The Variational Formulation of the Fokker--Planck Equation

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.

Example of a displacement convex functional of first order

“…Given an initial measure µ 0 ∈ P(S 1 ) and time-step τ > 0, let us consider the variational scheme or minimizing movement scheme [12,1,2] recursively defined by µ τ 0 = µ 0 and…”

“…The existence of gradient flows in Wasserstein metric is based on variational schemes or minimizing movements [12,1,2], first considered in this setting by Jordan, Kinderlehrer and Otto. The application of the scheme is greatly simplified when the functional considered is displacement convex.…”

Chemical Reactions as $\Gamma$-Limit of Diffusion

“…The weak formulation (2.8) also shows that a solution u ε can be interpreted as a gradient flow of the quadratic energy 1 2 a ε (u, u) with respect to the L 2 (D; γ ε ) distance. Another gradient-flow structure for the solutions of the same problem could be obtained by a different choice of energy functional and distance: for example, as proved in [16], Fokker-Planck equations like (2.6) can be interpreted also as the gradient flow of the relative entropy functional…”

“…Initiated by the work of Otto [16,21] and extended in many directions since, this framework provides an appealing variational structure for very general diffusion processes.…”

A new family of transportation costs with applications to reaction–diffusion and parabolic equations with boundary conditions

“…More precisely, one of the most surprising achievements of [8,10,11] has been that many evolution equations of the form d dt ρ(t) = div ∇ρ(t) + ρ(t)∇V + ρ(t)(∇W * ρ(t) , can be seen as gradient flows of some entropy functional on the space of probability measures with respect to the Wasserstein distance: W 2 (µ, ν) = inf |x − y| 2 dγ(x, y) : π 1# γ = µ, π 2# γ = ν .…”