This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou–Benamou formula for the Kantorovich metric ${\mathbb{W}}_2$. Such metrics appear naturally in discretisations of ${\mathbb{W}}_2$-gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to ${\mathbb{W}}_2$, 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.

本文讨论了由Kanamovich度量的Benamou–Benamou公式的空间离散化定义的动态最优运输度量。 ${\mathbb{w\; ^}}_2$。这样的度量自然出现在对${\mathbb{w\; ^}}_2$耗散PDE的梯度流动配方。但是，最近显示，这些指标通常不会收敛到${\mathbb{w\; ^}}_2$，除非对离散网格施加强的几何约束。在本文中，我们证明了在一维周期性设置中，离散的运输度量收敛到具有非平凡有效迁移率的限制运输度量。该迁移率敏感地取决于网格的几何形状以及离散级别的非局部迁移率。我们的结果量化了离散运输可以在多大程度上利用网格中的微结构来降低运输成本。