Exponential Ergodicity in Relative Entropy and $L^2$-Wasserstein Distance for non-equilibrium partially dissipative Kinetic SDEs

TL;DR

This paper establishes exponential ergodicity in relative entropy and Wasserstein distance for non-equilibrium kinetic SDEs with partial dissipation, extending previous equilibrium results to more general, non-gradient force scenarios.

Contribution

It introduces new ergodicity results for non-equilibrium kinetic SDEs without explicit invariant measures, using hypercontractivity and inequalities, and extends findings to McKean-Vlasov and particle systems.

Findings

Exponential ergodicity in relative entropy for general kinetic SDEs.

Exponential ergodicity in the $L^2$-Wasserstein distance.

Uniform convergence rates for mean-field particle systems.

Abstract

In this paper, we derive exponential ergodicity in relative entropy for general kinetic SDEs under a partially dissipative condition. It covers non-equilibrium situations where the forces are not of gradient type and the invariant measure does not have an explicit density, extending previous results set in the equilibrium case. The key argument is to establish the hypercontractivity of the associated semigroup, which follows from its hyperboundedness and its $L^2$-exponential ergodicity. Moreover, we obtain exponential ergodicity in the $L^2$-Wasserstein distance by combining Talagrand's inequality with a log-Harnack inequality. These results are further extended to the McKean-Vlasov setting and to the associated mean-field interacting particle systems, with convergence rates that are uniform in the number of particles in the latter case, under small nonlinear perturbations.