  We show weak existence and uniqueness in law for a general class of
stochastic differential equations in $\mathbb{R}^d$, $d\ge 1$, with prescribed
sub-invariant measure $\widehat{\mu}$. The dispersion and drift coefficients of
the stochastic differential equation are allowed to be degenerate and
discontinuous, and locally unbounded, respectively. Uniqueness in law is
obtained via $L^1(\mathbb{R}^d,\widehat{\mu})$-uniqueness in a subclass of
continuous Markov processes, namely right processes that have $\widehat{\mu}$
as sub-invariant measure and have continuous paths for $\widehat{\mu}$-almost
every starting point. Weak existence is obtained for a broader class via the
martingale problem.
