Inverse problems for ergodicity of Markov chains

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