Space-Time HWI Inequalities and the Wasserstein Current for Kinetic Equations

We prove a space-time analogue of the HWI inequality of Otto-Villani for controlled kinetic equations with a semiconvex confining potential. It bounds the time-averaged relative entropy by spatial and conditional velocity transport distances, the velocity Fisher information, an intrinsic action of the path, and a boundary term given by the time endpoint values of the Wasserstein current. The key step is a differential comparison inequality for the Wasserstein current, which captures the transfer of dissipation from velocity to position. Along the Langevin semigroup, this inequality yields a kinetic evolution variational inequality involving the second time derivative of the squared spatial Wasserstein distance and a damping term. Under a compatible spatial Talagrand's transport inequality, we obtain a boundary-free space-time logarithmic Sobolev inequality on every interval $[0,T]$, $T>0$, with the optimal $T^{-2}$ short-time scaling of the constant. As a consequence, we also establish entropy decay for every friction coefficient and every initial density of finite entropy, with optimal cubic order at short times and exponential decay at long times.