DOMAINS OF ATTRACTION OF INVARIANT DISTRIBUTIONS OF THE INFINITE ATLAS MODEL

• 成果类型: Article
• 署名作者: Banerjee, Sayan; Budhiraja, Amarjit
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1570
• 发表日期: 2022
• 页码: 1610-1646
• 关键词: brownian particles; DIFFUSIONS; systems; CONVERGENCE; MOTIONS

摘要

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model does not have a unique stationary distribution and in fact for every a >= 0, pi(a) := circle times(infinity)(i=1) Exp(2 + ia) is a stationary measure for the gap process. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure pi(a) if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to pi(a) weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of pi(a) for each a >= 0. The cases a = 0 and a > 0 are qualitatively different as is seen from the analysis and the sufficient conditions that we provide. Proofs are based on the analysis of synchronous couplings, namely, couplings of the ranked particle systems started from different initial configurations, but driven using the same set of Brownian motions.