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 R. 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 segment onto μ− and the cost of the transportation to be minimized is the integral of the lengths of the individual paths. This variational model has received a lot of attention because of its remarkable mathematical properties [1], [10]. From the economical viewpoint the Monge-Kantorovich problem is rather unrealistic. In most transportation networks, the aggregation of particles on common routes is preferable to individual straight ones. Thus the local structure of human-designed distribution systems doesn’t look as a set of straight wires but rather like a tree. This branching structure is observable in communication networks [5], drainage networks [7], pipelines [4] and in many natural systems like the blood circulation in mammals, the river basins and the trees. The design of functionals for mass transportation by branched structures was first addressed in [5] as a discrete graph optimization problem with prescribed sources and well points. Recently, continuous models have been proposed for this same setting [9], [8] and [3]. We will describe in the sequel these models in a more detailed way. They all define a cost functional for the transportation between μ and μ−. The optimal value for this functional yields a distance between μ and μ−. Our aim here is to compare this new distance with the so called Wasserstein distance associated with the Monge-Kantorovich model. This distance on probability measures owes its importance to the fact that, on compact domains, it gives a metric to the topology of weak convergence. Given two probability measures μ and μ− with support in a compact domain C ⊂ R this distance is obtained by minimizing the Monge-Kantorovich functional ∫ C×C c(x, y)π(dx, dy) among all probability measures π on C × C whose marginal measures are exactly μ and μ−. We denote by Π(μ, μ−) this set of probabilities Π(μ, μ−) = {π ∈ P(C × C) : X ] π = μ and X− ] π = μ−}, where X± are the projections of C × C onto C, i.e. X(x, y) = x and X−(x, y) = y. The function c : C×C is a given cost function and its semicontinuity is sufficient for the existence of an optimal measure π0 ∈ Π(μ, μ−) which is called optimal transport plan. When c(x, y) = |x− y| the minimum value of this problem is denoted by W1(μ, μ−) and it defines a distance over the spaceP(C) of probability measures on ∗CMLA, ENS Cachan, 61, Av. du Président Wilson 94235 Cachan Cedex France, [email protected] †Scuola Normale Superiore, Piazza dei Cavalieri, 7 56126 Pisa Italy, [email protected]