Spectral Analysis of hypoelliptic random walks

TL;DR

This paper investigates the spectral properties of hypoelliptic random walks on manifolds, establishing uniform convergence bounds and linking discrete walks to continuous diffusions as step size diminishes.

Contribution

It provides the first spectral analysis of hypoelliptic random walks, demonstrating uniform convergence bounds and the connection to hypoelliptic diffusions in the small step limit.

Findings

Uniform bounds on convergence rate independent of step size h

Convergence to hypoelliptic diffusion as h approaches zero

Spectral properties characterized for hypoelliptic Markov chains

Abstract

We study the spectral theory of a reversible Markov chain associated to a hypoelliptic random walk on a manifold M. This random walk depends on a parameter h which is roughly the size of each step of the walk. We prove uniform bounds with respect to h on the rate of convergence to equilibrium, and the convergence when h goes to zero to the associated hypoelliptic diffusion.