  We give a variational formulation for
$-\log\mathbb{E}_\nu\left[e^{-f}|\mathcal{F}_t\right]$ 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.
