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, by first constructing a sub-Markovian \(C_0\) -semigroup of contractions with respect to \(\widehat{\mu }\) and then applying generalized Dirichlet form theory.
