  We consider classical solutions to the kinetic Fokker-Planck equation on a
bounded domain $\mathcal O \subset~\mathbb{R}^d$ 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 $D = \mathcal O \times \mathbb{R}^d$. Furthermore, a Harnack
inequality, as well as a maximum principle, is 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 $D$ is 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.
