The Bou\'e--Dupuis formula and the exponential hypercontractivity in the Gaussian space

TL;DR

This paper extends the Boué--Dupuis variational formula for Wiener functionals, demonstrating its validity under relaxed conditions and linking it to exponential hypercontractivity and the logarithmic Sobolev inequality in Gaussian space.

Contribution

It generalizes the Boué--Dupuis formula to broader conditions and connects it to fundamental inequalities in Gaussian analysis.

Findings

Proves the Boué--Dupuis formula under weaker integrability conditions.

Shows the formula implies exponential hypercontractivity of the Ornstein--Uhlenbeck semigroup.

Establishes the link to the logarithmic Sobolev inequality in 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. Bou\'e 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 Bou\'e--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.