The infinite Atlas process: Convergence to equilibrium

TL;DR

This paper studies the semi-infinite Atlas process, a system of Brownian particles with a unique stationary measure, and proves convergence to this equilibrium from various initial configurations using new convergence rate estimates.

Contribution

It establishes the attractiveness of the translation invariant stationary measure for a broad class of initial conditions and introduces a novel convergence rate estimate for large finite systems.

Findings

The translation invariant measure is globally attractive for certain initial configurations.

A new estimate on convergence rate to equilibrium is developed.

Convergence results hold for large finite particle systems.

Abstract

The semi-infinite Atlas process is a one-dimensional system of Brownian particles, where only the leftmost particle gets a unit drift to the right. Its particle spacing process has infinitely many stationary measures, with one distinguished translation invariant reversible measure. We show that the latter is attractive for a large class of initial configurations of slowly growing (or bounded) particle densities. Key to our proof is a new estimate on the rate of convergence to equilibrium for the particle spacing in a triangular array of finite, large size systems.