On the quasi-stationary distribution for some randomly perturbed transformations of an interval

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.