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Pseudo-Differ. Oper. Appl."],"published-print":{"date-parts":[[2026,12]]},"abstract":"Abstract<\/jats:title>\n \n Mat\u00e9rn 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 Mat\u00e9rn fields over tori through the pseudo-differential calculus. We first establish that a Mat\u00e9rn process on the\n d<\/jats:italic>\n -dimensional torus has sample paths in\n \n \n $$C^{\\nu ^-}_{\\textrm{loc}}$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n loc<\/mml:mtext>\n \n \u03bd<\/mml:mi>\n -<\/mml:mo>\n <\/mml:msup>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for every\n \n \n $$\\nu >0$$<\/jats:tex-math>\n \n \n \u03bd<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and no more, exactly as in the Euclidean case. Our main results concern symbols whose order varies with position. Given an order function\n \n \n $$m \\in C^\\infty (\\mathbb {T}^d;\\mathbb {R})$$<\/jats:tex-math>\n \n \n m<\/mml:mi>\n \u2208<\/mml:mo>\n \n C<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \n \n T<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n \u037e<\/mml:mo>\n R<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n we construct a Gaussian field whose sample-path H\u00f6lder exponent is\n \n \n $$H(x) = m(x)-d\/2$$<\/jats:tex-math>\n \n \n H<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n m<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n -<\/mml:mo>\n d<\/mml:mi>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n at each point, and we prove that the existence threshold\n \n \n $$m(x)>d\/2$$<\/jats:tex-math>\n \n \n m<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n ><\/mml:mo>\n d<\/mml:mi>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is local: the regularity of the field near a point depends on\n m<\/jats:italic>\n 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 Mat\u00e9rn spectral density by\n \n \n $$|k|^{-2}$$<\/jats:tex-math>\n \n \n \n |<\/mml:mo>\n k<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n -<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n produces a superposition of ordinary Mat\u00e9rn fields of higher smoothness.\n <\/jats:p>","DOI":"10.1007\/s11868-026-00840-9","type":"journal-article","created":{"date-parts":[[2026,10,20]],"date-time":"2026-10-20T07:45:58Z","timestamp":1792482358000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Pseudo-differential operators and generalized random fields over tori"],"prefix":"10.1007","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-0800-5692","authenticated-orcid":false,"given":"Nicolas","family":"Escobar","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,10,20]]},"reference":[{"key":"840_CR1","unstructured":"Ru\u00dfwurm, M., Klemmer, K., Rolf, E., Zbinden, R., Tuia, D.: Geographic location encoding with spherical harmonics and sinusoidal representation networks. 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