## The Annals of Probability

### Optimal surviving strategy for drifted Brownian motions with absorption

#### Abstract

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 $\mathbb{R}_{+}$, 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 $K\to\infty$) 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.

#### Article information

Source
Ann. Probab., Volume 46, Number 3 (2018), 1597-1650.

Dates
Revised: July 2017
First available in Project Euclid: 12 April 2018

https://projecteuclid.org/euclid.aop/1523520025

Digital Object Identifier
doi:10.1214/17-AOP1211

Mathematical Reviews number (MathSciNet)
MR3785596

Zentralblatt MATH identifier
06894782

#### Citation

Tang, Wenpin; Tsai, Li-Cheng. Optimal surviving strategy for drifted Brownian motions with absorption. Ann. Probab. 46 (2018), no. 3, 1597--1650. doi:10.1214/17-AOP1211. https://projecteuclid.org/euclid.aop/1523520025

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