OPTIMAL SURVIVING STRATEGY FOR DRIFTED BROWNIAN MOTIONS WITH ABSORPTION

• 成果类型: Article
• 署名作者: Tang, Wenpin; Tsai, Li-Cheng
• 署名单位: University of California System; University of California Berkeley; Columbia University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1211
• 发表日期: 2018
• 页码: 1597-1650
• 关键词: particle-systems; atlas models; ranks

摘要

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 -> infinity) maximizes the total number of surviving particles, with approximately 4/root pi root 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.