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

Abstract

We prove the second order differentiation formula along geodesics in finite-dimensional RCD(K, N ) spaces. Our approach strongly relies on the approximation of W2-geodesics by entropic interpolations and, in order to implement this approximation procedure, on the proof of new (even in the smooth setting) estimates for such interpolations.

An entropic interpolation proof of the HWI inequality

“…Let us now derive a criterion, in terms of the endpoint marginals µ 0 and µ 1 , for the existence of regular functions f ε , g ε with well-defined Fisher information solving the Schrödinger system (2.6). As noticed in [12], [13] the regularity (smoothness and integrability) of µ 0 (resp. µ 1 ) is inherited by f ε (resp.…”

“…and, as it is not difficult to see (e.g. following the computations carried out in [13] which fit to Setting 1), these PDEs are satisfied along (ρ ε t , ϑ ε t ). Finally, as in the Hamiltonian p represents a momentum density, it is natural to set p t := ν t v t .…”

“…Secondly, r ε is bounded from above since r ε ∈ C ∞ (X). These two facts imply that f ε , g ε ≥ c ′ > 0, as shown in [12]. As a consequence the proof of the HWI inequality is easier and the parallelism with the heuristic proof of Otto-Villani [23] is stronger, since no 'δ-argument' in Lemma 3.7 and Lemma 3.11 is needed; in particular, Lemma 3.7 is already known to hold by [12] and thus no proof is required.…”

“…Namely: -all the regularity and integrability results concerning Schrödinger potentials and entropic interpolations mentioned in Section 2 as well as the dynamic representation of the entropic cost; -the regularizing and contraction properties of (T t ); -the existence of 'good' cut-off functions; -the Benamou-Brenier formula and the Bochner-Lichnerowicz-Weitzenböck inequality. The reader is addressed to [13], [14] for the first point and to [10] for all the others.…”

Multi-marginal entropy-transport with repulsive cost

“…Second-order Calculus on RCD spaces: N. Gigli and L. Tamanini [16] studied the entropictransport problem on a class of metric spaces with (Riemannian) Ricci curvature bounded from below (2-marginals case, c(x 1 , x 2 ) = d(x 1 , x 2 ) 2 ). The entropic regularization procedure was crucial for establishing a second-order differential structure in that setting.…”