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×d covariance matrix Σ, and a d×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 [Ann. Probab. 22 (1994) 680–702] 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, Ben Tahar and Yaacoubi [Stochastics Stochastics Rep. 68 (2000) 229–253, Math. Methods Oper. Res. 56 (2002) 243–258] gave sufficient conditions on (θ, Σ, R) for positive recurrence for d=3; Bramson, Dai and Harrison [Ann. Appl. Probab. 20 (2009) 753–783] 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.