Quasistationary distributions for one-dimensional diffusions with singular boundary points

TL;DR

This paper characterizes when quasistationary distributions exist for one-dimensional diffusions with singular boundary points, extending previous results to cases with singular behavior at boundaries and certain absorption at zero.

Contribution

It provides new conditions for the existence of quasistationary distributions in diffusions with singular boundary points, broadening the scope of prior work.

Findings

Existence of quasistationary distributions linked to the spectrum of the generator.

Extension of previous results to singular boundary behaviors.

Conditions for absorption at zero ensuring quasistationary distributions.

Abstract

In the present work we characterize the existence of quasistationary distributions for diffusions on $(0,\infty)$ allowing singular behavior at $0$ and $\infty$. If absorption at 0 is certain, we show that there exists a quasistationary distribution as soon as the spectrum of the generator is strictly positive. This complements results of Collet et al. (Ann. Probab. 2009) and Kolb and Steinsaltz (Ann. Probab. 2012) for $0$ being a regular boundary point and extends results by Collet et al. (Ann. Probab. 2009) on singular diffusions.