Brownian Processes for Monte Carlo Integration on Compact Lie Groups

This article proposes a Monte Carlo approach for the evaluation of integrals of smooth functions defined on compact Lie groups. The approach is based on the ergodic property of Brownian processes in compact Lie groups. The article provides an elementary proof of this property and obtains the following results. It gives the rate of almost sure convergence of time averages along with a “large deviations” type upper bound and a central limit theorem. It derives probability of error bounds for uniform approximation of the paths of Brownian processes using two numerical schemes. Finally, it describes generalization to compact Riemannian manifolds.