explosion of a chain
Explosion is the phenomenon, possible only on an infinite state space, where a continuous-time Markov chain makes infinitely many jumps in a finite amount of time. It is the central well-posedness issue of continuous-time chains: when you build a chain by specifying rates, you must check that the resulting process is actually defined for all time, and explosion is exactly what can go wrong.
Recall the construction: from state i the chain waits an exponential holding time of rate q_i, then jumps according to the jump chain. Let J_n be the time of the n-th jump and define the explosion time zeta = lim_n J_n = sum_n (holding time at the n-th visited state). If the holding rates q_i grow fast enough along the realized path, these exponential holding times can have a finite sum almost surely, so zeta < infinity: the chain has taken infinitely many jumps by the finite time zeta and is, in the naive sense, 'undefined' afterwards. Whether this happens depends on the rates: a useful sufficient condition for NON-explosion is that the holding rates are bounded, sup_i q_i < infinity, or more sharply (Reuter's criterion for the explicit chain) that sum 1/q_{i_n} diverges along reachable paths. For a pure birth process with birth rates lambda_n, non-explosion is exactly sum 1/lambda_n = infinity; if the birth rates grow like n^(1+epsilon) the population reaches infinity in finite time.
Explosion matters because it forces a modeling decision and breaks naive formulas. If a chain explodes, the matrix exponential e^(tQ) need not be stochastic (its rows sum to less than 1, the deficit being P(already exploded)), the forward and backward equations can have several solutions, and 'the' chain is not unique without a rule for what happens at zeta. The standard convention is the minimal process: kill the chain at zeta (send it to a cemetery state). One can instead 'restart' it (with a boundary distribution), giving a different, also valid, non-minimal chain. The honest summary: conservativeness of Q (rows sum to zero) is NOT the same as non-explosion; you must verify a growth condition on the rates.
A pure birth process with birth rate lambda_n = n^2 in state n. Since sum 1/n^2 < infinity, the holding times sum to a finite (random) time almost surely: the population explodes to infinity in finite time. Contrast with lambda_n = n (the Yule process), where sum 1/n diverges and the chain does NOT explode.
Whether sum 1/lambda_n converges decides explosion for a pure birth process: faster-than-linear birth rates blow up in finite time.
Conservativeness (Q rows sum to zero) does NOT imply non-explosion. You must check a growth condition: bounded exit rates suffice, and for pure birth, non-explosion is exactly sum 1/lambda_n = infinity. When a chain explodes, e^(tQ) is sub-stochastic and 'the' chain is non-unique without a boundary rule.