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Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations. (English) Zbl 1136.60317

Summary: 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.

MSC:

60F05 Central limit and other weak theorems
34A45 Theoretical approximation of solutions to ordinary differential equations
34F05 Ordinary differential equations and systems with randomness
60G15 Gaussian processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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