The Lévy Area Process for the Free Brownian Motion

In this paper, we construct a Lévy area process for the free Brownian motion and in this way, a typical geometric rough path (in the sense of T. Lyons (1998, Rev. Mat. Iberoamer.14, 215–310)), lying above the free Brownian path. Thus, the general results of Lyons on differential equations driven by rough paths may be applied to the free Brownian motion case.