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

Abstract

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.