Variational calculus on Wiener space with respect to conditional expectations

We give a variational formulation for $-\log {\mathbb{E}}_{\nu }\left[e^{-f}\mathrm{\mid }{\mathcal{F}}_t\right]$ for a large class of measures $\nu$. We give a refined entropic characterization of the invertibility of some perturbations of the identity. We also discuss the attainability of the infimum in the variational formulation and obtain a Pr'ekopa-Leindler theorem for conditional expectations.