Canonical Brownian motion on the diffeomorphism group of the circle (Q1865326)

The main aim of this article is to present an alternative construction (to the one given by Malliavin in 1999) of the Brownian motion \((g_t)\) on the group \(G\) of \(C^\infty\)-diffeomorphisms on \(\mathbb{S}^1\), equipped with the so-called \(H^{3/2}\) metric. The author proceeds by defining a natural Brownian motion \((b_t^n)\) living on \(H_n\), \((H_n)\) being an exhaustive increasing subsequence of finite-dimensional subspaces of the Lie algebra of \(G\) (identified with \(C^\infty (\mathbb{S}^1,\mathbb{R}))\), and then by proving that \((g_t):= \lim_n \exp(b_t^n)\) exists and has a good version. As an application, the invariance of the law of each \(g_t\) under the adjoint action of \(\mathbb{S}^1\) is proved.