Homogenisation of one-dimensional discrete optimal transport

This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou-Benamou formula for the Kantorovich metric W-2. Such metrics appear naturally in discretisations of W-2-gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to W-2, unless strong geometric constraints are imposed on the discrete mesh. In this paper we prove that, in a 1-dimensional periodic setting, discrete transport metrics converge to a limiting transport metric with a non-trivial effective mobility. This mobility depends sensitively on the geometry of the mesh and on the non-local mobility at the discrete level. Our result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport.