# On Properties of the Generalized Wasserstein Distance

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## Abstract

The Wasserstein distances *W* _{ p } (*p* \({\geqq}\) 1), defined in terms of a solution to the Monge–Kantorovich problem, are known to be a useful tool to investigate transport equations. In particular, the Benamou–Brenier formula characterizes the square of the Wasserstein distance *W* _{2} as the infimum of the kinetic energy, or action functional, of all vector fields transporting one measure to the other. Another important property of the Wasserstein distances is the Kantorovich–Rubinstein duality, stating the equality between the distance *W* _{1}(*μ, ν*) of two probability measures *μ, ν* and the supremum of the integrals in d(*μ* −
*ν*) of Lipschitz continuous functions with Lipschitz constant bounded by one. An intrinsic limitation of Wasserstein distances is the fact that they are defined only between measures having the same mass. To overcome such a limitation, we recently introduced the generalized Wasserstein distances \({W_p^{a,b}}\), defined in terms of both the classical Wasserstein distance *W* _{ p } and the total variation (or *L* ^{1}) distance, see (Piccoli and Rossi in Archive for Rational Mechanics and Analysis 211(1):335–358, 2014). Here *p* plays the same role as for the classic Wasserstein distance, while *a* and *b* are weights for the transport and the total variation term. In this paper we prove two important properties of the generalized Wasserstein distances: (1) a generalized Benamou–Brenier formula providing the equality between \({W_2^{a,b}}\) and the supremum of an action functional, which includes a transport term (kinetic energy) and a source term; (2) a duality *à la* Kantorovich–Rubinstein establishing the equality between \({W_1^{1,1}}\) and the flat metric.

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