Exponential Convergence to Equilibrium for Kinetic Fokker-Planck Equations
Exponential Convergence to Equilibrium for Kinetic Fokker-Planck Equations
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DOI：
10.1080/03605302.2011.648039
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发表时间：
2010-09
影响因子：
1.9
通讯作者：
S. Calogero
中科院分区：
文献类型：
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作者：
S. Calogero
研究了一类具有非平凡扩散矩阵和周期边界条件的线性动力学Fokker-Planck方程。在几何环境下对问题进行了形式化描述，并在黎曼流形上用微积分的形式研究了收敛到平衡的速度问题。在速度场、能量函数和扩散矩阵的显式几何假设下，证明了整体正则解在时间上以指数速度收敛到平衡点。这一结果被维拉尼最近提出的修正的熵泛函的时间导数的估计所证明。对于空间齐次解，主要定理的假设归结为Bakry和Emery发现的对数Sobolev不等式有效性的曲率有界条件。这一结果适用于低温区的相对论Fokker-Planck方程，其指数趋于平衡的趋势以前是未知的。
A class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered. After formulating the problem in a geometric setting, the question of the rate of convergence to equilibrium is studied within the formalism of differential calculus on Riemannian manifolds. Under explicit geometric assumptions on the velocity field, the energy function and the diffusion matrix, it is shown that global regular solutions converge in time to equilibrium with exponential rate. The result is proved by estimating the time derivative of a modified entropy functional, as recently proposed by Villani. For spatially homogeneous solutions the assumptions of the main theorem reduce to the curvature bound condition for the validity of logarithmic Sobolev inequalities discovered by Bakry and Emery. The result applies to the relativistic Fokker-Planck equation in the low temperature regime, for which exponential trend to equilibrium was previously unknown.