{
  "abstract": "For a closed subset $C$ of Euclidean space, we study the diffusion whose drift pulls the state towards its nearest point in $C$, with noise intensity $T$. We let $T$ tend to zero and time tend to infinity. Convexity of $C$ is equivalent to a pathwise bound of the distance from $C$ by the norm of the process of Ornstein and Uhlenbeck. For arbitrary closed sets the bound holds in law. For compact convex $C$, we describe the invariant law by the curvature measures and the intrinsic volumes of $C$. The limit points of the law of the state form the orbit of reflected Brownian motion in $C$ started from the projection of the initial law. So the two limits commute only when this projection is the uniform law on $C$. Unless $C$ is a point, the mixing time is of order $1/T$ up to a logarithm. For the control problem the two limits of the value function commute for every compact convex set and every bounded Lipschitz terminal cost. A companion paper treats finite disjoint unions of convex sets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35145",
  "author_total": 1,
  "authors": [
    {
      "name": "Narayana Darapaneni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7msmp5exfn931.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35145",
  "fulltext_url": null,
  "id": "7msmp5exfn931",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35145",
  "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.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7224003638",
  "pdf_url": "https://arxiv.org/pdf/2610.35145",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7msmp5exfn931.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35145",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35145"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Zero temperature versus long time for diffusions driven by metric projections onto convex sets",
  "title_text": "Zero temperature versus long time for diffusions driven by metric projections onto convex sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/7msmp5exfn931",
  "venue": "arXiv",
  "versions": []
}