A general framework for SPDE-based stationary random fields
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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 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 a wide class of linear SPDEs is proposed and proven. Their covariance are then obtained through their associated spectral measure. We also present a result that relates the covariance in the case of a White Noise source term with that of a generic case through convolution. Using these results, we obtain a variety of SPDE-based stationary random fields. In particular, well-known results regarding the Matérn Model and models with Markovian behavior 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 easily be controlled. Models with a fractional evolution in time are introduced and described, and we thereby obtain a large class of spatio-temporal models which separate regularity over space and time without separability or symmetry conditions. We also obtain results concerning stationary solutions for physically inspired models, such as solutions for the heat equation, the advection-diffusion equation, some Langevin's equations and the wave equation.
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