Exponential convergence to quasi-stationary distribution for absorbed one-dimensional diffusions with killing

TL;DR

This paper establishes criteria for exponential convergence to a unique quasi-stationary distribution in absorbed one-dimensional diffusions with killing, using probabilistic and coupling methods rather than spectral theory.

Contribution

It provides new probabilistic criteria for exponential convergence to quasi-stationary distributions in one-dimensional diffusions with killing, applicable under broad boundary conditions.

Findings

Criteria for exponential convergence in total variation.

Applicability to diffusions with singular measures and jumps.

Exponential ergodicity results for the Q-process.

Abstract

This article studies the quasi-stationary behaviour of absorbed one-dimensional diffusion processes with killing on $[0,\infty)$. We obtain criteria for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. Our approach is based on probabilistic and coupling methods, contrary to the classical approach based on spectral theory results. Our general criteria apply in the case where $\infty$ is entrance and 0 either regular or exit, and are proved to be satisfied under several explicit assumptions expressed only in terms of the speed and killing measures. We also obtain exponential ergodicity results on the $Q$-process. We provide several examples and extensions, including diffusions with singular speed and killing measures, general models of population dynamics, drifted Brownian motions and some one-dimensional processes with jumps.