  We derive an asymptotic log-Harnack inequality for nonlinear monotone SPDE
driven by possibly degenerate multiplicative noise. Our main tool is the
asymptotic coupling by the change of measure. As an application, we show that,
under certain monotone and coercive conditions on the coefficients, the
corresponding Markov semigroup is asymptotically strong Feller, asymptotic
irreducibility, and possesses a unique and thus ergodic invariant measure. The
results are applied to highly degenerate finite-dimensional or
infinite-dimensional diffusion processes.
