{
  "abstract": "We analyze diffusion processes ${(μ_t)}_{t\\geq 0}$ on the $2$-Wasserstein space $\\mathscr P_2$ over $\\mathbb R^d$, which can be classified as Ornstein--Uhlenbeck processes by means of their Fukushima decomposition. Characteristically for this class, $ | μ_t|_2^2-|μ_0|_2^2-2ct+2\\int_0 ^t|μ_s|_2^2\\,d s,\\, t\\geq 0,$ is a martingale, where $c\\in(0,\\infty)$ is a constant and $|μ_t|_2:=(\\int_{\\mathbb R^d}x^{\\text{T}} xμ_t(d x ))^{1/2}$ denotes the second moment. Making a particular choice from this class, we obtain a process ${(μ_t)}_{t\\geq 0}$ such that all moments exist and ${μ_t(θ)}-{μ_0(θ)}-\\int_0 ^tμ_s\\big(Δθ-\\langle \\text{id}_{\\mathbb R^d},\\nabla θ\\rangle\\big)d s$, $t\\geq 0$, is a martingale for each multivariate polynomial $θ:\\mathbb R^d\\to\\mathbb R$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.14917",
  "author_total": 2,
  "authors": [
    {
      "name": "Martin Grothaus"
    },
    {
      "name": "Simon Wittmann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/909k7mr054g72.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.14917",
  "fulltext_url": null,
  "id": "909k7mr054g72",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.14917",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7164901181",
  "pdf_url": "https://arxiv.org/pdf/2606.14917",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/909k7mr054g72.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:35Z",
      "native_id": "2606.14917",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.14917"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Ornstein$-$Uhlenbeck process on $\\mathscr P_2$ with a volatility operator",
  "title_text": "The Ornstein−Uhlenbeck process on 𝒫_2 with a volatility operator",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:53Z",
  "url": "https://research.draconitic.com/paper/909k7mr054g72",
  "venue": "arXiv",
  "versions": []
}