We establish a Brenier-type theorem for iterated Wasserstein spaces. The classical prototype is the Rüschendorf–Rachev theorem characterizing optimal L2 couplings by subgradients of convex functions, with Brenier’s polar-factorization theorem providing the corresponding Monge-map formulation in the absolutely continuous setting.

Specifically, for a separable Hilbert space H and N ≥ 1, we construct a full-support probability Λ on the N-times iterated Wasserstein space over H that is transport regular: for all probabilities P and Q in this space, with P absolutely continuous with respect to Λ, the optimal transport from P to Q for the squared Wasserstein-2 distance is unique and of Monge type.

In the first non-classical case, N = 2, we show that optimal transports are given as the pushforward by the Wasserstein-2 gradient, or Lions derivative, of an L-convex function. To establish the result for general N, we develop new adapted notions of Lions lift, L-convexity, and Lions derivative. 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-type theorem for AW2 and characterize optimal couplings for the squared AW2 distance through convex functionals on the space of L2 processes.