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September 2009 Quasi-stationary distributions and diffusion models in population dynamics
Patrick Cattiaux, Pierre Collet, Amaury Lambert, Servet Martínez, Sylvie Méléard, Jaime San Martín
Ann. Probab. 37(5): 1926-1969 (September 2009). DOI: 10.1214/09-AOP451


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 L2 of the reference measure for the killed process.


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Patrick Cattiaux. Pierre Collet. Amaury Lambert. Servet Martínez. Sylvie Méléard. Jaime San Martín. "Quasi-stationary distributions and diffusion models in population dynamics." Ann. Probab. 37 (5) 1926 - 1969, September 2009.


Published: September 2009
First available in Project Euclid: 21 September 2009

zbMATH: 1176.92041
MathSciNet: MR2561437
Digital Object Identifier: 10.1214/09-AOP451

Primary: 92D25
Secondary: 37A30 , 60J60 , 60J70 , 60J85 , 60K35

Keywords: Birth–death process , convergence rate , entrance boundary at infinity , Generalized Feller diffusion , logistic growth , Population dynamics , Q-process , quasi-stationary distribution , Yaglom limit

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 5 • September 2009
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