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A natural notion of Ornstein-Uhlenbeck processes with applications to simulated annealing

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We consider Ornstein-Uhlenbeck processes (OU-processes) related to hypoelliptic diffusion 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-type processes by adding 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 the Markov process $ X $ starting at the 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 Markov process associated to $ – \log p(1,x) $ is called the OU-process related to the given hypoelliptic diffusion on $ G $. We prove the global strong existence of this OU-process on $ G $. The Poincare inequality for a large class of potentials $U$ is then shown by perturbation methods and used to obtain a hypoelliptic equivalent of the standard result on cooling schedules for simulated annealing. The relation between local results on $ \mathcal{L} $ and global results for the constructed OU-process is widely used in this study.

Those new simulated annealing algorithms use less independent Brownian motions than space dimensions. Several numerical examples demonstrating our results are presented.

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