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Asymptotic behavior of differential equations drivenby periodic andrandom processes with slowly decaying correlations

Published online by Cambridge University Press:  15 November 2005

Renaud Marty
Renaud Marty*
Affiliation:
Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France; marty@cict.fr
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Abstract

We consider a differential equation with a random rapidly varying coefficient.The random coefficient is aGaussian process with slowly decaying correlations and compete with a periodic component. In theasymptotic framework corresponding to the separation of scales present in theproblem, we prove that the solution of the differential equationconverges in distribution to the solution of a stochastic differential equationdriven by a classical Brownian motion in some cases, by a fractional Brownianmotion in other cases. The proofs of these results are based on the Lyons theory ofrough paths. Finally we discuss applications in two physical situations.

Keywords

Limit theoremsstationary processesrough paths.

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 165 - 184
Copyright
© EDP Sciences, SMAI, 2005

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