QUASI-STATIONARY DISTRIBUTIONS AND DIFFUSION MODELS IN POPULATION DYNAMICS

Cattiaux P; Collet, P.; Lambert, A; Martínez, S.; Méléard S; San Martin, J

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-8at the origin, and the diffusion to have an entrance boundary at +8. 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 +8 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 +8 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. © Institute of Mathematical Statistics, 2009.

Más información

Título según WOS: QUASI-STATIONARY DISTRIBUTIONS AND DIFFUSION MODELS IN POPULATION DYNAMICS
Título según SCOPUS: Quasi-stationary distributions and diffusion models in population dynamics
Título de la Revista: ANNALS OF PROBABILITY
Volumen: 37
Número: 5
Editorial: INST MATHEMATICAL STATISTICS
Fecha de publicación: 2009
Página de inicio: 1926
Página final: 1969
Idioma: English
URL: http://projecteuclid.org/euclid.aop/1253539860
DOI:

10.1214/09-AOP451

Notas: ISI, SCOPUS