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1. Hello this has some LaTeX
- Creator:
- Surname, Given N.
- Depositor:
- jenniferwhitney@cunet.carleton.ca
- Date Uploaded:
- 09/10/2025
- Date Modified:
- 09/10/2025
- Rights Statement:
- In Copyright
- Abstract with a label:
- We begin by investigating the uniqueness of an optimal mass transport problem with $N$ marginals for $N\geq 3$ by transforming it into a lower marginal optimal transportation problem. Specifically, for a family of probability spaces $\{(X_k,\mathcal{B}_{X_k},\mu_k)\}_{k=1}^N$ and a cost function $c: X_1\times\cdots\times X_N\to \mathbb{R}$, we consider \[\tag{\textbf{MKP}}\label{T1} \inf_{\lambda\in\Pi(\mu_1,\ldots,\mu_N)}\int_{\prod_{k=1}^N X_k}c\,d\lambda. \] Then, for each ordered subset $\mathcal{P} of \{1,...,N\}$, we create a new cost function $c_\mathcal{P}$ corresponding to the original cost function $c$ defined on $\prod_{k=1}^p X_{i_k}$. This new cost function $c_\mathcal{P}$ possesses many of the features of the original cost $c$, while having the property that any optimal plan $\lambda$ of \eqref{T1} restricted to $\prod_{k=1}^p X_{i_k}$ is also an optimal plan to the problem \[\label{T2} \inf_{\tau\in\Pi(\mu_{i_1},\ldots\mu_{i_p})}\int_{\prod_{k=1}^p X_{i_k}}c_{\mathcal{P}}\,d\tau. \] Our main contribution is to demonstrate that, for appropriate choices of the index set $\mathcal{P}$, one can recover the optimal plans of \eqref{T1} from \eqref{T2}. We examine situations in which the problem \eqref{T1} admits a unique solution depending on the uniqueness of the solution for the lower marginal problems of the form \eqref{T2}. This allows us to establish numerous uniqueness results for multi-marginal problems, even when the unique optimal plan is not necessarily induced by a map. To achieve this, we extensively employ disintegration theorems and the $c$-extremality notions. Furthermore, we demonstrate several new applications to illustrate the applicability of this approach. Next, we deal with the case $N=2$. For $X,Y\subseteq\mathbb{R}^{n+1}$, for Borel probability spaces $(X,\mathcal{B}_X,\mu)$ and $(Y,\mathcal{B}_Y,\nu)$ and the cost $c:X\times Y\to\mathbb{R}$, we study \[\tag{\textbf{2-MKP}}\label{T3} \inf\left\{\int_{X\times Y} c(x,y)\,d\lambda\ :\ \lambda \in\Pi(\mu,\nu) \right\}. \] We first consider an optimal transport problem with a multi-layers target space for the cost $c(x,y)=h(x-y)$ with $h$ being strictly convex and differentiable. Namely, we assume \[ X=\overline{X}\times\{\overline{x}\},\quad\text{and}\quad Y=\bigcup_{k=1}^K \left(\overline{Y}_{k}\times \{\overline{y}_k\}\right), \] We also assume that $\mu|_{\overline{X}}\ll\mathcal{L}^n$, but $\mu\perp\mathcal{L}^{n+1} $. We show that for $K\geq 2$, the solution to \eqref{T3} is unique but it concentrates on the graph of several maps. Secondly, for general closed subsets $X\subseteq \mathbb{R}^{n+1}$, we consider the case where the first marginal of the form \[ \int_X f(x)\,d\mu(x)=\int_X f(x)\alpha(x)\,d\mathcal{L}^{n+1}(x)+\int_{X_0} f(x_0)\, d S(x_0),\quad \forall f\in C_b(X), \] where $X_0\subseteq X$ is an $n$-dimensional manifold.