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Wiener soccer and its generalization. (English) Zbl 0890.60075
Summary: 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 by S. Kozlov, J. Pitman and M. Yor, Probab. Theory Appl. 37, No. 3, 550-553 (1992); translation from Teor. Veroyatn. Primen. 37, No. 3, 562-564 (1992; Zbl 0773.60079)]. 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.
MSC:
60J65 Brownian motion
60J35 Transition functions, generators and resolvents
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