Isotropic Q-fractional Brownian motion on the sphere: regularity and fast simulation

TL;DR

This paper introduces isotropic Q-fractional Brownian motion on spheres, analyzes its regularity, and develops spectral and simulation methods with proven convergence and efficiency.

Contribution

It extends Gaussian fields to fractional Brownian motion on spheres, providing new regularity results and efficient spectral simulation techniques.

Findings

Sample paths exhibit specific Hölder regularity depending on parameters.

Spectral approximation converges strongly and almost surely.

Simulation methods are computationally efficient and accurate.

Abstract

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 H\"older 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 accuracy and computational complexity are numerically tested, the latter matching the complexity of simulating a Q-Wiener process if allowing for a temporal error.