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 $H_\tau.$ We show that solutions of the Hamiltonian system for $H_\tau$ converge to a solution of the Hamiltonian system for $H$ as $\tau$ converges to zero. We provide sufficient conditions on $H$ to carry out this process.