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Brownian couplings, convexity, and shy-ness

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Kendall, Wilfrid S.. (2009) Brownian couplings, convexity, and shy-ness. Electronic communications in probability, Vol.14 (No.7). pp. 66-80. ISSN 1083-589X

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Official URL: http://dx.doi.org/10.1214/ECP.v14-1417

Abstract

Benjamini, Burdzy, and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying at least a given positive distance away from each other for all time. Among other results, they showed that no shy couplings could exist for reflected Brownian motions in C-2 bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al. (2007) result (a) to all bounded convex domains (whether planar and smooth or not) whose boundaries contain no line segments, (b) to all bounded convex planar domains regardless of further conditions on the boundary.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Markov processes
Journal or Publication Title:	Electronic communications in probability
Publisher:	University of Washington. Dept. of Mathematics
ISSN:	1083-589X
Date:	12 February 2009
Volume:	Vol.14
Number:	No.7
Number of Pages:	15
Page Range:	pp. 66-80
Identification Number:	10.1214/ECP.v14-1417
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
References:	R. Atar and K. Burdzy. On nodal lines of Neumann eigenfunctions. Electronic Communications in Probability, 7:129–139, 2002. MR1937899 R. Atar and K. Burdzy. On Neumann eigenfunctions in lip domains. Journal of the American Mathematical Society, 17(2):243–265 (electronic), 2004. MR2051611 R. F. Bass and E. P. Hsu. The semimartingale structure of reflecting Brownian motion. Proc. Amer. Math. Soc., 108(4):1007–1010, 1990. MR1007487 I. Benjamini, K. Burdzy, and Z.-Q. Chen. Shy couplings. Probability Theory and Related Fields, 137 (3-4):345–377, March 2007. MR2278461 K. Burdzy and W. S. Kendall. Efficient Markovian couplings: examples and counterexamples. The Annals of Applied Probability, 10(2):362–409, May 2000. MR1768241 M. Émery. Stochastic calculus in manifolds. Universitext. Springer-Verlag, Berlin, 1989. With an appendix by P.-A. Meyer. MR1030543 M. Émery. On certain almost Brownian filtrations. Ann. Inst. H. Poincaré Probab. Statist., 41(3): 285–305, 2005. K. Itô. Stochastic differentials. Applied Mathematics and Optimization, 1:374–381, 1975. MR0388538 J. Jost, W. S. Kendall, U. Mosco, M. Röckner, and K. T. Sturm. New Directions in Dirichlet Forms, volume 8 of Studies in Advanced Mathematics. American Mathematical Society, Providence, RI, 1998. MR1652277 W. S. Kendall. Probability, convexity, and harmonic maps with small image I: Uniqueness and fine existence. Proceedings of the London Mathematical Society (Third Series), 61:371–406, 1990. MR1063050 W. S. Kendall. Convexity and the hemisphere. The Journal of the London Mathematical Society (Second Series), 43:567–576, 1991. MR1113394 W. S. Kendall. Coupling all the Lévy stochastic areas of multidimensional Brownian motion. The Annals of Probability, 35(3):935–953, May 2007. MR2319712 M. N. Pascu and M. E. Gageonea. On a conjecture of Laugesen and Morpurgo, 2008. URL http://arxiv.org/abs/0807.4726. D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer- Verlag, Berlin, 1991. H. Tanaka. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J., 9(1):163–177, 1979. MR0529332
URI:	http://wrap.warwick.ac.uk/id/eprint/28384