Renewal dynamical approach for non-minimal quasi-stationary distributions of one-dimensional diffusions

TL;DR

This paper introduces a renewal dynamical approach to analyze non-minimal quasi-stationary distributions of one-dimensional diffusions, establishing conditions for convergence and existence of Yaglom limits.

Contribution

It develops a novel renewal transform method linking convergence of quasi-stationary distributions to lifetime moment growth rates.

Findings

Convergence of the renewal transform characterizes quasi-stationary distributions.

A necessary condition for Yaglom limits is identified.

The approach provides new insights into the structure of non-minimal distributions.

Abstract

We consider quasi-stationary distributions for one-dimensional diffusions via the renewal dynamical approach. We show that convergence of the iterative renewal transform to quasi-stationary distributions is equivalent to a condition on the moment growth rate of the lifetime, which is at the same time a necessary condition for the existence of Yaglom limits.