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

Lorenzo Portinale, Filippo Quattrocchi

TL;DR

This work establishes discrete-to-continuum Gamma-convergence for dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with energy densities of linear growth or flow-based form. It extends previous superlinear-growth results to linear-growth settings, enabling the homogenization of boundary-value transport problems and revealing that the limit energy is governed by a cell-formula $f_\mathrm{hom}$ whose geometry is dictated by the underlying graph. The key contributions include a limsup and liminf analysis, separation of linear-growth and flow-based cases, and a detailed cell-problem analysis with explicit examples showing how $f_\mathrm{hom}$ can yield $W_1$-type limits and crystalline norms. The results have implications for understanding scaling limits of $1$-Wasserstein transport, and they illuminate how graph geometry shapes the homogenized cost, supported by embedded-graph examples and visualizations.

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.

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

TL;DR

This work establishes discrete-to-continuum Gamma-convergence for dynamical optimal transport on -periodic graphs with energy densities of linear growth or flow-based form. It extends previous superlinear-growth results to linear-growth settings, enabling the homogenization of boundary-value transport problems and revealing that the limit energy is governed by a cell-formula whose geometry is dictated by the underlying graph. The key contributions include a limsup and liminf analysis, separation of linear-growth and flow-based cases, and a detailed cell-problem analysis with explicit examples showing how can yield -type limits and crystalline norms. The results have implications for understanding scaling limits of -Wasserstein transport, and they illuminate how graph geometry shapes the homogenized cost, supported by embedded-graph examples and visualizations.

Abstract

We prove discrete-to-continuum convergence for dynamical optimal transport on -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.
Paper Structure (17 sections, 10 theorems, 190 equations, 2 figures)

This paper contains 17 sections, 10 theorems, 190 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General framework: continuous and discrete transport problems
  3. The continuous setting: transport problems on the torus
  4. The discrete framework: transport problems on periodic graphs
  5. Statement and proof of the main result
  6. Proof of the limsup inequality
  7. Proof of the liminf inequality
  8. Case 1: $F$ with linear growth
  9. Case 2: F is flow-based
  10. About the lower semicontinuity of $\mathbb{MA}$
  11. Analysis of the cell problem with examples
  12. Discrete $1$-Wasserstein distance
  13. General properties of $f_\mathrm{hom}$
  14. Embedded graphs
  15. One-dimensional case with nearest-neighbor interaction
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume that either $F$ satisfies the linear growth condition, i.e. for some constant $C<\infty$, or that $F$ does not depend on the $\rho$-variable (flow-based type). Then, as $\varepsilon \to 0$, the discrete functionals $\mathcal{MA}_\varepsilon$$\Gamma$-converge to the continuous functional $\mathbb{MA}_\mathrm{hom}$ with respect to weak convergence.

Figures (2)

  • Figure 1: Example of $\mathbb{Z}^d$-periodic graph embedded in $\mathbb{R}^d$
  • Figure 2: Examples of graphs in $\mathbb{R}^2$ and corresponding unit balls for $f_\mathrm{hom}$

Theorems & Definitions (42)

  • Theorem 1.1: Main result
  • Definition 2.1: Continuity equation
  • Definition 2.3: Action functional
  • Remark 2.4
  • Remark 2.5: Lack of compatible compactness
  • Remark 2.6: Superlinear growth
  • Remark 2.7
  • Remark 2.9
  • Definition 2.10: Discrete energy functional
  • Remark 2.11
  • ...and 32 more