A variational representation and Pr\'ekopa's theorem for Wiener functionals
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.
Keywords
Cite
@article{arxiv.1505.02479,
title = {A variational representation and Pr\'ekopa's theorem for Wiener functionals},
author = {Yuu Hariya},
journal= {arXiv preprint arXiv:1505.02479},
year = {2015}
}
Comments
In the second version, an inaccurate description of Pr\'ekopa's theorem in the introduction is corrected; the assumption of Theorem 1.1 is improved; three papers are added into the list of references