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：

---

马尔可夫链的遍历性反问题

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。我们的标准是根据链的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性来确定的。同时，这些实用标准还应用于一些示例，包括一类特殊的单胎生育过程和几个多维模型。