts: Varadhan Asymptotics and Path-Law Singularity",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/a97w6e21j7g4p",
  "venue": "arXiv",
  "versions": []
}