  In the spirit of the macroscopic crowd motion models with hard congestion
(i.e. a strong density constraint $\rho\leq 1$) introduced by Maury {\it et
al.} some years ago, we analyze a variant of the same models where diffusion of
the agents is also taken into account. From the modeling point of view, this
means that individuals try to follow a given spontaneous velocity, but are
subject to a Brownian diffusion, and have to adapt to a density constraint
which introduces a pressure term affecting the movement. From the PDE point of
view, this corresponds to a modified Fokker-Planck equation, with an additional
gradient of a pressure (only living in the saturated zone $\{\rho=1\}$) in the
drift. The paper proves existence and some estimates, based on optimal
transport techniques.
