Unbalanced optimal total variation transport problems and generalized Wasserstein barycenters

In this paper, we establish a Kantorovich duality for unbalanced optimal total variation transport problems. As consequences, we recover a version of duality formula for partial optimal transports established by Caffarelli and McCann; and we also get another proof of Kantorovich–Rubinstein theorem for generalized Wasserstein distance $\widetilde {W}_1^{a,b}$ proved before by Piccoli and Rossi. Then we apply our duality formula to study generalized Wasserstein barycenters. We show the existence of these barycenters for measures with compact supports. Finally, we prove the consistency of our barycenters.

Duality results for iterated function systems with a general family of branches

Given [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] compact metric spaces, we consider two iterated function systems [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are contractions. Let [Formula: see text] be the set of probabilities [Formula: see text] with [Formula: see text]-marginal being holonomic with respect to [Formula: see text] and [Formula: see text]-marginal being holonomic with respect to [Formula: see text]. Given [Formula: see text] and [Formula: see text], let [Formula: see text] be the set of probabilities in [Formula: see text] having [Formula: see text]-marginal [Formula: see text] and [Formula: see text]-marginal [Formula: see text]. Let [Formula: see text] be the relative entropy of [Formula: see text] with respect to [Formula: see text] and [Formula: see text] be the relative entropy of [Formula: see text] with respect to [Formula: see text]. Given a cost function [Formula: see text], let [Formula: see text]. We will prove the duality equation: [Formula: see text] In particular, if [Formula: see text] and [Formula: see text] are single points and we drop the entropy, the equation above can be rewritten as the Kantorovich duality for the compact spaces [Formula: see text] and a continuous cost function [Formula: see text].