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December 2015 Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations
Annika Lang, Christoph Schwab
Ann. Appl. Probab. 25(6): 3047-3094 (December 2015). DOI: 10.1214/14-AAP1067


Isotropic Gaussian random fields on the sphere are characterized by Karhunen–Loève expansions with respect to the spherical harmonic functions and the angular power spectrum. The smoothness of the covariance is connected to the decay of the angular power spectrum and the relation to sample Hölder continuity and sample differentiability of the random fields is discussed. Rates of convergence of their finitely truncated Karhunen–Loève expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample generation via fast Fourier transforms on the sphere are indicated. The relevance of the results on sample regularity for isotropic Gaussian random fields and the corresponding lognormal random fields on the sphere for several models from environmental sciences is indicated. Finally, the stochastic heat equation on the sphere driven by additive, isotropic Wiener noise is considered, and strong convergence rates for spectral discretizations based on the spherical harmonic functions are proven.


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Annika Lang. Christoph Schwab. "Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations." Ann. Appl. Probab. 25 (6) 3047 - 3094, December 2015.


Received: 1 May 2013; Revised: 1 May 2014; Published: December 2015
First available in Project Euclid: 1 October 2015

zbMATH: 1328.60126
MathSciNet: MR3404631
Digital Object Identifier: 10.1214/14-AAP1067

Primary: 41A25 , 60G17 , 60G60 , 60H15 , 65C30 , 65N30
Secondary: 33C55 , 60G15 , 60H35

Keywords: Gaussian random fields , isotropic random fields , Karhunen–Loève expansion , Kolmogorov–Chentsov theorem , sample differentiability , sample Hölder continuity , spectral Galerkin methods , spherical harmonic functions , Stochastic partial differential equations , strong convergence rates

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.25 • No. 6 • December 2015
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