Aim of this talk is to present the second order differentiation formula along geodesics in RCD*(K,N) spaces with K allowed to be negative and N finite. This formula is new even in the context of Alexandrov spaces. In establishing such main theorem, we derive few auxiliary results which are interesting on their own, in particular:

- we obtain a wide range of new properties for the solutions of the dynamic Schroedinger problem, as for instance equiboundedness of the densities along the entropic interpolations and equi-Lipschitz continuity of the Schroedinger potentials;

- in accordance with the smooth case, we prove that the viscous solution of the Hamilton-Jacobi equation can be obtained, in the context of compact RCD*(K,N) spaces, via a vanishing viscosity method.

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