Coupling Theory, Optimal Transport, and Strassen's Theorem Beyond Regular Orders

Many results in probability (most famously, Strassen's theorem on stochastic domination), characterize some relationship between probability distributions in terms of the existence of a particular structured coupling between them. Optimal transport, and in particular Kantorovich duality, provides a framework for formally unifying these results, but the standard duality theory requires topological conditions that are not satisfied in some settings. In this work, we investigate the extent to which Kantorovich duality still provides meaningful connections between distributional relations and their coupling counterparts, in the topologically irregular setting. Towards this end, we show that Strassen's theorem ``nearly holds'' for topologically irregular orders but that the full theorem admits counterexamples. The core of the proof is a novel technical result in optimal transport, which shows that the Kantorovich dual problem is well-behaved for optimal transport problems whose cost functions can be written as a non-increasing limit of lower semi-continuous functions.