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

TL;DR

This paper investigates how conformal transformations affect Brownian motions on Heisenberg groups, revealing their connections to conditioned Brownian motions and bridges on related geometric structures.

Contribution

It demonstrates that conformal maps like the Cayley transform and inversion relate Brownian motions on Heisenberg groups to conditioned processes on spheres and bridges, extending stochastic analysis on these groups.

Findings

Cayley transform maps Brownian paths to time-changed Brownian motions on CR spheres.

Inversion of Brownian motion yields a Brownian bridge conditioned at the origin.

Results connect stochastic processes on Heisenberg groups with geometric transformations.

Abstract

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.