The Annals of Probability

Covering Problems for Brownian Motion on Spheres

Peter Matthews

Abstract

Bounds are given on the mean time taken by a strong Markov process to visit all of a finite collection of subsets of its state space. These bounds are specialized to Brownian motion on the surface of the unit sphere $\Sigma_p$ in $R^p$. This leads to bounds on the mean time taken by this Brownian motion to come within a distance $\varepsilon$ of every point on the sphere and bounds on the mean time taken to come within $\varepsilon$ of every point or its opposite. The second case is related to the Grand Tour, a technique of multivariate data analysis that involves a search of low-dimensional projections. In both cases the bounds are asymptotically tight as $\varepsilon \rightarrow 0$ on $\Sigma_p$ for $p \geq 4$.

Article information

Source
Ann. Probab., Volume 16, Number 1 (1988), 189-199.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991894

Digital Object Identifier
doi:10.1214/aop/1176991894

Mathematical Reviews number (MathSciNet)
MR920264

Zentralblatt MATH identifier
0638.60014

JSTOR