January 31, 2024 | Pierre Monmarché, Zhenjie Ren, and Songbo Wang
This paper explores time-uniform log-Sobolev inequalities (LSI) for solutions of time-inhomogeneous Fokker–Planck equations, which are crucial for obtaining time-uniform propagation of chaos estimates. The authors present two main results: a high-diffusivity regime result and a bounded-Lipschitz perturbation argument. These results are then applied to McKean–Vlasov equations, leading to sharp uniform-in-time propagation of chaos estimates in smooth cases and global propagation of chaos for vortex interactions with quadratic confinement potential on the whole space. The paper also establishes global gradient and Hessian estimates for non-attractive logarithmic and Riesz singular interactions, which are of independent interest. The proofs involve detailed analysis of the Markov semigroup and coupling arguments, providing a comprehensive framework for understanding the propagation of chaos in various settings.This paper explores time-uniform log-Sobolev inequalities (LSI) for solutions of time-inhomogeneous Fokker–Planck equations, which are crucial for obtaining time-uniform propagation of chaos estimates. The authors present two main results: a high-diffusivity regime result and a bounded-Lipschitz perturbation argument. These results are then applied to McKean–Vlasov equations, leading to sharp uniform-in-time propagation of chaos estimates in smooth cases and global propagation of chaos for vortex interactions with quadratic confinement potential on the whole space. The paper also establishes global gradient and Hessian estimates for non-attractive logarithmic and Riesz singular interactions, which are of independent interest. The proofs involve detailed analysis of the Markov semigroup and coupling arguments, providing a comprehensive framework for understanding the propagation of chaos in various settings.