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Subelliptic Random Walks on Riemannian Manifolds and Their Convergence to Equilibrium

Davide Tramontana

TL;DR

This work constructs a global $(h,\rho)$-subelliptic diffusion on a closed Riemannian manifold driven by a subelliptic operator $A$, and proves convergence to a stationary distribution using a detailed spectral analysis of the associated Markov operator $S_{h,\rho}$. The authors deploy Fefferman–Phong localization to reduce $A$ locally to a universal model $\tilde{A}$, then patch these local models into a global diffusion via a finite subunit atlas. They establish a lower bound for the Markov kernel, perform a high/low frequency decomposition to obtain Weyl-type spectral estimates, and derive a Weyl-type infinitesimal generator $\mathsf{A}_{0,\rho}$, leading to a discrete spectrum near eigenvalue 1 and a spectral gap $g_{h,\rho}$ of order $h^2$. Consequently, the diffusion converges to the equilibrium distribution at an exponential rate $e^{-k g_{h,\rho}}$ in total variation, with the rate improving as $h\to0$. The results connect subelliptic diffusion on manifolds to Gibbs–Metropolis-type dynamics, providing quantitative, geometry-aware convergence guarantees governed by the subellipticity order $\varepsilon$.

Abstract

The aim of this work is to study the convergence to equilibrium of an $(h,ρ)$-subelliptic random walk on a closed, connected Riemannian manifold $(M,g)$ associated with a subelliptic second-order differential operator $A$ on $M$. In such a random walk, $h$ roughly represents the step size and $ρ$ the speed at which it is carried out. To construct the random walk and prove the convergence result, we employ a technique due to Fefferman and Phong, which reduces the problem to the study of a constant-coefficient operator $\tilde{A}$ that is locally equivalent to our second-order subelliptic operator $A$, in the sense that the diffusion generated by $\tilde{A}$ induces a local diffusion for $A$. By using the compactness of $M$ this local diffusion can be lifted to a global diffusion, and the convergence result is then obtained via the spectral theory of the associated Markov operator.

Subelliptic Random Walks on Riemannian Manifolds and Their Convergence to Equilibrium

TL;DR

This work constructs a global -subelliptic diffusion on a closed Riemannian manifold driven by a subelliptic operator , and proves convergence to a stationary distribution using a detailed spectral analysis of the associated Markov operator . The authors deploy Fefferman–Phong localization to reduce locally to a universal model , then patch these local models into a global diffusion via a finite subunit atlas. They establish a lower bound for the Markov kernel, perform a high/low frequency decomposition to obtain Weyl-type spectral estimates, and derive a Weyl-type infinitesimal generator , leading to a discrete spectrum near eigenvalue 1 and a spectral gap of order . Consequently, the diffusion converges to the equilibrium distribution at an exponential rate in total variation, with the rate improving as . The results connect subelliptic diffusion on manifolds to Gibbs–Metropolis-type dynamics, providing quantitative, geometry-aware convergence guarantees governed by the subellipticity order .

Abstract

The aim of this work is to study the convergence to equilibrium of an -subelliptic random walk on a closed, connected Riemannian manifold associated with a subelliptic second-order differential operator on . In such a random walk, roughly represents the step size and the speed at which it is carried out. To construct the random walk and prove the convergence result, we employ a technique due to Fefferman and Phong, which reduces the problem to the study of a constant-coefficient operator that is locally equivalent to our second-order subelliptic operator , in the sense that the diffusion generated by induces a local diffusion for . By using the compactness of this local diffusion can be lifted to a global diffusion, and the convergence result is then obtained via the spectral theory of the associated Markov operator.

Paper Structure

This paper contains 24 sections, 40 theorems, 462 equations.

Table of Contents

  1. Introduction
  2. Subelliptic Calculus on Manifolds
  3. Subelliptic Operators on Manifolds and Main Hypotheses
  4. Subunit Vector Fields
  5. h-Subelliptic Random Walks
  6. Preliminary Comments
  7. The Oleinik and Radkevich Local Diffusion
  8. CZ-localization
  9. The Subunit Geometry
  10. Subunit Cubes on Manifolds
  11. Construction of a Global (h,rho)-subelliptic Diffusion
  12. Basic Properties of the (h,rho)-subelliptic Markov Operator
  13. Auxiliary Results in Preparation to the Diffusion Theorems
  14. A Lower Bound for the Markov Kernel
  15. Decomposition in High and Low Frequency Part
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $A$ be a second-order differential operator of the form where $a_{ij},b_k,d \in C^\infty(Q^{\ast\ast},\mathbb{R})$ for every $i,j,k=1,...,n$ and $\mathsf{A}_2(x)=(a_{ij}(x))_{i,j}$ is symmetric and positive semidefinite for any $x \in Q^{\ast\ast}$. Then, if $A$ is subelliptic of order $\varepsilon$$(0<\varepsilon<1)$, there exists a universal block where $\rho>0$ is sufficiently small, $c_1

Theorems & Definitions (120)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 110 more