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

Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations