We consider continuous-time (not necessarily finite) Markov chains on discrete spaces and identify a curvature-dimension inequality, the condition $C{D}_{\mathrm{\Upsilon }}(\kappa ,\infty )$, which serves as a natural analogue of the classical Bakry-Émery condition $CD(\kappa ,\infty )$ 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 $C{D}_{\mathrm{\Upsilon }}$ 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 $C{D}_{\mathrm{\Upsilon }}$ condition is also shown to be compatible with the diffusive setting, in the sense that corresponding hybrid processes enjoy a tensorization property.

我们考虑离散空间上的连续时间（不一定是有限的）马尔可夫链，并确定曲率维不等式，条件 $C{d}_{\mathrm{\Upsilon }}（\kappa ，\infty ）$，它是经典Bakry-Émery条件的自然类似物 $Cd（\kappa ，\infty ）$在几个方面。特别是，它是针对经典方法量身定制的，该经典方法是通过计算和估计沿马尔可夫链生成器产生的热流的熵的二次时间导数来证明修正的对数Sobolev不等式。我们证明曲率界在$C{d}_{\mathrm{\Upsilon }}$在张量下保留它们，讨论与其他离散曲率概念的链接，并考虑各种示例，包括完整图形，超立方体和出生死亡过程。我们进一步考虑幂类型熵，并本着同样的精神确定导致贝克纳不等式的自然CD条件。这$C{d}_{\mathrm{\Upsilon }}$ 在相应的混合过程享有张量特性的意义上，该条件也显示与扩散设置兼容。