Triple and simultaneous collisions of competing Brownian particles

Abstract

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.