Fluctuations of the Atlas model from inhomogeneous stationary profiles

TL;DR

This paper analyzes the equilibrium fluctuations of the inhomogeneous Atlas model, revealing a stochastic PDE limit with fractional Brownian motion regularity and distinct edge behavior, including divergence of variance near the lower boundary.

Contribution

It introduces a new SPDE framework for inhomogeneous Atlas model fluctuations and characterizes the non-Gaussian limit of the lowest particle's position.

Findings

Fluctuations converge to a SPDE driven by space-time noise.

The fluctuation process has H"older regularity similar to fractional Brownian motion with Hurst 1/4.

Variance diverges at the lower edge, leading to a non-Gaussian limit for the lowest particle.

Abstract

The infinite Atlas model describes the evolution of a countable collection of Brownian particles on the real line, where the lowest particle is given a drift of $\gamma \in [0,\infty)$. We study equilibrium fluctuations for the Atlas model when the system of particles starts from an inhomogeneous stationary profile with exponentially growing density. We show that the appropriately centered and scaled occupation measure of the particle positions, with suitable translations, viewed as a space-time random field, converges to a limit given by a certain stochastic partial differential equation (SPDE). The initial condition for this equation is given by a Brownian motion, the equation is driven by an additive space-time noise that is white in time and colored in space, and the linear operator governing the evolution is the infinitesimal generator of a geometric Brownian motion. We use this SPDE to also characterize the fluctuations of the ranked particle positions with a suitable centering and scaling. Our results describe the behavior of the particles in the bulk and one finds that the Gaussian process describing the asymptotic fluctuations has the same H\"{o}lder regularity as a fractional Brownian motion with Hurst parameter $1/4$. One finds that, unlike the setting of a homogeneous profile (Dembo and Tsai (2017)), the behavior on the lower edge of the particle system is very different from the bulk behavior and in fact the variance of the Gaussian limit diverges to $\infty$ as one approaches the lower edge. Indeed, our results show that, with the gaps between particles given by one of the inhomogeneous stationary distributions, the lowest particle, started from $0$, with a linear in time translation, converges in distribution to an explicit non-Gaussian limit as $t\to \infty$.