论文标题
马尔可夫内核及其应用的运输不平等
Transportation inequalities for Markov kernels and their applications
论文作者
论文摘要
我们研究了马尔可夫内核在公制空间$ x $上的功能不平等与概率空间上的运输距离不平等的关系。在Hellinger上扩展了Luise和Savaré的结果 - 对于$ RCD（k，\ infty）$公制空间的特定热量半群的坎多维奇收缩不平等，我们表明，更普遍地，这种收缩不平等等于反向庞加莱的不平等。 We also adapt the "dynamic dual" formulation of the Hellinger--Kantorovich distance to define a new family of divergences on $\mathcal{P}(X)$ which generalize the Rényi divergence, and we show that contraction inequalities for these divergences are equivalent to the reverse logarithmic Sobolev and Wang Harnack inequalities.我们讨论了应用程序的应用，包括马尔可夫过程与平衡的收敛性，以及在有限和无限维组中热内核测量的准不变性。
We study the relationship between functional inequalities for a Markov kernel on a metric space $X$ and inequalities of transportation distances on the space of probability measures $\mathcal{P}(X)$. Extending results of Luise and Savaré on Hellinger--Kantorovich contraction inequalities for the particular case of the heat semigroup on an $RCD(K,\infty)$ metric space, we show that more generally, such contraction inequalities are equivalent to reverse Poincaré inequalities. We also adapt the "dynamic dual" formulation of the Hellinger--Kantorovich distance to define a new family of divergences on $\mathcal{P}(X)$ which generalize the Rényi divergence, and we show that contraction inequalities for these divergences are equivalent to the reverse logarithmic Sobolev and Wang Harnack inequalities. We discuss applications including results on the convergence of Markov processes to equilibrium, and on quasi-invariance of heat kernel measures in finite and infinite-dimensional groups.
