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arXiv · 2311.17284

Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs

Lorenzo Portinale·Filippo Quattrocchi
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Abstract

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.

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Lorenzo Portinale, Filippo Quattrocchi. 2023-11-28. Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs. https://doi.org/10.1017/s0956792524000810

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