Extremes of Spherical Fractional Brownian Motion

TL;DR

This paper derives the asymptotic behavior of the probability that a fractional Brownian motion on an N-dimensional sphere exceeds a high threshold, focusing on entire spheres and geodesic discs.

Contribution

It provides the first asymptotic analysis of excursion probabilities for spherical fractional Brownian motion with respect to high thresholds.

Findings

Asymptotic formulas for excursion probabilities as u approaches infinity.

Results applicable to both entire spheres and geodesic discs.

Enhanced understanding of extremes for spherical Gaussian fields.

Abstract

Let $\{B_\beta (x), x \in \mathbb{S}^N\}$ be a fractional Brownian motion on the $N$-dimensional unit sphere $\mathbb{S}^N$ with Hurst index $\beta$. We study the excursion probability $\mathbb{P}\{\sup_{x\in T} B_\beta(x) > u \}$ and obtain the asymptotics as $u\to \infty$, where $T$ can be the entire sphere $\mathbb{S}^N$ or a geodesic disc on $\mathbb{S}^N$.