A Brenier Theorem on $(P_2(...P_2(H)...),W_2)$ and Applications to Adapted Transport

Abstract

We develop Brenier theorems on iterated Wasserstein spaces. For a separable Hilbert space $H$ and $N\ge 1$, we construct a full-support probability $\Lambda$ on $P_2^N(H)=P_2(...P_2(H)...)$ that is transport regular: for every $Q$ with finite second moment, transporting $\Lambda$ to $Q$ with cost $W_2^2$ admits a unique optimizer, and this optimizer is of Monge type. The analysis rests on a characterization of optimal couplings on $P_2(H)$ and, more generally, on $P_2^N(H)$ 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 $A{W}_2$: our results yield a first Brenier theorem for $A{W}_2$ and characterize $A{W}_2^2$-optimal couplings through convex functionals on the space of $L_2$-processes.