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 examines whether these desirable properties transfer to manifold domains through rigorous analysis of Matérn processes on tori using pseudo-differential operator theory. We establish that processes on $d$-dimensional tori require smoothness parameter $\nu >3d\mathrm{/}2$ to achieve regularity $C_{\text{loc}}^{(\nu -3d\mathrm{/}2{)}^{-}}$, revealing a dimension-dependent threshold that contrasts with the Euclidean requirement of merely $\nu >0$. Our proof employs the Cardona-Martínez theory of pseudo-differential operators, providing new analytical tools to the study of random fields over manifolds. We also introduce the canonical-Matérn process, a parameter family that achieves regularity $C_{\text{loc}}^{(\nu -3d\mathrm{/}2+2{)}^{-}}$, gaining two orders of smoothness over standard Matérn processes.