Gaussian random fields: with and without covariances

We begin with isotropic Gaussian random fields, and show how the Bochner-Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Matérn processes on Euclidean space, spheres, manifolds and graphs, using Bessel potentials and stochastic partial differential equations (SPDEs).

Citation:
Bingham NH, Symons TL. Gaussian random fields: with and without covariances. Theory Probab Math Stat. 2022;106:27-40.

Keywords:
Bessel potential; Covariance function; Gaussian Markov random field; Gaussian random field; Mat´ern process; Numerical linear algebra; Precision matrix; Sparseness; Stochastic partial differential equation

Abstract:
We begin with isotropic Gaussian random fields, and show how the Bochner-Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Matérn processes on Euclidean space, spheres, manifolds and graphs, using Bessel potentials and stochastic partial differential equations (SPDEs).