Domain of attraction of quasi-stationary distribution for one-dimensional diffusions

TL;DR

This paper investigates the conditions under which a unique quasi-stationary distribution exists and attracts all initial distributions for one-dimensional diffusions killed at zero, with specific boundary conditions.

Contribution

It provides a necessary and sufficient condition for the existence and uniqueness of the quasi-stationary distribution in this setting.

Findings

Existence of a unique quasi-stationary distribution under certain boundary conditions.

The quasi-stationary distribution attracts all initial distributions.

Characterization of the boundary conditions for quasi-stationarity.

Abstract

We study quasi-stationarity for one-dimensional diffusions killed at 0, when 0 is a regular boundary and $+\infty$ is an entrance boundary. We give a necessary and sufficient condition for the existence of exactly one quasi-stationary distribution, and we also show that this distribution attracts all initial distributions.