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Optimal surviving strategy for drifted Brownian motions with absorption. (English) Zbl 1429.60075
Summary: We study the “up the river” problem formulated by D. Aldous [“Up to the river”, Preprint], where a unit drift is distributed among a finite collection of Brownian particles on \(\mathbb{R}_{+}\), which are annihilated once they reach the origin. Starting \(K\) particles at \(x=1\), we prove Aldous’ conjecture [loc. cit.] that the “push-the-laggard” strategy of distributing the drift asymptotically (as \(K\rightarrow\infty\)) maximizes the total number of surviving particles, with approximately \(\frac{4}{\sqrt{\pi}}\sqrt{K}\) 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.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
35Q70 PDEs in connection with mechanics of particles and systems of particles
82C22 Interacting particle systems in time-dependent statistical mechanics
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