论文标题
通过下面有RICCI曲率的完整对数Sobolev不平等
Complete Logarithmic Sobolev inequalities via Ricci curvature bounded below
论文作者
论文摘要
我们证明，对于对称Markov Semigroup，RICCI曲率从下面界定的非阳性常数与有限的$ L_ \ infty $ - 混合时间相结合，这意味着修改的Log-Sobolev不平等。对于具有光谱差距和有限varopoulos尺寸的Markov Semigroups，这种$ L_ \ Infty $ - 混合时间估计总是存在。我们的结果适用于Carlen和Maas最近引入的非共同RICCI曲率界限的非共性量子Markov半群。作为一种应用，我们证明了紧凑的Riemannian歧管上的热半群允许其所有矩阵值扩展的均匀修饰的对数 - 贝贝尔不等式。
We prove that for a symmetric Markov semigroup, Ricci curvature bounded from below by a non-positive constant combined with a finite $L_\infty$-mixing time implies the modified log-Sobolev inequality. Such $L_\infty$-mixing time estimates always hold for Markov semigroups that have spectral gap and finite Varopoulos dimension. Our results apply to non-ergodic quantum Markov semigroups with noncommutative Ricci curvature bounds recently introduced by Carlen and Maas. As an application, we prove that the heat semigroup on a compact Riemannian manifold admits a uniform modified log-Sobolev inequality for all its matrix-valued extensions.
