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

Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises