Hostname: page-component-669899f699-8p65j Total loading time: 0 Render date: 2025-04-26T09:48:22.684Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixing rates for Brownian motion in a convex polyhedron

Published online by Cambridge University Press:  14 July 2016

Peter Matthews
Peter Matthews*
Affiliation:
University of Maryland Baltimore County
*
∗Postal address: Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD 21228, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

For Brownian motion on a convex polyhedral subset of a sphere or torus, the rate of convergence in distribution to uniformity is studied. The main result is a method to take a Markov coupling on the full sphere or torus and create a faster coupling on the convex polyhedral subset. Upper bounds on variation distance are computed, and applications are discussed.

Keywords

COUPLINGREFLECTING BROWNIAN MOTIONVARIATION DISTANCEVOLUME OF A POLYHEDRON
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 2 , June 1990 , pp. 259 - 268
Copyright
Copyright © Applied Probability Trust 1990 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Research supported by the National Security Agency under Grant Number MDA 904-88-H-2014.

References

Barlow, R. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehard and Winston, New York.Google Scholar
Chen, M. F. and Li, C. F. (1989) Coupling methods for multidimensional diffusion processes. Ann. Prob. 17, 151–177.Google Scholar
Diaconis, P. (1988) Group Representations in Probability and Statistics. IMS, Hayward, CA.CrossRefGoogle Scholar
Dyer, M. and Frieze, A. (1988) On the complexity of computing the volume of a polyhedron. SIAM J. Comput. 17, 967–974.Google Scholar
Dyer, M., Frieze, A. and Kannan, R. (1988) A random polynomial time algorithm for approximating the volume of convex bodies. Preprint.Google Scholar
Harrison, J. and Williams, R. (1987) Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Prob. 15, 115–137.Google Scholar
Hsu, P. (1986) Brownian exit distribution of a ball. Seminar on Stochastic Processes, 1985, ed. Cinlar, E., Chung, K. L. and Getoor, R. K., Birkhaüser, Boston, 108–116.CrossRefGoogle Scholar
Ito, K. and Mckean, H. (1965) Diffusion Processes and their Sample Paths. Academic Press, New York.Google Scholar
Karlin, S. and Taylor, H. (1981) A Second Course in Stochastic Processes, Academic Press, New York.Google Scholar
Kendall, W. S. (1986) Nonnegative Ricci curvature and the Brownian coupling property. Stochastics 19, 111–129.Google Scholar
Lindvall, T. and Rogers, L. (1986) Coupling of multidimensional diffusions by reflection. Ann. Prob. 14, 860–872.Google Scholar
Matthews, P. (1989) Generating a random linear extension of a partial order. UMBC Technical Report.Google Scholar
Pitman, J. and Yor, M. (1986) Asymptotic laws of planar Brownian motion. Ann. Prob. 14, 733–779.Google Scholar
Sinclair, A. and Jerrum, M. (1989) Approximate counting, uniform generation, and rapidly mixing Markov chains. Information and Computing 82, 93–133.CrossRefGoogle Scholar