The strong link between the Optimal Transport Problem (OTP) with quadratic cost and convex analysis through its dual problem is the key point to prove the celebrated Brenier theorem (1991). The main goal of this talk is to show how to recover the same picture at the ``higher level'' of laws of random measures $\mathcal{P}_2(\mathcal{P}_2(\mathbb{R}^d))$ (or simply random measures), despite the lack of smoothness of the cost and the fact that $\mathcal{P}_2(\mathbb{R}^d)$ is a positively curved space.

The presented approach is based on the notion of total convexity and of their total subdifferential, which we will show to characterize optimal Kantorovich potentials. Furthermore, exploiting the Lions' lifting in the Hilbert space of square-integrable random variables, we identify a class of regular measures on $\mathcal{P}_2(\mathcal{P}_2(\mathbb{R}^d))$ for which the OTP has a unique solution for every target random measure which is of Monge-type, proving a version of the Brenier theorem at this higher level. Finally, we show a recipe to build such regular measures, which are push-forward of suitable non-degenerate Gaussian measures on the Lions' lifted space.

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(In collaboration with Giuseppe Savaré.)