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May 2018 Optimal surviving strategy for drifted Brownian motions with absorption
Wenpin Tang, Li-Cheng Tsai
Ann. Probab. 46(3): 1597-1650 (May 2018). DOI: 10.1214/17-AOP1211

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.

Citation

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Wenpin Tang. Li-Cheng Tsai. "Optimal surviving strategy for drifted Brownian motions with absorption." Ann. Probab. 46 (3) 1597 - 1650, May 2018. https://doi.org/10.1214/17-AOP1211

Information

Received: 1 December 2015; Revised: 1 July 2017; Published: May 2018
First available in Project Euclid: 12 April 2018

zbMATH: 06894782
MathSciNet: MR3785596
Digital Object Identifier: 10.1214/17-AOP1211

Subjects:
Primary: 60K35
Secondary: 35Q70, 82C22

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 3 • May 2018
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