Ornstein-uhlenbeck Processes on Lie Groups

We consider Ornstein-Uhlenbeck processes (OU-processes) related to hypoelliptic diffusion on finite-dimensional Lie groups: let 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 Poincaré 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 Poincaré 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 L and global results for the constructed OU-process is widely used in this study. 1. Preparations from functional analysis We consider a finite-dimensional, connected Lie group G with Lie algebra g, its right-invariant Haar measure μ and a family of left-invariant vector fields V1, . . . , Vd ∈ g. We assume that Hörmander’s condition holds, i.e. the sub-algebra generated by V1, . . . , Vd coincides with g. We consider furthermore a stochastic basis (Ω,F ,P) with a d-dimensional Brownian motion B and the Lie group valued process