  In this article, we investigate some of the fine properties of the value
function associated to an optimal control problem in the Wasserstein space of
probability measures. Building on new interpolation and linearisation formulas
for non-local flows, we prove semiconcavity estimates for the value function,
and establish several variants of the so-called sensitivity relations which
provide connections between its superdifferential and the adjoint curves
stemming from the maximum principle. We subsequently make use of these results
to study the propagation of regularity for the value function along optimal
trajectories, as well as to investigate sufficient optimality conditions and
optimal feedbacks for mean-field optimal control problems.
