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.

Consider a system of infinitely many Brownian particles on the real line. At any moment, these particles can be ranked from the bottom upward. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. The gaps between consecutive particles form the (infinite-dimensional) gap process. We find a stationary distribution for the gap process. We also show that if the initial value of the gap process is stochastically larger than this stationary distribution, this process converges back to this distribution as time goes to infinity. This continues the work by Pal and Pitman (Ann. Appl. Probab. 18 (2008) 2179–2207). Also, this includes infinite systems with asymmetric collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov (Ann. Inst. H. Poincare 52 (2016) 323–354). Nous considérons un système infini de particules browniennes sur la droite réelle. À tout moment ces particules peuvent être ordonnées de façon croissante. Chaque particule se déplace suivant un mouvement brownien dont les coefficients de dérive et de diffusion dépendent du rang de la particule. Les distances entre les particules successives forment le processus (infini dimensionnel) des écarts. Nous trouvons une mesure stationnaire du processus des écarts. Nous montrons aussi que si la distribution initiale du processus des écarts domine stochastiquement la distribution stationnaire, le processus converge vers cette distribution en grand temps. Ce travail poursuit donc l’étude de Pal et Pitman (Ann. Appl. Probab. 18 (2008) 2179–2207). Il inclut aussi le cas des systèmes infinis avec collisions asymétriques, similaire au cas fini de Karatzas et Shkolnikov (Ann. Inst. H. Poincare 52 (2016) 323–354).

Consider the infinite Atlas model: a semi-infinite collection of particles driven by independent standard Brownian motions with zero drifts, except for the bottom-ranked particle which receives unit drift. We derive a continuum one-parameter family of product-of-exponentials stationary gap distributions, with exponentially growing density at infinity. This result shows that there are infinitely many stationary gap distributions for the Atlas model, and hence resolves a conjecture of Pal and Pitman (2008) [PP08] in the negative. This result is further generalized for infinite systems of competing Brownian particles with generic rank-based drifts.

We study models of regulatory breakup, in the spirit of Strong and Fouque [Ann. Finance 7 (2011) 349–374] but with a fluctuating number of companies. An important class of market models is based on systems of competing Brownian particles: each company has a capitalization whose logarithm behaves as a Brownian motion with drift and diffusion coefficients depending on its current rank. We study such models with a fluctuating number of companies: If at some moment the share of the total market capitalization of a company reaches a fixed level, then the company is split into two parts of random size. Companies are also allowed to merge, when an exponential clock rings. We find conditions under which this system is nonexplosive (i.e., the number of companies remains finite at all times) and diverse, yet does not admit arbitrage opportunities.