Young-Heon Kim

We study two variational problems encoding the idea of optimally approximating a probability measure $\rho$ via a lower-dimensional set $\Sigma$ of bounded complexity.In both problems we quantify the performance of $\Sigma$ in approximating $\rho$ via the average of the $p$\textsuperscript{th}-power of the distance to $\Sigma$, $\mathbb{E}_{\omega\sim\rho}$. This yields an optimal transport (OT) interpretation in terms of minimizing the \emph{Monge-Kantorovich $p$-cost} (also called the \emph{Wasserstein $p$-cost}) over measures $\nu$ supported on $\Sigma$. This perspective is essential in framing some of the applications of our work. However, for the theory, it turns out that we can eschew the language of OT in favor of more geometric perspectives.The two problems differ in the way we quantify the ``complexity'' of the approximating sets. In the first problem (Part I) we consider a class of constraints defined in terms of parametrizations $f$ of $\Sigma$. The particular class we choose is the set of $W^{k,q}(\mathbb{R}^m;\mathbb{R}^n)$ Sobolev norms, where $kq>m$ and $n>m$. We build up general theory with these assumptions plus a few technical hypotheses. The generality of the framework allows us to make connections to manifold learning problems with noisy data and certain generative machine learning problems with regularization.One of the key concepts we develop along the way is the ``gradient'' of $\mathbb{E}_{\omega\sim\rho}$; it has the form of a vector field that we call the \emph{barycenter field}. We use this to develop nontrivial numerical methods for simulating the problem. The barycenter field does not rely on the Sobolev structure, so we can use it in studying the second problem as well.In the second problem (Part II) we restrict ourselves to $(m=1)$-dimensional approximations $\Sigma$ and use a fully parametrization-independent constraint, the Hausdorff 1-measure $\mathcal{H}^1(\Sigma)$. This yields optimizers with richer topological structure than those of the Sobolev problem. Via analysis of the barycenter field, we are able to mostly resolve an open problem regarding the topological characterization of optimizers, proving that for general target dimension $n\geq 2$ and objective exponent $p\in\{2\}\cup(\frac{3+\sqrt{5}}{2},\infty)$ optimizers are finite binary trees. We expect the result also holds for $p\in(2,\frac{3+\sqrt{5}}{2}]$ but were unable to show it with our methods.

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In this work, we introduce and study a class of convex functionals on pairs ofprobability measures, the linear transfers, which have a structure that com-monly arises in the dual formulations of many well-studied variational prob-lems. We show that examples of linear transfers include a large number ofwell-known transport problems, including the weak, stochastic, martingale,and cost-minimising transports. Further examples include the balayage ofmeasures, and ergodic optimisation of expanding dynamical systems, amongothers. We also introduce an extension of the linear transfers, the convextransfers, and show that they include the relative entropy functional andp-powers (p ≥ 1) of linear transfers.We study the properties of linear and convex transfers and show that theinf-convolution operation preserves their structure. This allows dual formu-lations of transport-entropy and other related inequalities, to be computedin a systematic fashion.Motivated by connections of optimal transport to the theory of Aubry-Mather and weak KAM for Hamiltonian systems, we develop an analog inthe setting of linear transfers. We prove the existence of an idempotentoperator which maps into the set of weak KAM solutions, an idempotentlinear transfer that plays the role of the Peierls barrier, and we identifyanalogous objects in this setting such as the Mather measures and the Aubryset. We apply this to the framework of ergodic optimisation in the holonomiccase.

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In the first part of this thesis, we study the structure of solutions to the optimal martingale transport problem, when the marginals lie in higher dimensional Euclidean spaces (ℝ^d, d ≥ 2). The problem has been extensively studied in one-dimensional space (ℝ), but few results have been shown in higher dimensions. In this thesis, we propose two conjectures and provide key ideas that lead to solutions in important cases.In the second part, we study the structure of solutions to the optimal subharmonic martingale transport problem, again when the marginals lie in higher dimensional Euclidean spaces. First, we show that this problem has an equivalent formulation in terms of the celebrated Skorokhod embedding problem in probability theory. We then describe the fine structure of the solution provided the marginals are radially symmetric. The general case remains unsolved, and its potential solution calls for a deeper understanding of harmonic analysis and Brownian motion in higher dimensional spaces.

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