Stationary gap distributions for infinite systems of competing Brownian particles

Abstract

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.