Stochastic and geometric analysis of two infinite-dimensional groups

Abstract

The two groups I studied in this dissertation are Diff(S ¹), the group of orientation-preserving C ^∞-diffeomorphisms of the circle, and Sp(∞), an infinite-dimensional symplectic group arising from certain symplectic representation of the group Diff(S¹). In Chapter 1, I constructed Brownian motion on Diff(S¹) associated with a very strong metric of the Lie algebra diff(S¹). In Chapter 2, I first studied the relationship between Diff(S ¹) and Sp(∞) and found that they are not isomorphic with each other, then I constructed a Brownian motion on the group Sp(∞). In Chapter 3, I computed the Ricci curvature of the group Sp(∞) associated with a certain inner product on the Lie algebra [special characters omitted](∞).