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 ≥ 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 E (cid:104) e F ( B ) (cid:105) 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 R d , and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the d -dimensional Gaussian space.