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Non-Markovian maximal couplings and a vertical reflection principle on a class of sub-Riemannian manifolds

Liangbing Luo, Robert W. Neel

TL;DR

The paper develops non-Markovian, maximal couplings for Brownian motions on a class of sub-Riemannian manifolds by exploiting global isometries and vertical reflection principles. It gives explicit vertical couplings for the 3D Heisenberg group and its higher-dimensional non-isotropic variants, plus $ ilde{SL}(2)$, $SL(2)$, and $SU(2)$, plus a non-isotropic Heisenberg family, and establishes a reflection structure that reduces coupling times to hitting times of a vertical surface. Across these spaces, the authors derive explicit or exponential tail bounds for coupling times and obtain vertical gradient estimates for the heat semigroup; for the Heisenberg group the vertical coupling is sharp, while for the compact cases (SU(2)) and the hyperbolic cases (SL(2), ilde{SL}(2)) they provide uniform exponential decay bounds. A two-stage coupling strategy handles points on different vertical fibers by combining a horizontal coupling with the vertical reflection, yielding global estimates for total variation distance and gradient bounds, with implications for heat-semigroup inequalities and functional inequalities on these spaces. The work highlights how maximal couplings tied to space symmetries can yield tractable, uniform, and sharp results in the sub-Riemannian setting, guiding further study of comparison geometry and diffusion on Carnot groups.

Abstract

We develop an approach to constructing non-Markovian, non-co-adapted couplings for sub-Riemannian Brownian motions in sub-Riemannian manifolds with large symmetry groups by treating the specific cases of the three-dimensional Heisenberg group, higher-dimensional non-isotropic Heisenberg groups, SL(2,R) and its universal cover, and SU(2). Our primary focus is on the situation when the processes start from two points on the same vertical fiber, since in general Markovian or co-adapted couplings cannot give the sharp rate for the coupling time in this case. Non-Markovian couplings of this type on sub-Riemannian manifolds were first introduced by Banerjee-Gordina-Mariano, for the three-dimensional Heisenberg group, and were more recently extended by Bénéfice to SL(2,R) and SU(2), using a detailed consideration of the Brownian bridge. In contrast, our couplings are based on global isometries of the space, giving couplings that are maximal, as well as making the construction relatively simple and uniform across different manifolds. The coupled processes satisfy a reflection principle with respect to their coupling time, so that the coupling time reduces to the hitting time for one component of the Brownian motion, which is useful in explicitly bounding the tail probability of the coupling time. Further, it's natural to use this coupling as the second stage of a two-stage coupling when considering points on different vertical fibers. We estimate the coupling time in these various situations and give applications to inequalities for the heat semigroup.

Non-Markovian maximal couplings and a vertical reflection principle on a class of sub-Riemannian manifolds

TL;DR

The paper develops non-Markovian, maximal couplings for Brownian motions on a class of sub-Riemannian manifolds by exploiting global isometries and vertical reflection principles. It gives explicit vertical couplings for the 3D Heisenberg group and its higher-dimensional non-isotropic variants, plus , , and , plus a non-isotropic Heisenberg family, and establishes a reflection structure that reduces coupling times to hitting times of a vertical surface. Across these spaces, the authors derive explicit or exponential tail bounds for coupling times and obtain vertical gradient estimates for the heat semigroup; for the Heisenberg group the vertical coupling is sharp, while for the compact cases (SU(2)) and the hyperbolic cases (SL(2), ilde{SL}(2)) they provide uniform exponential decay bounds. A two-stage coupling strategy handles points on different vertical fibers by combining a horizontal coupling with the vertical reflection, yielding global estimates for total variation distance and gradient bounds, with implications for heat-semigroup inequalities and functional inequalities on these spaces. The work highlights how maximal couplings tied to space symmetries can yield tractable, uniform, and sharp results in the sub-Riemannian setting, guiding further study of comparison geometry and diffusion on Carnot groups.

Abstract

We develop an approach to constructing non-Markovian, non-co-adapted couplings for sub-Riemannian Brownian motions in sub-Riemannian manifolds with large symmetry groups by treating the specific cases of the three-dimensional Heisenberg group, higher-dimensional non-isotropic Heisenberg groups, SL(2,R) and its universal cover, and SU(2). Our primary focus is on the situation when the processes start from two points on the same vertical fiber, since in general Markovian or co-adapted couplings cannot give the sharp rate for the coupling time in this case. Non-Markovian couplings of this type on sub-Riemannian manifolds were first introduced by Banerjee-Gordina-Mariano, for the three-dimensional Heisenberg group, and were more recently extended by Bénéfice to SL(2,R) and SU(2), using a detailed consideration of the Brownian bridge. In contrast, our couplings are based on global isometries of the space, giving couplings that are maximal, as well as making the construction relatively simple and uniform across different manifolds. The coupled processes satisfy a reflection principle with respect to their coupling time, so that the coupling time reduces to the hitting time for one component of the Brownian motion, which is useful in explicitly bounding the tail probability of the coupling time. Further, it's natural to use this coupling as the second stage of a two-stage coupling when considering points on different vertical fibers. We estimate the coupling time in these various situations and give applications to inequalities for the heat semigroup.
Paper Structure (25 sections, 36 theorems, 166 equations)

This paper contains 25 sections, 36 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Summary of results
  4. Acknowledgements
  5. The 3-dimensional Heisenberg group
  6. Geometry and submersion structure of $\mathbb{H}$
  7. A maximal vertical coupling
  8. An efficient horizontal coupling
  9. Coupling on $\widetilde{\operatorname{SL}(2)}$
  10. The geometry of $\widetilde{\operatorname{SL}(2)}$ and $\operatorname{SL}(2)$
  11. Vertical coupling on $\widetilde{\operatorname{SL}(2)}$
  12. Horizontal coupling
  13. Coupling on $\operatorname{SL}(2)$
  14. Vertical coupling on $\operatorname{SL}(2)$
  15. Horizontal coupling on $\operatorname{SL}(2)$
  16. ...and 10 more sections

Key Result

Theorem 1

Let $M$ be the Heisenberg group (of any dimension, and possibly non-isotropic), $\operatorname{SL}(2)$ or its universal cover, or $\operatorname{SU}(2)$, and let $q$ and $\tilde{q}$ be two points on the same vertical fiber with vertical displacement $2a$ for some $a>0$. Then there exists a hypersurf is an $M$-Brownian motion from $\tilde{q}$ that couples with $B_t$ at time $\sigma_a$. Moreover, th

Theorems & Definitions (71)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Remark 1
  • Theorem 5
  • proof
  • Theorem 6
  • Theorem 7
  • ...and 61 more