Optimal surviving strategy for drifted Brownian motions with absorption

We study the .Up the River. problem formulated by Aldous (2002), where a unit drift is distributed among a finite collection of Brownian particles on R+, which are annihilated once they reach the origin. Starting K particles at x=1, we prove Aldous. conjecture [Aldous (2002)] that the .push-the-laggard. strategy of distributing the drift asymptotically (as K..) maximizes the total number of surviving particles, with approximately 4..K... surviving particles. We further establish the hydrodynamic limit of the particle density, in terms of a two-phase partial differential equation (PDE) with a moving boundary, by utilizing certain integral identities and coupling techniques.