From unbalanced optimal transport to the Camassa-Holm equation
by
Amphithéâtre Léon Motchane
IHES
We present an extension of the Wasserstein L2 distance to the space of positive Radon measures as an infimal convolution between the Wasserstein L2 metric and the Fisher-Rao metric. In the work of Brenier, optimal transport has been developed in its study of the incompressible Euler equation. For the Wasserstein-Fisher-Rao metric, the corresponding fluid dynamic equation is known as the Camassa-Holm equation (at least in dimension 1), originally introduced as a geodesic flow on the group of diffeomorphisms. This point of view provides an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L 2 type of cone metric. As a direct consequence, we describe a new polar factorization on the automorphism group of half-densities which can be seen as a constrained version of Brenier’s theorem. The main application consists in writing the Camassa-Holm equation on S^1 as a particular case of the incompressible Euler equation on a group of homeomorphisms of R^2 that preserve a radial density which has a singularity at 0, the cone point.