摘要：This paper concerns a variational representation formula for Wiener functionals. Let B={Bt}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 EeF(B) 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 eF(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 Rd, and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the d-dimensional Gaussian space.
