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Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs (2311.17284v1)

Published 28 Nov 2023 in math.OC, cs.NA, math.AP, and math.NA

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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