Invariant Measures and Evolution Equations for Markov Processes Characterized Via Martingale Problems

We extend Echeverria's criterion for invariant measures for a Markov process characterized via martingale problems to the case where the state space of the Markov process is a complete separable metric space. Essentially, the only additional conditions required are a separability condition on the operator occurring in the martingale problem and the well-posedness of the martingale problem in the class of progressively measurable solutions (as opposed to well-posedness in the class of r.c.l.l. solutions, i.e. solutions with paths that are right continuous and have left limits, in the locally compact case). Uniqueness of the solution to the (measure valued) evolution equation for the distribution of the Markov process (as well as a perturbed equation) is also proved when the test functions are taken from the domain of the operator of the martingale problem.