The optimal transport problem defines a notion of distance in the space of probability measures over a manifold, the Wasserstein space. In his thesis, McCann discovered that this space is a length space: the distance between probability measures is given by the length of minimizing geodesics called displacement interpolants or Wasserstein geodesics. In 2000, Otto defined a (purely formal) Riemannian calculus allowing the computation of tangent vectors to displacement interpolants and the computation of Hessians of functionals along these geodesics. In this talk, I will present an Eulerian calculus on Wasserstein space, which extends the Otto calculus from a purely Riemannian setting to general Lagrangians. This Eulerian calculus allows for the computation of derivatives and Hessians of functionals involving derivatives of densities, resolving a question of Villani. New first order displacement convex functionals are presented. Finally, I will show how this calculus can be made rigorous via the DiPerna-Lions theory of renormalized solutions. This talk is based on my thesis and ongoing joint work with Almut Burchard.
