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Non co-adapted couplings of Brownian motions on free, step 2 Carnot groups

Magalie Bénéfice

TL;DR

This work extends non co-adapted Brownian couplings from the Heisenberg group to the free step-$2$ Carnot groups $\mathbb{G}_n$, constructing an explicit coupling whose coupling time tail decays with a bound depending on the horizontal distance and the initial Lévy-area data. The authors combine a reflection-coupling phase on the horizontal coordinates with a Karhunen–Loève/Gaussian decomposition and inverse-Wishart-based coupling of the Lévy areas to achieve a successful coupling on $\mathbb{G}_n$, and then lift these results to all homogeneous step-$2$ Carnot groups. They derive total variation bounds for the associated heat semigroup and establish gradient estimates for both the horizontal and vertical components, including Cheng–Yau-type estimates for harmonic functions, by comparing the first coupling time with exit times from domains. These results generalize prior Heisenberg-based coupling techniques and provide tools for analytic inequalities (Harnack, Poincaré, Sobolev) in broad subRiemannian contexts with explicit constants.

Abstract

On the free, step $2$ Carnot groups of rank $n$ $\mathbb{G}_n$, the subRiemannian Brownian motion consists in a $\mathbb{R}^n$-Brownian motion together with its $\frac{n(n-1)}{2}$ L{é}vy areas. In this article we construct an explicit successful non co-adapted coupling of Brownian motions on $\mathbb{G}_n$. We use this construction to obtain gradient inequalities for the heat semi-group on all the homogeneous step $2$ Carnot groups. Comparing the first coupling time and the first exit time from a domain, we also obtain gradient inequalities for harmonic functions on $\mathbb{G}_n$. These results generalize the coupling strategy by Banerjee, Gordina and Mariano on the Heisenberg group.

Non co-adapted couplings of Brownian motions on free, step 2 Carnot groups

TL;DR

This work extends non co-adapted Brownian couplings from the Heisenberg group to the free step- Carnot groups , constructing an explicit coupling whose coupling time tail decays with a bound depending on the horizontal distance and the initial Lévy-area data. The authors combine a reflection-coupling phase on the horizontal coordinates with a Karhunen–Loève/Gaussian decomposition and inverse-Wishart-based coupling of the Lévy areas to achieve a successful coupling on , and then lift these results to all homogeneous step- Carnot groups. They derive total variation bounds for the associated heat semigroup and establish gradient estimates for both the horizontal and vertical components, including Cheng–Yau-type estimates for harmonic functions, by comparing the first coupling time with exit times from domains. These results generalize prior Heisenberg-based coupling techniques and provide tools for analytic inequalities (Harnack, Poincaré, Sobolev) in broad subRiemannian contexts with explicit constants.

Abstract

On the free, step Carnot groups of rank , the subRiemannian Brownian motion consists in a -Brownian motion together with its L{é}vy areas. In this article we construct an explicit successful non co-adapted coupling of Brownian motions on . We use this construction to obtain gradient inequalities for the heat semi-group on all the homogeneous step Carnot groups. Comparing the first coupling time and the first exit time from a domain, we also obtain gradient inequalities for harmonic functions on . These results generalize the coupling strategy by Banerjee, Gordina and Mariano on the Heisenberg group.
Paper Structure (23 sections, 20 theorems, 137 equations)

This paper contains 23 sections, 20 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Organisation of the paper
  5. The free, step $2$ Carnot groups
  6. SubRiemannian structure on the free, step $2$ Carnot groups
  7. Definition of an equivalent pseudo-distance
  8. Subelliptic Brownian motion
  9. The non co-adapted coupling strategy for $\mathbb G_n$
  10. The Brownian motions start from the same fiber
  11. General case
  12. Coupling on homogeneous Carnot groups
  13. Homogeneous, step $2$ Carnot groups
  14. Coupling results
  15. Comparison between the first coupling time and the first exit time from a domain
  16. ...and 8 more sections

Key Result

Theorem 1.1

Set $n\geq 2$. Let $g=(x,z)$, $\tilde{g}=(\tilde{x},\tilde{z})$ be two points in $\mathbb G_n$ and $\zeta\in \mathfrak{so}(n)$ such that $g^{-1}\star \tilde{g}=(\tilde{x}-x,\zeta)$. There exists a successful coupling of Brownian motions $\left(\mathbb{B}_t,\tilde{\mathbb{B}}_t\right)_t$ on $\mathbb where:

Theorems & Definitions (47)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 2.1
  • Remark 5
  • Proposition 3.1
  • Remark 6
  • ...and 37 more