Variational calculus on Wiener space with respect to conditional expectations

Abstract: 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.