Isotropic Q-fractional Brownian motion on the sphere: regularity and fast simulation (2410.19649v2)

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.

• The paper derives bounds on the Hölder regularity of Q-fBm using the spatial covariance operator and Hurst parameter.
• It employs a spectral method using the Karhunen-Loeve expansion to achieve both strong and almost sure convergence in simulations.
• Comparative analysis of circulant embedding and CRMD reveals efficient simulation techniques with distinct computational trade-offs.

Isotropic $Q$-Fractional Brownian Motion on the Sphere: Regularity and Fast Simulation

This paper presents a comprehensive paper on isotropic $Q$-fractional Brownian motion ($Q$-fBm) on $d$-dimensional spheres, focusing on its regularity properties as well as efficient simulation techniques. The authors explore the intersection of Gaussian random fields (GRFs) on spheres with fractional Brownian motion (fBm) to extend the capabilities of stochastic partial differential equations (SPDEs) modeling.

Key Contributions

The concept of $Q$-fBm is introduced as an extension of isotropic Gaussian random fields and $Q$-Wiener processes on spheres. The paper investigates the spatial and temporal regularity of these processes, yielding results dependent on the spatial covariance operator $Q$ and the Hurst parameter $H$. The main contributions are detailed as follows:

1. Regularity Analysis: The authors derive bounds on the Hölder regularity of $Q$-fBm in both space and time. The spatial regularity is dependent on the properties of the covariance operator $Q$, while the temporal regularity is bounded by $H$. This expands theoretical understanding of fractional processes on spherical domains.
2. Spectral Method for Simulation: A spectral method is employed to approximate $Q$-fBm on spheres, supporting both strong and almost sure convergence. The utility of the Karhunen-Loeve expansion is highlighted in achieving efficient transformation and representation of these processes.
3. Simulation Techniques: Two methods for simulating the temporal component of $Q$-fBm are compared—circulant embedding (CE) and conditionalized random midpoint displacement (CRMD). CE operates with a computational complexity of $O(N \log N)$, while CRMD has linear complexity $O(N)$ and is preferred for faster computation with approximation trade-offs.

Numerical Results

The paper numerically evaluates the temporal convergence and computational complexity of $Q$-fBm simulation methods. It demonstrates how circulant embedding matches the complexity of $Q$-Wiener processes when allowing temporal error. The strong error analysis is complemented by empirical convergence observations, validating the theoretical rates derived.

Implications and Future Directions

The investigation into $Q$-fBm opens up new avenues for advancing simulations in domains requiring stochastic modeling on spherical geometry. This is particularly relevant for environmental science and astrophysics where global models often align naturally with spherical domains.

Future research could explore:

• Higher Dimensional Spheres: Extending these methods to more generalized manifolds could prove fruitful, harnessing the potential in diverse application areas.
• Enhanced Algorithms: Developing improved algorithms for simulating $Q$-fBm with better accuracy and reduced computational overhead.
• Application in SPDEs: Implementing the $Q$-fBm framework into practical SPDE models to assess real-world efficacy.

In conclusion, this paper successfully enhances the understanding and simulation of fractional Brownian motion on spheres. The fusion of mathematical theory with numerical experimentation provides a robust foundation for future explorations in SPDEs and stochastic modeling across various disciplines.