Space-Time Log-Sobolev Inequality and Hypocoercive Hypercontractivity for Underdamped Langevin Dynamics

We study hypercontractivity for the underdamped Langevin dynamics with a convex confining potential. Unlike in the overdamped case, the noise acts only on the velocity variable, so the usual argument based on the logarithmic Sobolev inequality (LSI) does not apply. Nevertheless, we prove that, when the spatial marginal satisfies an LSI with constant $\rho$ and the friction parameter is of order $\sqrt{\rho}$, the underdamped Langevin semigroup improves integrability over the kinetic time scale $t \sim \rho^{-1/2}$. The proof rests on a novel space-time logarithmic Sobolev inequality, derived from a controlled version of the entropy decay estimate, which captures how dissipation in the velocity variable is transferred to the position variable. Combining this space-time LSI with a duality argument based on a forward/backward interpolation of the underdamped Langevin semigroup yields the desired hypocoercive hypercontractivity estimate. As a corollary, we obtain decay of the R\'enyi divergence at the sharp hypocoercive rate $\mathcal{O}(\sqrt{\rho})$.