This paper deals with dynamical optimal transport metrics defined by 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 nonlocal mobility at the discrete level. Loosely speaking, the result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport.We start by informally introducing the main objects of study in this paper and present the main result. For more formal definitions we refer to Section 2 below.Continuous optimal transport. Let P(S) (resp. M (S)) denote the set of Borel probability measures (resp. signed measures) on a Polish space (S, d). We will work on the one-dimensional torus S 1 = R/Z and use the convention that arithmetic operations are understood modulo 1.

We consider finite-volume approximations of Fokker-Planck equations on bounded convex domains in R d and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker-Planck equation via the method of Evolutionary Γ-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalising the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.

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This paper deals with the large-scale behaviour of dynamical optimal transport on Z d -periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density.Our homogenisation result is derived from a Γ-convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs. Contents 42 9. Analysis of the cell problem 57 Appendix A. The Kantorovich-Rubinstein metric on signed measures 66 Appendix B. Norms on curves in the space of measures 68 Appendix C. Domain property of convex functions 68 Appendix D. Notation 68 References 70