  We have proved in a previous paper that a space-time Brownian motion
conditioned to remain in a Weyl chamber associated to an affine Kac-Moody Lie
algebra is distributed as the radial part process of a Brownian sheet on the
compact real form of the underlying finite dimensional Lie algebra, the radial
part being defined considering the coadjoint action of a loop group on the dual
of a centrally extended loop algebra. We present here a very brief proof of
this result based on a time inversion argument and on elementary stochastic
differential calculus.
