A general framework for variational calculus on Wiener space

TL;DR

This paper develops a comprehensive variational calculus framework on Wiener space, enabling new insights into stochastic differential equations, measure perturbations, and inequalities like Prékopa-Leindler.

Contribution

It introduces a general approach to variational formulations for a broad class of measures on Wiener space, linking perturbations, entropy, and existence results.

Findings

Derived a variational formula for negative log expectations under various measures

Established strong existence results for stochastic differential equations using the framework

Proved a Prékopa-Leindler inequality for the measure 

Abstract

We provide a framework to derive a variational formulation for $-\log\mathbb{E}_\nu\left[e^{-f}\right]$ for a large class of measures $\nu$. 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 Pr\'ekopa-Leindler theorem for the measure $\nu$.