  We prove the convergence of a Wasserstein gradient flow of a free energy in
an inhomogeneous media. Both the energy and media can depend on the spatial
variable in a fast oscillatory manner. In particular, we show that the gradient
flow structure is preserved in the limit which is expressed in terms of an
effective energy and Wasserstein metric. The gradient flow and its limiting
behavior is analyzed through an energy dissipation inequality (EDI). The result
is consistent with asymptotic analysis in the realm of homogenization. However,
we note that the effective metric is in general different from that obtained
from the Gromov-Hausdorff convergence of metric spaces. We apply our framework
to a linear Fokker-Planck equation but we believe the approach is robust enough
to be applicable in a broader context.
