Multiple Collisions in Systems of Competing Brownian Particles

Abstract

Consider a finite system of competing Brownian particles on the real line. Each particle moves as a Brownian motion, with drift and diffusion coefficients depending only on its current rank relative to the other particles. We find a sufficient condition for a.s. absence of a total collision (when all particles collide) and of other types of collisions, say of the three lowest-ranked particles. This continues the work of Ichiba, Karatzas, Shkolnikov (2013) and Sarantsev (2015).

34 pages. Keywords: Reflected Brownian motion, competing Brownian particles, asymmetric collisions, named particles, ranked particles, triple collisions, multiple collisions, Skorohod problem, positive orthant, squared Bessel process, stochastic comparison