arXiv subjects

Luca Tamanini

Publications and source records attributed to Luca Tamanini.

21 records · Page 2Linked to original sources

arXiv2018

Benamou-Brenier and duality formulas for the entropic cost on $RCD^*(K,N)$ spaces

In this paper we prove that, within the framework of $RCD^*(K,N)$ spaces with $N < \infty$, the entropic cost (i.e. the minimal value of the Schrödinger problem) admits: - a threefold dynamical variational representation, in the spirit of the Benamou-Brenier formula for the Wasserstein distance; - a Hamilton-Jacobi-Bellman dual representation, in line with Bobkov-Gentil-Ledoux and Otto-Villani results on the duality between Hamilton-Jacobi and continuity equation for optimal transport; - a Kantorovich-type duality formula, where the Hopf-Lax semigroup is replaced by a suitable `entropic' counterpart. We thus provide a complete and unifying picture of the equivalent variational representations of the Schrödinger problem (still missing even in the Riemannian setting) as well as a perfect parallelism with the analogous formulas for the Wasserstein distance.

math.AP↗

arXiv2018

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

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. 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. Finally, the techniques adopted in this paper can be used to show that in the $RCD$ setting the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, in accordance with the smooth case. With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality.

math.AP↗

arXiv2017

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

Aim of this paper is to prove the second order differentiation formula along geodesics in compact $RCD^*(K,N)$ spaces with $N<\infty$. This formula is new even in the context of Alexandrov spaces. 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. In particular we obtain: - equiboundedness of the densities along the entropic interpolations, - equi-Lipschitz continuity of the Schrödinger potentials, - a uniform weighted L2 control of the Hessian of such potentials. Finally, the techniques used in this paper can be 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.

math.AP↗