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Pseudo-Differential Operators and Generalized Random Fields over Tori

Nicolas Escobar-Velasquez

TL;DR

The paper analyzes Matérn Gaussian random fields on the torus using a pseudo-differential operator framework, revealing a dimension-dependent threshold: on a $d$-torus, achieving local Hölder regularity $C^{(\nu-3d/2)^-}_{loc}$ requires $\nu > 3d/2$, unlike the Euclidean case. It employs Cardona-Martínez theory to establish regularity results and introduces the canonical-Matérn process, a three-parameter extension that attains $C^{(\nu-3d/2+2)^-}_{loc}$, gaining two orders of smoothness. The work also develops a canonical field with explicit dimension-dependent covariance behavior, provides a discretization scheme whose discrete model converges to the continuous limit, and proves an exponentially convergent convolution representation for the canonical-Matérn covariance, enabling efficient computation. Together, these results refine the understanding of spatial modeling on manifolds and offer a practical, geometry-aware alternative to standard Matérn models in periodic domains.

Abstract

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 $ν> 3d/2$ to achieve regularity $C^{(ν-3d/2)^-}_{\text{loc}}$, revealing a dimension-dependent threshold that contrasts with the Euclidean requirement of merely $ν> 0$. 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 $C^{(ν-3d/2+2)^-}_{\text{loc}}$, gaining two orders of smoothness over standard Matérn processes.

Pseudo-Differential Operators and Generalized Random Fields over Tori

TL;DR

The paper analyzes Matérn Gaussian random fields on the torus using a pseudo-differential operator framework, revealing a dimension-dependent threshold: on a -torus, achieving local Hölder regularity requires , unlike the Euclidean case. It employs Cardona-Martínez theory to establish regularity results and introduces the canonical-Matérn process, a three-parameter extension that attains , gaining two orders of smoothness. The work also develops a canonical field with explicit dimension-dependent covariance behavior, provides a discretization scheme whose discrete model converges to the continuous limit, and proves an exponentially convergent convolution representation for the canonical-Matérn covariance, enabling efficient computation. Together, these results refine the understanding of spatial modeling on manifolds and offer a practical, geometry-aware alternative to standard Matérn models in periodic domains.

Abstract

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 -dimensional tori require smoothness parameter to achieve regularity , revealing a dimension-dependent threshold that contrasts with the Euclidean requirement of merely . 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 , gaining two orders of smoothness over standard Matérn processes.

Paper Structure

This paper contains 21 sections, 20 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matérn Operator
  4. Function Spaces
  5. Pseudo-Differential Operators
  6. Generalized Random Fields
  7. Regularity of Matérn Processes on Tori
  8. Canonical Field and Canonical-Matérn Process
  9. The Canonical Field
  10. Discretization and Numerical Implementation
  11. Canonical-Matérn Process
  12. Convolution Structure and Computational Methods
  13. Conclusion
  14. Abstract Theory for GeRFs on Tori
  15. Kernel Representation Theory
  16. ...and 6 more sections

Key Result

Proposition 2.2

The dual space $\mathcal{S}'(\mathbb{T}^d)$ consists of continuous linear functionals on $\mathcal{S}(\mathbb{T}^d)$, characterized as sequences $\{\mu_k\}_{k \in \mathbb{Z}^d}$ with polynomial growth:

Theorems & Definitions (43)

  • Definition 2.1: Test Function Space
  • Proposition 2.2: Dual Space
  • Definition 2.3: Pseudo-differential Operators
  • Definition 2.4: Generalized Random Field
  • Definition 2.5: Stationarity and Spectral Measure
  • Definition 3.1: Local Hölder and almost-Hölder spaces
  • Proposition 3.2
  • Proposition 3.3: Path regularity of Torus Matérn Processes
  • proof
  • Remark 3.4
  • ...and 33 more