 Nonexplosion of a continuous-time Markov chain

A continuous-time Markov chain is nonexplosive if it makes only finitely many jumps on every bounded time interval. A uniformly bounded total jump intensity permits domination by a Poisson process. More generally, if a population process stays in a finite random set of states on each finite interval and every individual state has finite exit rate, localizing to fixed finite sets gives the same conclusion. In an open migration process, external Poisson immigration bounds the total population while internal moves preserve it and exits reduce it, giving this localization.