We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretisation for the equation under study.

翻译：我们研究了一类具有空间张量迁移率的非局部偏微分方程，该方程由无穷局部化图上的非局部动力学渐近导出。我们的策略依赖于两个方程的变分结构，分别对应黎曼和芬斯勒梯度流。更精确地说，我们证明了图上非局部交互作用方程的弱解收敛到欧氏空间中上述具有张量迁移率的非局部交互作用方程类的弱解。这凸显了图的一个有趣性质：它可作为所研究方程的一种潜在空间离散化方法。