A variational representation and Prékopa's theorem for Wiener functionals

Abstract

In 1998, Boué 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 Prékopa's theorem for Wiener functionals, which is then applied to formulate the Brascamp-Lieb inequality in the framework of Wiener spaces.

In the second version, an inaccurate description of Prékopa's theorem in the introduction is corrected; the assumption of Theorem 1.1 is improved; three papers are added into the list of references