 Solution

After leaving zero, the jump chain avoids every future reset with probabilityThis probability is positive exactly when ${\prod }_{i\ge 1}(1+{\rho }_i)<\infty$. If $q>0$, the probability of ever returning to zero is less than one, so the irreducible chain is transient. If $q=0$, a reset to zero occurs almost surely after each departure; by the Strong Markov property this happens repeatedly, so the chain is recurrent. Therefore