### 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 |
---|---|

Pages (from-to) | 641-657 |

Number of pages | 17 |

Journal | Foundations of Physics |

Volume | 31 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 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

**Noise corrections to stochastic trace formulas.** / Palla, Gergely; Vattay, G.; Voros, André; Søndergaard, Niels; Dettmann, Carl Philip.

Research output: Contribution to journal › Article

*Foundations of Physics*, vol. 31, no. 4, pp. 641-657. https://doi.org/10.1023/A:1017569010085

}

TY - JOUR

T1 - Noise corrections to stochastic trace formulas

AU - Palla, Gergely

AU - Vattay, G.

AU - Voros, André

AU - Søndergaard, Niels

AU - Dettmann, Carl Philip

PY - 2001/4

Y1 - 2001/4

N2 - 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 ℒn. 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.

AB - 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 ℒn. 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.

UR - http://www.scopus.com/inward/record.url?scp=0035612337&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035612337&partnerID=8YFLogxK

U2 - 10.1023/A:1017569010085

DO - 10.1023/A:1017569010085

M3 - Article

VL - 31

SP - 641

EP - 657

JO - Foundations of Physics

JF - Foundations of Physics

SN - 0015-9018

IS - 4

ER -