  We present three sets of results for the stationary distribution of a
two-dimensional semimartingale reflecting Brownian motion (SRBM) that lives in
the nonnegative quadrant. The SRBM data can equivalently be specified by three
geometric objects, an ellipse and two lines, in the two-dimensional Euclidean
space. First, we revisit the variational problem (VP) associated with the SRBM.
Building on Avram, Dai and Hasenbein (2001), we show that the value of the VP
at a point in the quadrant is equal to the optimal value of a linear function
over a convex domain. Depending on the location of the point, the convex domain
is either D(1) or D(2) or D(1) cap D(2), where each D(i), i = 1, 2, can easily
be described by the three geometric objects. Our results provide a geometric
interpretation for the value function of the VP and allow one to see
geometrically when one edge of the quadrant has influence on the optimal path
traveling from the origin to a destination point. Second, we provide a
geometric condition that characterizes the existence of a product form
stationary distribution. Third, we establish exact tail asymptotics of two
boundary measures that are associated with the stationary distribution; a key
step in our proof is to sharpen two asymptotic inversion lemmas in Dai and
Miyazawa (2011) that allow one to infer the exact tail asymptotic of a boundary
measure from the singularity of its moment generating function.
