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

Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial spatio-temporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, graph based models. This article describes how the stochastic partial differential equation approach generalising Matérn models via space projections connects with several of these approaches, each connection being useful in different situations. In addition an overview main ideas, some important extensions, theory, applications, other recent developments are discussed. The methods include both Markovian non-Markovian models, non-Gaussian fields, non-stationary space–time on arbitrary manifolds, practical computational considerations.