Diffeomorphisms of the circle and Brownian motions on an infinite-dimensional symplectic group

TL;DR

This paper explores the embedding of the circle's diffeomorphism group into an infinite-dimensional symplectic group, demonstrating non-surjectivity, and constructs a Brownian motion on this symplectic group, extending previous mathematical frameworks.

Contribution

It provides the first explicit embedding of f(S^1) into sp(), proves the embedding is not surjective, and constructs a Brownian motion on sp(), advancing understanding of infinite-dimensional symplectic groups.

Findings

Embedding of f(S^1) into sp() is not surjective.

Constructed a Brownian motion on sp().

Extended previous work on infinite-dimensional symplectic groups.

Abstract

An embedding of the group $\Diff(S^{1})$ of orientation preserving diffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplectic group, $\Sp(\infty)$, is studied. The authors prove that this embedding is not surjective. A Brownian motion is constructed on $\Sp(\infty)$. This study is motivated by recent work of H. Airault, S. Fang and P. Malliavin.