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A variational perspective on the dissipative Hamiltonian structure of the Vlasov-Fokker-Planck equation

Sangmin Park

TL;DR

The paper advances a geometric understanding of the Vlasov-Fokker-Planck equation by treating it as a damped Hamiltonian flow in the 2-Wasserstein space. It introduces a time-discrete coordinate-wise minimizing movement scheme that alternates velocity and position updates via a pair of fibered Wasserstein distances, leveraging geodesic-convexity of the energy $\mathcal{H}=\mathcal{V}+\mathcal{W}+\mathcal{U}$ to ensure well-posedness and discrete energy dissipation. Under Lipschitz gradients of the confinement and interaction potentials, the discrete solutions converge to the distributional solution of VFP as the step size vanishes, and the scheme yields a robust variational proof of existence of weak solutions. The work connects the dissipative Hamiltonian structure to a rigorous, constructive discretization, providing discrete analogues of energy decay and convergence rates that align with the continuous hypocoercive behavior. This framework opens doors to stable, structure-preserving simulations of kinetic systems in high dimensions and enhances understanding of convergence to equilibrium from a geometric vantage point.

Abstract

The Vlasov-Fokker-Planck equation describes the evolution of the probability density of the position and velocity of particles under the influence of external confinement, interaction, friction, and stochastic force. It is well-known that this equation can be formally seen as a dissipative Hamiltonian system in the Wasserstein space of probability measures. In order to better understand this geometric formalism, we introduce a time-discrete variational scheme, solutions of which converge to the solution of the Vlasov-Fokker-Planck equation as time step vanishes; in particular, this provides a new proof of the existence of a weak solution to the equation. The variational scheme combines the symplectic Euler scheme and the (degenerate) implicit steepest descent, and updates the probability density at each iteration first in the velocity variable then in the position variable. The algorithm leverages the geometric structure of the equation, and has several desirable properties. Energy functionals involved in each variational problem are geodesically-convex, which implies the unique solvability of the problem. Furthermore, the correct dissipation of the Hamiltonian is observed at the discrete level up to higher order errors controlled by the second moments.

A variational perspective on the dissipative Hamiltonian structure of the Vlasov-Fokker-Planck equation

TL;DR

The paper advances a geometric understanding of the Vlasov-Fokker-Planck equation by treating it as a damped Hamiltonian flow in the 2-Wasserstein space. It introduces a time-discrete coordinate-wise minimizing movement scheme that alternates velocity and position updates via a pair of fibered Wasserstein distances, leveraging geodesic-convexity of the energy to ensure well-posedness and discrete energy dissipation. Under Lipschitz gradients of the confinement and interaction potentials, the discrete solutions converge to the distributional solution of VFP as the step size vanishes, and the scheme yields a robust variational proof of existence of weak solutions. The work connects the dissipative Hamiltonian structure to a rigorous, constructive discretization, providing discrete analogues of energy decay and convergence rates that align with the continuous hypocoercive behavior. This framework opens doors to stable, structure-preserving simulations of kinetic systems in high dimensions and enhances understanding of convergence to equilibrium from a geometric vantage point.

Abstract

The Vlasov-Fokker-Planck equation describes the evolution of the probability density of the position and velocity of particles under the influence of external confinement, interaction, friction, and stochastic force. It is well-known that this equation can be formally seen as a dissipative Hamiltonian system in the Wasserstein space of probability measures. In order to better understand this geometric formalism, we introduce a time-discrete variational scheme, solutions of which converge to the solution of the Vlasov-Fokker-Planck equation as time step vanishes; in particular, this provides a new proof of the existence of a weak solution to the equation. The variational scheme combines the symplectic Euler scheme and the (degenerate) implicit steepest descent, and updates the probability density at each iteration first in the velocity variable then in the position variable. The algorithm leverages the geometric structure of the equation, and has several desirable properties. Energy functionals involved in each variational problem are geodesically-convex, which implies the unique solvability of the problem. Furthermore, the correct dissipation of the Hamiltonian is observed at the discrete level up to higher order errors controlled by the second moments.
Paper Structure (18 sections, 31 theorems, 308 equations)

This paper contains 18 sections, 31 theorems, 308 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Summary of results
  4. Related works
  5. The Wasserstein distance with fixed marginals
  6. Motivation: symplectic Euler scheme in Euclidean spaces
  7. Definition and basic properties
  8. Metric slope with respect to the Wasserstein distance with fixed marginals
  9. Partial subdifferential calculus in the Wasserstein space
  10. Convexity and partial subdifferentials of the entropy functional
  11. Partial subdifferential of the linear functionals
  12. Partial subdifferential of the sum of the functionals
  13. A minimizing movement scheme for the Vlasov-Fokker-Planck equation
  14. Existence of minimizing movements: compactness of the discrete solutions
  15. Convergence of the discrete solutions to the weak solution of the PDE
  16. ...and 3 more sections

Key Result

Proposition 2.2

Given $\sigma\in\mathscr{P}_2(\mathbb{R}^d)$, each $(\mathscr{P}_2^v(\mathbb{R}^{2d};\sigma),W_{2,v})$ is a complete metric space.

Theorems & Definitions (77)

  • Remark 2.1: Relationship of $W_{2,v},W_{2,x}$ with $W_2$
  • Proposition 2.2: Completeness of $(\mathscr{P}_2^v,W_{2,v})$
  • proof
  • Definition 2.3
  • Theorem 2.4: Benamou-Brenier formula for $W_{2,v}$
  • proof
  • Proposition 2.5: A refined Arzelà-Ascoli Theorem AGS
  • Proposition 2.6: Lower semicontinuity of $W_{2,v}$ with respect to the narrow topology
  • proof
  • Proposition 2.7: Optimal transport map with fixed marginals
  • ...and 67 more