Degenerate stochastic differential equations and super-Markov chains

We consider diffusions corresponding to the generator for continuous with yi nonnegative. We show uniqueness for the corresponding martingale problem under certain non-degeneracy conditions on bi, yi and present a counter-example when these conditions are not satisfied. As a special case, we establish uniqueness in law for some classes of super-Markov chains with state dependent branching rates and spatial motions