This paper concerns a variational representation formula for Wiener functionals. Let be a standard d-dimensional Brownian motion. Boué and Dupuis (1998) showed that, for any bounded measurable functional of B up to time 1, the expectation admits a variational representation in terms of drifted Brownian motions. In this paper, with a slight modification of insightful reasoning by Lehec (2013) allowing also to be a functional of B over the whole time interval, we prove that the Boué–Dupuis formula holds true provided that both and are integrable, relaxing conditions in earlier works. We also show that the formula implies the exponential hypercontractivity of the Ornstein–Uhlenbeck semigroup in , and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the d-dimensional Gaussian space.
The research of Y. Hariya was supported in part by JSPS KAKENHI Grant Number 17K05288.
The authors gratefully acknowledge the valuable comments of the anonymous referee, especially on the literature on quasi-invariant measures for dispersive partial differential equations. They are grateful to Professor Shigeki Aida for bringing Section 8.1 of  to their attention as referred to in Remark 2.12(2). They thank Professor Ali Süleyman Üstünel for his interest in their work: he kindly sent them a reprint of  as referred to just after Lemma 2.4, as soon as an earlier version of the paper was posted on arXiv; they were also informed of his former Ph.D. student’s papers on arXiv, as referred to in Remark 1.2(3). Their thanks also go to the anonymous referee of , one of whose comments motivated them to do the study in Section 3.
"The Boué–Dupuis formula and the exponential hypercontractivity in the Gaussian space." Electron. Commun. Probab. 27 1 - 13, 2022. https://doi.org/10.1214/22-ECP461