Localisation for constrained transports II: applications

Abstract

We present a range of applications of localisation for constrained transports for pairs of probability measures in order with respect to a lattice cone. These examples comprise irreducible convex paving for martingale transports in infinite-dimensional spaces, irreducible convex paving for submartingale transports, localisation of the Monge--Kantorovich problem, pavings for harmonic transports, pavings for solutions of martingale problems. Furthermore, we consider approximating martingale transports and characterise the polar sets in this setting, identifying them as the trivial polar sets -- the sets whose projections have vanishing measures. This provides a negative answer to a question of Cox