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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.09423 |
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| _version_ | 1866915613035724800 |
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| author | Escobar-Velasquez, Nicolas |
| author_facet | Escobar-Velasquez, Nicolas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09423 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pseudo-Differential Operators and Generalized Random Fields over Tori Escobar-Velasquez, Nicolas Statistics Theory 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. |
| title | Pseudo-Differential Operators and Generalized Random Fields over Tori |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2511.09423 |