  We study reflecting Brownian motion with drift constrained to a wedge in the
plane. Our first set of results provide necessary and sufficient conditions for
existence and uniqueness of a solution to the corresponding submartingale
problem with drift, and show that its solution possesses the Markov and Feller
properties. Next, we study a version of the problem with absorption at the
vertex of the wedge. In this case, we provide a condition for existence and
uniqueness of a solution to the problem and some results on the probability of
the vertex being reached.
