The variational formulation of the Fokker-Planck equation
The variational formulation of the Fokker-Planck equation
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DOI：
10.1137/s0036141096303359
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发表时间：
1998-01-01
影响因子：
2
通讯作者：
Otto, F
中科院分区：
文献类型：
--
作者：
Jordan, R;Kinderlehrer, D;Otto, F
Fokker-Planck方程或前向Kolmogorov方程描述了与伊藤随机微分方程相关的随机过程的概率密度的演化。它适用于各种各样的时间依赖系统，其中随机性发挥了作用。本文研究了漂移项由势梯度给出的Fokker-Planck方程。对于一类广泛的潜力，我们构建了一个时间离散，迭代变分方案，其解决方案收敛到的福克-普朗克方程的解决方案。这种迭代方案的主要新奇在于，时间步长由Wasserstein度量的概率措施。这个公式使我们能够揭示一个有吸引力的，以前未探索的，福克-普朗克方程和相关的自由能泛函之间的关系。也就是说，我们证明了动态可以被视为梯度通量，或最陡的下降，相对于Wasserstein度量的自由能。
The Fokker-Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of time-dependent systems in which randomness plays a role. In this paper, we are concerned with Fokker-Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a time discrete, iterative variational scheme whose solutions converge to the solution of the Fokker-Planck equation. The major novelty of this iterative scheme is that the time-step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the Fokker-Planck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric.