Approximation of quasi-stationary distributions for 1-dimensional killed diffusions with unbounded drifts

TL;DR

This paper develops an approximation method for the quasi-stationary distribution of 1D killed diffusions with unbounded drifts, using a Fleming-Viot particle system, with applications to biological models.

Contribution

It introduces a novel approximation approach for quasi-stationary distributions of diffusions with unbounded drifts, extending previous methods to more complex boundary behaviors.

Findings

The method effectively approximates the limiting distribution for complex diffusions.

Numerical applications demonstrate the approach's accuracy on biological models.

Abstract

The long time behavior of an absorbed Markov process is well described by the limiting distribution of the process conditioned to not be killed when it is observed. Our aim is to give an approximation's method of this limit, when the process is a 1-dimensional It\^o diffusion whose drift is allowed to explode at the boundary. In a first step, we show how to restrict the study to the case of a diffusion with values in a bounded interval and whose drift is bounded. In a second step, we show an approximation method of the limiting conditional distribution of such diffusions, based on a Fleming-Viot type interacting particle system. We end the paper with two numerical applications : to the logistic Feller diffusion and to the Wright-Fisher diffusion with values in $]0,1[$ conditioned to be killed at 0.