Homogenization of Wasserstein gradient flows

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.