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

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.

This talk is part of the Probability series.