Donsker theorem for rough paths


Rough paths were introduced by Lyons in 1998 to cope with the non-continuity of the Itô map (the map which sends a sample-path of the Brownian motion to the solution of an SDE driven by this path). The so-called enriched Brownian motion is the couple made by the Brownian itself and its Lévy area. It is thus a natural question to enrich the Donsker theorem and see if the adjunction of this new component does change the convergence rate we established earlier. We prove in this article that the rate is unchanged whatever the integrability of the random variables involved in the random walk. How ever, the more integrable they are, the more Hölderian are their sample-paths hence the richer is the convergence.

This paper is the sequel of, in which we worked in infinite dimension.