| 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 [2]. We regularize H via Moreau-Yosida approximation to get Htau and denote by mutau a solution of system with the new Hamiltonian Htau. Suppose Htau converges to H as tau tends to zero. We show mutau converges to mu and mu 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. |
