Positive recurrence of reflecting Brownian motion in three dimensions

Maury Bramson, J. G. Dai, J. M. Harrison

Research output: Contribution to journalArticle

17 Scopus citations

Abstract

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 θ, a nonsingular d × d covariance matrix ∑, and a d × 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- 258], 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.

Original languageEnglish (US)
Pages (from-to)753-783
Number of pages31
JournalAnnals of Applied Probability
Volume20
Issue number2
DOIs
StatePublished - Apr 2010

Keywords

  • Diffusion approximation
  • Fluid model
  • Heavy traffic
  • Queueing networks
  • Reflecting Brownian motion
  • Skorohod problem
  • Strong Markov process
  • Transience

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