Ornstein-Uhlenbeck processes on Lie groups

We consider Ornstein-Uhlenbeck processes (OU-processes) associated to hypo-elliptic diffusion processes on finite-dirnensional Lie groups: let L be a hypo-elliptic, 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(l, 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 Poincare 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 hypo-elliptic diffusion on G. We prove the global strong existence of these OU-type processes on G under an integrability assumption on U. The Poincare inequality for a large class of potentials U is then shown by a perturbation technique. These results are applied to obtain a hypo-elliptic equivalent of standard results on cooling schedules for simulated annealing on compact homogeneous spaces M. (c) 2008 Elsevier Inc. All rights reserved