Remote access

How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2294  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax

Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems

About this Title

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)
Published electronically: September 27, 2010
MathSciNet review: 2808856
MSC: Primary 37Kxx, 49-XX; Secondary 35Qxx, 53Dxx

View full volume PDF

View other years and numbers:

Table of Contents


  • 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


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 $\mathbb{R}^D$.

References [Enhancements On Off] (What's this?)