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This paper proves an equality in law between the invariant measure of areflected system of Brownian motions and a vector of point-to-line last passagepercolation times in a discrete random environment. A consequence describes thedistribution of the all-time supremum of Dyson Brownian motion with drift. Afinite temperature version relates the point-to-line partition functions of twodirected polymers, with an inverse-gamma and a Brownian environment, andgeneralises Dufresne's identity. Our proof introduces an interacting system ofBrownian motions with an invariant measure given by a field of point-to-linelog partition functions for the log-gamma polymer.

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