A Brownian Motion on the Diffeomorphism Group of the Circle

Let Diff(S ¹) be the group of orientation preserving C ^?∞? diffeomorphisms of S ¹. In 1999, P. Malliavin and then in 2002, S. Fang constructed a canonical Brownian motion associated with the H ^3/2 metric on the Lie algebra diff(S ¹). The canonical Brownian motion they constructed lives in the group Homeo(S ¹) of Hölderian homeomorphisms of S ¹, which is larger than the group Diff(S ¹). In this paper, we present another way to construct a Brownian motion that lives in the group Diff(S ¹), rather than in the larger group Homeo(S ¹).