论文标题
一种概率的方法，用于凸出$（ϕ）$ - 马尔可夫链的熵衰减
A probabilistic approach to convex $(ϕ)$-entropy decay for Markov chains
论文作者
论文摘要
我们研究了连续时间马尔可夫链和相关凸sobolev不平等的熵功能的指数耗散，包括MLSI和Beckner不平等。我们提出了一种结合了Bakryémery方法和耦合论点的方法，我们将其用作离散Bochner身份的概率替代方案。该方法非常适合在非扰动环境中工作，我们获得了超过高温/弱相互作用状态的随机步行的新估计值。在此框架中，我们还表明，沿着半群的瓦斯汀距离的指数收缩意味着MLSI。我们还重新审视了经常获得新的不平等现象，有时会改善最著名的常数。特别是，我们分析了零范围的动力学，铁杆和伯努利 - 拉普拉斯模型以及居里·魏斯和伊辛模型的Glauber动力学。
We study the exponential dissipation of entropic functionals for continuous time Markov chains and the associated convex Sobolev inequalities, including MLSI and Beckner inequalities. We propose a method that combines the Bakry Émery approach and coupling arguments, which we use as a probabilistic alternative to the discrete Bochner identities. The method is well suited to work in a non perturbative setting and we obtain new estimates for interacting random walks beyond the high temperature/weak interaction regime. In this framework, we also show that the exponential contraction of the Wasserstein distance along the semigroup implies MLSI. We also revisit classical examples often obtaining new inequalities and sometimes improving on the best known constants. In particular, we analyse the zero range dynamics, hardcore and Bernoulli-Laplace models and the Glauber dynamics for the Curie Weiss and Ising model.
