Research Article

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

Marty, Renaud

Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France; marty@cict.fr

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.

(Received May 6 2004)

(Revised November 8 2004)

(Revised February 9 2005)

(Online publication November 15 2005)

Key Words:

• Limit theorems;
• stationary processes;
• rough paths.

Mathematics Subject Classification:

• 34F05;
• 60F05;
• 60G15