A Brownian Motion on the Group of Diffeomorphisms of the Circle

TL;DR

This paper introduces a new construction of Brownian motion specifically within the group of smooth, orientation-preserving diffeomorphisms of the circle, expanding the understanding of stochastic processes on infinite-dimensional groups.

Contribution

It provides an alternative method to construct Brownian motion directly in the smooth diffeomorphism group, differing from previous approaches that used less regular groups.

Findings

Constructed Brownian motion in the smooth diffeomorphism group

Established properties of the new stochastic process

Compared with previous models on less regular groups

Abstract

The canonical Brownian motion that P. Malliavin in 1999 and then S. Fang in 2002 constructed lives in the group of Holderian homeomorphisms of the circle. In this paper, we present another way to construct a Brownian motion that lives exactly in the group of orientation preserving smooth diffeomorphisms of the circle.