{
  "abstract": "Matérn covariance functions are ubiquitous in spatial statistics, valued for their interpretable parameters and well-understood sample path properties in Euclidean settings. This paper extends the theory of Matérn fields over tori through the pseudo-differential calculus. We first establish that a Matérn process on the d-dimensional torus has sample paths in Cloc- for every ν>0 and no more, exactly as in the Euclidean case. Our main results concern symbols whose order varies with position. Given an order function m∈C∞(Td;R) we construct a Gaussian field whose sample-path Hölder exponent is H(x)=m(x)-d/2 at each point, and we prove that the existence threshold m(x)>d/2 is local: the regularity of the field near a point depends on m only through its germ at that point. Finally we examine the canonical field, which supplies a direct check on both the threshold and the exponent. We use it to prove a rigidity statement: weighting the Matérn spectral density by |k|-2 produces a superposition of ordinary Matérn fields of higher smoothness.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Biostatistics, Epidemiology and Informatics, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104 USA."
  ],
  "article_type": "research article",
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Nicolas Escobar",
      "orcid": "0009-0006-0800-5692"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9hk9b7t0fhy48.bib",
  "cited_by": 0,
  "doi": "10.1007/s11868-026-00840-9",
  "fulltext_url": "https://europepmc.org/articles/PMC13683494",
  "id": "9hk9b7t0fhy48",
  "issns": [
    "1662-9981",
    "1662-999X"
  ],
  "landing_url": "https://europepmc.org/article/MED/42908762",
  "language": "eng",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7223010607",
  "pdf_url": "https://europepmc.org/articles/PMC13683494?pdf=render",
  "pmcid": "PMC13683494",
  "pmid": "42908762",
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9hk9b7t0fhy48.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:10Z",
      "native_id": "MED:42908762",
      "source": "europepmc",
      "url": "https://europepmc.org/article/MED/42908762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Point processes and geometric inequalities",
      "subfield": "general"
    }
  ],
  "title": "Pseudo-differential operators and generalized random fields over tori.",
  "title_text": "Pseudo-differential operators and generalized random fields over tori.",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:20:10Z",
  "url": "https://research.draconitic.com/paper/9hk9b7t0fhy48",
  "venue": "Journal of pseudo-differential operators and applications",
  "versions": []
}