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

We develop the theory of a metric, which we call the v-based Wasserstein metric and denote by W-v, on the set of probability measures P(X) on a domain X subset of R-m. This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure v and is relevant in particular for the case when v is singular with respect to m-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The v-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to v; 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 v, 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-v, 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 v, the v-based Wasserstein metric interpolates between the usual quadratic Wasserstein metric (obtained when v is a Dirac mass) and a metric associated with the uniquely defined generalized geodesics obtained when v is sufficiently regular (eg, absolutely continuous with respect to Lebesgue). When v concentrates on a lower dimensional submanifold of R-m, we prove that the variational problem in the definition of the v-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 v 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.