Ornstein-Uhlenbeck Processes on Lie Groups

TL;DR

This paper constructs and analyzes Ornstein-Uhlenbeck processes on Lie groups associated with hypoelliptic diffusions, establishing inequalities and existence results, and applying them to simulated annealing on homogeneous spaces.

Contribution

It introduces a new class of OU-processes on Lie groups linked to hypoelliptic diffusions, proving their existence and Poincaré inequalities, and extends results to simulated annealing on homogeneous spaces.

Findings

Established Poincaré inequality for the natural OU-process on Lie groups.

Proved global strong existence of OU-processes under integrability conditions.

Applied results to hypoelliptic simulated annealing on compact homogeneous spaces.

Abstract

We consider Ornstein-Uhlenbeck processes (OU-processes) associated to hypoelliptic diffusion processes on finite-dimensional Lie groups: let $ \mathcal{L} $ be a hypoelliptic, left-invariant ``sum of the squares''-operator on a Lie group $ G $ with associated Markov process $ X $, then we construct OU-processes by adding negative horizontal gradient drifts of functions $ U $. In the natural case $ U(x) = - \log p(1,x) $, where $ p(1,x) $ is the density of the law of $ X $ starting at identity $ e $ at time $ t =1 $ with respect to the right-invariant Haar measure on $G$, we show the Poincar\'e inequality by applying the Driver-Melcher inequality for ``sum of the squares'' operators on Lie groups. The resulting Markov process is called the natural OU-process associated to the hypoelliptic diffusion on $ G $. We prove the global strong existence of these OU-type processes on $ G $ under an integrability assumption on $U$. The Poincar\'e inequality for a large class of potentials $U$ is then shown by a perturbation technique. These results are applied to obtain a hypoelliptic equivalent of standard results on cooling schedules for simulated annealing on compact homogeneous spaces $M$.