Electronic Communications in Probability

Brownian couplings, convexity, and shy-ness

Wilfrid Kendall

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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.

Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 7, 66-80.

Dates
Accepted: 12 February 2009
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465234716

Digital Object Identifier
doi:10.1214/ECP.v14-1417

Mathematical Reviews number (MathSciNet)
MR2481667

Subjects
Primary: 60J65: Brownian motion [See also 58J65]

Keywords
Brownian motion coupling

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Kendall, Wilfrid. Brownian couplings, convexity, and shy-ness. Electron. Commun. Probab. 14 (2009), paper no. 7, 66--80. doi:10.1214/ECP.v14-1417. https://projecteuclid.org/euclid.ecp/1465234716


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References

  • Atar, Rami; Burdzy, Krzysztof. On nodal lines of Neumann eigenfunctions. Electron. Comm. Probab. 7 (2002), 129–139 (electronic).
  • Atar, Rami; Burdzy, Krzysztof. On Neumann eigenfunctions in lip domains. J. Amer. Math. Soc. 17 (2004), no. 2, 243–265 (electronic).
  • Bass, Richard; Hsu, Elton. The semimartingale structure of reflecting Brownian motion. Proc. Amer. Math. Soc. 108 (1990), no. 4, 2, 1007–1010. 1990.
  • Benjamini, Itai; Burdzy, Krzysztof; Chen, Zhen-Qing. Shy couplings. Probab. Theory Related Fields 137 (2007), no. 3-4, 345–377.
  • Burdzy, Krzysztof; Kendall, Wilfrid S. Efficient Markovian couplings: examples and counterexamples. Ann. Appl. Probab. 10 (2000), no. 2, 362–409.
  • Émery, Michel. Stochastic calculus in manifolds. With an appendix by P.-A. Meyer. Universitext. Springer-Verlag, Berlin, 1989. x+151 pp. ISBN: 3-540-51664-6
  • Émery, Michel. On certain almost Brownian filtrations. Ann. Inst. H. Poincaré Probab. Statist. 40 (2005), no. 3, 285–305.
  • Itô, Kiyoshi Stochastic differentials. Appl. Math. Optim. 1 (1974/75), no. 4, 374–381.
  • Jost, Jürgen; Kendall, Wilfrid; Mosco, Umberto; Röckner, Michael; Sturm, Karl-Theodor. New directions in Dirichlet forms. AMS/IP Studies in Advanced Mathematics, 8. American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 1998. xiv+277 pp. ISBN: 0-8218-1061-8
  • Kendall, Wilfrid S. Probability, convexity, and harmonic maps with small image. I. Uniqueness and fine existence. Proc. London Math. Soc. (3) 61 (1990), no. 2, 371–406.
  • Kendall, Wilfrid S. Convexity and the hemisphere. J. London Math. Soc. (2) 43 (1991), no. 3, 567–576.
  • Kendall, Wilfrid S. Coupling all the Lévy stochastic areas of multidimensional Brownian motion. Ann. Probab. 35 (2007), no. 3, 935–953.
  • Pascu, Mihai N.; Gageonea, Maria E. On a conjecture of Laugesen and Morpurgo. http://arxiv.org/abs/0807.4726
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Springer-Verlag, Berlin, 1991. x+553 pp. ISBN: 3-540-52167-4
  • Tanaka, Hiroshi. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979), no. 1, 163–177.