One-dimensional Brownian particle systems with rank dependent drifts

TL;DR

This paper investigates the long-term behavior of rank-dependent drift systems of Brownian particles, characterizing stationary distributions and analyzing a specific infinite system with a unique drift pattern.

Contribution

It provides a characterization of stationary distributions for finite systems and studies the dynamics of an infinite system with a particular drift rule.

Findings

Finite systems have stationary laws given by independent exponential distributions.

In the infinite system with only the minimum particle drifting upward, exponential spacings remain stationary.

Conjectures related to the behavior of such systems are discussed.

Abstract

We study interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. Our main objective has been to study the long range behavior of the spacings between the Brownian motions arranged in increasing order. For finitely many Brownian motions interacting in this manner, we characterize drifts for which the family of laws of the vector of spacings is tight, and show its convergence to a unique stationary joint distribution given by independent exponential distributions with varying means. We also study one particular countably infinite system, where only the minimum Brownian particle gets a constant upward drift, and prove that independent and identically distributed exponential spacings remain stationary under the dynamics of such a process. Some related conjectures in this direction have also been discussed.