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

10 pages; 0 figure; Updated to match the published version

References in corpus (1)

• Atlas models of equity markets

Cited by in corpus (1)

• The infinite Atlas process: Convergence to equilibrium