NOISY ONE-DIMENSIONAL MAPS NEAR A CRISIS .1. WEAK GAUSSIAN WHITE AND COLORED NOISE

We study one-dimensional single-humped maps near the boundary crisis a t fully developed chaos in the presence of additive weak Gaussian whit e noise. By means of a new perturbation-like method the quasi-invarian t density is calculated from the invariant density at the crisis in th e absence of noise. In the precritical regime, where the deterministic map may show periodic windows, a necessary and sufficient condition f or the validity of this method is derived. From the quasi-invariant de nsity we determine the escape rate, which has the form of a scaling la w and compares excellently with results from numerical simulations. We find that deterministic transient chaos is stabilized by weak noise w henever the maximum of the map is of order z > 1. Finally, we extend o ur method to more general maps near a boundary crisis and to multiplic ative as well as colored weak Gaussian noise. Within this extended cla ss of noises and for single-humped maps with any fixed order z > 0 of the maximum, in the scaling law for the escape rate both the critical exponents and the scaling function are universal.