Persistence probabilities of spherical fractional Brownian motion

TL;DR

The paper determines the persistence probability decay for spherical fractional Brownian motion (SFBM) on the unit sphere, showing that as the threshold $\varepsilon\to0$, $\mathbb{P}(\sup_{\eta\in\mathbb{S}_{d-1}} S_H(\eta) < \varepsilon) = \varepsilon^{(d-1)/H + o(1)}$ for $0<H\le 1/2$. The authors develop a detailed RKHS framework for SFBM via integral representations with generalized Legendre (Gegenbauer) polynomials, and they extract sharp Legendre coefficient asymptotics, notably for functions with endpoint singularities such as arccos. A central part of the argument combines geometric comparison (Toponogov) with probabilistic comparison (Slepian) to obtain a sharp lower bound, and an RKHS shift technique together with occupation-time results for SFBM to obtain the matching upper bound. The work also yields a by-product: precise asymptotics for coefficients in expansions of endpoint-singular functions in re-scaled Gegenbauer polynomials, generalizing prior results. This advances understanding of boundary effects and persistence on curved, compact index sets and relates spherical results to the Euclidean Molchan framework.

Abstract

We compute the rate of decay of the persistence probabilities of spherical fractional Brownian motion, which was defined by Lévy (1965) and Istas (2005). The rate resembles the Euclidean case treated in Molchan (1999). As a by-product we consider the coefficients of series representations of functions with algebraic endpoint singularities in terms of re-scaled Gegenbauer polynomials, which partly generalises Sidi (2009).

Figures (1)

• Figure 1: Comparison Triangles

Theorems & Definitions (54)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Proposition 2.5
• proof
• Definition 2.6
• Lemma 2.7