Mathematics – Probability
Scientific paper
2007-11-15
Mathematics
Probability
revised version, to appear in Journal of functional analysis
Scientific paper
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 $M$.
Baudoin Fabrice
Hairer Martin
Teichmann Josef
No associations
LandOfFree
Ornstein-Uhlenbeck Processes on Lie Groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Ornstein-Uhlenbeck Processes on Lie Groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Ornstein-Uhlenbeck Processes on Lie Groups will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-215102