On properties of the Generalized Wasserstein distance

Understand

The Wasserstein distances $W_p$ ($p\geq 1$), defined in terms of 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 moving one measure to the other.
• Another important property of the Wasserstein distances is the Kantorovich-Rubinstein duality stating the equality between the distance $W_1$ and the supremum of the integrals 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.