A Brief on Optimal Transport

TL;DR

This paper provides an overview of optimal transport theory, covering foundational topology and measure theory, key concepts like couplings, the Kantorovich problem, and Wasserstein distance, with detailed insights from Villani's work.

Contribution

It offers a comprehensive introduction to optimal transport, including existence theorems and metric properties, based on Villani's foundational chapters.

Findings

Existence of optimal transport plans established

Wasserstein distance metrizes probability measures on compact domains

Detailed exposition of Kantorovich problem and couplings

Abstract

The presentation covers prerequisite results from Topology and Measure Theory. This is then followed by an introduction into couplings and basic definitions for optimal transport. The Kantrorovich problem is then introduced and an existence theorem is presented. Following the setup of optimal transport is a brief overview of the Wasserstein distance and a short proof of how it metrizes the space of probability measures on a COMPACT domain. This presentation is a detailed examination of Villani's "Optimal Transport: Old and New" chapters 1-4 and part of 6.