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?
Best Answer
OK - so you are talking about the ergodicity of a Markov chain with respect to a finite stationary measure. One general result you should be aware of is that in this situation ergodicity of the time shift in the path space (this is essentially the definition you use - you just refer to the corresponding ergodic theorem) is equivalent to "irreducibility" (absence of non-trivial invariant subsets) of the state space, see Different uses of the word "ergodic"
However it does not help much. The point is that the question you ask is essentially the same as asking about a "general method" to decide whether a given probability measure preserving transformation is ergodic. For, any such transformation can be considered as a "deterministic" Markov chain (all transition probabilities are delta measures). On the other hand, as I have just mentioned, Markov chain ergodicity is equivalent to ergodicity of the deterministic time shift on its path space.
Now, there are no general results in ergodic theory which, to quote your question "can help to easily establish ergodicity". Indeed, any textbook in ergodic theory opens with the standard set of examples (irrational rotation, Bernoulli shift, Gauss transformation, hyperbolic toral endomorphism), ergodicity of each of which is established in its own way. Unfortunately, these approaches are very far from being universal, and ergodicity of certain well-known transformations is a very hard problem (the most famous example being, arguably, Boltzmann's ergodic hypothesis).
EDIT (additional comments) Harris recurrence is in a sense an artifact which does not have anything to do with ergodicity. It appeared at a very early stage of the theory (its predecessor is Doeblin's condition) and was geared at establishing (fast) convergence in total variation to a stationary distribution. Therefore, absolute continuity of transition probabilities (understood literally or in a somewhat weaker form) was a necessary ingredient of this setup. Markov chains with singular transition probabilities (in particular, deterministic Markov chains corresponding to measure preserving transformations) were in this probabilistic context considered as something utterly exotic. I would recommend the books by Foguel MR0261686 (in the first place) and Rosenblatt MR0329037 (more specific) for an "ergodic theory" approach to Markov chains. Both these books contain the statement I had mentioned.