Hypocoercive relaxation to equilibrium for some kinetic models via a third order differential inequality

TL;DR

This paper establishes a novel third order differential inequality approach to demonstrate exponential convergence to equilibrium in certain kinetic models where traditional methods fail due to the lack of Poincaré inequalities.

Contribution

It introduces a third order differential inequality framework to analyze hypocoercive relaxation in kinetic models with velocity-dependent randomness.

Findings

Derived explicit convergence rates to equilibrium

Applied the method to kinetic Fokker-Planck dynamics

Extended results to piecewise deterministic evolutions

Abstract

This paper deals with the study of some particular kinetic models, where the randomness acts only on the velocity variable level. Usually, the Markovian generator cannot satisfy any Poincar\'e's inequality. Hence, no Gronwall's lemma can easily lead to the exponential decay of Ft (the L2 norm of a test function along the semi-group). Nevertheless for the kinetic Fokker-Planck dynamics and for a piecewise deterministic evolution we show that Ft satisfies a third order differential inequality which gives an explicit rate of convergence to equilibrium.