Comparison of distances between measures

The problem of optimal transportation between a set of sources and a set of wells has become recently the object of new mathematical models generalizing the Monge-Kantorovich problem. These models are more realistic as they predict the observed branching structure of communication networks. They also define new distances between measures. The question arises of how these distances compare to the classical Wasserstein distance obtained by the Monge-Kantorovich problem. In this paper we show sharp inequalities between the dα distance induced by branching transport paths and the classical Wasserstein distance over probability measures in a compact domain of Rm . The problem of the optimal mass transportation was introduced by Monge in the 18th century. Kantorovich gave it a first rigorous mathematical treatment. In the Monge-Kantorovich model, two probability measures µ+ and µ− (the source and target mass distributions) are given. Each particle of µ+ travels on a straight line se...

• Type: Journal
• Year: 2007
• Where: APPML
• Authors: Jean-Michel Morel, Filippo Santambrogio
• Keywords: APPML 2007, Branching Structure, Classical Wasserstein Distance, Monge-Kantorovich Problem