### Abstract

We review studies of an evolution operator ℒ for a discrete Langevin equation with a strongly hyperbolic classical dynamics and a Gaussian noise. The leading eigenvalue of ℒ yields a physically measurable property of the dynamical system, the escape rate from the repeller. The spectrum of the evolution operator 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 ℒ^{n}. Using an integral representation of the evolution operator 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.

Original language | English |
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Pages (from-to) | 641-657 |

Number of pages | 17 |

Journal | Foundations of Physics |

Volume | 31 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 1 2001 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Foundations of Physics*,

*31*(4), 641-657. https://doi.org/10.1023/A:1017569010085