# Wasserstein Hamiltonian flows

## TL;DR

This paper develops a framework for Hamiltonian flows in the space of probability densities using the Wasserstein metric, unifying classical equations like Vlasov and Schrödinger equations under this formalism.

## Contribution

It introduces a novel Hamiltonian flow formalism in density space with Wasserstein metric, connecting various classical equations.

## Key findings

- Reformulation of Vlasov, Schrödinger, and Schrödinger bridge equations as Hamiltonian flows in density space.
- Derivation of Euler-Lagrange equations in the Wasserstein density space.
- Establishment of kinetic Hamiltonian flows in the $L^2$-Wasserstein metric tensor.

## Abstract

We establish kinetic Hamiltonian flows in density space embedded with the $L^2$-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, Schr{\"o}dinger equation and Schr{\"o}dinger bridge problem, can be rewritten as the formalism of Hamiltonian flows in density space.