Stationary distributions of the Atlas model

Abstract

In this article we study the Atlas model, which consists of Brownian particles on $ \mathbb{R} $, independent except that the Atlas (i.e., lowest ranked) particle $ X_{(1)}(t) $ receives drift $ \gamma dt $, $ \gamma \in \mathbb{R} $. For any fixed shape parameter $ a>2\gamma _- $, we show that, up to a shift $ \frac{a} {2}t $, the entire particle system has an invariant distribution $ \nu _a $, written in terms an explicit Radon-Nikodym derivative with respect to the Poisson point process of density $ ae^{a\xi } d\xi $. We further show that $ \nu _a $ indeed has the product-of-exponential gap distribution $ \pi _a $ derived in [ST17]. As a simple application, we establish a bound on the fluctuation of the Atlas particle $ X_{(1)}(t) $ uniformly in $ t $, with the gaps initiated from $ \pi _a $ and $ X_{(1)}(0)=0 $.