论文标题
超额合同，对数Sobolev不平等和Talagrand的成本不平等的稳定性
Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality
论文作者
论文摘要
我们提供了纳尔逊超额收缩不平等，对数Sobolev的不平等以及Talagrand的运输成本不平等的赤字估计，这是在限制下，即投入是半持续的，半持续的，半gog的convex，或半gog的concove。特别是，我们对对数Sobolev不平等的结果补充了Eldan，Lehec和Shenfeld最近获得的结果，该结果是对具有较小协方差的输入的赤字估计值。同样，我们对Talagrand的运输成本不平等的结果补充，对于大量的半循环输入，我们改善了Mikulincer最近证明的赤字估计值。我们将使用基于Fokker-Planck方程的流量单调性方案来获得我们对超收缩率的赤字估计，并且我们对对数Sobolev不平等的赤字估计将作为推论得出。对于Talagrand的不平等，我们使用了最佳的运输论点。我们框架的一个吸引人的特征是鲁棒性，这使我们能够得出与汉密尔顿 - 贾科比方程，庞加莱不平等以及贝克纳的不平等相关的过度转化性不平等的赤字估计。
We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a deficit estimate recently proved by Mikulincer. Our deficit estimates for hypercontractivity will be obtained by using a flow monotonicity scheme built on the Fokker--Planck equation, and our deficit estimates for the logarithmic Sobolev inequality will be derived as a corollary. For Talagrand's inequality, we use an optimal transportation argument. An appealing feature of our framework is robustness and this allows us to derive deficit estimates for the hypercontracivity inequality associated with the Hamilton--Jacobi equation, the Poincaré inequality, and for Beckner's inequality.
