Second order differentiation formula on $RCD^*(K,N)$ spaces

Understand

Aim of this paper is to prove the second order differentiation formula for $H^{2,2}$ functions along geodesics in $RCD^*(K,N)$ spaces with $N < \infty$.

• This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity.
• We establish this result by showing that $W_2$-geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolation.
• In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own.