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

Abstract

This paper concerns a variational representation formula for Wiener functionals. Let $B=\{B_t{\}}_{t\ge 0}$ be a standard $d$-dimensional Brownian motion. Boué and Dupuis (1998) showed that, for any bounded measurable functional $F(B)$ of $B$ up to time $1$, the expectation $\mathbb{E}\left[e^{F(B)}\right]$ 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 $F(B)$ to be a functional of $B$ over the whole time interval, we prove that the Boué--Dupuis formula holds true provided that both $e^{F(B)}$ and $F(B)$ are integrable, relaxing conditions in earlier works. We also show that the formula implies the exponential hypercontractivity of the Ornstein--Uhlenbeck semigroup in ${\mathbb{R}}^d$, and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the $d$-dimensional Gaussian space.

15 pages: newly added reference [9] by Chandra et al. (arXiv:2006.15933); also added is a corollary (Corollary 2.1) to Theorem 1.1, in which the case of bounded drifts is treated

References in corpus (2)

• Phase transitions for ${\varphi }_3^4$
• A variational representation and Prékopa's theorem for Wiener functionals