Stratified Monge–Kantorovich Optimal Transport Problems

Abstract. The topology and uniqueness of optimal plans in the Monge–Kantorovich optimization problem are central topics of research in this field. In this paper, we study Monge–Kantorovich optimal transport problems while relaxing the condition of absolute continuity for the marginal measures. Let [Formula: see text], where [Formula: see text] and [Formula: see text] are Borel probability spaces, and [Formula: see text] is a cost function. The goal is to minimize the integral of [Formula: see text] over all couplings [Formula: see text] in [Formula: see text], the set of all couplings of [Formula: see text] and [Formula: see text]. Inspired by a shape recognition problem in computer vision, Gangbo and McCann demonstrated for the quadratic cost [Formula: see text] that Kantorovich solutions are unique, even when Monge solutions fail to exist. This occurs when [Formula: see text] is allowed to charge [Formula: see text]-dimensional subsets of [Formula: see text]. Motivated by their findings, this work has two main objectives. First, we explore an optimal transport problem with a multilayered target space for a cost function [Formula: see text], where [Formula: see text] is strictly convex. Specifically, we consider [Formula: see text] and [Formula: see text] structured as follows: [Formula: see text], and [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] are distinct real numbers. The restriction of [Formula: see text] to [Formula: see text] is absolutely continuous with respect to the [Formula: see text]-dimensional Lebesgue measure, but [Formula: see text] is singular relative to the [Formula: see text]-dimensional Lebesgue measure. For [Formula: see text], the problem reduces to a standard Monge–Kantorovich transport problem with a unique solution concentrated on a single map. We show that for [Formula: see text], the solution remains unique but concentrates on the graph of several maps. Second, we consider a general setting where [Formula: see text], and [Formula: see text] has a marginal form that combines [Formula: see text]-dimensional and [Formula: see text]-dimensional components. Specifically, [Formula: see text] can be expressed as a sum of two parts: one absolutely continuous with respect to the [Formula: see text]-dimensional Lebesgue measure and another supported on an [Formula: see text]-dimensional manifold [Formula: see text], with a measure [Formula: see text] that is absolutely continuous with respect to the [Formula: see text]-dimensional Lebesgue measure in each coordinate chart of [Formula: see text]. This setting can be interpreted as a two-layer problem where [Formula: see text] charges both [Formula: see text]-dimensional and [Formula: see text]-dimensional subsets.