Interpolation inequalities and curvature condition via p-Wasserstein spaces
- Martin Kell (MPI MiS, Leipzig)
The proof of Borell–Brascamp–Lieb (BBL) inequality for Riemannian manifolds by CorderoErausquin-McCann-Schmuckenschläger, and later for Finsler manifolds by Ohta, let Lott-Villani and Sturm to a new notion of a lower bound on the generalized Ricci curvature for general metric measure spaces, called curvature dimension. Both, the BBL inequality and the curvature condition, rely on geodesics in the 2-Wasserstein space, which was a natural candidate because of its connection to convex analysis in the Euclidean setting.
Based on Ohta's proof we show how to prove the BBL inequality via geodesics in the p-Wasserstein spaces for any p>1. Following Lott-Villani-Sturm, a new curvature condition can be defined via convexities along geodesics in the p-Wasserstein space and many known results like Poincaré, Bishop-Gromov follow by similar arguments.
As a "vertical dual" one can use the recent theory of the q-Cheeger energy (q is the Hölder conjugate of p) developed by Ambrosio-Gigli-Savaré to even get a q-Laplacian comparison, which however is equivalent to the usual one in the smooth setting. In a second talk given later we will study the gradient flow of the q-Cheeger energy, called q-heat flow, and use the "duality" and curvature condition to identify it with the gradient flow of the (3-p)-Renyi entropy (classical entropy in case p=2) in the p-Wasserstein space.
If times permits the Orlicz-Wasserstein space is introduced and the necessary adjustments to get the interpolation inequality and curvature condition are shown. However, by now there is no "vertical dual" to the theory of Orlicz-Wasserstein spaces, in particular there is no Orlicz-Cheeger energy and no Orlicz-Laplacian.