We provide a framework to derive a variational formulation for $-\log\mathbb{E}_ν\left[e^{-f}\right]$ for a large class of measures $ν$. We use a family of perturbations of the identity $(W^u)$ whose invertibility we characterize thanks to entropy. This yields results of strong existence for various stochastic differential equations. We also discuss the attainability of the infimum in the variational formulation and we derive a Prékopa-Leindler theorem for the measure $ν$.