Calculus of Variations and Geometric Measure Theory

G. Conforti - L. Tamanini

A formula for the time derivative of the entropic cost and applications

created by tamanini1 on 23 Dec 2019
modified on 08 Jun 2022


Published Paper

Inserted: 23 dec 2019
Last Updated: 8 jun 2022

Journal: J. Funct. Anal.
Year: 2019


In the recent years the Schrödinger problem has gained a lot of attention because of the connection, in the small-noise regime, with the Monge-Kantorovich optimal transport problem. Its optimal value, the entropic cost $\mathscr{C}_T$, is here deeply investigated. In this paper we study the regularity of $\mathscr{C}_T$ with respect to the parameter $T$ under a curvature condition and explicitly compute its first and second derivative. As applications:

- we determine the large-time limit of $\mathscr{C}_T$ and provide sharp exponential convergence rates; we obtain this result not only for the classical Schrödinger problem but also for the recently introduced Mean Field Schrödinger problem $[3]$;

- we improve the Taylor expansion of $T \mapsto T\mathscr{C}_T$ around $T=0$ from the first to the second order.