  We study the problem of homogenization for inertial particles moving in a
periodic 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. To achieve this we use a formal
multiple scale expansion in the scale parameter. This expansion relies on the
hypo-ellipticity of the underlying diffusion. An expression for the diffusivity
tensor is found and various of its properties studied. In particular, an
expansion in terms of the non-dimensional particle relaxation time $\tau$ (the
Stokes number) is shown to co-incide with the known result for passive
(non-inertial) tracers in the singular limit $\tau \to 0$. This requires the
solution of a singular perturbation problem, achieved by means of a formal
multiple scales expansion in $\tau.$ Incompressible and potential fields are
studied, as well as fields which are neither, and theoretical findings are
supported by numerical simulations.
