Ornstein-Uhlenbeck Processes on Lie Groups

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 $M$.