论文标题
圆柱域中的动力学福克 - 普兰克方程的概率研究
A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains
论文作者
论文摘要
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 $。此外，在$ d $上提供了harnack的不平等以及最大原则，用于该动力学福克克 - 普兰克方程的解决方案，以及存在相关吸收的兰格文流程的平滑过渡密度。该过渡密度显示出满足显式高斯上限。最后，还研究了该过渡密度在$ d $的边界处的连续性和积极性。所有这些结果尤其对于研究兰格文扩散过程的行为至关重要。
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.
