  We study the Bellman equation in the Wasserstein space arising in the study
of mean field control problems, namely stochastic optimal control problems for
McKean-Vlasov diffusion processes.Using the standard notion of viscosity
solution \`a la Crandall-Lions extended to our Wasserstein setting,we prove a
comparison result under general conditions on the drift and reward
coefficients, whichcoupled with the dynamic programming principle, implies that
the value function is the unique viscosity solution of the Master Bellman
equation.This is the first uniqueness result in such a second-order context.
The classical arguments used in the standard cases of equations in
finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces
do not extend to the present framework, due to the awkward nature of the
underlying Wasserstein space. The adopted strategy is based on
finite-dimensional approximations of the value function obtained in terms of
the related cooperative $n$-player game, and on the construction of a smooth
gauge-type function, built starting from a regularization of a sharp estimate
of the Wasserstein metric; such a gauge-type function is used to generate
maxima/minima through a suitable extension of the Borwein-Preiss generalization
of Ekeland's variational principle on the Wasserstein space.
