Pseudo-Differential Operators and Generalized Random Fields over Tori

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