We study continuous mappings on the Heisenberg group that up to a time change preserve horizontal Brownian motion. It is proved that only harmonic morphisms possess this property.
Projections of Poisson cut-outs in the Heisenberg group and the visual $3$-sphere
Conformal transforms and Doob's h-processes on Heisenberg groups
Couplings in L^p distance of two Brownian motions and their L{é}vy area
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