Decomposable stationary distribution of a multidimensional SRBM

We focus on the stationary distribution of a multidimensional semimartingale reflecting Brownian motion (SRBM) on a nonnegative orthant. Assuming that the stationary distribution exists and is decomposable-equal to the product of two marginal distributions, we prove that these marginal distributions are the stationary distributions of some lower dimensional SRBMs, whose data can be explicitly computed through that of the original SRBM. Thus, under the decomposability condition, the stationary distribution of a high dimensional SRBM can be computed through those of lower dimensional SRBMs. Next, we derive necessary and sufficient conditions for some classes of SRBMs to satisfy the decomposability condition.