# Quasi-stationary distributions and diffusion models in population dynamics

Abstract : 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 $- \infty$ at the origin, and the diffusion to have an entrance boundary at $+\infty$. 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 $+ \infty$ 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 $+\infty$ 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.
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https://hal.archives-ouvertes.fr/hal-00138521
Contributor : Amaury Lambert Connect in order to contact the contributor
Submitted on : Monday, January 26, 2009 - 7:06:56 PM
Last modification on : Wednesday, December 9, 2020 - 3:17:13 PM
Long-term archiving on: : Thursday, September 23, 2010 - 5:00:33 PM

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### Citation

Patrick Cattiaux, Pierre Collet, Amaury Lambert, Servet Martinez, Sylvie Méléard, et al.. Quasi-stationary distributions and diffusion models in population dynamics. 2007. ⟨hal-00138521v3⟩

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