Isotropic Q-fractional Brownian motion on the sphere: regularity and fast simulation
Preprint, 2024
As an extension of isotropic Gaussian random fields and Q-Wiener processes on d-dimensional spheres, isotropic Q-fractional Brownian motion is introduced and sample Hölder regularity in space-time is shown depending on the regularity of the spatial covariance operator Q and the Hurst parameter H. The processes are approximated by a spectral method in space for which strong and almost sure convergence are shown. The underlying sample paths of fractional Brownian motion are simulated by circulant embedding or conditionalized random midpoint displacement. Temporal convergence and computational complexity are numerically tested, the latter matching the complexity of simulating a Q-Wiener process if allowing for a temporal error.
strong convergence
d-dimensional sphere
FFT
spherical harmonic functions
Fractional Brownian motion
Gaussian processes
spectral methods
circulant embedding
conditionalized random midpoint displacement
Karhunen–Loève expansion
Author
Annika Lang
Chalmers, Mathematical Sciences, Applied Mathematics and Statistics
Björn Müller
Chalmers, Mathematical Sciences, Applied Mathematics and Statistics
Efficient approximation methods for random fields on manifolds
Swedish Research Council (VR) (2020-04170), 2021-01-01 -- 2024-12-31.
Time-Evolving Stochastic Manifolds (StochMan)
European Commission (EC) (EC/HE/101088589), 2023-09-01 -- 2028-08-31.
Subject Categories (SSIF 2011)
Mathematics
Computational Mathematics
Geometry
Probability Theory and Statistics
Mathematical Analysis
DOI
10.48550/arXiv.2410.19649
Related datasets
Code to “Isotropic Q-fractional Brownian motion on the sphere: regularity and fast simulation” [dataset]
DOI: 10.5281/zenodo.14529834