We establish strong Feller property and irreducibility for the transition semigroup associated to a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove uniqueness of the invariant measure under very mild assumptions. The drift of the equation diverges exactly where the noise coefficient vanishes, resulting in a competition between the dissipative effects and the degeneracy of the noise. The main idea is to introduce a mathematical method to measure the accumulation of the solution towards the potential barriers, allowing to give rigorous meaning to the inverse of the noise operator even in the degenerate case. If the singularity of the drift and the degeneracy of the noise are suitably balanced, the dynamics are shown to stabilise for large times. From the mathematical point of view, the results provide a first generalisation of the classical work by Peszat & Zabczyk [1] to the case of degenerate multiplicative diffusions. From the application perspective, the models cover interesting scenarios in physics, in the context of evolution of relative concentrations of mixtures, under the influence of thermodynamically-relevant potentials of Flory-Huggins type.

[1] Peszat, S., Zabczyk, J.: Strong Feller property and irreducibility for diffusions on Hilbert spaces. The Annals of Probability, 157–172, (1995).