The Annals of Probability

Covering Problems for Brownian Motion on Spheres

Peter Matthews

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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

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

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G17: Sample path properties 60E15: Inequalities; stochastic orderings 58G32

Brownian motion Grand Tour hitting time sphere covering rapid mixing


Matthews, Peter. Covering Problems for Brownian Motion on Spheres. Ann. Probab. 16 (1988), no. 1, 189--199. doi:10.1214/aop/1176991894.

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