EXPONENTIALITY OF FIRST PASSAGE TIMES OF CONTINUOUS TIME MARKOV CHAINS

Abstract. Let (X,Px) be a continuous time Markov chain with finite or countable state space S and let T be its first passage time in a subset D of S. It is well known that if µ is a quasi-stationary distribution relatively to T, then this time is exponentially distributed under Pµ. However, quasi-stationarity is not a necessary condition. In this paper, we determine more general conditions on an initial distribution µ for T to be exponentially distributed under Pµ. We show in addition how quasi-stationary distributions can be expressed in terms of any initial law which makes the distribution of T exponential. We also study two examples in branching processes where exponentiality does imply quasi-stationarity. 1