Théorème limite pour une équation différentielle à coefficient aléatoire à mémoire longue

We consider an ordinary differential equation depending on a small parameter and with a long-range random coefficient. We establish that the solution of this ordinary differential equation converges to the solution of a stochastic differential equation driven by a fractional Brownian motion. The index of the fractional Brownian motion depends on the asymptotic behavior of the covariance function of the random coefficient. The proof of the convergence uses the T. Lyons theory of “rough paths”. To cite this article: R. Marty, C. R. Acad. Sci. Paris, Ser. I 338 (2004).