Differential Forms on Wasserstein Space and Infinite-Dimensional Hamiltonian Systems

作者：Gangbo Wilfrid^*; Kim Hwa Kil; Pacini Tommaso

来源：Memoirs of the American Mathematical Society, 2011, 211(993): 1-+.

DOI：10.1090/S0065-9266-2010-00610-0

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.

• 出版日期2011-5