离散时间马氏链的泛函不等式及遍历性

Functional inequalities and ergodicity are important branches of stochastic analysis, and they have a wide range of applications. The project is mainly devoted to studying two problems. Firstly, we intend to investigate weak Poincaré inequalities for discrete-time Markov chains by using a generalized Cheeger's method. Since the transition kernel of discrete-time Markov chains may not be non-negative definite, the new defined Cheeger constants should be closely related to both the positive and the negative spectral points. Moreover, the effect of periodicity on weak Poincaré inequalities will also be considered. Secondly, combining a probabilistic method with Poincaré inequalities, we aim to study the geometric ergodicity for non-reversible discrete-time Markov chains on general state spaces, and then discuss the quantitative relationship between spectral gap and spectral radius.

泛函不等式及遍历性是随机分析理论的重要研究分支，有着广泛的应用。本项目主要研究以下两个方面的问题：首先，我们拟利用推广的Cheeger常数法研究离散时间马氏链的弱Poincaré不等式。由于离散时间马氏链的转移核不一定是非负定的，因此新定义的Cheeger常数既要考虑正谱点又要考虑负谱点。进一步，我们拟研究马氏链的周期性对弱Poincaré不等式的影响。其次，我们计划采用概率方法并结合Poincaré不等式研究一般状态空间离散时间不可逆马氏链的几何遍历性，进而讨论谱隙和谱半径之间的定量关系。

本项目利用三种方法研究了离散时间马氏链的遍历速度：首先，利用矩估计方法和更新公式，研究了一般状态空间可逆马氏链在全变差距离下的几何遍历速度和一致遍历速度；当马氏链不可逆时，采用Foster-Lyapunov条件，探讨了其在V范数下的收敛速度；接着，通过离散时间的泛函不等式，并结合截断方法，研究了一般状态空间不可逆马氏链在全变差距离下的各种遍历速度。截断方法的使用减弱了对马氏链的初分布的可积性要求。