  We study the stochastic processes that are images of Brownian motions on
Heisenberg group H2n+1 under conformal maps. In particular, we obtain that
Cayley transform maps Brownian paths in H2n+1 to a time changed Brownian motion
on CR sphere S2n+1 conditioned to be at its south pole at a random time. We
also obtain that the inversion of Brownian motion on H2n+1 started from x\not=
0, is up to time change, a Brownian bridge on H2n+1 conditioned to be at the
origin.
