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 (Z + -indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on 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 the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ashe) on R + with Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a 1 4 -Fractional Brownian Motion ( 1 4 -fbm). In particular, we prove a conjecture of Pal and Pitman [17] about the asymptotic Gaussian fluctuation of the ranked particles.Date: November 9, 2018. 2010 Mathematics Subject Classification. Primary 60K35; Secondary 60H15, 82C22.

Equilibrium fluctuations for a discrete Atlas model

“…is a continuous martingale of quadratic variation 2t ∞ 0 ( f (u)) 2 du. Uniqueness of martingale solutions to the stochastic heat equation with Neumann and Robin boundary condition can be found, for instance, in [3] and [2, Proposition 2.7], respectively. Theorem 1.…”

Infinite systems of competing Brownian particles

“…. = 1, this is called the infinite Atlas model, which was studied in [27,8]. The term Atlas stands for the bottom particle, which moves as a Brownian motion with drift 1 (as long as it does not collide with other particles) and "supports other particles on its shoulders".…”

The infinite Atlas process: Convergence to equilibrium

“…As for the behavior of the leading particle of atlas ∞ , [5] verifies [15,Conj. 3], that starting with spacing at the translation invariant equilibrium law µ…”