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SUMMARY:A natural notion of Ornstein-Uhlenbeck processes with applications
  to simulated annealing - Josef Teichmann (TU Vienna)
DTSTART:20080212T140000Z
DTEND:20080212T150000Z
UID:TALK10302@talks.cam.ac.uk
CONTACT:Norros I.
DESCRIPTION:We consider Ornstein-Uhlenbeck processes (OU-processes) relate
 d to\nhypoelliptic diffusion on finite-dimensional Lie groups: let $ \\mat
 hcal{L} $\nbe a hypoelliptic\, left-invariant ``sum of the squares''-opera
 tor on a Lie\ngroup $ G $ with associated Markov process $ X $\, then we c
 onstruct OU-type\nprocesses by adding horizontal gradient drifts of functi
 ons $ U $.  In the\nnatural case $ U(x) = - \\log p(1\,x) $\, where $ p(1\
 ,x) $ is the density of\nthe law of the Markov process $ X $ starting at t
 he identity $ e $ at time $\nt =1 $ with respect to the right-invariant Ha
 ar measure on $G$\, we show the\nPoincar\\'e inequality by applying the Dr
 iver-Melcher inequality for ``sum of\nthe squares'' operators on Lie group
 s.\n\nThe Markov process associated to $ - \\log p(1\,x) $ is called the O
 U-process\nrelated to the given hypoelliptic diffusion on $ G $. We prove 
 the global\nstrong existence of this OU-process on $ G $. The Poincare ine
 quality for\na large class of potentials $U$ is then shown by perturbation
  methods and\nused to obtain a hypoelliptic equivalent of the standard res
 ult on cooling\nschedules for simulated annealing. The relation between lo
 cal results on $\n\\mathcal{L} $ and global results for the constructed OU
 -process is widely\nused in this study.\n\nThose new simulated annealing a
 lgorithms use less independent Brownian\nmotions than space dimensions. Se
 veral numerical examples demonstrating our\nresults are presented.\n\n\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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