A unifying approach to non-minimal quasi-stationary distributions for one-dimensional diffusions

TL;DR

This paper introduces a unified method to analyze convergence to non-minimal quasi-stationary distributions in one-dimensional diffusions, focusing on tail behavior and applying it to Kummer diffusions with negative drifts.

Contribution

It presents a novel approach linking convergence to tail behavior through the first hitting uniqueness property, applicable to a broad class of diffusions.

Findings

Established a method reducing convergence analysis to tail behavior.

Applied the approach to Kummer diffusions with negative drifts.

Identified initial distributions that converge to each non-minimal quasi-stationary distribution.

Abstract

Convergence to non-minimal quasi-stationary distributions for one-dimensional diffusions is studied. We give a method of reducing the convergence to the tail behavior of the lifetime via a property which we call the first hitting uniqueness. We apply the results to Kummer diffusions with negative drifts and give a class of initial distributions converging to each non-minimal quasi-stationary distribution.