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, 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.