Forbidden transitions in Markovian systems

An important result in the theory of Markov chains in continuous time is the Lévy dichotomy, that a transition probability is either always or never zero. This was proved by Ornstein for autonomous chains, in which the transition probabilities are invariant under time shifts, and this paper considers the situation when this invariance is not assumed. The problem was solved for finite chains in joint work with David Williams in 1973, but the case of a countable infinity of states is much deeper. By analysing the family of relations on the state space that codify the zeros of the transition probabilities, it proves possible to develop a general theory that generalizes Ornstein's theorem, though many questions remain open.