Stratified Monge-Kantorovich optimal transport problems

Abstract

In this paper, we investigate Monge-Kantorovich problems for which the absolute continuity of marginals is relaxed. For $X,Y\subseteq {\mathbb{R}}^{n+1}$ let $(X,{\mathcal{B}}_X,\mu )$ and $(Y,{\mathcal{B}}_Y,\nu )$ be two Borel probability spaces, $c:X\times Y\to \mathbb{R}$ be a cost function, and consider the problem \begin{align*}\tag{MKP}\label{MKPEQ} \inf\left\{\int_{X\times Y} c(x,y)\,dλ :\ λ\inΠ(μ,ν) \right\}. \end{align*} Inspired by the seminal paper \cite{GANGBOMCCANN2} with applications in shape recognition problem, we first consider \eqref{MKPEQ} for the cost $c(x,y)=h(x-y)$ with $h$ strictly convex defined on the multi-layers target space \begin{align*} X=\overline{X}\times\{\overline{x}\},\quad\text{and}\quad Y=\bigcup_{k=1}^K \left(\overline{Y}_{k}\times \{\overline{y}_k\}\right), \end{align*} where $\overline{X},{\overline{Y}}_k\subseteq {\mathbb{R}}^n$ for $k\in \{1,\dots ,K\},$ $\overline{x}\in \mathbb{R}$, and $\{{\overline{y}}_1,...,{\overline{y}}_K\}\subseteq \mathbb{R}$. Here, we assume that $μ|_\overline{X}\ll\mathcal{L}^n$ (the Lebesgue measure on ${\mathbb{R}}^n$), but $\mu$ is singular w.r.t. ${\mathcal{L}}^{n+1}$. When $K=1$, this translates to the standard \eqref{MKPEQ} for which the unique solution is concentrated on a map. We show that for $K\ge 2,$ the solution is still unique but it concentrates on the graph of several maps. Next, we study \eqref{MKPEQ} for a closed subset $X\subseteq {\mathbb{R}}^{n+1}$ and its $n$-dimensional submanifold $X_0$ with the first marginal of the form \begin{align*} \int_X f(x)\,dμ(x)=\int_X f(x)α(x)\,d\mathcal{L}^{n+1}(x)+\int_{X_0} f(x_0)\,d S(x_0),\ \ \forall f\in C_b(X). \end{align*} Here, $S$ is a measure on $X_0$ such that $S\ll {\mathcal{L}}^n$ on each coordinate chart of $X_0$. This can be seen as a two-layers problem as the measure $\mu$ charges both $n$- and $n+1$-dimensional subsets.