Dissipative Hamiltonian structure of the kinetic Fokker-Planck equation and its connections to sampling

Abstract: The kinetic Fokker-Planck equation (KFP) describes the evolution of the probability density of the position and velocity of particles under the influence of external confinement, 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. Moreover, the geometric structure has possible connections to the conjectured optimal convergence rates of underdamped Langevin Monte Carlo (ULMC), a sampling algorithm known to empirically outperform the (standard) Langevin Monte Carlo. This talk will focus on a time-discrete variational scheme for KFP which we introduce to more rigorously understand the geometric structure.
We will begin by introducing the optimal transport problem and the Wasserstein distance, as well as the techniques of gradient flows which form the basis of our variational scheme. After highlighting the connections to ULMC, we will discuss how the proposed variational scheme is (i) consistent with the dissipative Hamiltonian structure, and (ii) (geodesically)-convex at each iteration.