{
  "abstract": "Let $(U_t)_{t\\in\\mathbb R}$ be a stationary Ornstein--Uhlenbeck process in $\\mathbb R^d$ whose coordinate processes $(U_t^{(i)})_{t\\in\\mathbb R}$, $1\\le i\\le d$, are independent, centered, and have covariance function $\\mathbb E [U_s^{(i)}U_t^{(i)}]=e^{-|t-s|/2}$. For every $T>0$, we obtain an explicit formula for the probability that the convex hull $\\mathrm{conv}\\{U_t:0\\le t\\le T\\}$ contains the origin. We obtain a similar formula for a circular Ornstein--Uhlenbeck loop whose covariance is the normalized $T$-periodization of this kernel. For each fixed dimension $d$, we determine the precise long-time asymptotics of the non-absorption probabilities, which decay as explicit constants times $T^{d-1} e^{-T/2}$ as $T\\to\\infty$. We also show that, as $d\\to\\infty$, the absorption probabilities in both models undergo a transition from $0$ to $1$ around $T=2d$, with a common Gaussian limiting profile on the scale $\\sqrt d$. We also obtain explicit formulas for the probability that $σZ$ belongs to the convex hull of Brownian motion or Brownian bridge on $[0,1]$ in $\\mathbb R^d$, where $Z$ is an independent standard Gaussian vector and $σ\\ge0$. The large-$σ$ asymptotics recover the known formulas for the expected volumes of these convex hulls. Finally, we compute the conic intrinsic volumes of the closed positive hulls of deterministic Ornstein--Uhlenbeck spirals and their circular analogues in Hilbert spaces. Their Grassmann angles coincide with the corresponding Ornstein--Uhlenbeck absorption probabilities. After suitable rescaling, the small-time limits of these conic intrinsic volumes recover the formulas of Gao--Vitale and Gao for the intrinsic volumes of the Brownian body and its Brownian-bridge analogue.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37911",
  "author_total": 1,
  "authors": [
    {
      "name": "Zakhar Kabluchko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7yv5mcy6e4hyb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37911",
  "fulltext_url": null,
  "id": "7yv5mcy6e4hyb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37911",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.37911",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7yv5mcy6e4hyb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.37911",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37911"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Ornstein--Uhlenbeck spirals: Absorption Probabilities and Conic Intrinsic Volumes",
  "title_text": "Ornstein--Uhlenbeck spirals: Absorption Probabilities and Conic Intrinsic Volumes",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/7yv5mcy6e4hyb",
  "venue": "arXiv",
  "versions": []
}