We prove discrete-to-continuum convergence for dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with energy density having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of 1-Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.

翻译：本文证明在能量密度具有无穷远处线性增长条件下，$\mathbb{Z}^d$周期图上的动态最优传输具有离散-连续收敛性。该结果回答了Gladbach、Kopfer、Maas与Portinale（《变分法与偏微分方程》62(5), 2023）提出的未解决问题，该文献仅在超线性增长这一更强假设下证明了离散边界值动态传输问题的收敛行为。本研究将现有理论拓展至重要案例类别，如1-Wasserstein传输问题的标度极限。与二次情形类似，图的几何结构对极限代价函数的构造具有关键影响——本文末章通过可视化示例对此进行了探讨。