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 Σₚ in Rᵖ. This leads to bounds on the mean time taken by this Brownian motion to come within a distance ε of every point on the sphere and bounds on the mean time taken to come within ε 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 ε → 0 on Σₚ for p ≥ 4.
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Peter Matthews (1988) studied this question.
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