Stationary distributions of the Atlas model

Abstract

In this article we study the Atlas model, which constitutes of Brownian particles on $ \mathbb{R} $, independent except that the Atlas (i.e., lowest ranked) particle $ X_{(1)}(t) $ receive 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 $ a e^{a\xi} d\xi $. We further show that $ \nu_a $ indeed has the product-of-exponential gap distribution $ \pi_a $ derived in Sarantsev and Tsai (2016). 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 $.