Viscosity Solutions of the Eikonal Equation on the Wasserstein Space

Dynamic programming equations for mean field control problems with a separable structure are Eikonal type equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity theory. We use Fourier representation of the Sobolev norms on the space of measures, together with the standard techniques from the finite dimensional theory to prove a comparison result among semi-continuous sub and super solutions, obtaining a unique characterization of the value function.