A positive recurrent reflecting Brownian motion with divergent fluid path

Consider a semimartingale reflecting Brownian motion Z whose state space is the d-dimensional non-negative orthant. The data for such a process are a drift vector θ, a non-singular 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 et al. (2000, 2002), 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 by-product, we find that the fluid-based criterion of Dupuis and Williams (1994) is not only necessary but also sufficient 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