{
  "abstract": "We construct hypocoercive Kolmogorov-type operators on the Euclidean motion group $\\mathrm{SE}(3)=\\mathbb{R}^3\\rtimes\\mathrm{SO}(3)$ using ZXZ Euler angles and study their long-time behaviour using the abstract hypocoercivity method. The operator $L=S-A$ decomposes into a symmetric part $S$, corresponding to a degenerate diffusion on the rotation component $\\mathrm{SO}(3)$, and an antisymmetric part $A$, which captures transport and torque effects induced by the Lie group structure. The associated invariant probability measure is $μ_Φ=γ_Φ\\otimesν_3$, where $γ_Φ$ is a probability measure on $\\mathbb{R}^3$ with density $Z(Φ)^{-1}e^{-Φ}$ with respect to the Lebesgue measure, and $ν_3$ is the normalized Haar measure on $\\mathrm{SO}(3)$. As a consequence, if $Φ\\in C^\\infty(\\mathbb{R}^3)$ is bounded from below, $γ_Φ$ satisfies a Poincaré inequality with constant $Λ>0$, and there exists $c<\\infty$ such that \\begin{equation*} |\\nabla_ξ^2Φ(ξ)| \\leq c\\bigl(1+|\\nabla_ξΦ(ξ)|\\bigr), \\quad ξ\\in\\mathbb{R}^3, \\end{equation*} then the semigroup generated by the closure of $L$ converges exponentially fast to equilibrium in $L^2(\\mathrm{SE}(3),μ_Φ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03005",
  "author_total": 2,
  "authors": [
    {
      "name": "Martin Grothaus"
    },
    {
      "name": "Andrea V. Hurtado-Quiceno"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3gay9mmk1rzp4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03005",
  "fulltext_url": null,
  "id": "3gay9mmk1rzp4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03005",
  "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.DG"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7220367730",
  "pdf_url": "https://arxiv.org/pdf/2610.03005",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3gay9mmk1rzp4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03005",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03005"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Langevin-type dynamics and hypocoercivity on $\\mathrm{SE}(3)$",
  "title_text": "Langevin-type dynamics and hypocoercivity on SE(3)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/3gay9mmk1rzp4",
  "venue": "arXiv",
  "versions": []
}