Gaussian random fields on the sphere and sphere cross line

TL;DR

This paper reviews Gaussian processes on spheres and sphere cross line, focusing on path continuity, smoothness, and spectral decay, with implications for spatio-temporal modeling.

Contribution

It provides a comprehensive review of the Dudley integral, Belyaev dichotomy, and the relationship between path smoothness and spectral decay in Gaussian fields on spheres.

Findings

Path continuity linked to spectral decay rates

Extension to spatio-temporal sphere cross line

Insights into Gaussian process regularity

Abstract

We review the Dudley integral for the Belyaev dichotomy applied to Gaussian processes on spheres, and discuss the approximate (or restricted) continuity of paths in the discontinuous case. We discuss also the spatio-temporal case, of sphere cross line. In the continuous case, we investigate the link between the smoothness of paths and the decay rate of the angular power spectrum, following Tauberian work of the first author, Malyarenko, and Lang and Schwab.