Consider the infinite-particle Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite system of otherwise independent Brownian particles, starting at a Poisson point process on the positive half-line. We show that, at the equilibrium density of 2, the limiting fluctuations of ranked particles are characterized by the additive stochastic heat equation with the Neumann boundary condition at zero. In particular, this characterization resolves a conjecture of Pal and Pitman (2008) about the asymptotic Gaussian fluctuations of ranked particles.

 

This is joint work with Amir Dembo.