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Conformal transforms and Doob's h-processes on Heisenberg groups

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.