Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems

By Wilfrid Gangbo, Hwa Kil Kim, T. Pacini. 2008

46 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0dc64e6e11848c87b85cfae2058ff6dd31106aa1

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In Applied Mathematics

1. Chang Liu
2. Jingwei Zhuo
3. Jun Zhu

In Geometry and Topology

1. Chang Liu
2. Jingwei Zhuo
3. Jun Zhu

Citations in the corpus

• On differentiability in the Wasserstein space and well-posedness for Hamilton–Jacobi equations (2019, unknown): “19 [14]) W2 2 (σt, σ t ) ≤ (2)‖id‖(2)%, 39”
• Understanding MCMC Dynamics as Flows on the Wasserstein Space (2019, background): “6; Gangbo et al. (2010), Sec.”