Extending the Wasserstein metric to positive measures

We define a metric in the space of positive finite positive measures that extends the 2-Wasserstein metric, i.e. its restriction to the set of probability measures is the 2-Wasserstein metric. We prove a dual and a dynamic formulation and extend the gradient flow machinery of the Wasserstein space. In addition, we relate the barycenter in this space to the barycenter in the Wasserstein space of the normalized measures.