Document ESAIM: P&S, May 2005, Vol. 9, pp. 165-184
DOI: 10.1051/ps:2005009
Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations
Renaud MartyLaboratoire de Statistique et Probabilités, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France; marty@cict.fr
(Received May 6, 2004. Revised November 8, 2004 and February 9, 2005.)
Abstract
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.
Mathematics Subject Classification. 34F05, 60F05, 60G15.
Key words: Limit theorems, stationary processes, rough paths.
© EDP Sciences, SMAI 2005