Stochastic Tools on Hilbert Manifolds: Interplay with Geometry and Physics

Abstract:

Projections via the action of a Hilbert Lie group of a class of semi-martingales (given by Itô fields) defined on Hilbert manifolds are investigated. Using Itô calculus, we show that the drift term arising in the projected process can be interpreted in terms of a regularised trace of the second fundamental form of the orbits. For group actions with finite dimensional orbit space, we introduce a notion of strongly harmonic functions resp. regularised Brownian motion, which project onto harmonic functions resp. onto Brownian motion, whenever the orbits are minimal (in a regularised sense). We relate this projection procedure of semi-martingales to the Faddeev-Popov procedure in gauge field theory.