We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in R d . These metrics are natural discrete counterparts to the Kantorovich metric W 2 , defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric W T in terms of W 2 , as the size of the mesh T tends to 0. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence of the transport metric.

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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 prove weak and strong versions of the coarea formula and the chain rule for distributional Jacobian determinants Ju for functions u in fractional Sobolev spaces W s,p (Ω), where Ω is a bounded domain in R n with smooth boundary. The weak forms of the formulae are proved for the range sp > n − 1, s > n−1 n , while the strong versions are proved for the range sp ≥ n, s ≥ n n+1 . We also provide a chain rule for distributional Jacobian determinants of Hölder functions and point out its relation to two open problems in geometric analysis.

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