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

TL;DR

This paper characterizes the existence and construction of quasi-stationary distributions for one-dimensional diffusions killed at zero, providing explicit conditions and methods for their identification and properties.

Contribution

It offers a necessary and sufficient condition for QSD existence and constructs all such distributions, also providing a drift-based criterion for R-positivity.

Findings

Necessary and sufficient condition for QSD existence

Explicit construction of all QSDs for the process

A drift-based criterion for R-positivity

Abstract

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.