Compactness in Adapted Weak Topologies

Over the years a number of topologies for the set of laws of stochastic processes have been proposed. Building on the weak topology they all aim to capture more accurately the temporal structure of the processes. In a parallel paper we show that all of these topologies (i.e. the information topology of Hellwig, the nested distance topology of Pflug-Pichler, the extended weak convergence of Aldous and a topology built from Lasalle's notion of a causal transference plan) are equal in finite discrete time. Regrettably, the simple characterization of compactness given by Prokhorov's theorem for the weak topology fails to be true in this finer topology. This phenomenon is closely related to the failure of a natural metric for this topology to be complete. For certain problems, a 'fix' consists in passing to the metric completion. Still, it also seems interesting to find out what compact sets look like in the uncompleted space. Here we give a characterization of compact sets in this adapted weak topology which is strongly reminiscent of the Arzel\`a-Ascoli theorem (with a dash of Prokhorov's theorem). The tools developed are also useful elsewhere. We give a different proof of the continuity of the conditionally independent gluing map of two measures with one marginal in common and in our companion paper the ideas developed here form the main non-algebraic ingredient in showing that the information topology introduced by Hellwig is equal to the nested weak topology of Pflug-Pichler.