Brownian path preserving mappings on the Heisenberg group

TL;DR

This paper characterizes continuous mappings on the Heisenberg group that preserve horizontal Brownian motion, showing that only harmonic morphisms have this property, thus linking geometric harmonicity with stochastic process preservation.

Contribution

It proves that harmonic morphisms are uniquely capable of preserving horizontal Brownian motion on the Heisenberg group.

Findings

Harmonic morphisms preserve horizontal Brownian motion.

Only harmonic morphisms have this property.

The result links harmonicity with stochastic invariance.

Abstract

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.