Potential Analysis

, Volume 34, Issue 1, pp 23-41

First online:

A Brownian Motion on the Diffeomorphism Group of the Circle

  • Mang WuAffiliated withDepartment of Mathematics, University of Connecticut Email author 

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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).


Diffeomorphism group Brownian motion Stochastic differential equation

Mathematics Subject Classifications (2010)

60H07 58J65 60J65