The Annals of Applied Probability

Positive recurrence of reflecting Brownian motion in three dimensions

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

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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 Σ, 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.

Article information

Ann. Appl. Probab., Volume 20, Number 2 (2010), 753-783.

First available in Project Euclid: 9 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]
Secondary: 60K25: Queueing theory [See also 68M20, 90B22] 60G42: Martingales with discrete parameter 90C33: Complementarity and equilibrium problems and variational inequalities (finite dimensions)

Reflecting Brownian motion transience Skorohod problem fluid model queueing networks heavy traffic diffusion approximation strong Markov process


Bramson, Maury; Dai, J. G.; Harrison, J. M. Positive recurrence of reflecting Brownian motion in three dimensions. Ann. Appl. Probab. 20 (2010), no. 2, 753--783. doi:10.1214/09-AAP631.

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