Moreau–Yosida approximation and convergence of Hamiltonian systems on Wasserstein space

In this paper, we study the stability property of Hamiltonian systems on the Wasserstein space. Let H be a given Hamiltonian satisfying certain properties. We regularize H using the Moreau–Yosida approximation and denote it by Hτ. We show that solutions of the Hamiltonian system for Hτ converge to a solution of the Hamiltonian system for H as τ converges to zero. We provide sufficient conditions on H to carry out this process.