White noise limits for discrete dynamical systems driven by fast deterministic dynamics

Dror Givon, Raz Kupferman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


We study a class of singularly perturbed dynamical systems that have fast and slow components, ε≪1 being the fast to slow timescale ratio. The fast components are governed by a strongly mixing discrete map, which is iterated at time intervals ε. The slow components are governed by a first-order finite-difference equation that uses a time step ε. As ε tends to zero, the fast components may be eliminated, giving rise to SDEs for the slow components. The emerging stochastic calculus is, in the general case, of neither Itô nor Stratonovich type, but depends on the correlation time of the mixing process.

Original languageAmerican English
Pages (from-to)385-412
Number of pages28
JournalPhysica A: Statistical Mechanics and its Applications
Issue number3-4
StatePublished - 15 Apr 2004

Bibliographical note

Funding Information:
We are grateful to Noam Berger for elucidating many of the technical aspects of this work, and to Jonathan Mattingly, Carlo Nitsch and Denis Talay for useful discussions. Special thanks to Andrew Stuart for pointing out the relation with Ref. [12] and for his comments on the manuscript. This research was supported in part by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, and in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the US Department of Energy under Contract DE-AC03-76-SF00098.


  • Mixing
  • SDEs
  • Scale separation


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