A Positive Recurrent Reflecting Brownian Motion with Divergent Fluid Path


Precise conditions are known for positive recurrence of semimartingale reflecting Brownian motion (SRBM) in 2 and 3 dimensions, with the argument in 3 dimensions being more involved than in 2 dimensions.  The setting in 4 and more dimensions is more complicated than in 3 dimensions and there are presently no general results.

Associated with each SRBM are fluid paths, which are solutions of deterministic equations corresponding to the random equations of the SRBM. A standard result of Dupuis and Williams states that when every fluid path associated with the SRBM is attracted to the origin, the SRBM is positive recurrent.  This result was employed by El Kharroubi et al. to give sufficient conditions for positive recurrence in 3 dimensions.  In a recent paper with Dai and Harrison, it was shown that the above fluid path behavior is also necessary for positive recurrence of the SRBM.

Here, we present a family of examples in 6 dimensions where the SRBM is positive recurrent but for which a linear fluid path diverges to infinity. These examples show, in particular, that the converse of the Dupuis-Williams result does not hold in 6 and more dimensions.  They also illustrate the difficulty in formulating conditions for positive recurrence of SRBM in higher dimensions.