Strong Local Nondeterminism of Spherical Fractional Brownian Motion

TL;DR

This paper investigates the properties of spherical fractional Brownian motion, establishing optimal spectral estimates and demonstrating its strong local nondeterminism, which enhances understanding of its fine-scale behavior on the sphere.

Contribution

It provides the first optimal estimates for the angular power spectrum of spherical fractional Brownian motion and proves its strong local nondeterminism.

Findings

Optimal estimates for the angular power spectrum.

Proof of strong local nondeterminism.

Insights into high-frequency behavior of the process.

Abstract

Let $B = \left\{ B\left( x\right),\, x\in \mathbb{S}^{2}\right\} $ be the fractional Brownian motion indexed by the unit sphere $\mathbb{S}^{2}$ with index $0<H\leq \frac{1}{2}$, introduced by Istas \cite{IstasECP05}. We establish optimal estimates for its angular power spectrum $\{d_\ell, \ell = 0, 1, 2, \ldots\}$, and then exploit its high-frequency behavior to establish the property of its strong local nondeterminism of $B$.