We consider a Markov chain $X_n^{\varepsilon}$ obtained by adding
small noise to a discrete time dynamical system and study the chain's
quasi-stationary distribution (qsd). The dynamics are given by iterating a
function $f:I \to I$ for some interval I when f has finitely many
fixed points, some stable and some unstable. We show that under some conditions
the quasi-stationary distribution of the chain concentrates around the stable
fixed points when $\varepsilon \to 0$. As a corollary, we obtain the result for
the case when f has a single attracting cycle and perhaps repelling
cycles and fixed points. In this case, the quasi-stationary distribution
concentrates on the attracting cycle. The result applies to the model of
population dependent branching processes with periodic conditional mean
function.
Publié le : 1998-02-14
Classification:
Large deviations,
logistic
map,
branching systems,
quasi-stationary
distribution,
60J80,
60F10
@article{1027961045,
author = {Klebaner, Fima C. and Lazar, Justin and Zeitouni, Ofer},
title = {On the quasi-stationary distribution for some randomly perturbed
transformations of an interval},
journal = {Ann. Appl. Probab.},
volume = {8},
number = {1},
year = {1998},
pages = { 300-315},
language = {en},
url = {http://dml.mathdoc.fr/item/1027961045}
}
Klebaner, Fima C.; Lazar, Justin; Zeitouni, Ofer. On the quasi-stationary distribution for some randomly perturbed
transformations of an interval. Ann. Appl. Probab., Tome 8 (1998) no. 1, pp. 300-315. http://gdmltest.u-ga.fr/item/1027961045/