 Solution

An irreducible continuous-time chain is recurrent when, after leaving any state, it returns there almost surely. Non-explosion ensures that returns of $X$ correspond exactly to returns of its embedded jump chain, so one is recurrent precisely when the other is.

The expected total occupation time of zero isEach visit contributes an independent holding time of mean $1/q_0$. Thus this expectation is $(1/q_0)$ times the expected number of visits. If the integral is infinite, the jump chain has infinitely many expected visits; for an irreducible chain this is equivalent to recurrence.