本文分成四部分. 第一部分, 我们给出了分解方法在谱隙估计中的抽象描述, 统一了 Chen 在 [11,第九章] 使用的分割方法和 [24] 的有限分割方法, 并将 [24] 的方法推广到一般状态空间的跳过程. 然后, 我们研究了这种分解方法的单调性和逼近性质. 第二部分, 我们考虑分解方法在一般状态空间跳过程的谱隙估计中的应用. 利用分解方法证明了可逆跳过程存在谱隙当且仅当以某 “小集” (如紧集) 为吸收态的 Dirichlet 第一特征值大于 0. 在此基础上, 用经典的 Lyapunov 条件, 给出了可逆跳过程的谱隙估计. 第三部分, 我们研究了开 Jackson 网络的指数遍历性.使用分解方法和对称化手法, 证明了它的指数遍历性等价于普通遍历性.更进一步, 我们给出了谱隙的上、下界估计, 其上界与下界只差一个常数. 第四部分, 我们研究了一般拟生灭过程的谱隙. 将状态空间按水平集分解, 构造了一个生灭过程和一列限制链, 得到了谱隙估计.并将结果应用到两类典型的排队论模型: 随机环境中的 M/M/1 模型和 M/M/c 同步休假模型.

This paper consists of four parts. In the first part, we give the abstract description of decomposition method in the estimation for spectral gap, which will unify the Chen's decomposition method in [11, Chapter 9] and finite decomposition method in [24]. By this decomposition, we can easily extend the finite decomposition method in [24] from finite Markov chain to the jump process on general state space. Then, we study the monotonicity and approximation for this decomposition method. In the second part, we consider the application of decomposition method to the jump process on general state space. By decomposition method, we prove that the spectral gap for reversible jump process exists if and only if the first Dirichlet eigenvalue for some petite set (such as compact set) as an absorbing set is greater than 0. Under this criterion, we give the estimation of spectral gap for reversible jump process via classic Lyapunov condition. In part three, we study the exponential ergodicity for open Jackson network. By decomposition method and symmetrization procedure, we prove that its exponential ergodicity is equivalent to the ordinary ergodicity. Moreover, we give the upper and lower bound of its spectral gap, where the upper bound and lower bound are different upto a constant. In the last part, we study the spectral gap for general quasi-birth and death process. By decomposing the state space according to the level set, and constructing a birth death process and a sequence of restricted processes on each level, we get its spectral gap estimation. Later, we apply our results to two kinds of classic queueing models: M/M/1 in random environment and M/M/c with synchronous vacation.