Noise corrections to stochastic trace formulas

We review studies of an evolution operator L for a discrete Langevin equati on with a strongly hyperbolic classical dynamics and a Gaussian noise. The leading eigenvalue of L yields a physically measurable property of the dyna mical system, the escape rate from the repeller. The spectrum of the evolut ion operator L in the weak noise limit can be computed in several ways. A m ethod using a local matrix representation of the operator allows to push th e corrections to the escape rate zip to order eight in the noise expansion parameter. These corrections then appear to form a divergent series. Actual ly, via a cumulant expansion, they relate to analogous divergent series for other quantities, the traces of the evolution operators L-n. Using an inte gral representation of the evolution operator L, ive then investigate the h igh order corrections to the latter traces. Their asymptotic behavior is fo und to be controlled by sub-dominant saddle points previously neglected in the perturbative expansion, and to be ultimately described by a kind of tra ce formula.