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An embedding of the group $\Diff(S^{1})$ of orientation preservingdiffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplecticgroup, $\Sp(\infty)$, is studied. The authors prove that this embedding is notsurjective. A Brownian motion is constructed on $\Sp(\infty)$. This study ismotivated by recent work of H. Airault, S. Fang and P. Malliavin. The Riccicurvature of the infinite-dimensional symplectic group is computed. The resultshows that in almost all directions, the Ricci curvature is negative infinity.

Brownian motion and Ricci curvature on an infinite dimensional symplectic group related to the diffeomorphism group of the circle