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 ${\mathcal{C}}_T$, is here deeply investigated. In this paper we study the regularity of ${\mathcal{C}}_T$ with respect to the parameter T under a curvature condition and explicitly compute its first and second derivative. As applications:

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we determine the large-time limit of ${\mathcal{C}}_T$ 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];

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

近年来，Schrödinger问题由于在小噪声状态下与Monge-Kantorovich最优运输问题的联系而备受关注。它的最优价值，熵成本 ${\mathcal{C}}_{Ť}$，在这里进行了深入调查。在本文中，我们研究了${\mathcal{C}}_{Ť}$相对于曲率条件下的参数T，并显式计算其一阶和二阶导数。作为应用程序：

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我们确定 ${\mathcal{C}}_{Ť}$并提供清晰的指数收敛速度；我们不仅对于经典的薛定ding问题，而且对于最近引入的均值场薛定ding问题[3]，都获得了该结果。

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我们提高了泰勒展开式 $Ť\mapsto Ť{\mathcal{C}}_{Ť}$ 大约 $Ť=0$ 从第一顺序到第二顺序。