Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems

Details Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems

2008-07-07

arxiv.org/abs/0807.1065v2

Mathematics

Analysis of PDEs

Version 2. Improved presentation, slight technical changes. To appear in Memoirs AMS

Scientific paper

Let M denote the space of probability measures on R^D endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in M was introduced by Ambrosio, Gigli and Savare'. In this paper we develop a calculus for the corresponding class of differential forms on 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 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 R^D.

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