  We study the problem of homogenization for inertial particles moving in a
time dependent random velocity field and subject to molecular diffusion. We
show that, under appropriate assumptions on the velocity field, the
large--scale, long--time behavior of the inertial particles is governed by an
effective diffusion equation for the position variable alone. This is achieved
by the use of a formal multiple scales expansion in the scale parameter. The
expansion relies on the hypoellipticity of the underlying diffusion. An
expression for the diffusivity tensor is found and various of its properties
are studied. The results of the formal multiscale analysis are justified
rigorously by the use of the martingale central limit theorem. Our theoretical
findings are supported by numerical investigations where we study the
parametric dependence of the effective diffusivity on the various
non--dimensional parameters of the problem.
