{
  "abstract": "We develop a variational formulation of the Ornstein-Uhlenbeck evolution equation based on a Gaussian divergence structure, in which the drift term is absorbed into the first variation of a weighted Dirichlet energy rather than treated as a lower-order perturbation. For the linear equation, the distributional, energy weak, and variational notions of solution coincide and determine the same unique solution. The same framework yields a variational formulation of the associated parabolic obstacle problem, for which we prove existence and then uniqueness under a standard additional regularity assumption on the obstacle. The same mechanism applies to nonlinear Gaussian gradient flows associated with convex integrands of standard $p$-growth. Finally, we extend the Gaussian-divergence formulation to the force-free kinetic Fokker-Planck equation with periodic position variables, establishing the same equivalence for the linear equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21003",
  "author_total": 2,
  "authors": [
    {
      "name": "Erica Ipocoana"
    },
    {
      "name": "Annalaura Rebucci"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/04p5xrat690z3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21003",
  "fulltext_url": null,
  "id": "04p5xrat690z3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21003",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213868858",
  "pdf_url": "https://arxiv.org/pdf/2609.21003",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/04p5xrat690z3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21003",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21003"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A variational approach to Ornstein-Uhlenbeck evolution equations and obstacle problems",
  "title_text": "A variational approach to Ornstein-Uhlenbeck evolution equations and obstacle problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/04p5xrat690z3",
  "venue": "arXiv",
  "versions": []
}