Exponentiality of first passage times of continuous time Markov chains

Let $(X,\p_x)$ 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 $\mu$ is a quasi-stationary distribution relatively to $T$, then this time is exponentially distributed under $\p_\mu$. However, quasi-stationarity is not a necessary condition. In this paper, we determine more general conditions on an initial distribution $\mu$ for $T$ to be exponentially distributed under $\p_\mu$. 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.