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Collision times and exit times from cones: A duality. (English) Zbl 0765.60083

The authors consider the first collision time \(\tau\) of three independent one-dimensional Wiener processes and show that \(\tau\) corresponds to the first exit time for Brownian motion in a cone in \(\mathbb{R}^ 2\). This duality is used to obtain the distribution of \(\tau\) from results of Spitzer and DeBlassie. In the case of equal infinitesimal variance, the distribution of \(\tau\) is determined in closed form by a more elementary approach. The duality is then used for the first exit time of Brownian motion in a cone of angle \(\pi/3\). Extensions to Markov processes and to \(n\) independent Wiener processes are also discussed.
Reviewer: W.Weil (Karlsruhe)

MSC:

60J65 Brownian motion
60G40 Stopping times; optimal stopping problems; gambling theory
60D05 Geometric probability and stochastic geometry
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