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 $p \geq 4$.
Publié le : 1988-01-14
Classification:
Brownian motion,
Grand Tour,
hitting time,
sphere covering,
rapid mixing,
60D05,
60G17,
60E15,
58G32
@article{1176991894,
author = {Matthews, Peter},
title = {Covering Problems for Brownian Motion on Spheres},
journal = {Ann. Probab.},
volume = {16},
number = {4},
year = {1988},
pages = { 189-199},
language = {en},
url = {http://dml.mathdoc.fr/item/1176991894}
}
Matthews, Peter. Covering Problems for Brownian Motion on Spheres. Ann. Probab., Tome 16 (1988) no. 4, pp. 189-199. http://gdmltest.u-ga.fr/item/1176991894/