Homogenisation of dynamical optimal transport on periodic graphs

Peter Gladbach; Eva Kopfer; Jan Maas; Lorenzo Portinale

doi:10.1007/s00526-023-02472-z

Homogenisation of dynamical optimal transport on periodic graphs

Calc Var Partial Differ Equ. 2023;62(5):143. doi: 10.1007/s00526-023-02472-z. Epub 2023 Apr 28.

Authors

Peter Gladbach¹, Eva Kopfer¹, Jan Maas², Lorenzo Portinale¹

Affiliations

¹ Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany.
² Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria.

Abstract

This paper deals with the large-scale behaviour of dynamical optimal transport on $Z^{d}$ -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 $Γ$ -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.

Keywords: 49J45; 65K10; 74Q10; Primary: 49Q22 Secondary: 49M25.