Uniform-in-time propagation of chaos for second order interacting particle systems

We study the long time behavior of second order particle systems interacting through global Lipschitz kernels. Combining hypocoercivity method in [37] and relative entropy method in [25], we are able to overcome the degeneracy of diffusion in position direction by controlling the relative entropy and relative Fisher information together. This implies the uniform-in-time propagation of chaos through the strong convergence of all marginals. Our method works at the level of Liouville equation and relies on the log Sobolev inequality of equilibrium of Vlasov-Fokker-Planck equation.