Equilibrium Fluctuations for a Discrete Atlas Model

TL;DR

This paper studies a discrete Atlas model with a particle source, demonstrating that its equilibrium fluctuations follow a stochastic heat equation with Neumann boundary conditions, and that the particle current converges to a fractional Brownian motion with H=1/4.

Contribution

It introduces a discrete Atlas model with a source at the origin and proves its equilibrium fluctuations are governed by a stochastic heat equation with boundary conditions.

Findings

Equilibrium fluctuations follow a stochastic heat equation with Neumann boundary conditions.

Particle current converges to a fractional Brownian motion with H=1/4.

Provides a rigorous connection between the discrete model and continuous stochastic PDEs.

Abstract

We consider a discrete version of the Atlas model, which corresponds to a sequence of zero-range processes on a semi-infinite line, with a source at the origin and a diverging density of particles. We show that the equilibrium fluctuations of this model are governed by a stochastic heat equation with Neumann boundary conditions. As a consequence, we show that the current of particles at the origin converges to a fractional Brownian motion of Hurst exponent H=1/4.