Quantitative Stability in Discrete Optimal Transport

Abstract: This work investigates several aspects related to quantitative stability in optimal transport, as well as uniqueness of the dual transport problem. Our main contributions are as follows. Chapter 1: Observations regarding the quantitative stability of optimal transport plans with respect to Wasserstein distance on the product space. Chapter 2: Extention of strong convexity inequalities for the Kantorovich functional to a larger class of source measures, using glueing arguments recently used for the quantitative stability of optimal transport maps. Chapters 3/4: A qualitative description of the behaviour of the fully discrete transport problem under perturbation of the support positions, as well as quantitative stability under uniqueness assumptions. Chapter 5: Extention of known uniqueness criteria for the dual transport problem. We show that when one marginal measure has Lipschitz-path connected support and the other has bounded support, the values of dual optimisers are unique up to a constant for a large family of costs, including $p$-costs for all $p>1$.