EQUILIBRIUM FLUCTUATION OF THE ATLAS MODEL

• 成果类型: Article
• 署名作者: Dembo, Amir; Tsai, Li-Cheng
• 署名单位: Stanford University; Stanford University; Columbia University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1171
• 发表日期: 2017
• 页码: 4529-4560
• 关键词: symmetric simple exclusion; brownian particle-systems; driven tracer particle; tagged particle; coefficients; DIFFUSIONS; collisions; EQUATIONS; ranks

摘要

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 -> infinity to the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ASHE) on 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.