In this talk we discuss recent advances towards Brenier-type results on iterated Wasserstein spaces $P_2^N(H) = P_2(...P_2(H)...)$ over a separable Hilbert space $H$. We construct a full-support probability measure $\Lambda$ in $P_2^N(H)$ that is transport-regular. A key ingredient is a novel characterisation of optimal couplings on $P_2(P_2(H))$ via convex potentials on the Lions lift, and, more generally, on $P_2^N(H)$ via a new adapted variant of the Lions lift that respects the nested structure. A primary motivation comes from adapted transport: here, our results yield a first Brenier theorem for the adapted Wasserstein distance. 

This talk is based on joint work with Mathias Beiglböck and Stefan Schrott.