Brownian Motions with One-Sided Collisions: The Stationary Case

TL;DR

This paper studies an infinite system of Brownian motions with one-sided reflections, demonstrating convergence to the Airy process in the stationary case and introducing a new universal cross-over process.

Contribution

It provides the first proof of convergence to the Airy process for this system with Poisson initial configurations and introduces a novel method that differs from previous approaches.

Findings

Convergence to the Airy process established for stationary initial conditions.

A new representation of finite-dimensional distributions of the Airy process.

Introduction of a universal cross-over process emerging from the method.

Abstract

We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process space-time stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finite-dimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the limit $t\to\infty$. This leads to a new universal cross-over process.