A probabilistic study of the kinetic Fokker–Planck equation in cylindrical domains

We consider classical solutions to the kinetic Fokker–Planck equation on a bounded domain O⊂Rd in position, and we obtain a probabilistic representation of the solutions using the Langevin diffusion process with absorbing boundary conditions on the boundary of the phase-space cylindrical domain D= O× R^d. Furthermore, a Harnack inequality, as well as a maximum principle, are provided on D for solutions to this kinetic Fokker–Planck equation, together with the existence of a smooth transition density for the associated absorbed Langevin process. This transition density is shown to satisfy an explicit Gaussian upper-bound. Finally, the continuity and positivity of this transition density at the boundary of D are also studied. All these results are in particular crucial to study the behavior of the Langevin diffusion process when it is trapped in a metastable state defined in terms of positions.