Institute of Mathematical Statistics, Annals of Probability, 5(37), 2009
DOI: 10.1214/09-aop451
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In this paper, we study quasi-stationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to $- ∞$ at the origin, and the diffusion to have an entrance boundary at $+∞$. These diffusions arise as images, by a deterministic map, of generalized Feller diffusions, which themselves are obtained as limits of rescaled birth--death processes. Generalized Feller diffusions take nonnegative values and are absorbed at zero in finite time with probability $1$. An important example is the logistic Feller diffusion. We give sufficient conditions on the drift near $0$ and near $+ ∞$ for the existence of quasi-stationary distributions, as well as rate of convergence in the Yaglom limit and existence of the $Q$-process. We also show that under these conditions, there is exactly one quasi-stationary distribution, and that this distribution attracts all initial distributions under the conditional evolution, if and only if $+∞$ is an entrance boundary. In particular this gives a sufficient condition for the uniqueness of quasi-stationary distributions. In the proofs spectral theory plays an important role on $L^2$ of the reference measure for the killed process.