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Brownian path preserving mappings on the Heisenberg group

Nikita Evseev

TL;DR

This work addresses which mappings between Heisenberg groups preserve horizontal Brownian motion up to a time change. By combining stochastic representations of harmonic functions, Itô calculus on the Heisenberg group, and harmonic morphism theory, the authors prove an equivalence: a map is Brownian path preserving if and only if it is a harmonic morphism. The results yield rigidity: any such map must have distortion 1 and, up to Möbius-type transformations, reduce to translations, rotations, or dilations; no nontrivial maps exist when the target dimension is too small. Overall, the paper extends classical Euclidean Brownian-motion characterizations to the sub-Riemannian Heisenberg setting, linking stochastic processes with geometric function theory on Carnot groups.

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.

Brownian path preserving mappings on the Heisenberg group

TL;DR

This work addresses which mappings between Heisenberg groups preserve horizontal Brownian motion up to a time change. By combining stochastic representations of harmonic functions, Itô calculus on the Heisenberg group, and harmonic morphism theory, the authors prove an equivalence: a map is Brownian path preserving if and only if it is a harmonic morphism. The results yield rigidity: any such map must have distortion 1 and, up to Möbius-type transformations, reduce to translations, rotations, or dilations; no nontrivial maps exist when the target dimension is too small. Overall, the paper extends classical Euclidean Brownian-motion characterizations to the sub-Riemannian Heisenberg setting, linking stochastic processes with geometric function theory on Carnot groups.

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.
Paper Structure (6 sections, 11 theorems, 60 equations)

This paper contains 6 sections, 11 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Heisenberg group.
  4. Horizontal Brownian motion on the Heisenberg group.
  5. Brownian motion and the Dirichlet problem
  6. Brownian path preserving mappings

Key Result

Lemma 2.1

Let $f\in C^2(\mathbb H^n; \mathbb R)$ and $W(t)$ be a horizontal Brownian motion in $\mathbb H^n$. Then

Theorems & Definitions (19)

  • Lemma 2.1: Itô formula
  • Lemma 2.2: McK1969
  • Theorem 2.3: O2003
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 9 more