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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 $C^{\nu^-}_{\mathrm{loc}}$ for every $\nu>0$ and no more, exactly as in the Euclidean case. Our main results concern symbols whose order varies with position. Given an order function $m \in C^\infty(\mathbb{T}^d;\mathbb{R})$ 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.

Submission history

From: Nicolás Escobar-Velásquez [view email]
[v1] Wed, 12 Nov 2025 15:37:23 UTC (17 KB)
[v2] Tue, 8 Sep 2026 13:48:02 UTC (45 KB)