Weak approximation of stochastic dynamics on Lie groups

We propose a new methodology for the creation of high weak order stochastic Lie-Runge-Kutta methods for time homogeneous SDE. More precisely, we establish sufficient conditions for a class of second order elliptic operators to generate a Feller semi-group on a general manifold of bounded geometry. We give an estimate of a Feller semi-group in terms of its infinitesimal generator, and we use that estimate to derive the order conditions for a weak second order Stochastic Lie Runge-Kutta method on a matrix Lie group.