Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions

TL;DR

This paper establishes a deep connection between the invariant measure of reflected Brownian motions and point-to-line last passage percolation times, revealing new insights into stochastic processes and directed polymers.

Contribution

It proves an equality in law linking invariant measures of reflected Brownian motions with point-to-line last passage percolation times, and introduces a novel interacting Brownian system with a log partition function invariant measure.

Findings

Invariant measure of reflected Brownian motions equals point-to-line last passage percolation times.

Distribution of Dyson Brownian motion's supremum with drift is characterized.

Finite temperature version relates two directed polymer models and generalizes Dufresne's identity.

Abstract

This paper proves an equality in law between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times in a discrete random environment. A consequence describes the distribution of the all-time supremum of Dyson Brownian motion with drift. A finite temperature version relates the point-to-line partition functions of two directed polymers, with an inverse-gamma and a Brownian environment, and generalises Dufresne's identity. Our proof introduces an interacting system of Brownian motions with an invariant measure given by a field of point-to-line log partition functions for the log-gamma polymer.