On Exponential Moments for Functionals Defined on the Loop Group

Abstract

We define the composition of a Schwartz distribution on a compact Lie group with the left‐invariant Brownian motion p _t , as a distribution on the Wiener–Lie space. In particular, the composition of p _t with the Dirac distribution will give an expression for the density of the law of p _t for t > 0. As an application, we get a simple condition for a functional F defined on the loop group in order to have exponential moments.