Subelliptic Random Walks on Riemannian Manifolds and Their Convergence to Equilibrium

TL;DR

This paper investigates how subelliptic random walks on Riemannian manifolds converge to equilibrium, employing spectral theory and local-to-global diffusion techniques to establish convergence results.

Contribution

It introduces a novel approach using Fefferman-Phong techniques to analyze convergence of subelliptic random walks on manifolds.

Findings

Proves convergence to equilibrium for subelliptic random walks.

Develops a method to relate local diffusions to global behavior.

Utilizes spectral theory to quantify convergence rates.

Abstract

The aim of this work is to study the convergence to equilibrium of an $(h,\rho)$-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 $\rho$ 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.