The Quasi-Stationary Distribution for Small Random Perturbations of Certain One-Dimensional Maps

We analyze the quasi-stationary distributions of the family of Markov chains {Xε n }, ε > 0, obtained from small non-local random perturbations of iterates of a map f : I → I on a compact interval. The class of maps considered is slightly more general than the class of one-dimensional Axiom A maps. Under certain conditions on the dynamics, we show that as ε → 0 the limit quasi-stationary distribution of the family of Markov chains is supported on the union of the periodic attractors of the map f . Moreover, we show that these conditions are satisfied by Markov chains obtained as perturbations of the logistic map f(x) = μx(1−x) by additive Gaussian noise and also by Markov chains that model density-dependent branching processes.