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| A classical problem in the theory of Markov processes is to determine the existence and uniqueness of an invariant probability measure. If existence and uniqueness hold then the question whether all transition probabilities converge to the invariant measure is of interest. It has long been known that the strong Feller property together with irreducibility is sufficient for all these properties
and in this case the transition probabilities do not only converge weakly but even in total variation. Unfortunately, the strong Feller property does not hold for many Markov processes with an infinite dimensional state space (like solutions of many SPDEs). We will show how
generalized asymptotic couplings can be used to prove uniqueness of an invariant measure and weak convergence of transition probabilities. |
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