Article Dans Une Revue Foundations of Physics Année : 2001

Noise corrections to stochastic trace formulas

G. Palla
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G. Vattay
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André Voros
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N. Søndergaard
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C.P. Dettmann
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Résumé

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.

Dates et versions

hal-00134275 , version 1 (01-03-2007)

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G. Palla, G. Vattay, André Voros, N. Søndergaard, C.P. Dettmann. Noise corrections to stochastic trace formulas. Foundations of Physics, 2001, 31, pp.641-657. ⟨hal-00134275⟩
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