The Annals of Applied Probability

Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations

Annika Lang and Christoph Schwab

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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.

Article information

Ann. Appl. Probab., Volume 25, Number 6 (2015), 3047-3094.

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

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Zentralblatt MATH identifier

Primary: 60G60: Random fields 60G17: Sample path properties 41A25: Rate of convergence, degree of approximation 60H15: Stochastic partial differential equations [See also 35R60] 65C30: Stochastic differential and integral equations 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Secondary: 60H35: Computational methods for stochastic equations [See also 65C30] 60G15: Gaussian processes 33C55: Spherical harmonics

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


Lang, Annika; Schwab, Christoph. Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations. Ann. Appl. Probab. 25 (2015), no. 6, 3047--3094. doi:10.1214/14-AAP1067.

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