The Boué–Dupuis formula and the exponential hypercontractivity in the Gaussian space

This paper concerns a variational representation formula for Wiener functionals. Let $B=\{ B_{t}\} _{t\ge 0}$ be standard $d$-dimensional Brownian motion. Bou\'e and Dupuis (1998) showed that, any bounded measurable functional $F(B)$ of $B$ up to time $1$, the expectation $\mathbb{E}\!\left[ e^{F(B)}\right] $ admits in terms drifted motions. In this paper, with slight modification insightful reasoning by Lehec (2013) allowing also over whole interval, we prove that Bou\'e--Dupuis holds true provided both $e^{F(B)}$ are integrable, relaxing conditions earlier works. We show implies exponential hypercontractivity Ornstein--Uhlenbeck semigroup $\mathbb{R}^{d}$, hence, due their equivalence, logarithmic Sobolev inequality Gaussian space.