Noise corrections to stochastic trace formulas

We review studies of an evolution operator ${\cal L}$ for a discrete Langevin equation with a strongly hyperbolic classical dynamics and a Gaussian noise. The leading eigenvalue of ${\cal L}$ yields a physically measurable property of the dynamical system, the escape rate from the repeller. The spectrum of the evolution operator ${\cal L}$ in the weak noise limit can be computed in several ways. A method using a local matrix representation of the operator allows to push the corrections to the escape rate up to order eight in the noise expansion parameter. These corrections then appear to form a divergent series. Actually, via a cumulant expansion, they relate to analogous divergent series for other quantities, the traces of the evolution operators ${\cal L}^n$. Using an integral representation of the evolution operator ${\cal L},$ we then investigate the high order corrections to the latter traces. Their asymptotic behavior is found to be controlled by sub-dominant saddle points previously neglected in the perturbative expansion, and to be ultimately described by a kind of trace formula.