论文标题
连续时间马尔可夫链的弯曲维度条件下的熵信息不平等
Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains
论文作者
论文摘要
在可逆连续时间马尔可夫连锁店的设置中，最近已证明$cd_υ$条件是在扩散环境中与Bakry-émery条件的一致类似物，因为在有限的维度下证明Li-yau不平等，并证明了修改后的对数Sobolev在正相的范围内，在正相的范围内。在本文中，我们研究了给出两者的情况，有限的尺寸项和正曲率结合。为此，我们介绍了$cd_υ（κ，f）$条件，其中尺寸项由所谓的$ cd $ function $ f $表示。我们得出了有关熵信息的功能不平等，我们将其称为熵信息不平等。此外，我们推断出熵信息不平等的应用，例如超包范围，Lipschitz函数的指数积分，有限直径界限以及著名的NASH不平等的修改版本。
In the setting of reversible continuous-time Markov chains, the $CD_Υ$ condition has been shown recently to be a consistent analogue to the Bakry-Émery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the $CD_Υ(κ,F)$ condition, where the dimension term is expressed by a so called $CD$-function $F$. We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropy-information inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.
