On the enhancement of diffusion by chaos, escape rates and stochastic instability

We consider stochastic perturbations of expanding maps of the interval wher e the noise can project the trajectory outside the interval. We estimate th e escape rate as a function of the amplitude of the noise and compare it wi th the purely diffusive case. This is done under a technical hypothesis whi ch corresponds to stability of the absolutely continuous invariant measure against small perturbations of the map. We also discuss in detail a case of instability and show how stability can be recovered by considering another invariant measure.