Fig. 1. Given two probability distributions over a discrete surface (left and right), our algorithm generates an interpolation that takes the geometric structure of the surface into account.We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finitedimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows.

The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or Computational Fluid Dynamic formulation, amounts to write the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. Several disretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space.In this article, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional one for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by Gladbach, Kopfer and Maas, as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien and Solomon fit in this framework.

We consider minimization problems for curves of measure, with kinetic and potential energy and a congestion penalization, as in the functionals that appear in Mean Field Games with a variational structure. We prove L 8 regularity results for the optimal density, which can be applied to the rigorous derivations of equilibrium conditions at the level of each agent's trajectory, via time-discretization arguments, displacement convexity, and suitable Moser iterations. Similar L 8 results have already been found by P.-L. Lions in his course on Mean Field Games, using a proof based on the use of a (very degenerate) elliptic equation on the dual potential (the value function) ϕ, in the case where the initial and final density were prescribed (planning problem). Here the strategy is highly different, and allows for instance to prove local-in-time estimates without assumptions on the initial and final data, and to insert a potential in the dynamics. $ ' & ' %´B t ϕ`| ∇ϕ| 2 2 " Vpxq`gpρq, B t ρ´∇¨pρ∇ϕq " 0, ϕpT, xq " Ψpxq, ρp0, xq " ρ 0 pxq.