The criterion for uniqueness of quasi-stationary distributions of Markov processes and their domain of attraction problem

We consider a Markov process $ X(t) $ on the nonnegative integers $E= S \cup \{0\}$, where $S=\{1,2,...\}$ is an irreducible class and 0 is an absorbing state. In this paper, we investigate conditions under which the quasi-stationary distribution for $X(t)$ exists and is unique, and any initial distribution supported in $S$ is in the domain of attraction of this quasi-stationary distribution. We further find five conditions which are equivalent to that the extinction time is uniformly bounded. As a consequence, we prove the van Doorn's conjecture in \cite{VD2012}. And we can greatly improve theorem 1 in \cite{VD2012}.