  We study the mean field Schr\"odinger problem (MFSP), that is the problem of
finding the most likely evolution of a cloud of interacting Brownian particles
conditionally on the observation of their initial and final configuration. Its
rigorous formulation is in terms of an optimization problem with marginal
constraints whose objective function is the large deviation rate function
associated with a system of weakly dependent Brownian particles. We undertake a
fine study of the dynamics of its solutions, including quantitative energy
dissipation estimates yielding the exponential convergence to equilibrium as
the the time between observations grows larger and larger, as well as a novel
class of functional inequalities involving the mean field entropic cost (i.e.
the optimal value in (MFSP)). Our strategy unveils an interesting connection
between forward backward stochastic differential equations and the Riemannian
calculus on the space of probability measures introduced by Otto, which is of
independent interest.
