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Probability Theory and Related Fields

, Volume 122, Issue 3, pp 341–388 | Cite as

Pathwise description of dynamic pitchfork bifurcations with additive noise

  • Nils Berglund
  • Barbara Gentz

Abstract.

 The slow drift (with speed ɛ) of a parameter through a pitchfork bifurcation point, known as the dynamic pitchfork bifurcation, is characterized by a significant delay of the transition from the unstable to the stable state. We describe the effect of an additive noise, of intensity σ, by giving precise estimates on the behaviour of the individual paths. We show that until time \(\) after the bifurcation, the paths are concentrated in a region of size \(\) around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval \(\), after which they are likely to stay close to the corresponding deterministic solution. We derive exponentially small upper bounds for the probability of the sets of exceptional paths, with explicit values for the exponents.

Keywords

Additive Noise Pitchfork Bifurcation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Nils Berglund
    • 1
  • Barbara Gentz
    • 2
  1. 1.Georgia Institute of Technology, Atlanta, GA 30332-0430, USA and Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany. Current address: Department of Mathematics, ETH Zürich, ETH Zentrum, 8092 Zürich, Switzerland. e-mail: berglund@math.ethz.chCH
  2. 2.Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany. e-mail: gentz@wias-berlin.deDE

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