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In this short note we show how the asymptotic strong Feller property (ASF)and local weak irreducibility can be established via generalized couplings. Wealso prove that a stronger form of ASF together with local weak irreducibilityimplies uniqueness of an invariant measure. The latter result is optimal in acertain sense and complements some of the corresponding results of Hairer,Mattingly (2008).

Asymptotic strong Feller property and local weak irreducibility via generalized couplings