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Variational structure of Fokker-Planck equations with variable mobility

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arxiv 2505.10676 v1 pith:WLFGRGRV submitted 2025-05-15 math.OC math.AP

Hailiang Liu , Athanasios E. Tzavaras This is my paper · ORCID
classification math.OCmath.AP
keywords equationsmobilityassociatedcapturingfokker-planckmatricesmetricoptimal
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Homogenization of Time-Discrete Gradient Flows

    math.AP 2026-07 conditional novelty 7.0 of 10

    For ε-periodic Fokker-Planck equations, the weighted-L2 minimizing-movement scheme commutes with homogenization and gives the effective equation, while the JKO scheme converges to a different limiting equation.

  2. A Unified Variational Framework for Optimal Transport with Lagrangian Costs

    math.OC 2026-07 conditional novelty 3.0 of 10

    For general Lagrangian costs on the torus, the paper re-proves the known equivalences among Lagrangian, Eulerian, convex-momentum, Hamilton-Jacobi, and Hamiltonian formulations of the transport distance.

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