Domains of attraction of invariant distributions of the infinite Atlas model

Abstract

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model does not have a unique stationary distribution and in fact for every $\mathit{a}\ge 0$, ${\mathit{\pi }}_{\mathit{a}}:={⨂}_{\mathit{i}=1}^{\infty }Exp(2+\mathit{i}\mathit{a})$ is a stationary measure for the gap process. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure ${\mathit{\pi }}_{\mathit{a}}$ if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to ${\mathit{\pi }}_{\mathit{a}}$ weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of ${\mathit{\pi }}_{\mathit{a}}$ for each $\mathit{a}\ge 0$. The cases $\mathit{a}=0$ and $\mathit{a}>0$ are qualitatively different as is seen from the analysis and the sufficient conditions that we provide. Proofs are based on the analysis of synchronous couplings, namely, couplings of the ranked particle systems started from different initial configurations, but driven using the same set of Brownian motions.