  We consider a continuous-time Markov chain with a finite or countable state
space. For a site y and subset H of the state space, the hitting time of y
under taboo H is defined to be infinite if the process trajectory hits H before
y, and the first hitting time of y otherwise. We investigate the probability
that such times are finite. In particular, if the taboo set is finite, an
efficient iterative scheme reduces the study to the known case of a singleton
taboo. A similar procedure applies in the case of finite complement of the
taboo set. The study is motivated by classification of branching processes with
finitely many catalysts.
  Keywords and phrases: Markov chain, hitting time, taboo probabilities,
catalytic branching process.
