Generalized couplings and convergence of transition probabilities

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.