A Brenier Theorem on $(\mathcal{P}_2 (\mathcal{P}_2(\mathbb{R}^d )), W_2 )$ and Applications to Adapted Transport

Abstract: Brenier's fundamental theorem characterizes optimal transport plans for measures $\mu, \nu$ on $\mathbb{R}^d$ and quadratic distance costs in terms of gradients of convex functions. In particular it guarantees the existence of optimal transport maps for measures which are absolutely continuous wrt Lebesgue measure. Our goal is to provide a version of this result for measures $P,Q$ on $\mathcal{P}_2(\mathbb{R}^d)$ and costs given by the squared Wasserstein distance $W_2^2(\mu, \nu)$. We characterize optimizers in terms of convexity of the Lions lift. This is based on an observation which seems to be of independent interest: the $c$-transform of a functional $\phi$, where $c(\mu, \nu)$ denotes maximal covariance of $\mu, \nu$ corresponds precisely to the Legendre transform of the Lions lift of $\phi$. Moreover we show that for typical $P \in\mathcal{P}_2(\mathbb{R}^d)$ the optimizer is unique and given by a transport map. In the absence of a canonical reference measure on $\mathcal{P}_2(\mathbb{R}^d)$ we use a topological notion to make `typical' precise. Specifically we show that the transport regular measures are of second Baire category. A particular motivation for our article stems from the theory of adapted transport where the adapted Wasserstein distance provides an adequate distance between stochastic processes. In contrast to other metrics, the adapted Wasserstein distance yields continuity of Doob-decomposition, optimal stopping and stochastic control problems. Based on our results for measures on $\mathcal{P}_2(\mathbb{R}^d)$ we obtain a first Brenier-type theorem for the adapted Wasserstein distance.