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

Mathias Beiglböck, Gudmund Pammer, Stefan Schrott

TL;DR

This work extends Brenier's theorem to iterated Wasserstein spaces by constructing transport-regular measures on $\mathcal{P}_2^N(H)$ and proving that optimal couplings are unique and Monge-type under a full-support reference law. Central to the approach is the Lions lift, extended to an adapted, $N$-stage setting, and its connection to maximal covariance cost via MC-convexity, enabling a duality framework and a concrete Monge characterization. The authors establish a Brenier-type theory for the adapted Wasserstein distance $AW_2$, showing that optimal adapted couplings correspond to gradients of adapted convex functionals on spaces of $L_2$-processes and that transport-regular measures are dense in the relevant spaces. They further develop probabilistic representations and an adapted transfer principle linking stochastic processes to iterated measure spaces, and apply the theory to the AW2 setting, providing a rigorous pathway to Monge solutions in adaptive transport problems encountered in stochastic optimization and finance. Overall, the paper builds a comprehensive convex-analytic foundation for adapted transport, with explicit constructions and broad implications for process-valued transport problems.

Abstract

We develop Brenier theorems on iterated Wasserstein spaces. For a separable Hilbert space $H$ and $N\geq 1$, we construct a full-support probability $Λ$ on $P_2^{N}(H)= P_2(... P_2(H)...)$ that is transport regular: for every $Q$ with finite second moment, transporting $Λ$ 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 $AW_2$: our results yield a first Brenier theorem for $AW_2$ and characterize $AW_2^2$-optimal couplings through convex functionals on the space of $L_2$-processes.

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

TL;DR

This work extends Brenier's theorem to iterated Wasserstein spaces by constructing transport-regular measures on and proving that optimal couplings are unique and Monge-type under a full-support reference law. Central to the approach is the Lions lift, extended to an adapted, -stage setting, and its connection to maximal covariance cost via MC-convexity, enabling a duality framework and a concrete Monge characterization. The authors establish a Brenier-type theory for the adapted Wasserstein distance , showing that optimal adapted couplings correspond to gradients of adapted convex functionals on spaces of -processes and that transport-regular measures are dense in the relevant spaces. They further develop probabilistic representations and an adapted transfer principle linking stochastic processes to iterated measure spaces, and apply the theory to the AW2 setting, providing a rigorous pathway to Monge solutions in adaptive transport problems encountered in stochastic optimization and finance. Overall, the paper builds a comprehensive convex-analytic foundation for adapted transport, with explicit constructions and broad implications for process-valued transport problems.

Abstract

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

Paper Structure

This paper contains 35 sections, 94 theorems, 301 equations.

Table of Contents

  1. Introduction
  2. Characterization of optimal transport maps via convex Lions lifts
  3. Fundamental theorem of optimal transport, $MC$-convexity and $L$-convexity
  4. Stochastic processes and Brenier theorems for the adapted Wasserstein distance
  5. Related literature
  6. Organization of the paper
  7. Lions lift and convexity of law invariant functions
  8. Recap on Lions lift and Lions derivative
  9. $\textup{MC}$-convexity
  10. Subdifferentials
  11. Brenier's theorem for measures on $\mathcal{P}_2(H)$
  12. Characterization of $\textup{MC}$-optimal couplings
  13. Regularization on $\mathbb{R}^d$ by Brownian sheet
  14. Regularization on $H$ by Brownian motion/standard Wiener process
  15. On transport-regularity and Baire category
  16. ...and 20 more sections

Key Result

Theorem 1.1

There exists a full support probability $\Lambda$ on $\mathcal{P}_2^N(H)$ which is transport regular. That is, for every probability $Q$ on $\mathcal{P}_2^N(H)$ with finite second moment admits a unique minimizer $\Pi^*\sim (X,Y)$ and $\Pi^*$ is of Monge type.

Theorems & Definitions (206)

  • Theorem 1.1: Monge solutions
  • Corollary 1.2
  • Theorem 1.3: Brenier's Theorem Br91AmGiSa08
  • Theorem 1.4: Brenier theorem on $\mathcal{P}_2(H)$
  • Theorem 1.5: $N$-iterated Brenier
  • Theorem 1.6
  • Theorem 1.7: Fundamental theorem of optimal transport
  • Theorem 1.8
  • Proposition 1.9
  • Theorem 1.10: Baire--Brenier theorem for $\mathcal{AW}_2$
  • ...and 196 more