Exponentiality of first passage times of continuous time Markov chains

TL;DR

This paper explores conditions under which the first passage time of a continuous-time Markov chain is exponentially distributed, extending beyond quasi-stationary distributions, with applications to branching processes.

Contribution

It identifies broader conditions for exponentiality of first passage times and relates quasi-stationary distributions to initial laws that produce exponential distributions.

Findings

Exponentiality of first passage times can occur without quasi-stationarity.

Quasi-stationary distributions can be characterized via initial laws leading to exponential passage times.

Examples in branching processes show exponentiality implies quasi-stationarity.

Abstract

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.