Variational calculus on Wiener space with respect to conditional expectations

TL;DR

This paper develops a variational framework on Wiener space to analyze conditional expectations, providing new insights into measure perturbations, invertibility, and a Prékopa-Leindler inequality for conditional expectations.

Contribution

It introduces a variational formulation for conditional expectations on Wiener space and offers a refined entropic characterization of measure perturbations and invertibility.

Findings

Established a variational formula for conditional expectations on Wiener space

Provided a refined entropic criterion for measure perturbation invertibility

Derived a Prékopa-Leindler inequality for 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.