Monge solutions and uniqueness in multi-marginal optimal transport: costs associated to graphs and a general condition
Creators: Vargas-Jiménez, Adolfo
Publisher: name: University of Alberta Library
Year: 2022

<p>This thesis is devoted to the proof of several results on the existence and uniqueness of Monge solutions to the multi-marginal optimal transportation problem. These results are found in Chapters \ref{Chapter3}, \ref{Chapter4} and \ref{Chapter5}, and represent joint work with Brendan Pass. The Chapters \ref{Chapter1} and \ref{Chapter2} are devoted to the introduction and preliminaries respectively. </p>

<p>In Chapter \ref{Chapter3} we study a multi-marginal optimal transportation problem with a cost function of the form $c(x<em>{1}, \ldots,x</em>{m})=\sum<em>{k=1}^{m-1}|x</em>{k}-x<em>{k+1}|^{2} + |x</em>{m}- F(x<em>{1})|^{2}$, where $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a given map. %is a map  from $X</em>{1}$ to $X<em>{1}$, $x</em>{k}\in \mathbb{R}^{n}$.<br>
 When $m=4$, $F$ is a positive multiple of the identity mapping, and the first and last marginals are absolutely continuous with respect to Lebesgue measure, we establish that any solution of the Kantorovich problem is induced by a map; the solution is therefore unique.  We go on to show that this result is sharp in a certain sense.  Precisely, we exhibit examples showing that Kantorovich solutions may concentrate on higher dimensional sets if any of the following hold: 1) $F$ is any linear mapping other than a positive scalar multiple of the identity, 2) the last marginal is not absolutely continuous with respect to Lebesgue measure, or 3) the number of marginals    $m \geq 5$, even when $F$ is the identity mapping. In the fourth chapter we study a multi-marginal optimal transport problem with cost $c(x<em>{1}, \ldots, x</em>{m})=\sum<em>{\{i,j\}\in P} |x</em>{i}- x<em>{j}|^{2}$, where $P\subseteq Q:=\{\{i,j\}: i, j \in \{1,2,...m\}, i \neq j\}$. We reformulate this problem by associating each cost of this type with a graph with $m$ vertices whose set of edges is indexed by $P$.  We then establish uniqueness and Monge solution results for two general classes of cost functions. Among many other examples, these classes encapsulate  the Gangbo and \&#39;{S}wi\c{e}ch cost \cite{GangboSwicech1998} and the cost $c(x</em>{1}, \ldots,x<em>{m})=\sum</em>{k=1}^{m-1}|x<em>{k}-x</em>{k+1}|^{2} + |x<em>{m}- x</em>{1}|^{2}$ when $m\leq 4$. In the final chapter we establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with respect to Lebesgue measure. When only the first marginal is assumed to be absolutely continuous, our condition is equivalent to the twist on splitting sets condition found in \cite{KimPass2014}.  In addition, it is satisfied by the special cost functions of Chapter \ref{Chapter3} and \ref{Chapter4} (found also in \cite{PassVargas2021, PassVargas21}), when absolute continuity is imposed on certain other collections of marginals.  We also present several  new examples of cost functions which violate the twist on splitting sets condition but satisfy the new condition introduced here; we therefore obtain Monge solution and uniqueness results for these cost functions, under  regularity conditions on an appropriate subset of the marginals.</p>