Domain of attraction of the quasi-stationary distribution for one-dimensional diffusions

Hanjun Zhang, Guoman He

Published 2014-09-29Version 1

In this paper, we study quasi-stationarity for one-dimensional diffusions killed at 0, when 0 is a regular boundary and $+\infty$ is an entrance boundary. We give a necessary and sufficient condition for the existence of exactly one quasi-stationary distribution, and that this distribution attracts all initial distributions. In particular, a novelty here is that we show that if the killed semigroup is intrinsically ultracontractive, then it is not only a sufficient condition ensuring the uniqueness of the quasi-stationary distribution, but also a necessary condition.