A Brenier Theorem on $(P_2 (...P_2(H)...), W_2 )$ and Applications to Adapted Transport
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arXiv
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| Format: | Preprint |
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2025
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| author | Beiglböck, Mathias Pammer, Gudmund Schrott, Stefan |
| author_facet | Beiglböck, Mathias Pammer, Gudmund Schrott, Stefan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03506 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Brenier Theorem on $(P_2 (...P_2(H)...), W_2 )$ and Applications to Adapted Transport Beiglböck, Mathias Pammer, Gudmund Schrott, Stefan Probability Functional Analysis Metric Geometry Optimization and Control 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. |
| title | A Brenier Theorem on $(P_2 (...P_2(H)...), W_2 )$ and Applications to Adapted Transport |
| topic | Probability Functional Analysis Metric Geometry Optimization and Control |
| url | https://arxiv.org/abs/2509.03506 |