Author:
Gladbach Peter,Kopfer Eva,Maas Jan,Portinale Lorenzo
Abstract
AbstractThis paper deals with the large-scale behaviour of dynamical optimal transport on$$\mathbb {Z}^d$$Zd-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$$\Gamma $$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.
Funder
H2020 European Research Council
Austrian Science Fund
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
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