Extremes of Spherical Fractional Brownian Motion

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 ${\mathbb{S}}^N$.