  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 {\theta}, a nonsingular d \times d covariance matrix {\Sigma}, 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 ({\theta}, 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 ({\theta},{\Sigma},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 {\theta} = (-1, -1, . >.
., -1)T, {\Sigma} = 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.
