A construction of infinite Brownian particle systems

TL;DR

This paper constructs infinite Brownian particle systems from specific initial conditions, including models related to TASEP, and introduces infinite-dimensional softly reflected Brownian motions with new intertwining relations.

Contribution

It provides a novel construction method for infinite Brownian particle systems starting from quasi-stationary conditions, extending finite-dimensional models to infinite dimensions.

Findings

Construction of infinite Brownian particle systems from quasi-stationary initial conditions

Introduction of infinite-dimensional softly reflected Brownian motions

Establishment of intertwining relations as infinite-dimensional Burke's Theorem analogues

Abstract

The paper identifies families of quasi-stationary initial conditions for infinite Brownian particle systems within a large class and provides a construction of the particle systems themselves started from such initial conditions. Examples of particle systems falling into our framework include Brownian versions of TASEP-like processes such as the diffusive scaling limit of the q-TASEP process. In this context the spacings between consecutive particles form infinite-dimensional versions of the softly reflected Brownian motions recently introduced in the finite-dimensional setting by O'Connell and Ortmann and are of independent interest. The proof of the main result is based on intertwining relations satisfied by the particle systems involved which can be regarded as infinite-dimensional analogues of the suitably generalized Burke's Theorem.