A general framework for SPDE-based stationary random fields

TL;DR

This paper develops a comprehensive theoretical framework for constructing stationary random fields using SPDEs, unifying existing models and introducing new spatio-temporal models with controllable properties.

Contribution

It provides a general approach to model stationary fields via linear SPDEs, including existence, uniqueness, and spectral properties, and introduces new models for spatio-temporal processes.

Findings

Unified framework for SPDE-based stationary fields

Recovery of known models like Matérn and Stein

Introduction of new spatio-temporal models with controllable properties

Abstract

This paper presents theoretical advances in the application of the Stochastic Partial Differential Equation (SPDE) approach in geostatistics. We show a general approach to construct stationary models related to a wide class of linear SPDEs, with applications to spatio-temporal models having non-trivial properties. Within the framework of Generalized Random Fields, a criterion for existence and uniqueness of stationary solutions for this class of SPDEs is proposed and proven. Their covariance are then obtained through their spectral measure. We present a result relating the covariance in the case of a White Noise source term with that of a generic case through convolution. Then, we obtain a variety of SPDE-based stationary random fields. In particular, well-known results regarding the Mat\'ern Model and Markovian models are recovered. A new relationship between the Stein model and a particular SPDE is obtained. New spatio-temporal models obtained from evolution SPDEs of arbitrary temporal derivative order are then obtained, for which properties of separability and symmetry can be controlled. We also obtain results concerning stationary solutions for physically inspired models, such as solutions to the heat equation, the advection-diffusion equation, some Langevin's equations and the wave equation.