POSITIVE RECURRENCE OF REFLECTING BROWNIAN MOTION IN THREE DIMENSIONS

• 成果类型: Article
• 署名作者: Bramson, Maury; Dai, J. G.; Harrison, J. M.
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities; University System of Georgia; Georgia Institute of Technology; Stanford University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/09-AAP631
• 发表日期: 2010
• 页码: 753-783
• 关键词: orthant

摘要

Consider a semimartingale reflecting Brownian motion (SRBM) Z whose state space is the d-dimensional nonnegative orthant. The data for such a process are a drift vector theta, a nonsingular d x d covariance matrix Sigma, and a d x d reflection matrix R that specifies the boundary behavior of Z. We say that Z is positive recurrent, or stable, if the expected time to hit an arbitrary open neighborhood of the origin is finite for every starting state. In dimension d = 2, necessary and sufficient conditions for stability are known, but fundamentally new phenomena arise in higher dimensions. Building on prior work by El Kharroubi, Ben Tahar and Yaacoubi [Stochastics Stochastics Rep. 68 (2000) 229-253, Math. Methods Oper Res. 56 (2002) 243-2581, we provide necessary and sufficient conditions for stability of SRBMs in three dimensions; to verify or refute these conditions is a simple computational task. As a byproduct, we find that the fluid-based criterion of Dupuis and Williams [Ann. Probab. 22 (1994) 680-702] is not only sufficient but also necessary for stability of SRBMs in three dimensions. That is, an SRBM in three dimensions is positive recurrent if and only if every path of the associated fluid model is attracted to the origin. The problem of recurrence classification for SRBMs in four and higher dimensions remains open.