{
  "abstract": "For every dimension $d\\ge1$, we determine the probability that a prescribed point in $\\mathbb R^d$ belongs to the convex hull over $[0,1]$ of a standard Brownian motion starting at the origin or of a Brownian bridge from $0$ to $0$. We obtain explicit one-dimensional integral formulas involving only standard special functions. For example, in dimension three the Brownian-motion absorption probability $p_3^B(r)$ for a prescribed point at distance $r>0$ from the origin is \\[ p_3^B(r) =\\frac{r}{\\sqrt2\\,π^{3/2}} \\int_0^\\infty \\left(\\fracπ{2\\sqrt{1+v^2}}-1\\right) \\mathrm{e}^{-r^2(1+v^2)/4} (K_1-K_0)\\!\\left(\\frac{r^2(1+v^2)}4\\right)\\,\\mathrm{d} v, \\] where $K_ν$ denotes the modified Bessel function of the second kind. For each fixed dimension $d\\ge1$ and each prescribed nonzero point in $\\mathbb R^d$, we derive precise asymptotics of the non-absorption probabilities for the convex hulls of Brownian motion on $[0,T]$ and a Brownian bridge from $0$ to $0$ over the same interval, as $T\\to\\infty$. The corresponding probabilities are asymptotic to explicit positive constants times $T^{-1/2}(\\log T)^{d-1}$ and $T^{-1}(\\log T)^{d-1}$, respectively. We also consider the first time $τ_d$ at which the growing convex hull of a standard Brownian motion in $\\mathbb R^d$ absorbs a prescribed point at distance one from the origin. We determine the Mellin and Laplace transforms of $τ_d$ explicitly. Although $\\mathbb E[τ_d]=\\infty$ in every dimension, we prove \\[ \\frac{\\logτ_d-2d}{2\\sqrt d} \\xrightarrow{\\mathrm d}N(0,1), \\qquad d\\to\\infty. \\] In particular, $τ_d^{1/d}\\to\\mathrm{e}^2$ in probability, as $d\\to\\infty$. Surprisingly, the same limit holds for $S_d^{1/d}$, where $S_d$ is the first time at which the Brownian convex hull contains the entire unit ball.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00381",
  "author_total": 1,
  "authors": [
    {
      "name": "Zakhar Kabluchko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bfp1f0862dfch.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00381",
  "fulltext_url": null,
  "id": "bfp1f0862dfch",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00381",
  "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/2611.00381",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bfp1f0862dfch.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00381",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00381"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Absorption Probabilities for Brownian Convex Hulls in Arbitrary Dimension",
  "title_text": "Absorption Probabilities for Brownian Convex Hulls in Arbitrary Dimension",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:05Z",
  "url": "https://research.draconitic.com/paper/bfp1f0862dfch",
  "venue": "arXiv",
  "versions": []
}