 Solution

When $\mu =\lambda$, an excursion after the jump from zero to one visits $n$ before its reset with probabilityThis tends to zero, so a catastrophe eventually occurs with probability one. Both the original chain and its jump chain are recurrent.

The expected number of jumps in a return cycle isThus the jump chain is null recurrent. But its holding time at $n\ge 1$ has mean $1/[\lambda (n+2)]$, and the initial holding time at zero has mean $1/\lambda$. The expected elapsed cycle time isThe continuous-time Markov chain is therefore positive recurrent. Its equilibrium distribution, computed as mean time in a state per mean cycle time, isThe entries sum to one by telescoping. The contrast is explained by progressively shorter holding times at the large states, not by a change in return probabilities.