{
  "abstract": "For every $x\\ge 0$, let $R^{x}=\\{ R^{x}_{t}\\} _{t\\ge 0}$ be a three-dimensional Bessel process started at $x$. Given $x>0$, with the aid of Pitman's theorem, a path transformation $T_{x}$ under which the process $T_{x}(R^{y})$ is distributed as $R^{x}$ for any $y>0$ is constructed. In fact, we prove that, for every $x,y>0$, the two three-dimensional processes \\begin{align*} \\bigl( R^{x},T_{y}(R^{x}),\\mathcal{P}(-R^{x})+x\\bigr) && \\text{and} && \\bigl( T_{x}(R^{y}),R^{y},\\mathcal{P}(-R^{y})+y\\bigr) \\end{align*} are identical in law, where $\\mathcal{P}$ is Pitman's transformation. In the course of our exploration, it will be revealed that, for any $x>0$, the process $\\mathcal{P}(-R^{x})+x$ has the same law as the standard three-dimensional Bessel process $R^{0}$, which is then applied to derive a disintegration formula for the law of $R^{x}$ in terms of Pitman's transformation. An application of the formula to three-dimensional Bessel bridges is also presented.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25205",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuu Hariya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fbjexsps3dtcv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25205",
  "fulltext_url": null,
  "id": "fbjexsps3dtcv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25205",
  "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",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7223043560",
  "pdf_url": "https://arxiv.org/pdf/2610.25205",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fbjexsps3dtcv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25205",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25205"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A relationship between three-dimensional Bessel processes in terms of Pitman's transformation",
  "title_text": "A relationship between three-dimensional Bessel processes in terms of Pitman's transformation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/fbjexsps3dtcv",
  "venue": "arXiv",
  "versions": []
}