  The asymptotic log-Harnack inequality is established for several different
models of stochastic differential systems with infinite memory: non-degenerate
SDEs, Neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems.
  As applications, the following properties are derived for the associated
segment Markov semigroups: asymptotic heat kernel estimate; uniqueness of the
invariant probability measure; asymptotic gradient estimate and hence,
asymptotically strong Feller property; and asymptotic irreducibilty.
