, Volume 34, Issue 1, pp 23-41

A Brownian Motion on the Diffeomorphism Group of the Circle

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Abstract

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).

This research was partially supported by NSF Grant DMS-0706784.