Positive recurrence for reflecting Brownian motion in higher dimensions

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. 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 in d = 2, but not in d ≥ 3. Fluid paths are solutions of deterministic equations that correspond to the random equations of the SRBM. A standard result of Dupuis and Williams (in Ann. Probab. 22:680-702, 1994) 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. (in Stoch. Stoch. Rep. 68:229-253, 2000; Math. Methods Oper. Res. 56:243-258, 2002) gave sufficient conditions involving fluid paths for positive recurrence of SRBM in d = 3. Here, we discuss two recent results regarding necessary conditions for positive recurrence of SRBM in d ≥ 3. Bramson et al. (in Ann. Appl. Probab. 20:753-783, 2010) showed that the conditions in El Kharroubi et al. (Math. Methods Oper. Res. 56:243-258, 2002) are, in fact, necessary in d = 3. On the other hand, Bramson (in Ann. Appl. Probab., to appear, 2011) provided a family of positive recurrent SRBMs, in d ≥ 6, with linear fluid paths that diverge to infinity. The latter result shows in particular that the converse of the Dupuis-Williams result does not hold.