We provide sufficient conditions for uniqueness of an invariant probability measure of a Markov kernel in terms of (generalized) couplings. Our main theorem generalizes previous results which require the state space to be Polish. We provide an example showing that uniqueness can fail if the state space is separable and metric (but not Polish) even though a coupling defined via a continuous and positive definite function exists.
Existence and uniqueness of invariant measures for SPDEs with two reflecting walls
Stationary flows and uniqueness of invariant measures
$Φ$-Entropy Inequality and Invariant Probability Measure for SDEs with Jump
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