Inserted: 26 oct 2017
Last Updated: 27 oct 2017
Main aim of this manuscript is to present a new interpolation technique for probability measures, different from Brenier-McCann's classical one, which is strongly inspired by the Schrödinger problem, an entropy minimization problem deeply related to optimal transport. By means of the solutions to the Schrödinger problem, we build an efficient approximation scheme, robust up to the second order. Such scheme allows us to prove the second order differentiation formula along geodesics in finite-dimensional $RCD^*$ spaces. This formula is new even in the context of Alexandrov spaces and we provide some applications.
The proof relies on new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain:
- equiboundedness of the densities along the entropic interpolations,
- local equi-Lipschitz continuity of the Schrödinger potentials,
- a uniform weighted $L^2$ control of the Hessian of such potentials.
These tools are very useful in the investigation of the geometric information encoded in entropic interpolations. The techniques used in this work can be also used to show that the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, in accordance with the smooth case.
Throughout the whole manuscript, several remarks on the physical interpretation of the Schrödinger problem and connections with the better known Schrödinger equation are pointed out. Hopefully, this will allow the reader to better understand the physical and probabilistic motivations of the problem as well as to connect them with the analytical and geometric nature of the dissertation.