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February 2022 A general framework for SPDE-based stationary random fields
Ricardo Carrizo Vergara, Denis Allard, Nicolas Desassis
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Bernoulli 28(1): 1-32 (February 2022). DOI: 10.3150/20-BEJ1317

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 of the solution in the case of a White Noise source term with the covariance in a generic case through convolution. Then, we obtain a variety of SPDE-based stationary random fields. In particular, well-known results regarding the Matérn 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.

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Ricardo Carrizo Vergara. Denis Allard. Nicolas Desassis. "A general framework for SPDE-based stationary random fields." Bernoulli 28 (1) 1 - 32, February 2022. https://doi.org/10.3150/20-BEJ1317

Information

Received: 1 July 2018; Revised: 1 October 2020; Published: February 2022
First available in Project Euclid: 10 November 2021

Digital Object Identifier: 10.3150/20-BEJ1317

Keywords: evolution equation , Generalized random fields , matérn model , space-time geostatistics , SPDE approach , spectral measure , symbol function

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 1 • February 2022
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