In this paper we study macroscopic
density equations
in which the diffusion coefficient
depends on a weighted spatial average of the density itself.
We show that large differences
(not present in the local density-dependence case)
appear between
the density equations that are derived from
different representations
of the Langevin
equation describing a system of interacting
Brownian particles.
Linear stability analysis
demonstrates that under some circumstances the density
equation interpreted like Ito
has pattern solutions, which
never appear for the
kinetic interpretation, which is the other one typically
appearing in the context of nonlinear diffusion processes.
We also introduce a discrete-time microscopic model of
particles that
confirms the results
obtained at the macroscopic density level.