Unique Quasi-Stationary Distribution, with a possibly stabilizing extinction

Aurélien Velleret

Published 2018-02-07Version 1

We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditioned to never be absorbed. The technique relies on a coupling procedure that is related to Doeblin's type conditions. The main novelty is that we modulate each coupling step depending both on a final horizon of time --for survival-- and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. For instance, these conditions apply to cases of quasi-ergodicity where the time needed for Doeblin's type condition to hold cannot be uniformly bounded over all initial conditions. The main arguments are both that the extinction prevents any escape towards infinity and that, compared to the time-scale of extinction, the process diffuses quicker. As an illustration, we describe three examples to which our results apply : a birth-and-death process confined by catastrophes ; a continuous-time pure jump process on $\mathbb{R}^d$ confined by its extinction rate ; a Brownian motion on $\mathbb{R}^2$ confined by its extinction when hitting obstacles.