Quantitative logarithmic Sobolev inequalities and stability estimates
Quantitative logarithmic Sobolev inequalities and stability estimates
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定量对数 Sobolev 不等式和稳定性估计
DOI：
10.3934/dcds.2016097
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
M. Ledoux
中科院分区：
文献类型：
--
作者：
M. Fathi;E. Indrei;M. Ledoux
本文对高斯测度的对数Sobolev不等式建立了一个改进的形式，该不等式仅限于满足Poincar不等式的概率密度。结果意味着赤字的二次Kantorovich-Wasserstein距离的下限。我们同样研究了Talagrand二次运输成本不等式中的亏损，这次是通过${\rm L}^1$-Kantorovich-Wasserstein距离，最佳的产品措施，并推导出对数Sobolev不等式中亏损的下界。在Bakry-Emery理论和相干态变换的背景下给出了应用。证明联合收割机工具，从半群和热核理论和最佳质量输运。
We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincar\'e inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an ${\rm L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context of the Bakry-\'Emery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.