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Differential Forms on Wasserstein Space and Infinite-Dimensional Hamiltonian Systems

Wilfrid Gangbo Georgia Institute of Technology, Atlanta, GA
Hwa Kil Kim Georgia Institute of Technology, Atlanta, GA
Tommaso Pacini Scuola Normale Superiore, Pisa, Italy
Available Formats:
Electronic ISBN: 978-1-4704-0610-3
Product Code: MEMO/211/993.E
List Price: $70.00 MAA Member Price:$63.00
AMS Member Price: $42.00 Click above image for expanded view Differential Forms on Wasserstein Space and Infinite-Dimensional Hamiltonian Systems Wilfrid Gangbo Georgia Institute of Technology, Atlanta, GA Hwa Kil Kim Georgia Institute of Technology, Atlanta, GA Tommaso Pacini Scuola Normale Superiore, Pisa, Italy Available Formats:  Electronic ISBN: 978-1-4704-0610-3 Product Code: MEMO/211/993.E  List Price:$70.00 MAA Member Price: $63.00 AMS Member Price:$42.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 2112011; 77 pp
MSC: Primary 35; 49; Secondary 53;

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 $\mathbb{R}^D$.

• Chapters
• 1. Introduction
• 2. The topology on $\mathcal {M}$ and a differential calculus of curves
• 3. The calculus of curves, revisited
• 4. Tangent and cotangent bundles
• 5. Calculus of pseudo differential 1-forms
• 6. A symplectic foliation of $\mathcal {M}$
• 7. The symplectic foliation as a Poisson structure
• A. Review of relevant notions of Differential Geometry
• Requests

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Volume: 2112011; 77 pp
MSC: Primary 35; 49; Secondary 53;

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 $\mathbb{R}^D$.

• Chapters
• 1. Introduction
• 2. The topology on $\mathcal {M}$ and a differential calculus of curves
• 3. The calculus of curves, revisited
• 4. Tangent and cotangent bundles
• 5. Calculus of pseudo differential 1-forms
• 6. A symplectic foliation of $\mathcal {M}$
• 7. The symplectic foliation as a Poisson structure
• A. Review of relevant notions of Differential Geometry
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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