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We constructed in a previous work the $\Phi^4_3$ measures on compactboundaryless $3$-dimensional Riemannian manifolds as some invariant probabilitymeasures of some Markovian dynamics. We prove in the present work that thesedynamics have unique invariant probability measures. This is done by using anexplicit coupling by change of measure that does not require any a prioriinformation on the support of the law of the solution to the dynamics. Thecoupling can be used to see that the semigroup generated by the dynamicssatisfies a Harnack-type inequality, which entails that the semigroup has thestrong Feller property.

Uniqueness of the $\Phi^4_3$ measures on closed Riemannian $3$-manifolds