Invariant-measure transfer between a jump chain and a CTMC

For holding rates $q_i$, a measure $\pi$ satisfies $\pi Q=0$ exactly when ${\mu }_i=q_i{\pi }_i$ is invariant for the jump chain. Thus ${\pi }_i={\mu }_i/q_i$ when normalization is possible.

Even when the jump chain has invariant probability $\mu$, the continuous-time invariant measure ${\mu }_i/q_i$ may have infinite mass if holding rates become too small, producing infinite mean return time.

A recurrent jump chain with infinite invariant measure $\mu$ can yield a positive-recurrent continuous-time chain when ${\sum }_i{\mu }_i/q_i<\infty$.