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Variational structure of Fokker-Planck equations with variable mobility

Hailiang Liu, Athanasios E. Tzavaras

TL;DR

This work develops a variational framework for Fokker–Planck equations with spatially varying mobility $A(x)$ by introducing a weighted Wasserstein distance $W_A$ under which the FP dynamics are gradient flows of the free energy $F(\rho)$. A key advance is the reduction of the $W_A$-based transport problem to a classical Monge problem via a Nash–Kuiper type isometric embedding $b$, yielding a cost $c(x,y)=|b(y)-b(x)|^2$ and a well-posed optimal map. The authors further formulate a time-discrete JKO-like scheme using $W_A$ to construct weak solutions, derive the Euler–Lagrange equations for the minimizers, and prove convergence in the linear-diffusion setting along with essential a priori and compactness estimates. The results connect high-friction limits, gradient-flow structure, and optimal transport in heterogeneous media, providing a constructive method for analysis and numerics of anisotropic diffusion processes with variable mobility.

Abstract

We study Fokker--Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced for which these equations are gradient flows. This metric is shown to emerge from an optimal control problem in the space of probability densities for a class of variable mobility matrices, with the cost function capturing the work dissipated via friction. Using the Nash-Kuiper isometric embedding theorem for Riemannian manifolds, we demonstrate the existence of optimal transport maps. Additionally, we construct a time-discrete variational scheme, establish key properties for the associated minimizing problem, and prove convergence to weak solutions of the associated Fokker-Planck equation.

Variational structure of Fokker-Planck equations with variable mobility

TL;DR

This work develops a variational framework for Fokker–Planck equations with spatially varying mobility by introducing a weighted Wasserstein distance under which the FP dynamics are gradient flows of the free energy . A key advance is the reduction of the -based transport problem to a classical Monge problem via a Nash–Kuiper type isometric embedding , yielding a cost and a well-posed optimal map. The authors further formulate a time-discrete JKO-like scheme using to construct weak solutions, derive the Euler–Lagrange equations for the minimizers, and prove convergence in the linear-diffusion setting along with essential a priori and compactness estimates. The results connect high-friction limits, gradient-flow structure, and optimal transport in heterogeneous media, providing a constructive method for analysis and numerics of anisotropic diffusion processes with variable mobility.

Abstract

We study Fokker--Planck equations with symmetric, positive definite mobility matrices capturing diffusion in heterogeneous environments. A weighted Wasserstein metric is introduced for which these equations are gradient flows. This metric is shown to emerge from an optimal control problem in the space of probability densities for a class of variable mobility matrices, with the cost function capturing the work dissipated via friction. Using the Nash-Kuiper isometric embedding theorem for Riemannian manifolds, we demonstrate the existence of optimal transport maps. Additionally, we construct a time-discrete variational scheme, establish key properties for the associated minimizing problem, and prove convergence to weak solutions of the associated Fokker-Planck equation.
Paper Structure (19 sections, 14 theorems, 187 equations)

This paper contains 19 sections, 14 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Gradient flow and the high-friction limit
  3. Gradient flow structure
  4. Fokker-Planck with variable mobility as a high-friction limit
  5. Maximal energy dissipation
  6. A weighted Wasserstein metric and optimal maps
  7. Preliminaries on optimal transportation
  8. The analog of Wasserstein for heterogeneous, anisotropic mobility
  9. Existence of optimal maps
  10. Variational time-discretization
  11. The a-priori estimates
  12. Euler-Lagrange equation
  13. Convergence
  14. A priori estimates and compactness
  15. A priori estimates
  16. ...and 4 more sections

Key Result

Proposition 3.1

Under hypotheses c0, abb, we have The minimization problem rv defines a weighted Wasserstein distance that satisfies with $d\mu_0 = \rho_0 dx$, $d\mu_1 = \rho_1 dx$ and $\Pi(\mu_0, \mu_1)$ defined in defpi.

Theorems & Definitions (27)

  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 17 more