Hamiltonian systems and the calculus of differential forms on the Wasserstein space

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Please use this identifier to cite or link to this item: http://hdl.handle.net/1853/29720

Title: Hamiltonian systems and the calculus of differential forms on the Wasserstein space
Author: Kim, Hwa Kil
Abstract: This thesis consists of two parts. In the first part, we study stability properties of Hamiltonian systems on the Wasserstein space. Let H be a Hamiltonian satisfying conditions imposed in the work of Ambrosio and Gangbo. We regularize H via Moreau-Yosida approximation to get H[subscript Tau] and denote by μ[subscript Tau] a solution of system with the new Hamiltonian H[subscript Tau] . Suppose H[subscript Tau] converges to H as τ tends to zero. We show μ[subscript Tau] converges to μ and μ is a solution of a Hamiltonian system which is corresponding to the Hamiltonian H. At the end of first part, we give a sufficient condition for the uniqueness of Hamiltonian systems. In the second part, we develop a general theory of differential forms on the Wasserstein space. Our main result is to prove an analogue of Green's theorem for 1-forms and show that every closed 1-form on the Wasserstein space is exact. If the Wasserstein space were a manifold in the classical sense, this result wouldn't be worthy of mention. Hence, the first cohomology group, in the sense of de Rham, vanishes.
Type: Dissertation
URI: http://hdl.handle.net/1853/29720
Date: 2009-06-01
Publisher: Georgia Institute of Technology
Subject: Hamiltonian systems
Differential forms
Wasserstein space
Hamiltonian systems
Differential forms
Department: Mathematics
Advisor: Committee Chair: Gangbo, Wilfrid; Committee Member: Loss, Michael; Committee Member: Pan, Ronghua; Committee Member: Swiech, Andrzej; Committee Member: Tannenbaum, Allen
Degree: Ph.D.

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