论文标题
马尔可夫连锁店离散环境中的曲率维度条件下的熵方法
The entropy method under curvature-dimension conditions in the spirit of Bakry-Émery in the discrete setting of Markov chains
论文作者
论文摘要
我们考虑在离散空间上的连续时间（不一定是有限的）马尔可夫链，并确定曲率差异不平等，条件$cd_υ（κ，\ infty）$，在几个方面是经典的Bakry-émery条件$ CD（κ，\ infty）$的天然类似物。特别是，它是根据经典的方法来量身定制的，该方法是通过计算和估算Markov链发电机生成的热流的熵的第二次衍生物，来证明修改的对数Sobolev不平等。我们证明，$cd_υ$的曲率界限在张力下保存下来，讨论指向其他离散曲率概念的链接，并考虑各种示例，包括完整的图表，超立方体和出生死亡过程。我们进一步考虑了力量类型的熵，并以同样的精神确定一种自然的CD条件，导致贝克纳不平等。 $cd_υ$条件也被证明与扩散设置兼容，从某种意义上说，相应的混合过程享有张力属性。
We consider continuous-time (not necessarily finite) Markov chains on discrete spaces and identify a curvature-dimension inequality, the condition $CD_Υ(κ,\infty)$, which serves as a natural analogue of the classical Bakry-Émery condition $CD(κ,\infty)$ in several respects. In particular, it is tailor-made to the classical approach of proofing the modified logarithmic Sobolev inequality via computing and estimating the second time derivative of the entropy along the heat flow generated by the generator of the Markov chain. We prove that curvature bounds in the sense of $CD_Υ$ are preserved under tensorization, discuss links to other notions of discrete curvature and consider a variety of examples including complete graphs, the hypercube and birth-death processes. We further consider power type entropies and determine, in the same spirit, a natural CD condition which leads to Beckner inequalities. The $CD_Υ$ condition is also shown to be compatible with the diffusive setting, in the sense that corresponding hybrid processes enjoy a tensorization property.
