Ornstein-Uhlenbeck processes on Lie groups

TY - JOUR

T1 - Ornstein-Uhlenbeck processes on Lie groups

AU - Baudoin, Fabrice

AU - Hairer, Martin

AU - Teichmann, Josef

PY - 2008/8/15

Y1 - 2008/8/15

N2 - We consider Ornstein-Uhlenbeck processes (OU-processes) associated to hypo-elliptic diffusion processes on finite-dimensional 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 (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 Poincaré 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 Poincaré 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.

AB - We consider Ornstein-Uhlenbeck processes (OU-processes) associated to hypo-elliptic diffusion processes on finite-dimensional 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 (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 Poincaré 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 Poincaré 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.

KW - Hypo-elliptic diffusion

KW - Lie group

KW - Simulated annealing

KW - Spectral gap

UR - http://www.scopus.com/inward/record.url?scp=46049112206&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2008.05.004

DO - 10.1016/j.jfa.2008.05.004

M3 - Journal article

AN - SCOPUS:46049112206

SN - 0022-1236

VL - 255

SP - 877

EP - 890

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 4

ER -