Singular Perturbations of Hamilton-Jacobi Equations in the Wasserstein Space

We study a singular perturbation problem for second-order Hamilton-Jacobi equations in the Wasserstein space. Specifically, we characterize the behavior of the solutions as the perturbation parameter $\epsilon$ tends to zero. The notion of solution we adopt is that of viscosity solutions in the sense of test functions on the Wasserstein space. Our proof utilizes the perturbed test function method, appropriately adapted to this setting. Finally, we highlight a connection with the homogenization of conditional slow-fast McKean-Vlasov stochastic differential equations.