A variational representation and Pr\'ekopa's theorem for Wiener functionals

TL;DR

This paper extends a variational representation for Wiener functionals to unbounded cases and applies it to establish a Prékopa-type theorem and Brascamp-Lieb inequality in Wiener spaces.

Contribution

It introduces a generalized variational representation for unbounded Wiener functionals and derives a Prékopa's theorem analogue for Wiener spaces.

Findings

Extended variational representation to unbounded functionals

Proved Prékopa's theorem analogue for Wiener functionals

Formulated Brascamp-Lieb inequality in Wiener space framework

Abstract

In 1998, Bou\'e and Dupuis proved a variational representation for exponentials of bounded Wiener functionals. Since their proof involves arguments related to the weak convergence of probability measures, the boundedness of functionals seems inevitable. In this paper, we extend the representation to unbounded functionals under a mild assumption on their integrability. As an immediate application of the extension, we prove an analogue of Pr\'ekopa's theorem for Wiener functionals, which is then applied to formulate the Brascamp-Lieb inequality in the framework of Wiener spaces.