The decomposition of stretched Brownian motion into Bass martingales

In previous work J. Backhoff-Veraguas, M. Beiglböck and the present authors showed that the notions of stretched Brownian motion and Bass martingale between two probability measures on Euclidean space coincide if and only if these two measures satisfy an irreducibility condition. Now we consider the general case, i.e. a pair of measures which are not necessarily irreducible. We show that there is a partition of Euclidean space into relatively open convex sets such that the stretched Brownian motion decomposes into a (possibly uncountable) family of Bass martingales on these sets. This partition coincides with the irreducible convex partition studied in previous work of H. De March and N. Touzi.