A formula for the time derivative of the entropic cost and applications
A formula for the time derivative of the entropic cost and applications
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DOI：
10.1016/j.jfa.2021.108964
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发表时间：
2019-12
期刊：
影响因子：
--
通讯作者：
Giovanni Conforti;Luca Tamanini
中科院分区：
文献类型：
--
作者：
Giovanni Conforti;Luca Tamanini
近年来，薛定谔问题由于在小噪声条件下与Monge-Kantorovich最优运输问题的联系而引起了人们的极大关注。本文对其最优值--熵成本CT值进行了深入的研究。本文研究了曲率条件下CT关于参数T的正则性，并显式地计算了它的一阶和二阶导数。作为应用：-我们确定了CT的大时间极限，并提供了精确的指数收敛速度；我们不仅对经典的薛定谔问题得到了这个结果，而且对最近引入的平均场薛定谔问题也得到了这个结果[3]；-我们把T-↦T-CT在T=0附近的泰勒展开式从一阶改进到了二阶。
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↦ T C T around T= 0 from the first to the second order.