{
  "abstract": "A valuation is a finitely additive function on the family of compact convex sets in $\\mathbb{R}^d$. We study non-negative infinitely divisible random valuations, with particular emphasis on monotone, $σ$-continuous models with independent increments along nested families. After separating the deterministic part, we show that the Lévy measure of such a valuation is generated by pairs $(F,r)$, where $F$ is a non-empty closed convex set and $r>0$, with each pair contributing $r\\mathbf{1}_{F\\cap K=\\emptyset}$. This yields a Poisson representation and an equivalent formulation through a pure-jump completely random measure on the space of closed convex sets. For stationary valuations, we derive a cylinder-Grassmannian representation of the Lévy measure. In the stationary isotropic case, we obtain a McMullen-type decomposition, at the level of one-dimensional distributions, into independent components stable under dilation of the argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19976",
  "author_total": 3,
  "authors": [
    {
      "name": "Andrii Ilienko"
    },
    {
      "name": "Ilya Molchanov"
    },
    {
      "name": "Tommaso Visonà"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1aw0z9a4ndr30.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19976",
  "fulltext_url": null,
  "id": "1aw0z9a4ndr30",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19976",
  "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.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7203903618",
  "pdf_url": "https://arxiv.org/pdf/2608.19976",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1aw0z9a4ndr30.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2608.19976",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19976"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Random valuations",
  "title_text": "Random valuations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/1aw0z9a4ndr30",
  "venue": "arXiv",
  "versions": []
}