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 $\mathbb{S}^N$.