The quasi-stationary distribution for small random perturbations of certain one-dimensional maps

We analyse the quasi-stationary distributions of the family of Markov chains {X-n(epsilon)}, epsilon > 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 epsilon --> 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) = mu x(1 - x) by additive Gaussian noise and also by Markov chains that model density-dependent branching processes. (C) 1999 Elsevier Science B.V. All rights reserved.