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A formula for the time derivative of the entropic cost and applications
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-02-22 , DOI: 10.1016/j.jfa.2021.108964
Giovanni Conforti , Luca Tamanini

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 CT, 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:

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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];

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we improve the Taylor expansion of TTCT around T=0 from the first to the second order.



中文翻译:

熵成本和应用的时间导数的公式

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 CŤ,在这里进行了深入调查。在本文中,我们研究了CŤ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 CŤ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 ŤŤCŤ 大约 Ť=0 从第一顺序到第二顺序。

更新日期:2021-03-16
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