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Journal Articles Journal of Evolution Equations Year : 2022

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

Abstract

We consider classical solutions of 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 dynamics. 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.

Dates and versions

hal-02974421 , version 1 (21-10-2020)

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Tony Lelièvre, Mouad Ramil, Julien Reygner. A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains. Journal of Evolution Equations, 2022, 22 (38), ⟨10.1007/s00028-022-00796-5⟩. ⟨hal-02974421⟩
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