Pseudo-Differential Operators and Generalized Random Fields over Tori

Abstract

Mat\'ern 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\'ern fields over tori through the pseudo-differential calculus. We first establish that a Mat\'ern 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 H\"older 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 Mat\'ern spectral density by $|k|^{-2}$ produces a superposition of ordinary Mat\'ern fields of higher smoothness.