A Brenier Theorem on (P2 (...P2(H)...), W2 ) and Applications to Adapted Transport

We develop Brenier theorems on iterated Wasserstein spaces. For a separable Hilbert space H and N≥ 1, we construct a full-support probability Lambda on P2N(H)= P2(... P2(H)...) that is transport regular: for every Q with finite second moment, transporting Lambda to Q with cost W22 admits a unique optimizer, and this optimizer is of Monge type. The analysis rests on a characterization of optimal couplings on P2(H) and, more generally, on P2N(H) via convex potentials on the Lions lift; in the latter case we employ a new adapted version of the lift tailored to the N-step structure. A key idea is a new identification between optimal-transport c-conjugation (with c given by maximal covariance) and classical convex conjugation on the lift. A primary motivation comes from the adapted Wasserstein distance AW2: our results yield a first Brenier theorem for AW2 and characterize AW22-optimal couplings through convex functionals on the space of L2-processes.