This paper deals with the large-scale behaviour of dynamical optimal transport on $\mathbb{Z}^d$-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 $\Gamma$-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.

翻译：本文涉及以 $mathbb ⁇ d$- 周期性图表 进行动态最佳运输的大规模行为, 以及一般低半连续和混凝土能量密度。 我们的主要贡献是同质化结果, 该结果描述了离散问题在连续最佳运输问题方面的实际行为。 有效的能源密度可以用一个单元格公式来明确表达, 这个公式是一个不依赖于离散图形和离散能量密度的局部几何的有限维维维度线编程问题。 我们的同质化结果来自一个用于测量曲线上动作功能的 $\ gamma$- convergence结果, 我们证明在能量密度的非常温和增长条件下。 我们在若干感兴趣的情况下对细胞公式进行了调查, 包括瓦瑟斯坦 距离的有限量离异化, 在那里发生了非三角限制行为 。