Multiple Collisions in Systems of Competing Brownian Particles

TL;DR

This paper investigates conditions under which a finite system of competing Brownian particles avoids total and partial collisions, extending previous research by providing new sufficient conditions for collision absence.

Contribution

It offers new sufficient conditions ensuring no total or partial collisions occur in systems of competing Brownian particles, advancing understanding of their collision behavior.

Findings

Identifies conditions preventing total collisions.

Establishes criteria avoiding collisions among the lowest-ranked particles.

Builds on prior work to generalize collision avoidance results.

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).