Stationary measure of a not-explosive continuous time Markov chain

Question

Let $X_n$ be a non-explosive continuous time Markov chain with a stationary measure. Is it possible that its embedded discrete time Markov chain does not have a stationary measure?

Answer 1

Let $X(t)$ be a continuous-time Markov chain on the nonnegative integers with generator matrix $$ Q_{ij} = \begin{cases} 1,& i\geqslant 0,j=i+1\\ 1,& i\geqslant 1, j=0\\ -1,& i=j=0\\ 0,& \text{otherwise}. \end{cases} $$ Clearly $X(t)$ is non-explosive as $\sup_{i,j\in S} Q_{ij}=1<\infty$. Consider the balance equations $$\sum_{i\in S,j\notin S}Q_{ij}= \sum_{i\in S,j\notin S}Q_{ji} $$ for the subsets $S=\{0\},\{0,1\},\ldots$ of the state space: $$ \pi_n = \sum_{i=n+1}^\infty \pi_i,\ i\geqslant 0.\tag1 $$ From $\sum_{i=0}^\infty \pi_i=1$ and setting $n=0$ in $(1)$ we find that $\pi_0=\frac12$, and iterating yields the recurrence $\pi_n = \frac12\pi_{n-1}$. It follows that $X(t)$ has the stationary distribution $\pi_n = \frac12^{n+1}$.

The embedded chain has transition matrix $$ P_{ij} = \begin{cases} \frac12,& j=0\\ \frac12,& i\geqslant 0, j=i+1\\ 0,& \text{otherwise}. \end{cases} $$ Let $\tau=\inf\{n>0: X_n=0 \mid X_0=0\}$. Then $$\mathbb E[\tau] = \sum_{n=0}^\infty \mathbb P(\tau>n) = \sum_{n=0}^\infty \frac12=\infty,$$ so the embedded chain is transient.