For continuous-time Markov chains, we provide criteria for non-ergodicity, non-algebraic ergodicity, non-exponential ergodicity, and non-strong ergodicity. For discrete-time Markov chains, criteria for non-ergodicity, non-algebraic ergodicity, and non-strong ergodicity are given. Our criteria are in terms of the existence of solutions to inequalities involving the Q-matrix (or transition matrix P in time-discrete case) of the chain. Meanwhile, these practical criteria are applied to some examples, including a special class of single birth processes and several multi-dimensional models.

中文翻译：

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马尔可夫链遍历性的逆问题

对于连续时间马尔可夫链，我们提供了非遍历性、非代数遍历性、非指数遍历性和非强遍历性的标准。对于离散时间马尔可夫链，给出了非遍历性、非代数遍历性和非强遍历性的准则。我们的标准是关于涉及链的Q矩阵（或时间离散情况下的转换矩阵P）的不等式的解的存在。同时，将这些实用标准应用于一些示例，包括一类特殊的单胎过程和几个多维模型。