Entropy Estimate for Degenerate SDEs with Applications to Nonlinear Kinetic Fokker-Planck Equations

TL;DR

This paper develops entropy estimates for degenerate stochastic differential equations and applies these results to establish entropy inequalities and exponential ergodicity for nonlinear kinetic Fokker-Planck equations.

Contribution

It introduces a novel entropy estimate framework for degenerate SDEs and applies it to nonlinear kinetic equations, advancing understanding of their long-term behavior.

Findings

Derived entropy cost inequality for degenerate diffusions

Established exponential ergodicity in entropy for kinetic Fokker-Planck equations

Connected initial distribution Wasserstein distance to entropy estimates

Abstract

The relative entropy for two different degenerate diffusion processes is estimated by using the Wasserstein distance of initial distributions and the difference between coefficients. As applications, the entropy cost inequality and exponential ergodicity in entropy are derived for distribution dependent stochastic Hamiltonian systems associated with nonlinear kinetic Fokker Planck equations.