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

TL;DR

This paper provides a probabilistic analysis of the kinetic Fokker-Planck equation in cylindrical domains, including boundary behavior, transition densities, and inequalities, to understand Langevin diffusion in metastable states.

Contribution

It introduces a probabilistic representation of solutions, establishes a Harnack inequality, and analyzes the transition density for the kinetic Fokker-Planck equation with absorbing boundaries.

Findings

Existence of a smooth transition density with Gaussian upper bounds

Harnack inequality and maximum principle for solutions

Continuity and positivity of the transition density at the boundary

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.