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Here the total holding rate isand therefore the jump chain has state-independent transitionsIt is an upward biased random walk on $\mathbb{Z}$. By the strong law of large numbers, $Y_n/n⟶1/3$ almost surely, so $Y_n\to +\infty$. The jump chain, and hence $X$, is transient.

Counting measure is invariant for this jump chain: for every $j$, its incoming mass is $1/3+2/3=1$. The invariant-measure transfer between a jump chain and a CTMC therefore gives an invariant measure proportional to $q_i^{-1}=3^{-|i|-1}$. Sincethe normalized invariant distribution of a continuous-time Markov chain isThis does not contradict transience because an invariant distribution of an explosive chain need not imply positive recurrence.

It remains to verify explosion. Starting from zero, the probability that the biased walk ever reaches $-k$ is $2^{-k}$. Once it reaches any state, its probability of returning after departure isso its expected total number of visits to that state, conditional on reaching it, is $1/(1-2/3)=3$. If $V_i$ is the number of visits to $i$, thenConsequently the expected sum of all holding times isThe nonnegative random variable $T_{\infty }$ is therefore finite almost surely. By explosion by summable holding times, $X$ makes infinitely many jumps by time $T_{\infty }$, so it is explosive.

For $i\ge 1$, the total holding rate is $(\lambda +1)2^i$, and the jump chain moves upward and downward with probabilitieswhile it moves from zero to one with probability one. Thus it is a reflected nearest-neighbour random walk. Recurrence is unaffected by the holding times, so it is recurrent exactly when $p\le q$. In the given range,For $\lambda >1$ the jump chain is transient.

The detailed-balance equations arewhere the formula also covers the boundary $i=0$. Hence every candidate invariant measure hasAt $\lambda =1$ this is summable and the chain is nonexplosive, givingFor $\lambda =2$ it is not summable. When $1<\lambda <2$, it is a summable formal solution of $\pi Q=0$, but the chain explodes, so it is not an invariant probability for the minimal process; this is precisely the caveat in invariant distribution of an explosive chain. Consequently

It remains to classify explosion. At $\lambda =1$, the recurrent jump chain visits zero infinitely often. The holding time at zero has rate one, so nonexplosion from recurrent visits to a slow state shows that the accumulated time is infinite almost surely.

For $\lambda >1$, let $V_i$ be the total number of visits of the transient jump chain to $i$. Using the stated uniform visit bound and writing $q_i=(\lambda +1)2^i$ for $i\ge 1$,because ${\sum }_i{2}^{-i}<\infty$. Conditional on the jump path, this is the expected sum of all holding times, so the total lifetime is finite almost surely. This is explosion of an upward-biased walk with geometrically increasing rates. Therefore