Two-Sided Infinite Systems of Competing Brownian Particles

Abstract

Two-sided infinite systems of Brownian particles with rank-dependent dynamics, indexed by all integers, exhibit different properties from their one-sided infinite counterparts, indexed by positive integers, and from finite systems. Consider the gap process, which is formed by spacings between adjacent particles. In stark contrast with finite and one-sided infinite systems, two-sided infinite systems can have one-or two-parameter family of stationary gap distributions, or the gap process weakly converging to zero as time goes to infinity.

Stationary gap distributions for infinite systems of competing Brownian particles

“…Infinite systems arise as natural models of large systems. Specifically, infinite systems of competing Brownian particles were first introduced in [PP08] for a special case of the infinite Atlas model, and later in [Shk11,IKS13] for the general case, as well as in [Sar15c] for two-sided systems X = (X n ) n∈Z . Existence and uniqueness were established in [Shk11, IKS13, Sar16a].…”

Long time behaviour and mean-field limit of Atlas models

“…Generalisations. Possible generalisations of the process defined by (2), on which we shall not elaborate, include rank-based models driven by Lévy processes [56,52], and second-order models, also called hybrid Atlas models, where the drift depends on both the rank and the index of a particle [28,15]. While this article focuses on systems with a finite but possibly large number of particles, countably infinite systems with rank-based evolution were also considered [47,9,53]: the order statistics of such infinite systems can be seen as a generalisation of Harris' Brownian motion [20], and precise estimates on the fluctuations of the bottom particle were recently obtained [11,23].…”

The infinite Atlas process: Convergence to equilibrium

“…For concreteness we focused on the atlas ∞ process, but a similar proof applies for systems of competing Brownian particles where σ 2 j ≡ 1, γ 1 > 0 and j → γ j is non-increasing and eventually zero. We further expect this to extend to some of the two-sided infinite systems considered in [20,Sec. 3], and that such an approach may help in proving the attractivity of µ (∞,2,a) ⋆ in the atlas ∞ system.…”