Electronic Journal of Probability

Triple and simultaneous collisions of competing Brownian particles

Andrey Sarantsev

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Consider a finite system of competing Brownian particles. They move as Brownian motions with drift and diffusion coefficients depending on their ranks. This includes the case of asymmetric collisions, when the local time of any collision is distributed unevenly between the two colliding particles, see Karatzas, Pal and Shkolnikov (2012). A triple collision occurs if three particles occupy the same site at a given moment. This is sometimes an undesirable phenomenon. Continuing the work of Ichiba, Karatzas and Shkolnikov (2013), we find necessary and sufficient condition for absense of triple collisions. We also prove sufficient conditions for absense of quadruple collisions, of quintuple collisions, and so on. Our method is reduction to reflected Brownian motion in the positive multidimesnional orthant hitting non-smooth parts of the boundary and, more generally, edges of the boundary of certain low dimension.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 29, 28 pp.

Accepted: 19 March 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J65: Brownian motion [See also 58J65] 60H10: Stochastic ordinary differential equations [See also 34F05]

Reflected Brownian motion competing Brownian particles triple collisions gap process asymmetric collisions

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Sarantsev, Andrey. Triple and simultaneous collisions of competing Brownian particles. Electron. J. Probab. 20 (2015), paper no. 29, 28 pp. doi:10.1214/EJP.v20-3279. https://projecteuclid.org/euclid.ejp/1465067135

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