the minimal process
The minimal process is the canonical continuous-time Markov chain you obtain from a given Q-matrix by running the jump-chain-plus-holding-times construction and killing the chain the instant it explodes. It is the standard answer to the non-uniqueness left open by explosion: among all chains whose local rates are Q, the minimal one is the one that does the least — it never re-enters from infinity. It is the 'default' CTMC and the unique solution of the Kolmogorov equations under non-explosion.
Construct it explicitly: starting at i, draw holding times and jumps from Q to get a sequence of states and jump times J_1 < J_2 < ..., set the explosion time zeta = lim_n J_n, and define X_t to follow this trajectory for t < zeta and X_t = (cemetery) for t >= zeta. The resulting transition function P_t^min(i,j) is minimal in the precise sense that for ANY other transition function P_t with generator Q, one has P_t^min(i,j) <= P_t(i,j) for all i, j, t — hence the name. It is the smallest nonnegative solution of the backward equation dP/dt = Q P with P_0 = I. When the chain is non-explosive (zeta = infinity almost surely from every start), the minimal process is honest (its rows sum to 1) and is the UNIQUE chain with generator Q, equal to e^(tQ). When the chain can explode, the minimal process is sub-stochastic and there exist other, larger, non-minimal chains that 'come back from infinity' according to a chosen boundary behavior.
The minimal process is why one can speak of 'the' chain with a given Q at all: it is a well-defined object for every Q-matrix, explosive or not. The honest caveat is exactly the converse: when explosion is possible, picking the minimal process is a modeling CHOICE, not forced by the rates. A queue whose customers arrive faster and faster might in reality bounce back from a full state; the minimal process instead simply stops, which may or may not be the physics you want. So: state your construction. If you only specify Q and explosion is possible, you have not specified a unique chain until you say 'minimal' (or give a boundary rule).
Take the explosive pure birth chain with lambda_n = n^2. Its minimal process climbs 0,1,2,... and is sent to the cemetery at the (finite) explosion time zeta; for t >= zeta it is undefined-as-a-state. A non-minimal alternative would instantly restart from some state after exploding — a genuinely different, also Markovian, chain with the same Q.
Faced with explosion, the minimal process simply stops; other chains with the same rates can re-enter from infinity.
Specifying Q does not specify a unique chain when explosion is possible — choosing the minimal process is a modeling decision. The minimal process is sub-stochastic exactly when the chain explodes, and equals e^(tQ) (rows summing to 1) precisely when the chain is non-explosive.