  We call a multidimensional distribution to be decomposable with respect to a
partition of two sets of coordinates if the original distribution is the
product of the marginal distributions associated with these two sets. We focus
on the stationary distribution of a multidimensional semimartingale reflecting
Brownian motion (SRBM) on a nonnegative orthant. An SRBM is uniquely determined
(in distribution) by its data that consists of a covariance matrix, a drift
vector, and a reflection matrix. Assume that the stationary distribution of an
SRBM exists. We first characterize two marginal distributions under the
decomposability assumption. We prove that they are the stationary distributions
of some lower dimensional SRBMs. We also identify the data for these lower
dimensional SRBMs. Thus, under the decomposability assumption, we can obtain
the stationary distribution of the original SRBM by computing those of the
lower dimensional ones. However, this characterization of the marginal
distributions is not sufficient for the decomposability. So, we next consider
necessary and sufficient conditions for the decomposability. We obtain those
conditions for several classes of SRBMs. These classes include SRBMs arising
from Brownian models of queueing networks that have two sets of stations with
feed-forward routing between these two sets. This work is motivated by
applications of SRBMs and geometric interpretations of the product form
stationary distributions.
