  We study the asymptotic properties of the stochastic Cahn-Hilliard equation
with the logarithmic free energy by establishing different dimension-free
Harnack inequalities according to various kinds of noises. The main
characteristics of this equation are the singularities of the logarithmic free
energy at 1 and --1 and the conservation of the mass of the solution in its
spatial variable. Both the space-time colored noise and the space-time white
noise are considered. For the highly degenerate space-time colored noise, the
asymptotic log-Harnack inequality is established under the so-called
essentially elliptic conditions. And the Harnack inequality with power is
established for non-degenerate space-time white noise.
