Existence and construction of quasi-stationary distributions for one-dimensional diffusions

In this paper, we study quasi-stationary distributions (QSDs) for one-dimensional diffusions killed at 0, when 0 is a regular boundary and $+\infty$ is a natural boundary. More precisely, we not only give a necessary and sufficient condition for the existence of a QSD, but we also construct all QSDs for one-dimensional diffusions. Moreover, we give a sufficient condition for $R$-positivity of the process killed at the origin. This condition is only based on the drift, which is easy to check.