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

Jordan, Richard; Kinderlehrer, David; Otto, Felix

SIAM J. Math. Anal. 29, No. 1, 1-17 (1998).

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.

Reviewer: S.Wedrychowicz (Rzeszów)

MSC:

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

Keywords:

Fokker-Planck equation; free energy; Wasserstein metric