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

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.