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We consider a stochastic partial differential equation with a logarithmicnonlinearity with singularities at $1$ and $-1$ and a constraint ofconservation of the space average. The equation, driven by a trace-classspace-time noise, contains a bi-Laplacian in the drift. We obtain existence ofsolution for equation with polynomial approximation of the nonlinearity.Tightness of this approximated sequence of solutions is proved, leading to alimit transition semi-group. We study the asymptotic properties of thissemi-group, showing the existence and uniqueness of invariant measure,asymptotic strong Feller property and topological irreducibility.

Asymptotic properties of stochastic Cahn-Hilliard equation with singular nonlinearity and degenerate noise