The SPDE approach for Gaussian and non-Gaussian fields: 10 years and\n still running

Abstract

Gaussian processes and random fields have a long history, covering multiple\napproaches to representing spatial and spatio-temporal dependence structures,\nsuch as covariance functions, spectral representations, reproducing kernel\nHilbert spaces, and graph based models. This article describes how the\nstochastic partial differential equation approach to generalising Mat\\'ern\ncovariance models via Hilbert space projections connects with several of these\napproaches, with each connection being useful in different situations. In\naddition to an overview of the main ideas, some important extensions, theory,\napplications, and other recent developments are discussed. The methods include\nboth Markovian and non-Markovian models, non-Gaussian random fields,\nnon-stationary fields and space-time fields on arbitrary manifolds, and\npractical computational considerations.\n