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 Cν ⁻loc C loc ν - for every ν >0 ν > 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 ∈ C^∞ (Tᵈ;R) m ∈ C ∞ ( T d ; R ) we construct a Gaussian field whose sample-path Hölder exponent is $$H(x) = m(x)-d/2$$ H ( x ) = m ( x ) - d / 2 at each point, and we prove that the existence threshold $$m(x)>d/2$$ 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 | k | - 2 produces a superposition of ordinary Matérn fields of higher smoothness.