  Two frameworks that have been used to characterize reflected diffusions
include stochastic differential equations with reflection and the so-called
submartingale problem. We introduce a general formulation of the submartingale
problem for (obliquely) reflected diffusions in domains with piecewise C^2
boundaries and piecewise continuous reflection vector fields. Under suitable
assumptions, we show that well-posedness of the submartingale problem is
equivalent to existence and uniqueness in law of weak solutions to the
corresponding stochastic differential equation with reflection. Our result
generalizes to the case of reflecting diffusions a classical result due to
Stroock and Varadhan on the equivalence of well-posedness of martingale
problems and well-posedness of weak solutions of stochastic differential
equations in d-dimensional Euclidean space. The analysis in the case of
reflected diffusions in domains with non-smooth boundaries is considerably more
subtle and requires a careful analysis of the behavior of the reflected
diffusion on the boundary of the domain. In particular, the equivalence can
fail to hold when our assumptions are not satisfied. The equivalence we
establish allows one to transfer results on reflected diffusions characterized
by one approach to reflected diffusions analyzed by the other approach. As an
application, we provide a characterization of stationary distributions of a
large class of reflected diffusions in convex polyhedral domains.
