{
  "abstract": "We investigate the dynamical behavior of a class of semilinear stochastic wave equations on a bounded smooth domain $\\Ocal\\subset\\R^d$ driven by additive trace-class noise, where the elastic response is governed by a \\emph{mixed local--nonlocal} operator $\\Acal=-θΔ+β(-Δ)^s$ with $s\\in(0,1)$. A fundamental challenge in this setting is that the local and nonlocal operators do not commute on bounded domains: the natural Dirichlet basis fails to diagonalize the restricted fractional Laplacian. Consequently, we first establish the \\emph{strong} convergence of the resulting non-diagonal, dense Galerkin approximation scheme. Leveraging these uniform energy bounds, we rigorously derive the associated Itô energy identity. In the defocusing regime ($\\varepsilon = +1$), this strong approximation yields global well-posedness on the energy-subcritical range, providing a unique probabilistically strong solution in the energy space $V \\times H$. Within this variational framework, we conduct an analysis of the Malliavin regularity, showing $(u(t), v(t)) \\in \\mathbb{D}^{1,2}(V) \\times \\mathbb{D}^{1,2}(H)$, and leverage fractional Sobolev embeddings to prove that the one-dimensional probability law of $u(t, x_0)$ is absolutely continuous via the Bouleau--Hirsch criterion. In stark contrast, for the focusing regime ($\\varepsilon = -1$), we establish local well-posedness and prove a rigorous dichotomy: under a negativity condition on the initial energy, either pathwise explosion occurs with positive probability in finite time, or the energy norm possesses an infinite second moment before an explicit critical time $T^*$. Finally, we observe how the dense Galerkin interaction matrices pose unique structural challenges for spatial statistical inference.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06883",
  "author_total": 2,
  "authors": [
    {
      "name": "Francisco Delgado-Vences"
    },
    {
      "name": "Jose Julian Pavon-Español"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2jd9h2e4a7nz1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06883",
  "fulltext_url": null,
  "id": "2jd9h2e4a7nz1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06883",
  "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.AP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7212043788",
  "pdf_url": "https://arxiv.org/pdf/2609.06883",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2jd9h2e4a7nz1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06883",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06883"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Strong Galerkin Approximation, Malliavin Regularity, and Blow-Up for a Mixed Local--Nonlocal Stochastic Wave Equation",
  "title_text": "Strong Galerkin Approximation, Malliavin Regularity, and Blow-Up for a Mixed Local--Nonlocal Stochastic Wave Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/2jd9h2e4a7nz1",
  "venue": "arXiv",
  "versions": []
}