  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.
