A Positive Recurrent Reflecting Brownian Motion with Divergent Fluid Path

Abstract

Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes with state space the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motion, and that reflect against the boundary in a specified manner. The data for such a process are a drift vector θ, a nonsingular d \times d covariance matrix Σ, and a d \times d reflection matrix R. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions for positive recurrence are easy to formulate for d = 2, but not for d > 2. Associated with the pair (θ, R) are fluid paths, which are solutions of deterministic equations corresponding to the random equations of the SRBM. A standard result of Dupuis and Williams [6] states that when every fluid path associated with the SRBM is attracted to the origin, the SRBM is positive recurrent. Employing this result, El Kharroubi et al. [7, 8] gave sufficient conditions on (θ,Σ,R) for positive recurrence for d = 3; Bramson et al. [2] showed that these conditions are, in fact, necessary. Relatively little is known about the recurrence behavior of SRBMs for d > 3. This pertains, in particular, to necessary conditions for positive recurrence. Here, we provide a family of examples, in d = 6, with θ = (-1, -1, . >. ., -1)T, Σ = I and appropriate R, that are positive recurrent, but for which a linear fluid path diverges to infinity. These examples show in particular that, for d >= 6, the converse of the Dupuis-Williams result does not hold.