A Wasserstein norm for signed measures, with application to non-local transport equation with source term

We introduce an optimal transportation interpretation of the Kantorovich norm on the space
of signed Radon measures with finite mass, based on the generalized Wasserstein distance for
measures with different masses.
With this new interpretation, we obtain new topological properties for this norm. We use
these tools to prove existence and uniqueness for solutions to non-local transport equations with
source terms, when the initial condition is a signed measure.