The Annals of Probability

Quasi-stationary distributions and diffusion models in population dynamics

Patrick Cattiaux, Pierre Collet, Amaury Lambert, Servet Martínez, Sylvie Méléard, and Jaime San Martín

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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 −∞ 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.

Article information

Source
Ann. Probab., Volume 37, Number 5 (2009), 1926-1969.

Dates
First available in Project Euclid: 21 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1253539860

Digital Object Identifier
doi:10.1214/09-AOP451

Mathematical Reviews number (MathSciNet)
MR2561437

Zentralblatt MATH identifier
1176.92041

Subjects
Primary: 92D25: Population dynamics (general)
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J60: Diffusion processes [See also 58J65] 60J85: Applications of branching processes [See also 92Dxx] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Keywords
Quasi-stationary distribution birth–death process population dynamics logistic growth generalized Feller diffusion Yaglom limit convergence rate Q-process entrance boundary at infinity

Citation

Cattiaux, Patrick; Collet, Pierre; Lambert, Amaury; Martínez, Servet; Méléard, Sylvie; San Martín, Jaime. Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab. 37 (2009), no. 5, 1926--1969. doi:10.1214/09-AOP451. https://projecteuclid.org/euclid.aop/1253539860


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