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rough paths.

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1 results

Asymptotic behavior of differential equations drivenby periodic andrandom processes with slowly decaying correlations
Renaud Marty
Journal:

ESAIM: Probability and Statistics / Volume 9 / February 2005

Published online by Cambridge University Press:

15 November 2005, pp. 165-184

Print publication:

February 2005
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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.