Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in L 1
Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in
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马尔可夫半群的 Log-Hessian 和偏差界以及 L 1 中的正则化效应
作者：
N. Gozlan;Xue;M. Madiman;Cyril Roberto;Paul
众所周知，一些重要的马尔可夫半群具有“正则化效应”——例如，布尔超立方体上的噪声算子或实线上的 Ornstein-Uhlenbeck 半群的超收缩性质，适用于 L ^ p 中的函数（当 p > 1 时）。Talagrand 在 1989 年推测，布尔超立方体上的噪声算子对于以下函数具有更微妙的正则化性质：可积，但这个猜想仍然存在。尽管如此，近年来，Eldan-Lee 和 Lehec 通过将 Ornstein-Uhlenbeck 半群的对数 Hessian 不等式与高斯测度下对数半凸函数的新偏差不等式相结合，证明了该猜想的高斯类比。在这项工作中，我们探讨了这种现象有多普遍的问题。具体来说，我们的第一个目标是探索这两个成分对于 ℝ n $\mathbb {R}^{n}$ 中的某些扩散半群的有效性，以及非负整数上的 M / M / ∞ $M/M/\infty $ 队列和正实数线上的拉盖尔半群的有效性。我们的第二个目标是证明这些设置的一维正则化效果，即使在这些成分无效的情况下也是如此。
It is well known that some important Markov semi-groups have a “regularization effect” – as for example th hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in L ^ p for p > 1. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in ℝ n $\mathbb {R}^{n}$ , as well as for the M / M / ∞ $M/M/\infty $ queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.