  We study the Fokker-Planck diffusion equation with diffusion coefficient
depending periodically on the space variable. Inside a periodic array of
inclusions the diffusion coefficient is reduced by a factor called the
diffusion magnitude. We find the upscaled equations obtained by taking both the
degeneration and the homogenization limits in which the diffusion magnitude and
the scale of the periodicity tends, respectively, to zero. Different behaviors,
classified as pure diffusion, diffusion with mass deposition, and absence of
diffusion, are found depending on the order in which the two limits are taken
and on the ratio between the size of the inclusions and the scale of the
periodicity.
