Constructive Krein-Rutman result for Kinetic Fokker-Planck equations in a domain

TL;DR

This paper proves well-posedness and spectral properties of the Kinetic Fokker-Planck equation in a domain with boundary reflections, establishing convergence to the principal eigenfunction using ultracontractivity.

Contribution

It introduces a constructive Krein-Rutman theorem for the KFP equation, demonstrating convergence to the first eigenfunction without mass conservation assumptions.

Findings

Existence and uniqueness of solutions in various spaces including Radon measures.

Construction of a principal eigenfunction and spectral gap.

Convergence of solutions to the first eigenfunction.

Abstract

We consider a general Kinetic Fokker-Planck (KFP) equation in a domain with Maxwell reflection condition on the boundary, not necessarily with conservation of mass. We establish the wellposedness in many spaces including Radon measures spaces, and in particular the existence and uniqueness of fundamental solutions. We also establish a Krein-Rutman theorem with constructive rate of convergence in an abstract setting that we use for proving that the solutions to the KFP equation converge toward the conveniently normalized first eigenfunction. Both results use the ultracontractivity of the associated semigroup in a fundamental way.