  We derive a closed equation for the empirical concentration of colloidal
particles in the presence of both hydrodynamic and direct interactions. The
ensemble average of our functional Langevin equation reproduces known
deterministic Dynamic Density Functional Theory (DDFT) [Rex and L\"owen, Phys.
Rev. Lett., 101(14):148302, 2008], and, at the same time, it also describes the
microscopic fluctuations around the mean behavior. We suggest separating the
ideal (non-interacting) contribution from additional corrections due to
pairwise interactions. We find that, for an incompressible fluid and in the
absence of direct interactions, the mean concentration follows Fick's law just
as for uncorrelated walkers. At the same time, the nature of the stochastic
terms in fluctuating DDFT is shown to be distinctly different for
hydrodynamically-correlated and uncorrelated walkers. This leads to striking
differences in the behavior of the fluctuations around Fick's law, even in the
absence of pairwise interactions. We connect our own prior work [A. Donev, T.
G. Fai, E. Vanden-Eijnden, J. Stat. Mech., P04004, 2014] on fluctuating
hydrodynamics of diffusion in liquids to the DDFT literature, and demonstrate
that the fluid cannot easily be eliminated from consideration if one wants to
describe the collective diffusion in colloidal suspensions.
