The Kinetic Fokker-Planck equation in a domain: Ultracontractivity, hypocoercivity and long-time asymptotic behavior

TL;DR

This paper studies the Kinetic Fokker-Planck equation in a bounded domain, proving ultracontractivity and hypocoercivity of the associated semigroup and operator, and establishing convergence rates to equilibrium.

Contribution

It introduces new results on ultracontractivity and hypocoercivity for the KFP equation with boundary conditions, providing explicit convergence rates.

Findings

Proved ultracontractivity of the semigroup.

Established hypocoercivity of the operator.

Derived explicit convergence rates to equilibrium.

Abstract

We consider the Kinetic Fokker-Planck (FKP) equation in a domain with Maxwell reflection condition on the boundary. We establish the ultracontractivity of the associated semigroup and the hypocoercivity of the associated operator. We deduce the convergence with constructive rate of the solution to the KFP equation towards the stationary state with same mass as the initial datum.