A Modified log-Harnack inequality and asymptotically strong Feller property

We introduce a new functional inequality, which is a modification of log-Harnack inequality established in [20] and [29], and prove that it implies the asymptotically strong Feller property (ASF). This inequality seems to generalize the criterion for ASF in [Proposition 3.12,14]. As a example, we show by an asymptotic coupling that 2D stochastic Navier-Stokes equation driven by highly degenerate but \emph{essentially elliptic} noises satisfies our modified log-Harnack inequality.