  We propose a new approach to studying classical solutions of the Bellman
equation and Master equation for mean field type control problems, using a
novel form of the "lifting" idea introduced by P.-L. Lions. Rather than
studying the usual system of Hamilton-Jacobi/Fokker-Planck PDEs using analytic
techniques, we instead study a stochastic control problem on a specially
constructed Hilbert space, which is reminiscent of a tangent space on the
Wasserstein space in optimal transport. On this Hilbert space we can use
classical control theory techniques, despite the fact that it is infinite
dimensional. A consequence of our construction is that the mean field type
control problem appears as a special case. Thus we preserve the advantages of
the lifiting procedure, while removing some of the difficulties. Our approach
extends previous work by two of the coauthors, which dealt with a deterministic
control problem for which the Hilbert space could be generic.
