I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking values in a general state space(say a Polish space).

Although I've learned many interesting things, such as the application of ergodic theory in number theory, I don't find anything which helps to solve my original problem, i.e. the ergodicity of Markov chain. I expected to find something which gives a sufficient and easy-to-verify condition on the ergodicity of the shift operator. But I only find such a condition(irreducibility) in case of Markov chain taking discrete values

Then as for the case of Markov chain taking values in a general state space, it seems to me that all the existing sufficient conditions for its ergodicity(such as the small set condition, drift condition) are done without mentioning any abstract setting of ergodic theory, such as in the book Markov chain and stochastic stability or in this recent paper. And it is difficult to verify these conditions in general.

So my questions are:

Are there any existing results in ergodic theory which can help to easily establish ergodicity of general Markov chain? My impression is that ergodic theory is powerful and has been developed for a long time, have I missed some important results?

If no such a result exists, then what is the more hopeful choice if one needs to prove the ergodicity of general Markov chain? One should stay with ergodic theory and try to find something applicable to Markov chain. Or one could completely forget the ergodic theory and only work hard to prove small set or drift conditions in one's own setting?

Maybe the answer to the second question is opinion-based. But please share your experience with me. I am a phd student and I would like to know if it is worth investing a lot of time in one of the direction. If you could give me some related advice, I will also be very thankful.

Thank you very much for your help.

I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking values in a general state space(say a Polish space).

Although I've learned many interesting things, such as the application of ergodic theory in number theory, I don't find anything which helps to solve my original problem, i.e. the ergodicity of Markov chain. I expected to find something which gives a sufficient and easy-to-verify condition on the ergodicity of the shift operator. But I only find such a condition(irreducibility) in case of Markov chain taking discrete values

Then as for the case of Markov chain taking values in a general state space, it seems to me that all the existing sufficient conditions for its ergodicity(such as the small set condition, drift condition) are done without mentioning any abstract setting of ergodic theory, such as in the book Markov chain and stochastic stability or in this recent paper. And it is difficult to verify these conditions in general.

So my questions are:

Are there any existing results in ergodic theory which can help to easily establish ergodicity of general Markov chain? My impression is that ergodic theory is powerful and has been developed for a long time, have I missed some important results?

If no such a result exists, then what is the more hopeful choice if one needs to prove the ergodicity of general Markov chain? One should stay with ergodic theory and try to find something applicable to Markov chain. Or one could completely forget the ergodic theory and only work hard to prove small set or drift conditions in one's own setting?

Maybe the answer to the second question is opinion-based. But please share your experience with me. I am a phd student and I would like to know if it is worth investing a lot of time in one of the direction. If you could give me some related advice, I will also be very thankful.

Thank you very much for your help.

Edition to make my question clear:

By "ergodicity of a Markov chain taking values in a general state space", I mean there is a Markov chain $(X_n)_{n\geq 1}$ with $X_n \in S$ and $S$ is a Polish space, suppose $\mu$ is a measure on $\mathcal{B}(S)$ and we know already this Markov chain is invariant with respect to $\mu$, then saying this chain is ergodic means we have for any $B \in \mathcal{B}(S)$, we have $$\dfrac{1}{n}\sum_{k=1}^n 1_B(X_n) \to \mu(B) \text{ almost surely}$$ By abstract ergodic theory I mean there is a measure space $(\Omega, \mathcal{B}, \mu)$ and a measure preserving transformation $T: \Omega \to \Omega$, i.e. $T^{-1}B \in \mathcal{B}$ and $\mu(T^{-1}B) = \mu(B)$, when $T$ is ergodic then we have theorems such as Birkhoff's ergodic theorem and a lot of others interesting results. I wish to find some results in this abstract setting such that the Markov chain's ergodicity is an application of the results. But I find nothing in this direction and all I know about how to prove ergodicity of general Markov chain don't use results of abstract ergodic theory. Is ergodic theory useful in proving ergodicity of general Markov chain?