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