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Suppose a CTMC is on the positive integer numbers whose underlying DTMC is irreducible and null recurrent. Must the CTMC be regular? Or is there a counter example?

To disprove it, I know simple random walk is an example of null recurrent. But it does not on positive integers. Another null recurrent DTMC is about Gambler’s Ruin Problem when it's fair. But since it has only finite states, the corresponding CTMC must be regular, so it won't serve as a counterexample.
On the other hand, if we try to prove it, maybe we could decompose the CTMC into periods of returning to original states and use strong law of large number. But I don't know how to show the expectation of each period is finite.

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    Let $T_k$ denote the time of kth transition. A CTMC is regular means $P(\lim_{k\to\infty}T_k=\infty)=1$. – 2017-02-26
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    Yeah, I looked it up and found notes with this term. Anyway, the state space does not matter at all, as long as it is countably infinite. – 2017-02-26
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    What comes to mind is the following: take the state space to be $\mathbb{N}_0$ for convenience, and then consider a chain which goes from $n$ to $n+1$ with rate $2^n$ and goes from $n$ to $0$ with the "smallest possible rate" that still gives null recurrence of the DTMC. The calculation here should be easy to do (because $P^k(0,0)$ is very easy to write down), and it should give you some intuition. Perhaps that intuition is faulty (I haven't studied this issue before), but it's a place to start anyway. – 2017-02-26

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