  We consider the coadjoint action of a Loop group of a compact group on the
dual of the corresponding centrally extended Loop algebra and prove that a
Brownian motion in a Cartan subalgebra conditioned to remain in an affine Weyl
chamber - which can be seen as a space time conditioned Brownian motion - is
distributed as the radial part process of a Brownian sheet on the underlying
Lie algebra.
