  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.
