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# memo_has_moved_text();Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems

Wilfrid Gangbo, Georgia Institute of Technology, Atlanta, Georgia, Hwa Kil Kim, Georgia Institute of Tecnology, Atlanta, Georgia and Tommaso Pacini, Mathematical Institute, Oxford, United Kingdom

Publication: Memoirs of the American Mathematical Society
Publication Year: 2011; Volume 211, Number 993
ISBNs: 978-0-8218-4939-2 (print); 978-1-4704-0610-3 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2010-00610-0
Published electronically: September 27, 2010
MathSciNet review: 2808856
MSC: Primary 37Kxx, 49-XX; Secondary 35Qxx, 53Dxx

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Chapters

• Chapter 1. Introduction
• Chapter 2. The topology on $M$ and a differential calculus of curves
• Chapter 3. The calculus of curves, revisited
• Chapter 4. Tangent and cotangent bundles
• Chapter 5. Calculus of pseudo differential 1-forms
• Chapter 6. A symplectic foliation of $M$
• Chapter 7. The symplectic foliation as a Poisson structure
• Appendix A. Review of relevant notions of differential geometry

### Abstract

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 we develop a calculus for the corresponding class of differential forms on $\mathcal{M}$. In particular we 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 $D=2d$ we 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 we emphasize the geometric viewpoint and the role played by certain diffeomorphism groups of $\mathbb{R}^D$.