Stationary Distributions of the Atlas Model

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 $X_{(1)}(0)=0$.