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

Nicolas Champagnat,Denis Villemonais

doi:10.30757/alea.v14-11

Abstract

This article studies the quasi-stationary behaviour of absorbed one-dimensional diffusion processes with killing on [0, ∞). 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 ∞ 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.

Highlights

This article studies the quasi-stationary behavior of general one-dimensional diffusion processes with killing in an interval E of R, absorbed at its finite boundaries
We recall that a quasi-stationary distribution for a continuous-time Markov process (Xt, t ≥ 0) on the state space E ∪ {∂}, is a probability measure α on E such that
Our goal is to give conditions ensuring the existence of a unique quasi-limiting distribution α on E, i.e. a probability measure α such that for all probability measures μ on E such that μ(E) > 0 and all A ⊂ E measurable, lim t→+∞

Summary

Introduction

This article studies the quasi-stationary behavior of general one-dimensional diffusion processes with killing in an interval E of R, absorbed at its finite boundaries. The case with killing is more complex and the existing results cover less general situations Steinsaltz and Evans (2007); Kolb and Steinsaltz (2012) These references are restricted to the study of solutions to SDEs with continuous absorption rate up to the boundary of E, and let the questions of uniqueness of the quasi-stationary distribution and of convergence in (1.1) open. The condition on the diffusion without killing comes from Champagnat and Villemonais (2016b) and covers most one-dimensional diffusions on [0, ∞) such that ∞ is an entrance boundary and a.s. absorbed in finite time at 0.

Absorbed diffusion processes with killing

Quasi-stationary distributions for diffusion processes with killing

Examples and comparison with existing results

Proof of the results of Section 3