expected hitting time with two absorbing states

Question

Consider a Markov chain in a finite space and with two absorbing states, each of which is accessible from the other, transient states. Is the expected number of transitions to reach any single absorbing state infinite?

Answer 1

Consider a chain with 3 states: $1,2,3$, the first and the last being absorbing and $2$ having transitions of $\frac12$ probability to each of the latter states. Then the expected times are $$ \begin{align} (0,1,\infty) &\text{ for reaching state }1 \\ (\infty,1,0) &\text{ for reaching state }3 \end{align} $$ Clearly, the expected hitting times from one absorbing set to another is infinite, but for all transient states it easily can be bounded.