Domains of attraction of invariant distributions of the infinite Atlas model
Domains of attraction of invariant distributions of the infinite Atlas model
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DOI：
10.1214/22-aop1570
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
Sayantan Banerjee;A. Budhiraja
中科院分区：
文献类型：
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作者：
Sayantan Banerjee;A. Budhiraja
无限阿特拉斯模型描述了一个由相互竞争的布朗粒子组成的可数系统，其中最低的粒子向上漂移一个单位，其余的则演变为标准布朗运动。无限大Atlas模型中粒子间隙的随机过程并不具有唯一的平稳分布，事实上，πa:=⊗∞i=1Exp（2 + ia）是间隙过程的平稳测度。当用该分布初始化间隙随机过程的时间平均律在大时限内弱收敛于π时，我们说间隙的初始分布是在平稳测度π的弱引力域中。我们给出了Atlas粒子初始间隙分布在π π引力弱域内的一般充分条件。从分析和我们提供的充分条件可以看出，a= 0和a >在性质上是不同的。证明是基于同步耦合的分析，即从不同初始构型开始的排列粒子系统的耦合，但使用同一组布朗运动驱动。
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, πa := ⊗∞ i=1Exp(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 πa if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to πa 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 πa 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.