Paracontrolled approach to the stochastic Cahn-Hilliard equation

We prove the existence and uniqueness of a local-in-time solution to the stochastic Cahn-Hilliard equation in space dimensions 1, 2, 3, 4 and 5. Our method relies on the Da Prato-Debussche trick for $d=4$ and paracontrolled distributions for $d=5.$