Electronic Communications in Probability

Wiener Soccer and Its Generalization

Yuliy Baryshnikov

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The trajectory of the ball in a soccer game is modelled by the Brownian motion on a cylinder, subject to elastic reflections at the boundary points (as proposed in [KPY]). The score is then the number of windings of the trajectory around the cylinder. We consider a generalization of this model to higher genus, prove asymptotic normality of the score and derive the covariance matrix. Further, we investigate the inverse problem: to what extent the underlying geometry can be reconstructed from the asymptotic score.

Article information

Electron. Commun. Probab., Volume 3 (1998), paper no. 1, 1-11.

Accepted: 17 November 1997
First available in Project Euclid: 2 March 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60J38

Wiener Process Brownian Motion

This work is licensed under aCreative Commons Attribution 3.0 License.


Baryshnikov, Yuliy. Wiener Soccer and Its Generalization. Electron. Commun. Probab. 3 (1998), paper no. 1, 1--11. doi:10.1214/ECP.v3-987. https://projecteuclid.org/euclid.ecp/1456935907

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  • Y. Colin de Verdiere, Reseaux electriques planaires I, Comm. Math. Helv., 69, 351-374 (1994)
  • Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley (1979).
  • S. Kozlov, J. Pitman, M. Yor, Wiener football, Probab. Theory Appl., 40, 530-533 (1993).
  • T. J. Lyons, H. P. McKean, Winding of the plane brownian motion, Adv. Math., 51, 212-225 (1984).
  • D. Mumford, Tata lectures on theta, II, Birkhauser, Boston, 1984.
  • J. Pitman, M. Yor, Asymptotic laws of planar Brownian motion, Ann. Probab. 14, 733-779 (1986) and 17, 965-1011 (1989).
  • H. Yanagihara, Stochastic determination of moduli of annular regions and tori, Ann. Probab. 14, 1404-1410 (1986).