Quasi-stationary distributions and diffusion models in population dynamics
Author
dc.contributor.author
Cattiaux, Patrick
Author
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Collet, Pierre
es_CL
Author
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Lambert, Amaury
es_CL
Author
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Martínez Aguilera, Servet
es_CL
Author
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Méléard, Sylvie
es_CL
Author
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San Martín Aristegui, Jaime
es_CL
Admission date
dc.date.accessioned
2013-12-26T20:22:42Z
Available date
dc.date.available
2013-12-26T20:22:42Z
Publication date
dc.date.issued
2009
Cita de ítem
dc.identifier.citation
The Annals of Probability 2009, Vol. 37, No. 5, 1926–1969
en_US
Identifier
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DOI: 10.1214/09-AOP451
Identifier
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https://repositorio.uchile.cl/handle/2250/125870
Abstract
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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 L2 of the reference measure for
the killed process.