Competing Brownian Particles

Abstract

Consider a finite system of N Brownian particles on the real line. Rank them from bottom to top: the (currently) lowest particle has rank 1, the second lowest has rank 2, etc., up to the top particle, which has rank N. The particle which has (currently) rank k moves as a Brownian motion with drift coefficient g_k and diffusion coefficient sigma_k^2. When two or more particles collide, they might exchange ranks; in this case, they exchange drift and diffusion coefficients. This model is called a system of competing Brownian particles. It was introduced in Banner, Fernholz, Karatzas (2005) for the purpose of financial modeling. Since then, it attracted a considerable amount of attention. We can also consider infinite systems of competing Brownian particles (with the lowest particle but no highest particle, that is, with ranks ranging from 1 to infinity). For both finite and infinite systems, the gap process is formed by the spacings (gaps) between adjacent particles. It is (N-1)-dimensional for a finite system with N particles and infinite-dimensional for an infinite system. We say that a triple collision has occurred if three or more particles occupy the same position at the same time. In this thesis, we prove several new results about these systems. In particular, we establish convergence results for the gap process of infinite systems, building on the work of Pal, Pitman (2008); and we find a necessary and sufficient condition for a.s. absence of triple collisions, continuing the research from Ichiba, Karatzas, Shkolnikov (2013).