Hamiltonian systems and the calculus of differential forms on the Wasserstein space
Kim, Hwa Kil
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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.