# Differential Forms on Wasserstein Space and Infinite-Dimensional Hamiltonian Systems

Share this page
*Wilfrid Gangbo; Hwa Kil Kim; Tommaso Pacini*

Let \(\mathcal{M}\) denote the space of probability measures on \(\mathbb{R}^D\) endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in \(\mathcal{M}\) was introduced by Ambrosio, Gigli, and Savaré. In this paper the authors develop a calculus for the corresponding class of differential forms on \(\mathcal{M}\). In particular they prove an analogue of Green's theorem for 1-forms and show that the corresponding first cohomology group, in the sense of de Rham, vanishes. For \(D=2d\) the authors then define a symplectic distribution on \(\mathcal{M}\) in terms of this calculus, thus obtaining a rigorous framework for the notion of Hamiltonian systems as introduced by Ambrosio and Gangbo. Throughout the paper the authors emphasize the geometric viewpoint and the role played by certain diffeomorphism groups of \(\mathbb{R}^D\).

#### Table of Contents

# Table of Contents

## Differential Forms on Wasserstein Space and Infinite-Dimensional Hamiltonian Systems

- Chapter 1. Introduction 18 free
- Chapter 2. The topology on M and a differential calculus of curves 512 free
- Chapter 3. The calculus of curves, revisited 1118
- Chapter 4. Tangent and cotangent bundles 1724
- Chapter 5. Calculus of pseudo differential 1-forms 2532
- Chapter 6. A symplectic foliation of M 4754
- Chapter 7. The symplectic foliation as a Poisson structure 5764
- Appendix A. Review of relevant notions of Differential Geometry 6370
- Bibliography 7582