  We consider the stochastic target problem of finding the collection of
initial laws of a mean-field stochastic differential equation such that we can
control its evolution to ensure that it reaches a prescribed set of terminal
probability distributions, at a fixed time horizon. Here, laws are considered
conditionally to the path of the Brownian motion that drives the system. We
establish a version of the geometric dynamic programming principle for the
associated reachability sets and prove that the corresponding value function is
a viscosity solution of a geometric partial differential equation. This
provides a characterization of the initial masses that can be almost-surely
transported towards a given target, along the paths of a stochastic
differential equation. Our results extend [16] to our setting.
