Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs
For mathematicians studying optimal transport on graphs, this extends convergence theory to linear growth cases, which was an open problem.
The paper proves discrete-to-continuum convergence for dynamical optimal transport on periodic graphs with linear growth energy density, extending previous results that required superlinear growth. This enables scaling limits of 1-Wasserstein transport problems.
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.