The free logarithmic Sobolev and the free transportation cost inequalities by time integrations
The free logarithmic Sobolev and the free transportation cost inequalities by time integrations
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自由对数索博列夫和自由运输成本不等式的时间积分
DOI：
10.1142/s0219025714500222
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Hiroaki Yoshida
中科院分区：
文献类型：
--
作者：
Aya Nemoto;Hiroaki Yoshida
本文利用时间积分重新讨论一维情形下对数Sobolev不等式和运输费用不等式的自由模拟。我们考虑自由Fokker-Planck方程的时间演化，计算2-Wasserstein距离的时间导数与最优质量输运，由此可以导出一些微分不等式。讨论了相对自由熵下的收敛性，通过时间积分得到了自由运输费用和自由对数Sobolev不等式.
In this paper, we shall revisit the free analogues of the logarithmic Sobolev and the transportation cost inequalities for one-dimensional case by time integrations. We consider time evolutions by the free Fokker–Planck equation and calculate the time derivative of the 2-Wasserstein distance with the optimal mass transportation, from which some differential inequalities can be derived. The convergence to the equilibrium in the relative free entropy is discussed, and the free transportation cost and the free logarithmic Sobolev inequalities can be obtained by time integrations.