{
  "abstract": "We consider the problem of estimating the background intensity of a Hawkes process when the triggering mechanism is known, treating the background as an unknown finite positive measure with a regular density. We propose a likelihood-based variational approach based on the Wasserstein-Fisher-Rao geometry. In this setting, the relevant first variations are naturally distribution-valued and therefore fall outside the classical gradient-flow formulation. We develop a distributional framework for the resulting dynamics and give them a finite-dimensional meaning through projections onto statistical manifolds. We identify the true background as the unique zero of a limiting generalized gradient and, under suitable stability, compactness, integrability and approximation conditions, establish a conditional consistency theorem for measurable estimators satisfying the empirical projected equations on events whose probabilities tend to one. Finally, we specialize the construction to Gaussian-mixture manifolds and obtain an explicit finite-dimensional scheme.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19588",
  "author_total": 2,
  "authors": [
    {
      "name": "Isaias Banales"
    },
    {
      "name": "Arturo Jaramillo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fmpht1ttk0wvv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19588",
  "fulltext_url": null,
  "id": "fmpht1ttk0wvv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19588",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7222154025",
  "pdf_url": "https://arxiv.org/pdf/2610.19588",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fmpht1ttk0wvv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:42Z",
      "native_id": "2610.19588",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19588"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation",
  "title_text": "Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:42Z",
  "url": "https://research.draconitic.com/paper/fmpht1ttk0wvv",
  "venue": "arXiv",
  "versions": []
}