Pseudo-differential operators and generalized random fields over tori.
- 1Department of Biostatistics, Epidemiology and Informatics, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104 USA.
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.
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BibTeX
@article{escobar2026pseudodifferential,
title = {{Pseudo-differential operators and generalized random fields over tori.}},
author = {Nicolas Escobar},
journal = {Journal of pseudo-differential operators and applications},
year = {2026},
month = oct,
doi = {10.1007/s11868-026-00840-9},
url = {https://europepmc.org/article/MED/42908762},
note = {CC BY},
}
RIS
TY - JOUR
TI - Pseudo-differential operators and generalized random fields over tori.
AU - Nicolas Escobar
PY - 2026
DA - 2026/10/20
JO - Journal of pseudo-differential operators and applications
DO - 10.1007/s11868-026-00840-9
UR - https://europepmc.org/article/MED/42908762
AB - 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.
SN - 1662-9981
SN - 1662-999X
ER -