Homogenisation of dynamical optimal transport on periodic graphs

TL;DR

This paper establishes a homogenisation theory for dynamical optimal transport on periodic graphs, linking discrete graph problems to continuous optimal transport via explicit cell formulas and $ ext{Gamma}$-convergence.

Contribution

It provides the first homogenisation result for dynamical optimal transport on periodic graphs, with explicit cell formulas depending on local geometry.

Findings

Explicit cell formula for effective energy density.

Homogenisation via $ ext{Gamma}$-convergence under mild conditions.

Non-trivial limits in finite-volume discretisations of Wasserstein distance.

Abstract

This paper deals with the large-scale behaviour of dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density. Our homogenisation result is derived from a $\Gamma$-convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs.