Regularity properties and simulations of Gaussian random fields on the Sphere cross Time

TL;DR

This paper investigates the regularity and simulation of Gaussian fields on spheres cross time, providing spectral decompositions, regularity analysis, and an efficient simulation method with proven accuracy.

Contribution

It introduces two spectral decompositions for Gaussian fields on spheres cross time, analyzes their regularity, and proposes a fast, accurate simulation method.

Findings

Spectral decompositions establish regularity in Sobolev spaces.

Simulation method is fast, efficient, and accurate in the L2 sense.

Regularity properties are characterized for Gaussian fields on spheres cross time.

Abstract

We study the regularity properties of Gaussian fields defined over spheres cross time. In particular, we consider two alternative spectral decompositions for a Gaussian field on $\mathbb{S}^d \times \mathbb{R}$. For each decomposition, we establish regularity properties through Sobolev and interpolation spaces. We then propose a simulation method and study its level of accuracy in the $L^2$ sense. The method turns to be both fast and efficient.