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

In the recent years the Schrodinger 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 C-T, is here deeply investigated. In this paper we study the regularity of C-T with respect to the parameter Tunder a curvature condition and explicitly compute its first and second derivative. As applications:- we determine the large-time limit of CTand provide sharp exponential convergence rates; we obtain this result not only for the classical Schrodinger problem but also for the recently introduced Mean Field Schrodinger problem [3];- we improve the Taylor expansion of T (sic). TCT around T= 0 from the first to the second order. (C) 2021 Elsevier Inc. All rights reserved.