  In this paper, we investigate the construction of a diffusion process whose
time-marginal densities are constrained to belong to a given set at all time.
The construction is obtained from a penalization approximation to the
constraint set, acting on the Wasserstein distance W2 to the constraint space.
Under some technical assumptions on the constraint space and the initial
distribution of the model, the penalization approximation yields to a
stochastic differential equation analogous to the Skorohod problem of reflected
diffusion.
