Space-Time HWI Inequalities and the Wasserstein Current for Kinetic Equations
Abstract
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.
Related
- Long-time behavior of first-order Hamilton-Jacobi equations in the Wasserstein space
- Path-Space Optimal Transport with Interactions: Kinetic Equations and Eikonal Structure
- Hypocoercivity-preserving space-time Galerkin methods for kinetic Fokker-Planck equations
- Sharp inequalities between Zolotarev and Wasserstein distances in 𝒫2(ℝd)
- Functional inequalities along Wasserstein geodesics
BibTeX
@misc{li2026spacetime,
title = {{Space-Time HWI Inequalities and the Wasserstein Current for Kinetic Equations}},
author = {Bowen Li and Jianfeng Lu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04831},
eprint = {2611.04831},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04831},
}
RIS
TY - UNPB
TI - Space-Time HWI Inequalities and the Wasserstein Current for Kinetic Equations
AU - Bowen Li
AU - Jianfeng Lu
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04831
UR - https://arxiv.org/abs/2611.04831
AB - 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.
ER -