On the existence and asymptotic behavior of weak solution of the kinetic Fokker-Planck equation in a bounded domain with absorbing boundary

We study here the kinetic Fokker-Planck equation in a bounded domain with absorbing boundary. We show the existence and the uniqueness of a weak solution to the initial-boundary problem and we associate it to a positive compact strongly continuous semigroup. Then, we establish that the weak solution satisfies the balanced exponential growth, meaning both exponential decay toward zero and a thermalization phenomenon.