The variational formulation of the Fokker-Planck equation. (English) Zbl 0915.35120

The authors consider the Fokker-Planck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, the authors 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 a previously unexplored, relationship between the Fokker-Planck equation and the associated free energy functional. Namely, the authors prove that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy with respect to the Wasserstein metric.


35R60 PDEs with randomness, stochastic partial differential equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35A15 Variational methods applied to PDEs
60J60 Diffusion processes
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