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 subset of 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 = O x 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.

Automated summary

This article focuses on classical solutions to the kinetic Fokker-Planck equation within a bounded domain O in position, utilizing the Langevin diffusion process with absorbing boundary conditions to obtain probabilistic representations of the solutions. Important results such as the Harnack inequality, maximum principle, and the smooth transition density for the absorbed Langevin process are provided on the phase-space cylindrical domain D = O x R-d. The study also examines the continuity and positivity of the transition density at the boundary of D.