Speaker: Gudmund Pammer
Title: Brenier’s Theorem for $\mathcal{P}_2(\dots \mathcal{P}_2(H) \dots )$ and Applications to Adapted Transport
Abstract: We establish a Brenier theorem for iterated Wasserstein spaces. For a separable Hilbert space $H$ and $N\ge1$, we construct a full-support probability $\Lambda\in\mathcal{P}_2^{N}(H)$ that is transport regular: for all $P,Q\in\mathcal{P}_2^{N}(H)$ with $P\ll\Lambda$, the $\mathcal{W}_2^2$-optimal transport from $P$ to $Q$ is unique and of Monge type. In the first non-classical case $N=2$ we show that optimal transports are given by the $\mathcal{W}_2$-gradient (Lions’ derivative) of an $L$-convex functional. For general $N$ we develop adapted notions of Lions’ lift, $L$-convexity and Lions’ derivative, which we apply to the adapted Wasserstein distance $\mathcal{AW}_2$, obtaining a first Brenier theorem for $\mathcal{AW}_2$.