Covering problems for Brownian motion on spheres (Q1099468)

Consider Brownian motion on the unit sphere \(\Sigma_ p\) and \({\mathbb{R}}^ p\), \(p\geq 3\), and let \(C_ 1(\epsilon,p)\) be the first time a Brownian path on \(\Sigma_ p\) has come within distance \(\epsilon\) of all points of \(\Sigma_ p\). Similarly, \(C_ 2(\epsilon,p)\) is defined if points on \(\Sigma_ p\) are replaced by antipodal pairs. For the mean times \(EC_ 1(\epsilon,p)\) and \(EC_ 2(\epsilon,p)\), asymptotic bounds are given (as \(\epsilon\to 0)\) which are tight for \(p\geq 4\). The bounds involve the expected hitting times of caps on \(\Sigma_ p\) and the number of caps needed to cover \(\Sigma_ p\). Their proof is based on a related more general result for strong Markov processes. The problem is motivated by a Grand Tour, a technique in multivariate data analysis.