O'Connell's process as a vicious Brownian motion

TL;DR

This paper explores a generalized noncolliding Brownian motion model, called the O'Connell process, which incorporates long-range killing terms and is connected to quantum Toda lattice eigenfunctions, extending the classical vicious Brownian motion framework.

Contribution

It introduces a new generalized process of Brownian motions with killing terms, linking it to the O'Connell process and quantum Toda lattice eigenfunctions.

Findings

Constructed the O'Connell process as a conditional killing Brownian motion.

Connected the process to eigenfunctions of the quantum Toda lattice.

Extended the classical vicious Brownian motion model.

Abstract

Vicious Brownian motion is a diffusion scaling limit of Fisher's vicious walk model, which is a system of Brownian particles in one dimension such that if two of them meet they kill each other. We consider the vicious Brownian motion conditioned never to collide with each other, and call it the noncolliding Brownian motion. This conditional diffusion process is equivalent to the eigenvalue process of a Hermitian-matrix-valued Brownian motion studied by Dyson. Recently O'Connell introduced a generalization of the noncolliding Brownian motion by using the eigenfunctions (the Whittaker functions) of the quantum Toda lattice in order to analyze a directed polymer model in 1+1 dimensions. We consider a system of one-dimensional Brownian motions with a long-ranged killing term as a generalization of the vicious Brownian motion and construct the O'Connell process as a conditional process of the killing Brownian motions to survive forever.