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

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 C_T, is here deeply investigated. In this paper we study the regularity of 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 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↦TC_T around T=0 from the first to the second order.