Recent advances in the periodic orbit theory of stochastically perturbed systems have permitted a calculation of the escape rate of a noisy chaotic map to order fifty in the noise strength. Comparison with the usual asymptotic expansions obtained from integrals, and with a previous calculation of the electrostatic potential of exactly selfsimilar fractal charge distributions, leads to a remarkably precise form for the late terms in the expansion, with parameters determined independently from the fractal repeller and the critical point of the map. Borel resummation gives a precise meaning to the asymptotic expansion, which can then be compared to the escape rate as computed by alternative methods.