Wasserstein Hamiltonian flows

Abstract

We establish kinetic Hamiltonian flows in density space embedded with the L2-Wasserstein metric tensor. We derive the Euler-Lagrange equation in density space, which introduces the associated Hamiltonian flows. We demonstrate that many classical equations, such as Vlasov equation, Schrödinger equation and Schrödinger bridge problem, can be rewritten as the formalism of Hamiltonian flows in density space.