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Semimartingale reflecting Brownian motions (SRBMs) are diffusion processeswith state space the d-dimensional nonnegative orthant, in the interior ofwhich the processes evolve according to a Brownian motion, and that reflectagainst the boundary in a specified manner. The data for such a process are adrift vector {\theta}, a nonsingular d \times d covariance matrix {\Sigma}, anda d \times d reflection matrix R. A standard problem is to determine under whatconditions the process is positive recurrent. Necessary and sufficientconditions for positive recurrence are easy to formulate for d = 2, but not ford > 2.  Associated with the pair ({\theta}, R) are fluid paths, which are solutionsof deterministic equations corresponding to the random equations of the SRBM. Astandard result of Dupuis and Williams [6] states that when every fluid pathassociated with the SRBM is attracted to the origin, the SRBM is positiverecurrent. Employing this result, El Kharroubi et al. [7, 8] gave sufficientconditions on ({\theta},{\Sigma},R) for positive recurrence for d = 3; Bramsonet 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 {\theta} = (-1, -1, . >.., -1)T, {\Sigma} = I and appropriate R, that are positive recurrent, but forwhich a linear fluid path diverges to infinity. These examples show inparticular that, for d >= 6, the converse of the Dupuis-Williams result doesnot hold.

A Positive Recurrent Reflecting Brownian Motion with Divergent Fluid Path