A Brownian Motion on the Diffeomorphism Group of the Circle

Let Diff(S1) be the group of orientation preserving C ∞  diffeomorphisms of S1. In 1999, P. Malliavin and then in 2002, S. Fang constructed a canonical Brownian motion associated with the H3/2 metric on the Lie algebra diff(S1). The canonical Brownian motion they constructed lives in the group Homeo(S1) of Hölderian homeomorphisms of S1, which is larger than the group Diff(S1). In this paper, we present another way to construct a Brownian motion that lives in the group Diff(S1), rather than in the larger group Homeo(S1).