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

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