The Variational Formulation of the Fokker--Planck Equation

TLDR

The Fokker–Planck equation describes the time evolution of probability densities for stochastic processes governed by Ito SDEs and applies to many time‑dependent systems where randomness is present. This work focuses on Fokker–Planck equations whose drift is the gradient of a potential. We develop a time‑discrete variational scheme for a broad class of potentials, where the time step is determined by the Wasserstein metric on probability measures, and show that its iterates converge to the solution of the Fokker–Planck equation. The formulation uncovers that the dynamics of such Fokker–Planck equations can be viewed as a gradient flow, or steepest‑descent, of the associated free‑energy functional with respect to the Wasserstein metric.

Abstract

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.