Equilibrium Fluctuation of the Atlas Model

Abstract: We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite ($ \mathbb{Z}+ $-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on $ \mathbb{R}+ $. In this context, we show that the joint law of ranked particles, after being centered and scaled by $t^{-1/4}$, converges as $t \to \infty$ to the Gaussian field corresponding to the solution of the additive stochastic heat equation on $\mathbb{R}_+$ with Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a $ \frac{1}{4} $-fractional Brownian motion. In particular, we prove a conjecture of Pal and Pitman (2008) about the asymptotic Gaussian fluctuation of the ranked particles.