The Annals of Probability

Equilibrium fluctuation of the Atlas model

Amir Dembo and Li-Cheng Tsai

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Abstract

We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite ($\mathbb{Z}_{+}$-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on $\mathbb{R}_{+}$. In this context, we show that the joint law of ranked particles, after being centered and scaled by $t^{-\frac{1}{4}}$, converges as $t\to\infty$ to the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ASHE) on $\mathbb{R}_{+}$ with the Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a fractional Brownian Motion (fBM). In particular, we prove a conjecture of Pal and Pitman [Ann. Appl. Probab. 18 (2008) 2179–2207] about the asymptotic Gaussian fluctuation of the ranked particles.

Article information

Source
Ann. Probab., Volume 45, Number 6B (2017), 4529-4560.

Dates
Received: March 2015
Revised: April 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1513069266

Digital Object Identifier
doi:10.1214/16-AOP1171

Mathematical Reviews number (MathSciNet)
MR3737917

Zentralblatt MATH identifier
06838126

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60H15: Stochastic partial differential equations [See also 35R60] 82C22: Interacting particle systems [See also 60K35]

Keywords
Equilibrium fluctuation fractional Brownian motion interacting particles reflected Brownian motion stochastic heat equation

Citation

Dembo, Amir; Tsai, Li-Cheng. Equilibrium fluctuation of the Atlas model. Ann. Probab. 45 (2017), no. 6B, 4529--4560. doi:10.1214/16-AOP1171. https://projecteuclid.org/euclid.aop/1513069266


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