{
  "abstract": "We developed a random-field theory for the Euler–Poisson–Darboux equation on Euclidean space under additive Gaussian forcing that was white in the evolution variable and spatially homogeneous. An explicit two-parameter Bessel Green kernel provided the basic deterministic propagator. Sharp high-frequency bounds for its integrated energy yielded a necessary and sufficient Dalang criterion for the existence of the stochastic convolution and showed that the singular Darboux coefficient did not change the wave-type spectral threshold. A dissipative energy identity for the Green kernel provided temporal increment estimates which, together with the spatial Fourier bounds, led to joint space–time Hölder regularity under a natural strengthened spectral moment. We also distinguished between the two initial traces: the Dalang condition guaranteed the mean-square displacement trace, whereas a mean-square zero-velocity trace required additional spectral regularity; finite total spectral mass provided a simple sufficient condition. Space–time white noise and Riesz-colored noise were treated explicitly. Finally, the mean solution was identified with the Darboux spherical mean, Brownian first-exit sampling, and the Pizzetti expansion.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Anwar Almualim",
      "orcid": "0000-0002-8374-9409"
    },
    {
      "affs": [
        0
      ],
      "name": "Fethi Bouzeffour",
      "orcid": "0000-0002-2743-2036"
    },
    {
      "affs": [
        0
      ],
      "name": "Mhamed Eddahbi",
      "orcid": "0000-0003-0889-3387"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2hart4pyhqrp1.bib",
  "cited_by": 0,
  "doi": "10.3390/axioms15100782",
  "fulltext_url": null,
  "id": "2hart4pyhqrp1",
  "issns": [
    "2075-1680"
  ],
  "landing_url": "https://www.mdpi.com/2075-1680/15/10/782",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2604|Differential Equations and Boundary Problems"
    ]
  ],
  "openalex": "W7222937169",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/2hart4pyhqrp1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:30Z",
      "native_id": "10.3390/axioms15100782",
      "source": "crossref",
      "url": "https://www.mdpi.com/2075-1680/15/10/782"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Differential Equations and Boundary Problems",
      "subfield": "general"
    }
  ],
  "title": "Stochastic Euler–Poisson–Darboux Equations and a Sharp Dalang Criterion",
  "title_text": "Stochastic Euler–Poisson–Darboux Equations and a Sharp Dalang Criterion",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:30Z",
  "url": "https://research.draconitic.com/paper/2hart4pyhqrp1",
  "venue": "Axioms",
  "versions": []
}