Comparison of viscosity solutions for a class of second-order PDEs on the Wasserstein space

Abstract.: We prove a comparison result for viscosity solutions of second-order parabolic partial differential equations in the Wasserstein space. The comparison is valid for semisolutions that are Lipschitz continuous in the measure in a Fourier-Wasserstein metric and uniformly continuous in time. The class of equations we consider is motivated by McKean-Vlasov control problems with common noise and filtering problems. The proof of comparison relies on a decomposition of the Wasserstein space and an application of Ishii’s lemma, which is tailor-made for the class of equations we consider.