Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations
Journal article, 2015

Isotropic Gaussian random fields on the sphere are characterized by Karhunen-Lo\`eve 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 H\"older continuity and sample differentiability of the random fields is discussed. Rates of convergence of their finitely truncated Karhunen-Lo\`eve expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample path 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.

Gaussian random fields

sample Hölder continuity

Karhunen-Loève expansion

isotropic random fields

spectral Galerkin methods

spherical harmonic functions

sample differentiability

Kolmogorov-Chentsov theorem

strong convergence rates

stochastic partial differential equations


Annika Lang

Chalmers, Mathematical Sciences, Mathematical Statistics

University of Gothenburg

Ch. Schwab

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Annals of Applied Probability

1050-5164 (ISSN)

Vol. 25 6 3047-3094

Subject Categories

Computational Mathematics

Probability Theory and Statistics


Basic sciences



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