Coupling Theory, Optimal Transport, and Strassen's Theorem for Eventual Domination

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 common framework for these results. We use this perspective to study "eventual domination", a class of orders arising naturally in stochastic processes (including in the analysis of branching random walks, Ising models, and diffusions), which does not satisfy the topological conditions required by standard optimal transport theory. More generally, we study the connection between distributional relations and their coupling counterparts for topologically irregular preorders (e.g. equivalence relations, partial orders).
To this end, we show that Strassen's theorem "nearly holds" in this topologically irregular setting but that the full theorem admits counterexamples, including for eventual domination. The core of the proof is a novel technical result in optimal transport, which shows that the Kantorovich dual problem is well-behaved for cost functions which can be written as a non-increasing limit of lower semi-continuous functions.