Transport Type Metrics on the Space of Probability Measures Involving Singular Base Measures

Abstract

We develop the theory of a metric, which we call the $$\nu $$ -based Wasserstein metric and denote by $$W_\nu $$ , on the set of probability measures $${\mathcal {P}}(X)$$ on a domain $$X \subseteq \mathbb {R}^m$$ . This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure $$\nu $$ and is relevant in particular for the case when $$\nu $$ is singular with respect to m-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The $$\nu $$ -based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to $$\nu $$ ; we also characterize it in terms of integrations of classical Wasserstein metric between the conditional probabilities when measures are disintegrated with respect to optimal transport to $$\nu $$ , and through limits of certain multi-marginal optimal transport problems. We also introduce a class of metrics which are dual in a certain sense to $$W_\nu $$ , defined relative to a fixed based measure $$\mu $$ , on the set of measures which are absolutely continuous with respect to a second fixed based measure $$\sigma $$ . As we vary the base measure $$\nu $$ , the $$\nu $$ -based Wasserstein metric interpolates between the usual quadratic Wasserstein metric (obtained when $$\nu $$ is a Dirac mass) and a metric associated with the uniquely defined generalized geodesics obtained when $$\nu $$ is sufficiently regular (eg, absolutely continuous with respect to Lebesgue). When $$\nu $$ concentrates on a lower dimensional submanifold of $$\mathbb {R}^m$$ , we prove that the variational problem in the definition of the $$\nu $$ -based Wasserstein metric has a unique solution. We also establish geodesic convexity of the usual class of functionals, and of the set of source measures $$\mu $$ such that optimal transport between $$\mu $$ and $$\nu $$ satisfies a strengthening of the generalized nestedness condition introduced in McCann and Pass (Arch Ration Mech Anal 238(3):1475–1520, 2020). We finally introduce a slight variant of the dual metric mentioned above in order to prove convergence of an iterative scheme to solve a variational problem arising in game theory.