Abstract
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.
Citation
Fima C. Klebaner. Justin Lazar. Ofer Zeitouni. "On the quasi-stationary distribution for some randomly perturbed transformations of an interval." Ann. Appl. Probab. 8 (1) 300 - 315, February 1998. https://doi.org/10.1214/aoap/1027961045
Information