Pseudo-differential operators and generalized random fields over tori

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 $C_{\text{loc}}^{-}$ 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 $m\in C^{\infty }(T^d;R)$ we construct a Gaussian field whose sample-path Hölder exponent is $H\left(x\right)=m\left(x\right)-d/2$ at each point, and we prove that the existence threshold $m\left(x\right)>d/2$ 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 ${\left|k\right|}^{-2}$ produces a superposition of ordinary Matérn fields of higher smoothness.