  We provide sufficient conditions for the uniqueness of an invariant measure
of a Markov process as well as for the weak convergence of transition
probabilities to the invariant measure. Our conditions are formulated in terms
of generalized couplings. We apply our results to several SPDEs for which
unique ergodicity has been proven in a recent paper by Glatt-Holtz, Mattingly,
and Richards and show that under essentially the same assumptions the weak
convergence of transition probabilities actually holds true.
