Survival probability of mutually killing Brownian motions and the O'Connell process

TL;DR

This paper explores the O'Connell process, a diffusive particle system linked to directed polymers, showing it can be viewed as mutually killing Brownian motions conditioned on survival, and connects it to noncolliding Brownian motion as a limit.

Contribution

It demonstrates that the O'Connell process without drift can be realized as mutually killing Brownian motions conditioned to survive forever, linking it to the Dyson model in a limiting case.

Findings

O'Connell process is a system of mutually killing Brownian motions conditioned on survival.

The process reduces to noncolliding Brownian motion (Dyson model) as the interaction length goes to zero.

Provides a physical interpretation of the O'Connell process in terms of survival-conditioned Brownian motions.

Abstract

Recently O'Connell introduced an interacting diffusive particle system in order to study a directed polymer model in 1+1 dimensions. The infinitesimal generator of the process is a harmonic transform of the quantum Toda-lattice Hamiltonian by the Whittaker function. As a physical interpretation of this construction, we show that the O'Connell process without drift is realized as a system of mutually killing Brownian motions conditioned that all particles survive forever. When the characteristic length of interaction killing other particles goes to zero, the process is reduced to the noncolliding Brownian motion (the Dyson model).